∫ x3∕2 ___________________________

3.498 \(\int \frac {x^{3/2}}{(a+b x^2)^2 (c+d x^2)^3} \, dx\)

Optimal. Leaf size=703 \[ -\frac {b^{7/4} (11 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {b^{7/4} (11 a d+b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}-\frac {b^{7/4} (11 a d+b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {b^{7/4} (11 a d+b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} (b c-a d)^4}-\frac {d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} (b c-a d)^4}+\frac {d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} (b c-a d)^4}-\frac {d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{7/4} (b c-a d)^4}-\frac {d \sqrt {x} (a d+23 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^3}-\frac {3 d \sqrt {x}}{4 \left (c+d x^2\right )^2 (b c-a d)^2}-\frac {\sqrt {x}}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

-1/8*b^(7/4)*(11*a*d+b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/(-a*d+b*c)^4*2^(1/2)+1/8*b^(7/4)*(

11*a*d+b*c)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/(-a*d+b*c)^4*2^(1/2)+1/64*d^(3/4)*(-3*a^2*d^2+22

*a*b*c*d+77*b^2*c^2)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(7/4)/(-a*d+b*c)^4*2^(1/2)-1/64*d^(3/4)*(-3*a

^2*d^2+22*a*b*c*d+77*b^2*c^2)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(7/4)/(-a*d+b*c)^4*2^(1/2)-1/16*b^(7

/4)*(11*a*d+b*c)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(3/4)/(-a*d+b*c)^4*2^(1/2)+1/16*b^(7/

4)*(11*a*d+b*c)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(3/4)/(-a*d+b*c)^4*2^(1/2)+1/128*d^(3/

4)*(-3*a^2*d^2+22*a*b*c*d+77*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(7/4)/(-a*d+b*c)

^4*2^(1/2)-1/128*d^(3/4)*(-3*a^2*d^2+22*a*b*c*d+77*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/2)*x^(1/

2))/c^(7/4)/(-a*d+b*c)^4*2^(1/2)-3/4*d*x^(1/2)/(-a*d+b*c)^2/(d*x^2+c)^2-1/2*x^(1/2)/(-a*d+b*c)/(b*x^2+a)/(d*x^

2+c)^2-1/16*d*(a*d+23*b*c)*x^(1/2)/c/(-a*d+b*c)^3/(d*x^2+c)

________________________________________________________________________________________

Rubi [A] time = 1.01, antiderivative size = 703, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {466, 471, 527, 522, 211, 1165, 628, 1162, 617, 204} \[ \frac {d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} (b c-a d)^4}-\frac {d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} (b c-a d)^4}+\frac {d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} (b c-a d)^4}-\frac {d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{7/4} (b c-a d)^4}-\frac {b^{7/4} (11 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {b^{7/4} (11 a d+b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}-\frac {b^{7/4} (11 a d+b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {b^{7/4} (11 a d+b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^4}-\frac {d \sqrt {x} (a d+23 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^3}-\frac {3 d \sqrt {x}}{4 \left (c+d x^2\right )^2 (b c-a d)^2}-\frac {\sqrt {x}}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully veriﬁed.

[In]

Int[x^(3/2)/((a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

(-3*d*Sqrt[x])/(4*(b*c - a*d)^2*(c + d*x^2)^2) - Sqrt[x]/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) - (d*(23*b*

c + a*d)*Sqrt[x])/(16*c*(b*c - a*d)^3*(c + d*x^2)) - (b^(7/4)*(b*c + 11*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[

x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^4) + (b^(7/4)*(b*c + 11*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])

/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^4) + (d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt

[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4)*(b*c - a*d)^4) - (d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*

d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4)*(b*c - a*d)^4) - (b^(7/4)*(b*c + 11*a*

d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(3/4)*(b*c - a*d)^4) + (b^(7/4)*(b

*c + 11*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(3/4)*(b*c - a*d)^4) + (

d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*

Sqrt[2]*c^(7/4)*(b*c - a*d)^4) - (d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*

d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(7/4)*(b*c - a*d)^4)

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /

; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di

st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[

{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b

]]))

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno

minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*

x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege

rQ[p]

Rule 471

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -

1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -

a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)

+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n

, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f

)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a

, b, c, d, e, f, n}, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[

((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d

)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)

*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub

st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;

FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +

c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S

imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},

x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[

(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /

; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {x^{3/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^4}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {c-11 d x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)}\\ &=-\frac {3 d \sqrt {x}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {4 c (2 b c+a d)-84 b c d x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{16 c (b c-a d)^2}\\ &=-\frac {3 d \sqrt {x}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {d (23 b c+a d) \sqrt {x}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {4 c \left (8 b^2 c^2+19 a b c d-3 a^2 d^2\right )-12 b c d (23 b c+a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{64 c^2 (b c-a d)^3}\\ &=-\frac {3 d \sqrt {x}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {d (23 b c+a d) \sqrt {x}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {\left (b^2 (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)^4}-\frac {\left (d \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c (b c-a d)^4}\\ &=-\frac {3 d \sqrt {x}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {d (23 b c+a d) \sqrt {x}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {\left (b^2 (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {a} (b c-a d)^4}+\frac {\left (b^2 (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {a} (b c-a d)^4}-\frac {\left (d \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{3/2} (b c-a d)^4}-\frac {\left (d \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{3/2} (b c-a d)^4}\\ &=-\frac {3 d \sqrt {x}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {d (23 b c+a d) \sqrt {x}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {\left (b^{3/2} (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {a} (b c-a d)^4}+\frac {\left (b^{3/2} (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {a} (b c-a d)^4}-\frac {\left (b^{7/4} (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}-\frac {\left (b^{7/4} (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}-\frac {\left (\sqrt {d} \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^{3/2} (b c-a d)^4}-\frac {\left (\sqrt {d} \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^{3/2} (b c-a d)^4}+\frac {\left (d^{3/4} \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{7/4} (b c-a d)^4}+\frac {\left (d^{3/4} \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{7/4} (b c-a d)^4}\\ &=-\frac {3 d \sqrt {x}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {d (23 b c+a d) \sqrt {x}}{16 c (b c-a d)^3 \left (c+d x^2\right )}-\frac {b^{7/4} (b c+11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {b^{7/4} (b c+11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {d^{3/4} \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} (b c-a d)^4}-\frac {d^{3/4} \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} (b c-a d)^4}+\frac {\left (b^{7/4} (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^4}-\frac {\left (b^{7/4} (b c+11 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^4}-\frac {\left (d^{3/4} \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} (b c-a d)^4}+\frac {\left (d^{3/4} \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} (b c-a d)^4}\\ &=-\frac {3 d \sqrt {x}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {d (23 b c+a d) \sqrt {x}}{16 c (b c-a d)^3 \left (c+d x^2\right )}-\frac {b^{7/4} (b c+11 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {b^{7/4} (b c+11 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {d^{3/4} \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} (b c-a d)^4}-\frac {d^{3/4} \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} (b c-a d)^4}-\frac {b^{7/4} (b c+11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {b^{7/4} (b c+11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {d^{3/4} \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} (b c-a d)^4}-\frac {d^{3/4} \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} (b c-a d)^4}\\ \end {align*}

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Mathematica [A] time = 2.39, size = 603, normalized size = 0.86 \[ \frac {-\frac {8 \sqrt {2} b^{7/4} (11 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{3/4}}+\frac {8 \sqrt {2} b^{7/4} (11 a d+b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{3/4}}-\frac {16 \sqrt {2} b^{7/4} (11 a d+b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac {16 \sqrt {2} b^{7/4} (11 a d+b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{a^{3/4}}+\frac {\sqrt {2} d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{7/4}}+\frac {\sqrt {2} d^{3/4} \left (3 a^2 d^2-22 a b c d-77 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{7/4}}+\frac {2 \sqrt {2} d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{c^{7/4}}-\frac {2 \sqrt {2} d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{c^{7/4}}-\frac {64 b^2 \sqrt {x} (b c-a d)}{a+b x^2}+\frac {8 d \sqrt {x} (a d-b c) (a d+15 b c)}{c \left (c+d x^2\right )}-\frac {32 d \sqrt {x} (b c-a d)^2}{\left (c+d x^2\right )^2}}{128 (b c-a d)^4} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^(3/2)/((a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

((-64*b^2*(b*c - a*d)*Sqrt[x])/(a + b*x^2) - (32*d*(b*c - a*d)^2*Sqrt[x])/(c + d*x^2)^2 + (8*d*(-(b*c) + a*d)*

(15*b*c + a*d)*Sqrt[x])/(c*(c + d*x^2)) - (16*Sqrt[2]*b^(7/4)*(b*c + 11*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[

x])/a^(1/4)])/a^(3/4) + (16*Sqrt[2]*b^(7/4)*(b*c + 11*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3

/4) + (2*Sqrt[2]*d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/

c^(7/4) - (2*Sqrt[2]*d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4

)])/c^(7/4) - (8*Sqrt[2]*b^(7/4)*(b*c + 11*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^

(3/4) + (8*Sqrt[2]*b^(7/4)*(b*c + 11*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4)

+ (Sqrt[2]*d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[

d]*x])/c^(7/4) + (Sqrt[2]*d^(3/4)*(-77*b^2*c^2 - 22*a*b*c*d + 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)

*Sqrt[x] + Sqrt[d]*x])/c^(7/4))/(128*(b*c - a*d)^4)

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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

Timed out

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giac [B] time = 2.32, size = 1217, normalized size = 1.73 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="giac")

[Out]

1/4*((a*b^3)^(1/4)*b^2*c + 11*(a*b^3)^(1/4)*a*b*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^

(1/4))/(sqrt(2)*a*b^4*c^4 - 4*sqrt(2)*a^2*b^3*c^3*d + 6*sqrt(2)*a^3*b^2*c^2*d^2 - 4*sqrt(2)*a^4*b*c*d^3 + sqrt

(2)*a^5*d^4) + 1/4*((a*b^3)^(1/4)*b^2*c + 11*(a*b^3)^(1/4)*a*b*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2

*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a*b^4*c^4 - 4*sqrt(2)*a^2*b^3*c^3*d + 6*sqrt(2)*a^3*b^2*c^2*d^2 - 4*sqrt(2)*a^

4*b*c*d^3 + sqrt(2)*a^5*d^4) - 1/32*(77*(c*d^3)^(1/4)*b^2*c^2 + 22*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3)^(1/4)*a^2

*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^6 - 4*sqrt(2)*a*b^3*c^5

*d + 6*sqrt(2)*a^2*b^2*c^4*d^2 - 4*sqrt(2)*a^3*b*c^3*d^3 + sqrt(2)*a^4*c^2*d^4) - 1/32*(77*(c*d^3)^(1/4)*b^2*c

^2 + 22*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3)^(1/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))

/(c/d)^(1/4))/(sqrt(2)*b^4*c^6 - 4*sqrt(2)*a*b^3*c^5*d + 6*sqrt(2)*a^2*b^2*c^4*d^2 - 4*sqrt(2)*a^3*b*c^3*d^3 +

sqrt(2)*a^4*c^2*d^4) + 1/8*((a*b^3)^(1/4)*b^2*c + 11*(a*b^3)^(1/4)*a*b*d)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x

+ sqrt(a/b))/(sqrt(2)*a*b^4*c^4 - 4*sqrt(2)*a^2*b^3*c^3*d + 6*sqrt(2)*a^3*b^2*c^2*d^2 - 4*sqrt(2)*a^4*b*c*d^3

+ sqrt(2)*a^5*d^4) - 1/8*((a*b^3)^(1/4)*b^2*c + 11*(a*b^3)^(1/4)*a*b*d)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x

+ sqrt(a/b))/(sqrt(2)*a*b^4*c^4 - 4*sqrt(2)*a^2*b^3*c^3*d + 6*sqrt(2)*a^3*b^2*c^2*d^2 - 4*sqrt(2)*a^4*b*c*d^3

+ sqrt(2)*a^5*d^4) - 1/64*(77*(c*d^3)^(1/4)*b^2*c^2 + 22*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3)^(1/4)*a^2*d^2)*log(

sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^4*c^6 - 4*sqrt(2)*a*b^3*c^5*d + 6*sqrt(2)*a^2*b^2*c^4*

d^2 - 4*sqrt(2)*a^3*b*c^3*d^3 + sqrt(2)*a^4*c^2*d^4) + 1/64*(77*(c*d^3)^(1/4)*b^2*c^2 + 22*(c*d^3)^(1/4)*a*b*c

*d - 3*(c*d^3)^(1/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^4*c^6 - 4*sqrt(2)*a

*b^3*c^5*d + 6*sqrt(2)*a^2*b^2*c^4*d^2 - 4*sqrt(2)*a^3*b*c^3*d^3 + sqrt(2)*a^4*c^2*d^4) - 1/2*b^2*sqrt(x)/((b^

3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(b*x^2 + a)) - 1/16*(15*b*c*d^2*x^(5/2) + a*d^3*x^(5/2) + 19*

b*c^2*d*sqrt(x) - 3*a*c*d^2*sqrt(x))/((b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*(d*x^2 + c)^2)

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maple [A] time = 0.03, size = 1094, normalized size = 1.56 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

1/2*b^2/(a*d-b*c)^4*x^(1/2)/(b*x^2+a)*a*d-1/2*b^3/(a*d-b*c)^4*x^(1/2)/(b*x^2+a)*c+11/8*b^2/(a*d-b*c)^4*(a/b)^(

1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*d+1/8*b^3/(a*d-b*c)^4*(a/b)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/

(a/b)^(1/4)*x^(1/2)+1)*c+11/8*b^2/(a*d-b*c)^4*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*d+1/8*

b^3/(a*d-b*c)^4*(a/b)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c+11/16*b^2/(a*d-b*c)^4*(a/b)^(1/4

)*2^(1/2)*ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))*d+1/16*b

^3/(a*d-b*c)^4*(a/b)^(1/4)/a*2^(1/2)*ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)*x^(

1/2)+(a/b)^(1/2)))*c+1/16*d^4/(a*d-b*c)^4/(d*x^2+c)^2/c*x^(5/2)*a^2+7/8*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(5/2)*a*

b-15/16*d^2/(a*d-b*c)^4/(d*x^2+c)^2*c*x^(5/2)*b^2+11/8*d^2/(a*d-b*c)^4/(d*x^2+c)^2*x^(1/2)*a*b*c-19/16*d/(a*d-

b*c)^4/(d*x^2+c)^2*x^(1/2)*b^2*c^2-3/16*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(1/2)*a^2+3/64*d^3/(a*d-b*c)^4/c^2*(c/d)

^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2-11/32*d^2/(a*d-b*c)^4/c*(c/d)^(1/4)*2^(1/2)*arctan(2^

(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b-77/64*d/(a*d-b*c)^4*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)

*b^2+3/64*d^3/(a*d-b*c)^4/c^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2-11/32*d^2/(a*d-b*c

)^4/c*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b-77/64*d/(a*d-b*c)^4*(c/d)^(1/4)*2^(1/2)*ar

ctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2+3/128*d^3/(a*d-b*c)^4/c^2*(c/d)^(1/4)*2^(1/2)*ln((x+(c/d)^(1/4)*2^(1/2

)*x^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))*a^2-11/64*d^2/(a*d-b*c)^4/c*(c/d)^(1/4)*2^

(1/2)*ln((x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))*a*b-77/128*d

/(a*d-b*c)^4*(c/d)^(1/4)*2^(1/2)*ln((x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*2^(1/2)*x^(1/2)

+(c/d)^(1/2)))*b^2

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maxima [A] time = 2.82, size = 889, normalized size = 1.26 \[ \frac {{\left (\frac {2 \, \sqrt {2} {\left (b c + 11 \, a d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b c + 11 \, a d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b c + 11 \, a d\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b c + 11 \, a d\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )} b^{2}}{16 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} - \frac {{\left (23 \, b^{2} c d^{2} + a b d^{3}\right )} x^{\frac {9}{2}} + {\left (35 \, b^{2} c^{2} d + 12 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {5}{2}} + {\left (8 \, b^{2} c^{3} + 19 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} \sqrt {x}}{16 \, {\left (a b^{3} c^{6} - 3 \, a^{2} b^{2} c^{5} d + 3 \, a^{3} b c^{4} d^{2} - a^{4} c^{3} d^{3} + {\left (b^{4} c^{4} d^{2} - 3 \, a b^{3} c^{3} d^{3} + 3 \, a^{2} b^{2} c^{2} d^{4} - a^{3} b c d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{5} d - 5 \, a b^{3} c^{4} d^{2} + 3 \, a^{2} b^{2} c^{3} d^{3} + a^{3} b c^{2} d^{4} - a^{4} c d^{5}\right )} x^{4} + {\left (b^{4} c^{6} - a b^{3} c^{5} d - 3 \, a^{2} b^{2} c^{4} d^{2} + 5 \, a^{3} b c^{3} d^{3} - 2 \, a^{4} c^{2} d^{4}\right )} x^{2}\right )}} - \frac {\frac {2 \, \sqrt {2} {\left (77 \, b^{2} c^{2} d + 22 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (77 \, b^{2} c^{2} d + 22 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (77 \, b^{2} c^{2} d + 22 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (77 \, b^{2} c^{2} d + 22 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{128 \, {\left (b^{4} c^{5} - 4 \, a b^{3} c^{4} d + 6 \, a^{2} b^{2} c^{3} d^{2} - 4 \, a^{3} b c^{2} d^{3} + a^{4} c d^{4}\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

1/16*(2*sqrt(2)*(b*c + 11*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*s

qrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(b*c + 11*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/

4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(b*c + 11*a*d)*log(sq

rt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(b*c + 11*a*d)*log(-sqrt(2)*a

^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))*b^2/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*

d^2 - 4*a^3*b*c*d^3 + a^4*d^4) - 1/16*((23*b^2*c*d^2 + a*b*d^3)*x^(9/2) + (35*b^2*c^2*d + 12*a*b*c*d^2 + a^2*d

^3)*x^(5/2) + (8*b^2*c^3 + 19*a*b*c^2*d - 3*a^2*c*d^2)*sqrt(x))/(a*b^3*c^6 - 3*a^2*b^2*c^5*d + 3*a^3*b*c^4*d^2

- a^4*c^3*d^3 + (b^4*c^4*d^2 - 3*a*b^3*c^3*d^3 + 3*a^2*b^2*c^2*d^4 - a^3*b*c*d^5)*x^6 + (2*b^4*c^5*d - 5*a*b^

3*c^4*d^2 + 3*a^2*b^2*c^3*d^3 + a^3*b*c^2*d^4 - a^4*c*d^5)*x^4 + (b^4*c^6 - a*b^3*c^5*d - 3*a^2*b^2*c^4*d^2 +

5*a^3*b*c^3*d^3 - 2*a^4*c^2*d^4)*x^2) - 1/128*(2*sqrt(2)*(77*b^2*c^2*d + 22*a*b*c*d^2 - 3*a^2*d^3)*arctan(1/2*

sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) +

2*sqrt(2)*(77*b^2*c^2*d + 22*a*b*c*d^2 - 3*a^2*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*

sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(77*b^2*c^2*d + 22*a*b*c*d^2 - 3*a^2

*d^3)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(77*b^2*c^2*d + 2

2*a*b*c*d^2 - 3*a^2*d^3)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/(b^4*c

^5 - 4*a*b^3*c^4*d + 6*a^2*b^2*c^3*d^2 - 4*a^3*b*c^2*d^3 + a^4*c*d^4)

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mupad [B] time = 7.01, size = 50125, normalized size = 71.30 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)/((a + b*x^2)^2*(c + d*x^2)^3),x)

[Out]

((x^(1/2)*(8*b^2*c^2 - 3*a^2*d^2 + 19*a*b*c*d))/(16*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (x^

(5/2)*(a^2*d^3 + 35*b^2*c^2*d + 12*a*b*c*d^2))/(16*c*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (b

*d*x^(9/2)*(a*d^2 + 23*b*c*d))/(16*c*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)))/(a*c^2 + x^2*(b*c^2

+ 2*a*c*d) + x^4*(a*d^2 + 2*b*c*d) + b*d^2*x^6) + 2*atan(((-(81*a^8*d^11 + 35153041*b^8*c^8*d^3 + 40174904*a*

b^7*c^7*d^4 + 11739420*a^2*b^6*c^6*d^5 - 1416184*a^3*b^5*c^5*d^6 - 787226*a^4*b^4*c^4*d^7 + 55176*a^5*b^3*c^3*

d^8 + 17820*a^6*b^2*c^2*d^9 - 2376*a^7*b*c*d^10)/(16777216*b^16*c^23 + 16777216*a^16*c^7*d^16 - 268435456*a^15

*b*c^8*d^15 + 2013265920*a^2*b^14*c^21*d^2 - 9395240960*a^3*b^13*c^20*d^3 + 30534533120*a^4*b^12*c^19*d^4 - 73

282879488*a^5*b^11*c^18*d^5 + 134351945728*a^6*b^10*c^17*d^6 - 191931351040*a^7*b^9*c^16*d^7 + 215922769920*a^

8*b^8*c^15*d^8 - 191931351040*a^9*b^7*c^14*d^9 + 134351945728*a^10*b^6*c^13*d^10 - 73282879488*a^11*b^5*c^12*d

^11 + 30534533120*a^12*b^4*c^11*d^12 - 9395240960*a^13*b^3*c^10*d^13 + 2013265920*a^14*b^2*c^9*d^14 - 26843545

6*a*b^15*c^22*d))^(1/4)*((-(81*a^8*d^11 + 35153041*b^8*c^8*d^3 + 40174904*a*b^7*c^7*d^4 + 11739420*a^2*b^6*c^6

*d^5 - 1416184*a^3*b^5*c^5*d^6 - 787226*a^4*b^4*c^4*d^7 + 55176*a^5*b^3*c^3*d^8 + 17820*a^6*b^2*c^2*d^9 - 2376

*a^7*b*c*d^10)/(16777216*b^16*c^23 + 16777216*a^16*c^7*d^16 - 268435456*a^15*b*c^8*d^15 + 2013265920*a^2*b^14*

c^21*d^2 - 9395240960*a^3*b^13*c^20*d^3 + 30534533120*a^4*b^12*c^19*d^4 - 73282879488*a^5*b^11*c^18*d^5 + 1343

51945728*a^6*b^10*c^17*d^6 - 191931351040*a^7*b^9*c^16*d^7 + 215922769920*a^8*b^8*c^15*d^8 - 191931351040*a^9*

b^7*c^14*d^9 + 134351945728*a^10*b^6*c^13*d^10 - 73282879488*a^11*b^5*c^12*d^11 + 30534533120*a^12*b^4*c^11*d^

12 - 9395240960*a^13*b^3*c^10*d^13 + 2013265920*a^14*b^2*c^9*d^14 - 268435456*a*b^15*c^22*d))^(1/4)*((((891*a^

9*b^7*d^15)/8192 + (77*b^16*c^9*d^6)/16 - (33367697*a*b^15*c^8*d^7)/8192 - (6291*a^8*b^8*c*d^14)/2048 - (10777

7537*a^2*b^14*c^7*d^8)/2048 - (83346257*a^3*b^13*c^6*d^9)/1024 - (39606577*a^4*b^12*c^5*d^10)/2048 + (7338751*

a^5*b^11*c^4*d^11)/4096 + (198309*a^6*b^10*c^3*d^12)/2048 + (5265*a^7*b^9*c^2*d^13)/256)*1i)/(b^13*c^17 - a^13

*c^4*d^13 + 13*a^12*b*c^5*d^12 + 78*a^2*b^11*c^15*d^2 - 286*a^3*b^10*c^14*d^3 + 715*a^4*b^9*c^13*d^4 - 1287*a^

5*b^8*c^12*d^5 + 1716*a^6*b^7*c^11*d^6 - 1716*a^7*b^6*c^10*d^7 + 1287*a^8*b^5*c^9*d^8 - 715*a^9*b^4*c^8*d^9 +

286*a^10*b^3*c^7*d^10 - 78*a^11*b^2*c^6*d^11 - 13*a*b^12*c^16*d) + (-(81*a^8*d^11 + 35153041*b^8*c^8*d^3 + 401

74904*a*b^7*c^7*d^4 + 11739420*a^2*b^6*c^6*d^5 - 1416184*a^3*b^5*c^5*d^6 - 787226*a^4*b^4*c^4*d^7 + 55176*a^5*

b^3*c^3*d^8 + 17820*a^6*b^2*c^2*d^9 - 2376*a^7*b*c*d^10)/(16777216*b^16*c^23 + 16777216*a^16*c^7*d^16 - 268435

456*a^15*b*c^8*d^15 + 2013265920*a^2*b^14*c^21*d^2 - 9395240960*a^3*b^13*c^20*d^3 + 30534533120*a^4*b^12*c^19*

d^4 - 73282879488*a^5*b^11*c^18*d^5 + 134351945728*a^6*b^10*c^17*d^6 - 191931351040*a^7*b^9*c^16*d^7 + 2159227

69920*a^8*b^8*c^15*d^8 - 191931351040*a^9*b^7*c^14*d^9 + 134351945728*a^10*b^6*c^13*d^10 - 73282879488*a^11*b^

5*c^12*d^11 + 30534533120*a^12*b^4*c^11*d^12 - 9395240960*a^13*b^3*c^10*d^13 + 2013265920*a^14*b^2*c^9*d^14 -

268435456*a*b^15*c^22*d))^(3/4)*(((-(81*a^8*d^11 + 35153041*b^8*c^8*d^3 + 40174904*a*b^7*c^7*d^4 + 11739420*a^

2*b^6*c^6*d^5 - 1416184*a^3*b^5*c^5*d^6 - 787226*a^4*b^4*c^4*d^7 + 55176*a^5*b^3*c^3*d^8 + 17820*a^6*b^2*c^2*d

^9 - 2376*a^7*b*c*d^10)/(16777216*b^16*c^23 + 16777216*a^16*c^7*d^16 - 268435456*a^15*b*c^8*d^15 + 2013265920*

a^2*b^14*c^21*d^2 - 9395240960*a^3*b^13*c^20*d^3 + 30534533120*a^4*b^12*c^19*d^4 - 73282879488*a^5*b^11*c^18*d

^5 + 134351945728*a^6*b^10*c^17*d^6 - 191931351040*a^7*b^9*c^16*d^7 + 215922769920*a^8*b^8*c^15*d^8 - 19193135

1040*a^9*b^7*c^14*d^9 + 134351945728*a^10*b^6*c^13*d^10 - 73282879488*a^11*b^5*c^12*d^11 + 30534533120*a^12*b^

4*c^11*d^12 - 9395240960*a^13*b^3*c^10*d^13 + 2013265920*a^14*b^2*c^9*d^14 - 268435456*a*b^15*c^22*d))^(1/4)*(

8192*a^2*b^22*c^22*d^5 - 2048*a*b^23*c^23*d^4 + 142592*a^3*b^21*c^21*d^6 - 1723648*a^4*b^20*c^20*d^7 + 9439232

*a^5*b^19*c^19*d^8 - 32966656*a^6*b^18*c^18*d^9 + 81665024*a^7*b^17*c^17*d^10 - 150731776*a^8*b^16*c^16*d^11 +

212486144*a^9*b^15*c^15*d^12 - 231069696*a^10*b^14*c^14*d^13 + 193363456*a^11*b^13*c^13*d^14 - 122330624*a^12

*b^12*c^12*d^15 + 55883776*a^13*b^11*c^11*d^16 - 16185344*a^14*b^10*c^10*d^17 + 1309696*a^15*b^9*c^9*d^18 + 12

05248*a^16*b^8*c^8*d^19 - 622592*a^17*b^7*c^7*d^20 + 145408*a^18*b^6*c^6*d^21 - 17152*a^19*b^5*c^5*d^22 + 768*

a^20*b^4*c^4*d^23))/(b^13*c^17 - a^13*c^4*d^13 + 13*a^12*b*c^5*d^12 + 78*a^2*b^11*c^15*d^2 - 286*a^3*b^10*c^14

*d^3 + 715*a^4*b^9*c^13*d^4 - 1287*a^5*b^8*c^12*d^5 + 1716*a^6*b^7*c^11*d^6 - 1716*a^7*b^6*c^10*d^7 + 1287*a^8

*b^5*c^9*d^8 - 715*a^9*b^4*c^8*d^9 + 286*a^10*b^3*c^7*d^10 - 78*a^11*b^2*c^6*d^11 - 13*a*b^12*c^16*d) - (x^(1/

2)*(16777216*b^27*c^25*d^4 + 100663296*a*b^26*c^24*d^5 - 1862270976*a^2*b^25*c^23*d^6 + 3970170880*a^3*b^24*c^

22*d^7 + 43464523776*a^4*b^23*c^21*d^8 - 366041628672*a^5*b^22*c^20*d^9 + 1452876496896*a^6*b^21*c^19*d^10 - 3

770791231488*a^7*b^20*c^18*d^11 + 7070048845824*a^8*b^19*c^17*d^12 - 10068131053568*a^9*b^18*c^16*d^13 + 11280

643522560*a^10*b^17*c^15*d^14 - 10296755748864*a^11*b^16*c^14*d^15 + 7971285237760*a^12*b^15*c^13*d^16 - 54298

06497792*a^13*b^14*c^12*d^17 + 3274973380608*a^14*b^13*c^11*d^18 - 1671068909568*a^15*b^12*c^10*d^19 + 6544750

01856*a^16*b^11*c^9*d^20 - 162426519552*a^17*b^10*c^8*d^21 + 7226785792*a^18*b^9*c^7*d^22 + 11707613184*a^19*b

^8*c^6*d^23 - 4677697536*a^20*b^7*c^5*d^24 + 842530816*a^21*b^6*c^4*d^25 - 72351744*a^22*b^5*c^3*d^26 + 235929

6*a^23*b^4*c^2*d^27)*1i)/(65536*(b^18*c^22 + a^18*c^4*d^18 - 18*a^17*b*c^5*d^17 + 153*a^2*b^16*c^20*d^2 - 816*

a^3*b^15*c^19*d^3 + 3060*a^4*b^14*c^18*d^4 - 8568*a^5*b^13*c^17*d^5 + 18564*a^6*b^12*c^16*d^6 - 31824*a^7*b^11

*c^15*d^7 + 43758*a^8*b^10*c^14*d^8 - 48620*a^9*b^9*c^13*d^9 + 43758*a^10*b^8*c^12*d^10 - 31824*a^11*b^7*c^11*

d^11 + 18564*a^12*b^6*c^10*d^12 - 8568*a^13*b^5*c^9*d^13 + 3060*a^14*b^4*c^8*d^14 - 816*a^15*b^3*c^7*d^15 + 15

3*a^16*b^2*c^6*d^16 - 18*a*b^17*c^21*d)))*1i) + (x^(1/2)*(9801*a^10*b^9*d^17 + 35532497*b^19*c^10*d^7 + 830454

702*a*b^18*c^9*d^8 - 285714*a^9*b^10*c*d^16 + 5434132341*a^2*b^17*c^8*d^9 + 7295711720*a^3*b^16*c^7*d^10 + 809

9206482*a^4*b^15*c^6*d^11 + 2987403540*a^5*b^14*c^5*d^12 - 117967102*a^6*b^13*c^4*d^13 - 113554584*a^7*b^12*c^

3*d^14 + 10537245*a^8*b^11*c^2*d^15))/(65536*(b^18*c^22 + a^18*c^4*d^18 - 18*a^17*b*c^5*d^17 + 153*a^2*b^16*c^

20*d^2 - 816*a^3*b^15*c^19*d^3 + 3060*a^4*b^14*c^18*d^4 - 8568*a^5*b^13*c^17*d^5 + 18564*a^6*b^12*c^16*d^6 - 3

1824*a^7*b^11*c^15*d^7 + 43758*a^8*b^10*c^14*d^8 - 48620*a^9*b^9*c^13*d^9 + 43758*a^10*b^8*c^12*d^10 - 31824*a

^11*b^7*c^11*d^11 + 18564*a^12*b^6*c^10*d^12 - 8568*a^13*b^5*c^9*d^13 + 3060*a^14*b^4*c^8*d^14 - 816*a^15*b^3*

c^7*d^15 + 153*a^16*b^2*c^6*d^16 - 18*a*b^17*c^21*d))) - (-(81*a^8*d^11 + 35153041*b^8*c^8*d^3 + 40174904*a*b^

7*c^7*d^4 + 11739420*a^2*b^6*c^6*d^5 - 1416184*a^3*b^5*c^5*d^6 - 787226*a^4*b^4*c^4*d^7 + 55176*a^5*b^3*c^3*d^

8 + 17820*a^6*b^2*c^2*d^9 - 2376*a^7*b*c*d^10)/(16777216*b^16*c^23 + 16777216*a^16*c^7*d^16 - 268435456*a^15*b

*c^8*d^15 + 2013265920*a^2*b^14*c^21*d^2 - 9395240960*a^3*b^13*c^20*d^3 + 30534533120*a^4*b^12*c^19*d^4 - 7328

2879488*a^5*b^11*c^18*d^5 + 134351945728*a^6*b^10*c^17*d^6 - 191931351040*a^7*b^9*c^16*d^7 + 215922769920*a^8*

b^8*c^15*d^8 - 191931351040*a^9*b^7*c^14*d^9 + 134351945728*a^10*b^6*c^13*d^10 - 73282879488*a^11*b^5*c^12*d^1

1 + 30534533120*a^12*b^4*c^11*d^12 - 9395240960*a^13*b^3*c^10*d^13 + 2013265920*a^14*b^2*c^9*d^14 - 268435456*

a*b^15*c^22*d))^(1/4)*((-(81*a^8*d^11 + 35153041*b^8*c^8*d^3 + 40174904*a*b^7*c^7*d^4 + 11739420*a^2*b^6*c^6*d

^5 - 1416184*a^3*b^5*c^5*d^6 - 787226*a^4*b^4*c^4*d^7 + 55176*a^5*b^3*c^3*d^8 + 17820*a^6*b^2*c^2*d^9 - 2376*a

^7*b*c*d^10)/(16777216*b^16*c^23 + 16777216*a^16*c^7*d^16 - 268435456*a^15*b*c^8*d^15 + 2013265920*a^2*b^14*c^

21*d^2 - 9395240960*a^3*b^13*c^20*d^3 + 30534533120*a^4*b^12*c^19*d^4 - 73282879488*a^5*b^11*c^18*d^5 + 134351

945728*a^6*b^10*c^17*d^6 - 191931351040*a^7*b^9*c^16*d^7 + 215922769920*a^8*b^8*c^15*d^8 - 191931351040*a^9*b^

7*c^14*d^9 + 134351945728*a^10*b^6*c^13*d^10 - 73282879488*a^11*b^5*c^12*d^11 + 30534533120*a^12*b^4*c^11*d^12

- 9395240960*a^13*b^3*c^10*d^13 + 2013265920*a^14*b^2*c^9*d^14 - 268435456*a*b^15*c^22*d))^(1/4)*((((891*a^9*

b^7*d^15)/8192 + (77*b^16*c^9*d^6)/16 - (33367697*a*b^15*c^8*d^7)/8192 - (6291*a^8*b^8*c*d^14)/2048 - (1077775

37*a^2*b^14*c^7*d^8)/2048 - (83346257*a^3*b^13*c^6*d^9)/1024 - (39606577*a^4*b^12*c^5*d^10)/2048 + (7338751*a^

5*b^11*c^4*d^11)/4096 + (198309*a^6*b^10*c^3*d^12)/2048 + (5265*a^7*b^9*c^2*d^13)/256)*1i)/(b^13*c^17 - a^13*c

^4*d^13 + 13*a^12*b*c^5*d^12 + 78*a^2*b^11*c^15*d^2 - 286*a^3*b^10*c^14*d^3 + 715*a^4*b^9*c^13*d^4 - 1287*a^5*

b^8*c^12*d^5 + 1716*a^6*b^7*c^11*d^6 - 1716*a^7*b^6*c^10*d^7 + 1287*a^8*b^5*c^9*d^8 - 715*a^9*b^4*c^8*d^9 + 28

6*a^10*b^3*c^7*d^10 - 78*a^11*b^2*c^6*d^11 - 13*a*b^12*c^16*d) + (-(81*a^8*d^11 + 35153041*b^8*c^8*d^3 + 40174

904*a*b^7*c^7*d^4 + 11739420*a^2*b^6*c^6*d^5 - 1416184*a^3*b^5*c^5*d^6 - 787226*a^4*b^4*c^4*d^7 + 55176*a^5*b^

3*c^3*d^8 + 17820*a^6*b^2*c^2*d^9 - 2376*a^7*b*c*d^10)/(16777216*b^16*c^23 + 16777216*a^16*c^7*d^16 - 26843545

6*a^15*b*c^8*d^15 + 2013265920*a^2*b^14*c^21*d^2 - 9395240960*a^3*b^13*c^20*d^3 + 30534533120*a^4*b^12*c^19*d^

4 - 73282879488*a^5*b^11*c^18*d^5 + 134351945728*a^6*b^10*c^17*d^6 - 191931351040*a^7*b^9*c^16*d^7 + 215922769

920*a^8*b^8*c^15*d^8 - 191931351040*a^9*b^7*c^14*d^9 + 134351945728*a^10*b^6*c^13*d^10 - 73282879488*a^11*b^5*

c^12*d^11 + 30534533120*a^12*b^4*c^11*d^12 - 9395240960*a^13*b^3*c^10*d^13 + 2013265920*a^14*b^2*c^9*d^14 - 26

8435456*a*b^15*c^22*d))^(3/4)*(((-(81*a^8*d^11 + 35153041*b^8*c^8*d^3 + 40174904*a*b^7*c^7*d^4 + 11739420*a^2*

b^6*c^6*d^5 - 1416184*a^3*b^5*c^5*d^6 - 787226*a^4*b^4*c^4*d^7 + 55176*a^5*b^3*c^3*d^8 + 17820*a^6*b^2*c^2*d^9

- 2376*a^7*b*c*d^10)/(16777216*b^16*c^23 + 16777216*a^16*c^7*d^16 - 268435456*a^15*b*c^8*d^15 + 2013265920*a^

2*b^14*c^21*d^2 - 9395240960*a^3*b^13*c^20*d^3 + 30534533120*a^4*b^12*c^19*d^4 - 73282879488*a^5*b^11*c^18*d^5

+ 134351945728*a^6*b^10*c^17*d^6 - 191931351040*a^7*b^9*c^16*d^7 + 215922769920*a^8*b^8*c^15*d^8 - 1919313510

40*a^9*b^7*c^14*d^9 + 134351945728*a^10*b^6*c^13*d^10 - 73282879488*a^11*b^5*c^12*d^11 + 30534533120*a^12*b^4*

c^11*d^12 - 9395240960*a^13*b^3*c^10*d^13 + 2013265920*a^14*b^2*c^9*d^14 - 268435456*a*b^15*c^22*d))^(1/4)*(81

92*a^2*b^22*c^22*d^5 - 2048*a*b^23*c^23*d^4 + 142592*a^3*b^21*c^21*d^6 - 1723648*a^4*b^20*c^20*d^7 + 9439232*a

^5*b^19*c^19*d^8 - 32966656*a^6*b^18*c^18*d^9 + 81665024*a^7*b^17*c^17*d^10 - 150731776*a^8*b^16*c^16*d^11 + 2

12486144*a^9*b^15*c^15*d^12 - 231069696*a^10*b^14*c^14*d^13 + 193363456*a^11*b^13*c^13*d^14 - 122330624*a^12*b

^12*c^12*d^15 + 55883776*a^13*b^11*c^11*d^16 - 16185344*a^14*b^10*c^10*d^17 + 1309696*a^15*b^9*c^9*d^18 + 1205

248*a^16*b^8*c^8*d^19 - 622592*a^17*b^7*c^7*d^20 + 145408*a^18*b^6*c^6*d^21 - 17152*a^19*b^5*c^5*d^22 + 768*a^

20*b^4*c^4*d^23))/(b^13*c^17 - a^13*c^4*d^13 + 13*a^12*b*c^5*d^12 + 78*a^2*b^11*c^15*d^2 - 286*a^3*b^10*c^14*d

^3 + 715*a^4*b^9*c^13*d^4 - 1287*a^5*b^8*c^12*d^5 + 1716*a^6*b^7*c^11*d^6 - 1716*a^7*b^6*c^10*d^7 + 1287*a^8*b

^5*c^9*d^8 - 715*a^9*b^4*c^8*d^9 + 286*a^10*b^3*c^7*d^10 - 78*a^11*b^2*c^6*d^11 - 13*a*b^12*c^16*d) + (x^(1/2)

*(16777216*b^27*c^25*d^4 + 100663296*a*b^26*c^24*d^5 - 1862270976*a^2*b^25*c^23*d^6 + 3970170880*a^3*b^24*c^22

*d^7 + 43464523776*a^4*b^23*c^21*d^8 - 366041628672*a^5*b^22*c^20*d^9 + 1452876496896*a^6*b^21*c^19*d^10 - 377

0791231488*a^7*b^20*c^18*d^11 + 7070048845824*a^8*b^19*c^17*d^12 - 10068131053568*a^9*b^18*c^16*d^13 + 1128064

3522560*a^10*b^17*c^15*d^14 - 10296755748864*a^11*b^16*c^14*d^15 + 7971285237760*a^12*b^15*c^13*d^16 - 5429806

497792*a^13*b^14*c^12*d^17 + 3274973380608*a^14*b^13*c^11*d^18 - 1671068909568*a^15*b^12*c^10*d^19 + 654475001

856*a^16*b^11*c^9*d^20 - 162426519552*a^17*b^10*c^8*d^21 + 7226785792*a^18*b^9*c^7*d^22 + 11707613184*a^19*b^8

*c^6*d^23 - 4677697536*a^20*b^7*c^5*d^24 + 842530816*a^21*b^6*c^4*d^25 - 72351744*a^22*b^5*c^3*d^26 + 2359296*

a^23*b^4*c^2*d^27)*1i)/(65536*(b^18*c^22 + a^18*c^4*d^18 - 18*a^17*b*c^5*d^17 + 153*a^2*b^16*c^20*d^2 - 816*a^

3*b^15*c^19*d^3 + 3060*a^4*b^14*c^18*d^4 - 8568*a^5*b^13*c^17*d^5 + 18564*a^6*b^12*c^16*d^6 - 31824*a^7*b^11*c

^15*d^7 + 43758*a^8*b^10*c^14*d^8 - 48620*a^9*b^9*c^13*d^9 + 43758*a^10*b^8*c^12*d^10 - 31824*a^11*b^7*c^11*d^

11 + 18564*a^12*b^6*c^10*d^12 - 8568*a^13*b^5*c^9*d^13 + 3060*a^14*b^4*c^8*d^14 - 816*a^15*b^3*c^7*d^15 + 153*

a^16*b^2*c^6*d^16 - 18*a*b^17*c^21*d)))*1i) - (x^(1/2)*(9801*a^10*b^9*d^17 + 35532497*b^19*c^10*d^7 + 83045470

2*a*b^18*c^9*d^8 - 285714*a^9*b^10*c*d^16 + 5434132341*a^2*b^17*c^8*d^9 + 7295711720*a^3*b^16*c^7*d^10 + 80992

06482*a^4*b^15*c^6*d^11 + 2987403540*a^5*b^14*c^5*d^12 - 117967102*a^6*b^13*c^4*d^13 - 113554584*a^7*b^12*c^3*

d^14 + 10537245*a^8*b^11*c^2*d^15))/(65536*(b^18*c^22 + a^18*c^4*d^18 - 18*a^17*b*c^5*d^17 + 153*a^2*b^16*c^20

*d^2 - 816*a^3*b^15*c^19*d^3 + 3060*a^4*b^14*c^18*d^4 - 8568*a^5*b^13*c^17*d^5 + 18564*a^6*b^12*c^16*d^6 - 318

24*a^7*b^11*c^15*d^7 + 43758*a^8*b^10*c^14*d^8 - 48620*a^9*b^9*c^13*d^9 + 43758*a^10*b^8*c^12*d^10 - 31824*a^1

1*b^7*c^11*d^11 + 18564*a^12*b^6*c^10*d^12 - 8568*a^13*b^5*c^9*d^13 + 3060*a^14*b^4*c^8*d^14 - 816*a^15*b^3*c^

7*d^15 + 153*a^16*b^2*c^6*d^16 - 18*a*b^17*c^21*d))))/((-(81*a^8*d^11 + 35153041*b^8*c^8*d^3 + 40174904*a*b^7*

c^7*d^4 + 11739420*a^2*b^6*c^6*d^5 - 1416184*a^3*b^5*c^5*d^6 - 787226*a^4*b^4*c^4*d^7 + 55176*a^5*b^3*c^3*d^8

+ 17820*a^6*b^2*c^2*d^9 - 2376*a^7*b*c*d^10)/(16777216*b^16*c^23 + 16777216*a^16*c^7*d^16 - 268435456*a^15*b*c

^8*d^15 + 2013265920*a^2*b^14*c^21*d^2 - 9395240960*a^3*b^13*c^20*d^3 + 30534533120*a^4*b^12*c^19*d^4 - 732828

79488*a^5*b^11*c^18*d^5 + 134351945728*a^6*b^10*c^17*d^6 - 191931351040*a^7*b^9*c^16*d^7 + 215922769920*a^8*b^

8*c^15*d^8 - 191931351040*a^9*b^7*c^14*d^9 + 134351945728*a^10*b^6*c^13*d^10 - 73282879488*a^11*b^5*c^12*d^11

+ 30534533120*a^12*b^4*c^11*d^12 - 9395240960*a^13*b^3*c^10*d^13 + 2013265920*a^14*b^2*c^9*d^14 - 268435456*a*

b^15*c^22*d))^(1/4)*((-(81*a^8*d^11 + 35153041*b^8*c^8*d^3 + 40174904*a*b^7*c^7*d^4 + 11739420*a^2*b^6*c^6*d^5

- 1416184*a^3*b^5*c^5*d^6 - 787226*a^4*b^4*c^4*d^7 + 55176*a^5*b^3*c^3*d^8 + 17820*a^6*b^2*c^2*d^9 - 2376*a^7

*b*c*d^10)/(16777216*b^16*c^23 + 16777216*a^16*c^7*d^16 - 268435456*a^15*b*c^8*d^15 + 2013265920*a^2*b^14*c^21

*d^2 - 9395240960*a^3*b^13*c^20*d^3 + 30534533120*a^4*b^12*c^19*d^4 - 73282879488*a^5*b^11*c^18*d^5 + 13435194

5728*a^6*b^10*c^17*d^6 - 191931351040*a^7*b^9*c^16*d^7 + 215922769920*a^8*b^8*c^15*d^8 - 191931351040*a^9*b^7*

c^14*d^9 + 134351945728*a^10*b^6*c^13*d^10 - 73282879488*a^11*b^5*c^12*d^11 + 30534533120*a^12*b^4*c^11*d^12 -

9395240960*a^13*b^3*c^10*d^13 + 2013265920*a^14*b^2*c^9*d^14 - 268435456*a*b^15*c^22*d))^(1/4)*((((891*a^9*b^

7*d^15)/8192 + (77*b^16*c^9*d^6)/16 - (33367697*a*b^15*c^8*d^7)/8192 - (6291*a^8*b^8*c*d^14)/2048 - (107777537

*a^2*b^14*c^7*d^8)/2048 - (83346257*a^3*b^13*c^6*d^9)/1024 - (39606577*a^4*b^12*c^5*d^10)/2048 + (7338751*a^5*

b^11*c^4*d^11)/4096 + (198309*a^6*b^10*c^3*d^12)/2048 + (5265*a^7*b^9*c^2*d^13)/256)*1i)/(b^13*c^17 - a^13*c^4

*d^13 + 13*a^12*b*c^5*d^12 + 78*a^2*b^11*c^15*d^2 - 286*a^3*b^10*c^14*d^3 + 715*a^4*b^9*c^13*d^4 - 1287*a^5*b^

8*c^12*d^5 + 1716*a^6*b^7*c^11*d^6 - 1716*a^7*b^6*c^10*d^7 + 1287*a^8*b^5*c^9*d^8 - 715*a^9*b^4*c^8*d^9 + 286*

a^10*b^3*c^7*d^10 - 78*a^11*b^2*c^6*d^11 - 13*a*b^12*c^16*d) + (-(81*a^8*d^11 + 35153041*b^8*c^8*d^3 + 4017490

4*a*b^7*c^7*d^4 + 11739420*a^2*b^6*c^6*d^5 - 1416184*a^3*b^5*c^5*d^6 - 787226*a^4*b^4*c^4*d^7 + 55176*a^5*b^3*

c^3*d^8 + 17820*a^6*b^2*c^2*d^9 - 2376*a^7*b*c*d^10)/(16777216*b^16*c^23 + 16777216*a^16*c^7*d^16 - 268435456*

a^15*b*c^8*d^15 + 2013265920*a^2*b^14*c^21*d^2 - 9395240960*a^3*b^13*c^20*d^3 + 30534533120*a^4*b^12*c^19*d^4

- 73282879488*a^5*b^11*c^18*d^5 + 134351945728*a^6*b^10*c^17*d^6 - 191931351040*a^7*b^9*c^16*d^7 + 21592276992

0*a^8*b^8*c^15*d^8 - 191931351040*a^9*b^7*c^14*d^9 + 134351945728*a^10*b^6*c^13*d^10 - 73282879488*a^11*b^5*c^

12*d^11 + 30534533120*a^12*b^4*c^11*d^12 - 9395240960*a^13*b^3*c^10*d^13 + 2013265920*a^14*b^2*c^9*d^14 - 2684

35456*a*b^15*c^22*d))^(3/4)*(((-(81*a^8*d^11 + 35153041*b^8*c^8*d^3 + 40174904*a*b^7*c^7*d^4 + 11739420*a^2*b^

6*c^6*d^5 - 1416184*a^3*b^5*c^5*d^6 - 787226*a^4*b^4*c^4*d^7 + 55176*a^5*b^3*c^3*d^8 + 17820*a^6*b^2*c^2*d^9 -

2376*a^7*b*c*d^10)/(16777216*b^16*c^23 + 16777216*a^16*c^7*d^16 - 268435456*a^15*b*c^8*d^15 + 2013265920*a^2*

b^14*c^21*d^2 - 9395240960*a^3*b^13*c^20*d^3 + 30534533120*a^4*b^12*c^19*d^4 - 73282879488*a^5*b^11*c^18*d^5 +

134351945728*a^6*b^10*c^17*d^6 - 191931351040*a^7*b^9*c^16*d^7 + 215922769920*a^8*b^8*c^15*d^8 - 191931351040

*a^9*b^7*c^14*d^9 + 134351945728*a^10*b^6*c^13*d^10 - 73282879488*a^11*b^5*c^12*d^11 + 30534533120*a^12*b^4*c^

11*d^12 - 9395240960*a^13*b^3*c^10*d^13 + 2013265920*a^14*b^2*c^9*d^14 - 268435456*a*b^15*c^22*d))^(1/4)*(8192

*a^2*b^22*c^22*d^5 - 2048*a*b^23*c^23*d^4 + 142592*a^3*b^21*c^21*d^6 - 1723648*a^4*b^20*c^20*d^7 + 9439232*a^5

*b^19*c^19*d^8 - 32966656*a^6*b^18*c^18*d^9 + 81665024*a^7*b^17*c^17*d^10 - 150731776*a^8*b^16*c^16*d^11 + 212

486144*a^9*b^15*c^15*d^12 - 231069696*a^10*b^14*c^14*d^13 + 193363456*a^11*b^13*c^13*d^14 - 122330624*a^12*b^1

2*c^12*d^15 + 55883776*a^13*b^11*c^11*d^16 - 16185344*a^14*b^10*c^10*d^17 + 1309696*a^15*b^9*c^9*d^18 + 120524

8*a^16*b^8*c^8*d^19 - 622592*a^17*b^7*c^7*d^20 + 145408*a^18*b^6*c^6*d^21 - 17152*a^19*b^5*c^5*d^22 + 768*a^20

*b^4*c^4*d^23))/(b^13*c^17 - a^13*c^4*d^13 + 13*a^12*b*c^5*d^12 + 78*a^2*b^11*c^15*d^2 - 286*a^3*b^10*c^14*d^3

+ 715*a^4*b^9*c^13*d^4 - 1287*a^5*b^8*c^12*d^5 + 1716*a^6*b^7*c^11*d^6 - 1716*a^7*b^6*c^10*d^7 + 1287*a^8*b^5

*c^9*d^8 - 715*a^9*b^4*c^8*d^9 + 286*a^10*b^3*c^7*d^10 - 78*a^11*b^2*c^6*d^11 - 13*a*b^12*c^16*d) - (x^(1/2)*(

16777216*b^27*c^25*d^4 + 100663296*a*b^26*c^24*d^5 - 1862270976*a^2*b^25*c^23*d^6 + 3970170880*a^3*b^24*c^22*d

^7 + 43464523776*a^4*b^23*c^21*d^8 - 366041628672*a^5*b^22*c^20*d^9 + 1452876496896*a^6*b^21*c^19*d^10 - 37707

91231488*a^7*b^20*c^18*d^11 + 7070048845824*a^8*b^19*c^17*d^12 - 10068131053568*a^9*b^18*c^16*d^13 + 112806435

22560*a^10*b^17*c^15*d^14 - 10296755748864*a^11*b^16*c^14*d^15 + 7971285237760*a^12*b^15*c^13*d^16 - 542980649

7792*a^13*b^14*c^12*d^17 + 3274973380608*a^14*b^13*c^11*d^18 - 1671068909568*a^15*b^12*c^10*d^19 + 65447500185

6*a^16*b^11*c^9*d^20 - 162426519552*a^17*b^10*c^8*d^21 + 7226785792*a^18*b^9*c^7*d^22 + 11707613184*a^19*b^8*c

^6*d^23 - 4677697536*a^20*b^7*c^5*d^24 + 842530816*a^21*b^6*c^4*d^25 - 72351744*a^22*b^5*c^3*d^26 + 2359296*a^

23*b^4*c^2*d^27)*1i)/(65536*(b^18*c^22 + a^18*c^4*d^18 - 18*a^17*b*c^5*d^17 + 153*a^2*b^16*c^20*d^2 - 816*a^3*

b^15*c^19*d^3 + 3060*a^4*b^14*c^18*d^4 - 8568*a^5*b^13*c^17*d^5 + 18564*a^6*b^12*c^16*d^6 - 31824*a^7*b^11*c^1

5*d^7 + 43758*a^8*b^10*c^14*d^8 - 48620*a^9*b^9*c^13*d^9 + 43758*a^10*b^8*c^12*d^10 - 31824*a^11*b^7*c^11*d^11

+ 18564*a^12*b^6*c^10*d^12 - 8568*a^13*b^5*c^9*d^13 + 3060*a^14*b^4*c^8*d^14 - 816*a^15*b^3*c^7*d^15 + 153*a^

16*b^2*c^6*d^16 - 18*a*b^17*c^21*d)))*1i)*1i + (x^(1/2)*(9801*a^10*b^9*d^17 + 35532497*b^19*c^10*d^7 + 8304547

02*a*b^18*c^9*d^8 - 285714*a^9*b^10*c*d^16 + 5434132341*a^2*b^17*c^8*d^9 + 7295711720*a^3*b^16*c^7*d^10 + 8099

206482*a^4*b^15*c^6*d^11 + 2987403540*a^5*b^14*c^5*d^12 - 117967102*a^6*b^13*c^4*d^13 - 113554584*a^7*b^12*c^3

*d^14 + 10537245*a^8*b^11*c^2*d^15)*1i)/(65536*(b^18*c^22 + a^18*c^4*d^18 - 18*a^17*b*c^5*d^17 + 153*a^2*b^16*

c^20*d^2 - 816*a^3*b^15*c^19*d^3 + 3060*a^4*b^14*c^18*d^4 - 8568*a^5*b^13*c^17*d^5 + 18564*a^6*b^12*c^16*d^6 -

31824*a^7*b^11*c^15*d^7 + 43758*a^8*b^10*c^14*d^8 - 48620*a^9*b^9*c^13*d^9 + 43758*a^10*b^8*c^12*d^10 - 31824

*a^11*b^7*c^11*d^11 + 18564*a^12*b^6*c^10*d^12 - 8568*a^13*b^5*c^9*d^13 + 3060*a^14*b^4*c^8*d^14 - 816*a^15*b^

3*c^7*d^15 + 153*a^16*b^2*c^6*d^16 - 18*a*b^17*c^21*d))) + (-(81*a^8*d^11 + 35153041*b^8*c^8*d^3 + 40174904*a*