∫ x5∕2 _________________________

3.481 \(\int \frac {x^{5/2}}{(a+b x^2) (c+d x^2)^3} \, dx\)

Optimal. Leaf size=628 \[ -\frac {a^{3/4} b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {a^{3/4} b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {a^{3/4} b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {a^{3/4} b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} (b c-a d)^3}+\frac {\left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} d^{3/4} (b c-a d)^3}-\frac {\left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} d^{3/4} (b c-a d)^3}-\frac {\left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} d^{3/4} (b c-a d)^3}+\frac {\left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{5/4} d^{3/4} (b c-a d)^3}+\frac {x^{3/2} (3 a d+5 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^2}+\frac {x^{3/2}}{4 \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

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

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

2)/a^(1/4))/(-a*d+b*c)^3*2^(1/2)-1/64*(-3*a^2*d^2+30*a*b*c*d+5*b^2*c^2)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/

4))/c^(5/4)/d^(3/4)/(-a*d+b*c)^3*2^(1/2)+1/64*(-3*a^2*d^2+30*a*b*c*d+5*b^2*c^2)*arctan(1+d^(1/4)*2^(1/2)*x^(1/

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

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

a*d+b*c)^3*2^(1/2)+1/128*(-3*a^2*d^2+30*a*b*c*d+5*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^(5/4)/d^(3/4)/(-a*d+b*c)^3*2^(1/2)-1/128*(-3*a^2*d^2+30*a*b*c*d+5*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^(5/4)/d^(3/4)/(-a*d+b*c)^3*2^(1/2)

________________________________________________________________________________________

Rubi [A] time = 0.76, antiderivative size = 628, 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, 579, 584, 297, 1162, 617, 204, 1165, 628} \[ \frac {\left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} d^{3/4} (b c-a d)^3}-\frac {\left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} d^{3/4} (b c-a d)^3}-\frac {\left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} d^{3/4} (b c-a d)^3}+\frac {\left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{5/4} d^{3/4} (b c-a d)^3}-\frac {a^{3/4} b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {a^{3/4} b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {a^{3/4} b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {a^{3/4} b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} (b c-a d)^3}+\frac {x^{3/2} (3 a d+5 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^2}+\frac {x^{3/2}}{4 \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully veriﬁed.

[In]

Int[x^(5/2)/((a + b*x^2)*(c + d*x^2)^3),x]

[Out]

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

b^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*(b*c - a*d)^3) - (a^(3/4)*b^(5/4)*ArcTan[1 + (

Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*(b*c - a*d)^3) - ((5*b^2*c^2 + 30*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^(5/4)*d^(3/4)*(b*c - a*d)^3) + ((5*b^2*c^2 + 30*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^(5/4)*d^(3/4)*(b*c - a*d)^3) - (a^(3/4)*

b^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*(b*c - a*d)^3) + (a^(3/4)*b^(5/

4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*(b*c - a*d)^3) + ((5*b^2*c^2 + 30*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^(5/4)*d^(3/4)*(b*

c - a*d)^3) - ((5*b^2*c^2 + 30*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^(5/4)*d^(3/4)*(b*c - a*d)^3)

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 297

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

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

Q[{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 579

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x

_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p +

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

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

e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 584

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy

mbol] :> Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,

f, g, m, p}, x] && IGtQ[n, 0]

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^{5/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^6}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=\frac {x^{3/2}}{4 (b c-a d) \left (c+d x^2\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (3 a-5 b x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)}\\ &=\frac {x^{3/2}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(5 b c+3 a d) x^{3/2}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (3 a (9 b c-a d)-b (5 b c+3 a d) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{16 c (b c-a d)^2}\\ &=\frac {x^{3/2}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(5 b c+3 a d) x^{3/2}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \left (\frac {32 a b^2 c x^2}{(b c-a d) \left (a+b x^4\right )}-\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\right ) x^2}{(b c-a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{16 c (b c-a d)^2}\\ &=\frac {x^{3/2}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(5 b c+3 a d) x^{3/2}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\left (2 a b^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c (b c-a d)^3}\\ &=\frac {x^{3/2}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(5 b c+3 a d) x^{3/2}}{16 c (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (a b^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^3}-\frac {\left (a b^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^3}-\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c \sqrt {d} (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c \sqrt {d} (b c-a d)^3}\\ &=\frac {x^{3/2}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(5 b c+3 a d) x^{3/2}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {(a b) \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 )}{2 (b c-a d)^3}-\frac {(a b) \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 )}{2 (b c-a d)^3}-\frac {\left (a^{3/4} b^{5/4}\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 )}{2 \sqrt {2} (b c-a d)^3}-\frac {\left (a^{3/4} b^{5/4}\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 )}{2 \sqrt {2} (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\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 d (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\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 d (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\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^{5/4} d^{3/4} (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\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^{5/4} d^{3/4} (b c-a d)^3}\\ &=\frac {x^{3/2}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(5 b c+3 a d) x^{3/2}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {a^{3/4} b^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {a^{3/4} b^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 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^{5/4} d^{3/4} (b c-a d)^3}-\frac {\left (5 b^2 c^2+30 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^{5/4} d^{3/4} (b c-a d)^3}-\frac {\left (a^{3/4} b^{5/4}\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 )}{\sqrt {2} (b c-a d)^3}+\frac {\left (a^{3/4} b^{5/4}\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 )}{\sqrt {2} (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\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^{5/4} d^{3/4} (b c-a d)^3}-\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\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^{5/4} d^{3/4} (b c-a d)^3}\\ &=\frac {x^{3/2}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(5 b c+3 a d) x^{3/2}}{16 c (b c-a d)^2 \left (c+d x^2\right )}+\frac {a^{3/4} b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {a^{3/4} b^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {\left (5 b^2 c^2+30 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^{5/4} d^{3/4} (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 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^{5/4} d^{3/4} (b c-a d)^3}-\frac {a^{3/4} b^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {a^{3/4} b^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 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^{5/4} d^{3/4} (b c-a d)^3}-\frac {\left (5 b^2 c^2+30 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^{5/4} d^{3/4} (b c-a d)^3}\\ \end {align*}

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Mathematica [A] time = 0.75, size = 544, normalized size = 0.87 \[ \frac {-32 \sqrt {2} a^{3/4} b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+32 \sqrt {2} a^{3/4} b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+64 \sqrt {2} a^{3/4} b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )-64 \sqrt {2} a^{3/4} b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )+\frac {\sqrt {2} \left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{5/4} d^{3/4}}-\frac {\sqrt {2} \left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{5/4} d^{3/4}}-\frac {2 \sqrt {2} \left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{c^{5/4} d^{3/4}}+\frac {2 \sqrt {2} \left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{c^{5/4} d^{3/4}}+\frac {32 x^{3/2} (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac {8 x^{3/2} (3 a d+5 b c) (b c-a d)}{c \left (c+d x^2\right )}}{128 (b c-a d)^3} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^(5/2)/((a + b*x^2)*(c + d*x^2)^3),x]

[Out]

((32*(b*c - a*d)^2*x^(3/2))/(c + d*x^2)^2 + (8*(b*c - a*d)*(5*b*c + 3*a*d)*x^(3/2))/(c*(c + d*x^2)) + 64*Sqrt[

2]*a^(3/4)*b^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 64*Sqrt[2]*a^(3/4)*b^(5/4)*ArcTan[1 + (Sqrt

[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - (2*Sqrt[2]*(5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sq

rt[x])/c^(1/4)])/(c^(5/4)*d^(3/4)) + (2*Sqrt[2]*(5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/

4)*Sqrt[x])/c^(1/4)])/(c^(5/4)*d^(3/4)) - 32*Sqrt[2]*a^(3/4)*b^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqr

t[x] + Sqrt[b]*x] + 32*Sqrt[2]*a^(3/4)*b^(5/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + (S

qrt[2]*(5*b^2*c^2 + 30*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^(5/

4)*d^(3/4)) - (Sqrt[2]*(5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sq

rt[d]*x])/(c^(5/4)*d^(3/4)))/(128*(b*c - a*d)^3)

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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^(5/2)/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

Timed out

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giac [B] time = 1.65, size = 963, normalized size = 1.53 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="giac")

[Out]

1/32*(5*(c*d^3)^(3/4)*b^2*c^2 + 30*(c*d^3)^(3/4)*a*b*c*d - 3*(c*d^3)^(3/4)*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^3*c^5*d^3 - 3*sqrt(2)*a*b^2*c^4*d^4 + 3*sqrt(2)*a^2*b*c^3*d

^5 - sqrt(2)*a^3*c^2*d^6) + 1/32*(5*(c*d^3)^(3/4)*b^2*c^2 + 30*(c*d^3)^(3/4)*a*b*c*d - 3*(c*d^3)^(3/4)*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^3*c^5*d^3 - 3*sqrt(2)*a*b^2*c^

4*d^4 + 3*sqrt(2)*a^2*b*c^3*d^5 - sqrt(2)*a^3*c^2*d^6) - 1/64*(5*(c*d^3)^(3/4)*b^2*c^2 + 30*(c*d^3)^(3/4)*a*b*

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

2)*a*b^2*c^4*d^4 + 3*sqrt(2)*a^2*b*c^3*d^5 - sqrt(2)*a^3*c^2*d^6) + 1/64*(5*(c*d^3)^(3/4)*b^2*c^2 + 30*(c*d^3)

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

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

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

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

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

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

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

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

(7/2) + 9*b*c^2*x^(3/2) - a*c*d*x^(3/2))/((b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*(d*x^2 + c)^2)

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maple [A] time = 0.02, size = 839, normalized size = 1.34 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)/(b*x^2+a)/(d*x^2+c)^3,x)

[Out]

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

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

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

/(d*x^2+c)^2*x^(3/2)*a^2*d^2+5/8/(a*d-b*c)^3/(d*x^2+c)^2*x^(3/2)*a*b*c*d-9/16/(a*d-b*c)^3/(d*x^2+c)^2*x^(3/2)*

b^2*c^2+3/64/(a*d-b*c)^3/c*d/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2-15/32/(a*d-b*c)^3/(

c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b-5/64/(a*d-b*c)^3*c/d/(c/d)^(1/4)*2^(1/2)*arctan(2

^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2+3/128/(a*d-b*c)^3/c*d/(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^2-15/64/(a*d-b*c)^3/(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-5/128/(a*d-b*c)^3*c/d/(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+3/64/(a*d-b*c)^3/c*d/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2-15/32/(a*d-b*c)^3/(c/d)

^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b-5/64/(a*d-b*c)^3*c/d/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/

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

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maxima [A] time = 2.61, size = 583, normalized size = 0.93 \[ -\frac {a b^{2} {\left (\frac {2 \, \sqrt {2} \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 {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \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 {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} + \frac {{\left (5 \, b^{2} c^{2} + 30 \, a b c d - 3 \, a^{2} d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \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 {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \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 {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{128 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )}} + \frac {{\left (5 \, b c d + 3 \, a d^{2}\right )} x^{\frac {7}{2}} + {\left (9 \, b c^{2} - a c d\right )} x^{\frac {3}{2}}}{16 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} + {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

-1/4*a*b^2*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/

(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/

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

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

*b^(3/4)))/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) + 1/128*(5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*(2

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

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

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

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

^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3) + 1/16*((5*b*c*d + 3*a*d^2)*x^(7/2) + (9*b*c^2 - a*c*d)*

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

2*a*b*c^3*d^2 + a^2*c^2*d^3)*x^2)

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mupad [B] time = 4.52, size = 31866, normalized size = 50.74 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)/((a + b*x^2)*(c + d*x^2)^3),x)

[Out]

2*atan(((((((27*a^20*b^4*d^20)/16 - (1107*a^19*b^5*c*d^19)/16 + (125*a^3*b^21*c^17*d^3)/16 - (31893*a^4*b^20*c

^16*d^4)/16 + (44481*a^5*b^19*c^15*d^5)/2 - (227605*a^6*b^18*c^14*d^6)/2 + (1382895*a^7*b^17*c^13*d^7)/4 - (27

23535*a^8*b^16*c^12*d^8)/4 + (1760163*a^9*b^15*c^11*d^9)/2 - (1361943*a^10*b^14*c^10*d^10)/2 + (1117215*a^11*b

^13*c^9*d^11)/8 + (2877545*a^12*b^12*c^8*d^12)/8 - (1026465*a^13*b^11*c^7*d^13)/2 + (744837*a^14*b^10*c^6*d^14

)/2 - (688489*a^15*b^9*c^5*d^15)/4 + (208665*a^16*b^8*c^4*d^16)/4 - (20115*a^17*b^7*c^3*d^17)/2 + (2295*a^18*b

^6*c^2*d^18)/2)*1i)/(b^14*c^16 + a^14*c^2*d^14 - 14*a^13*b*c^3*d^13 + 91*a^2*b^12*c^14*d^2 - 364*a^3*b^11*c^13

*d^3 + 1001*a^4*b^10*c^12*d^4 - 2002*a^5*b^9*c^11*d^5 + 3003*a^6*b^8*c^10*d^6 - 3432*a^7*b^7*c^9*d^7 + 3003*a^

8*b^6*c^8*d^8 - 2002*a^9*b^5*c^7*d^9 + 1001*a^10*b^4*c^6*d^10 - 364*a^11*b^3*c^5*d^11 + 91*a^12*b^2*c^4*d^12 -

14*a*b^13*c^15*d) - (x^(1/2)*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9

*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920

*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/

4)*(147456*a^19*b^4*c*d^20 + 17186816*a^3*b^20*c^17*d^4 - 201326592*a^4*b^19*c^16*d^5 + 1089601536*a^5*b^18*c^

15*d^6 - 3630694400*a^6*b^17*c^14*d^7 + 8402436096*a^7*b^16*c^13*d^8 - 14511243264*a^8*b^15*c^12*d^9 + 1970208

7680*a^9*b^14*c^11*d^10 - 21851799552*a^10*b^13*c^10*d^11 + 20194099200*a^11*b^12*c^9*d^12 - 15479078912*a^12*

b^11*c^8*d^13 + 9580707840*a^13*b^10*c^7*d^14 - 4594335744*a^14*b^9*c^6*d^15 + 1620770816*a^15*b^8*c^5*d^16 -

393216000*a^16*b^7*c^4*d^17 + 59375616*a^17*b^6*c^3*d^18 - 4718592*a^18*b^5*c^2*d^19))/(4096*(b^12*c^14 + a^12

*c^2*d^12 - 12*a^11*b*c^3*d^11 + 66*a^2*b^10*c^12*d^2 - 220*a^3*b^9*c^11*d^3 + 495*a^4*b^8*c^10*d^4 - 792*a^5*

b^7*c^9*d^5 + 924*a^6*b^6*c^8*d^6 - 792*a^7*b^5*c^7*d^7 + 495*a^8*b^4*c^6*d^8 - 220*a^9*b^3*c^5*d^9 + 66*a^10*

b^2*c^4*d^10 - 12*a*b^11*c^13*d)))*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^

3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 +

7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11)

)^(3/4) - (x^(1/2)*(81*a^11*b^8*d^9 + 625*a^3*b^16*c^8*d + 5976*a^10*b^9*c*d^8 + 15000*a^4*b^15*c^7*d^2 + 1335

00*a^5*b^14*c^6*d^3 + 538600*a^6*b^13*c^5*d^4 + 956550*a^7*b^12*c^4*d^5 + 583080*a^8*b^11*c^3*d^6 - 136260*a^9

*b^10*c^2*d^7))/(4096*(b^12*c^14 + a^12*c^2*d^12 - 12*a^11*b*c^3*d^11 + 66*a^2*b^10*c^12*d^2 - 220*a^3*b^9*c^1

1*d^3 + 495*a^4*b^8*c^10*d^4 - 792*a^5*b^7*c^9*d^5 + 924*a^6*b^6*c^8*d^6 - 792*a^7*b^5*c^7*d^7 + 495*a^8*b^4*c

^6*d^8 - 220*a^9*b^3*c^5*d^9 + 66*a^10*b^2*c^4*d^10 - 12*a*b^11*c^13*d)))*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*

c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^

6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 -

192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4) - (((((27*a^20*b^4*d^20)/16 - (1107*a^19*b^5*c*d^19)/16 + (125*

a^3*b^21*c^17*d^3)/16 - (31893*a^4*b^20*c^16*d^4)/16 + (44481*a^5*b^19*c^15*d^5)/2 - (227605*a^6*b^18*c^14*d^6

)/2 + (1382895*a^7*b^17*c^13*d^7)/4 - (2723535*a^8*b^16*c^12*d^8)/4 + (1760163*a^9*b^15*c^11*d^9)/2 - (1361943

*a^10*b^14*c^10*d^10)/2 + (1117215*a^11*b^13*c^9*d^11)/8 + (2877545*a^12*b^12*c^8*d^12)/8 - (1026465*a^13*b^11

*c^7*d^13)/2 + (744837*a^14*b^10*c^6*d^14)/2 - (688489*a^15*b^9*c^5*d^15)/4 + (208665*a^16*b^8*c^4*d^16)/4 - (

20115*a^17*b^7*c^3*d^17)/2 + (2295*a^18*b^6*c^2*d^18)/2)*1i)/(b^14*c^16 + a^14*c^2*d^14 - 14*a^13*b*c^3*d^13 +

91*a^2*b^12*c^14*d^2 - 364*a^3*b^11*c^13*d^3 + 1001*a^4*b^10*c^12*d^4 - 2002*a^5*b^9*c^11*d^5 + 3003*a^6*b^8*

c^10*d^6 - 3432*a^7*b^7*c^9*d^7 + 3003*a^8*b^6*c^8*d^8 - 2002*a^9*b^5*c^7*d^9 + 1001*a^10*b^4*c^6*d^10 - 364*a

^11*b^3*c^5*d^11 + 91*a^12*b^2*c^4*d^12 - 14*a*b^13*c^15*d) + (x^(1/2)*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*c^1

2 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b

^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 19

2*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*(147456*a^19*b^4*c*d^20 + 17186816*a^3*b^20*c^17*d^4 - 201326592*a

^4*b^19*c^16*d^5 + 1089601536*a^5*b^18*c^15*d^6 - 3630694400*a^6*b^17*c^14*d^7 + 8402436096*a^7*b^16*c^13*d^8

- 14511243264*a^8*b^15*c^12*d^9 + 19702087680*a^9*b^14*c^11*d^10 - 21851799552*a^10*b^13*c^10*d^11 + 201940992

00*a^11*b^12*c^9*d^12 - 15479078912*a^12*b^11*c^8*d^13 + 9580707840*a^13*b^10*c^7*d^14 - 4594335744*a^14*b^9*c

^6*d^15 + 1620770816*a^15*b^8*c^5*d^16 - 393216000*a^16*b^7*c^4*d^17 + 59375616*a^17*b^6*c^3*d^18 - 4718592*a^

18*b^5*c^2*d^19))/(4096*(b^12*c^14 + a^12*c^2*d^12 - 12*a^11*b*c^3*d^11 + 66*a^2*b^10*c^12*d^2 - 220*a^3*b^9*c

^11*d^3 + 495*a^4*b^8*c^10*d^4 - 792*a^5*b^7*c^9*d^5 + 924*a^6*b^6*c^8*d^6 - 792*a^7*b^5*c^7*d^7 + 495*a^8*b^4

*c^6*d^8 - 220*a^9*b^3*c^5*d^9 + 66*a^10*b^2*c^4*d^10 - 12*a*b^11*c^13*d)))*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^1

2*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*

a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10

- 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(3/4) + (x^(1/2)*(81*a^11*b^8*d^9 + 625*a^3*b^16*c^8*d + 5976*a^10*

b^9*c*d^8 + 15000*a^4*b^15*c^7*d^2 + 133500*a^5*b^14*c^6*d^3 + 538600*a^6*b^13*c^5*d^4 + 956550*a^7*b^12*c^4*d

^5 + 583080*a^8*b^11*c^3*d^6 - 136260*a^9*b^10*c^2*d^7))/(4096*(b^12*c^14 + a^12*c^2*d^12 - 12*a^11*b*c^3*d^11

+ 66*a^2*b^10*c^12*d^2 - 220*a^3*b^9*c^11*d^3 + 495*a^4*b^8*c^10*d^4 - 792*a^5*b^7*c^9*d^5 + 924*a^6*b^6*c^8*

d^6 - 792*a^7*b^5*c^7*d^7 + 495*a^8*b^4*c^6*d^8 - 220*a^9*b^3*c^5*d^9 + 66*a^10*b^2*c^4*d^10 - 12*a*b^11*c^13*

d)))*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c

^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a

^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4))/((((((27*a^20*b^4*d^2

0)/16 - (1107*a^19*b^5*c*d^19)/16 + (125*a^3*b^21*c^17*d^3)/16 - (31893*a^4*b^20*c^16*d^4)/16 + (44481*a^5*b^1

9*c^15*d^5)/2 - (227605*a^6*b^18*c^14*d^6)/2 + (1382895*a^7*b^17*c^13*d^7)/4 - (2723535*a^8*b^16*c^12*d^8)/4 +

(1760163*a^9*b^15*c^11*d^9)/2 - (1361943*a^10*b^14*c^10*d^10)/2 + (1117215*a^11*b^13*c^9*d^11)/8 + (2877545*a

^12*b^12*c^8*d^12)/8 - (1026465*a^13*b^11*c^7*d^13)/2 + (744837*a^14*b^10*c^6*d^14)/2 - (688489*a^15*b^9*c^5*d

^15)/4 + (208665*a^16*b^8*c^4*d^16)/4 - (20115*a^17*b^7*c^3*d^17)/2 + (2295*a^18*b^6*c^2*d^18)/2)*1i)/(b^14*c^

16 + a^14*c^2*d^14 - 14*a^13*b*c^3*d^13 + 91*a^2*b^12*c^14*d^2 - 364*a^3*b^11*c^13*d^3 + 1001*a^4*b^10*c^12*d^

4 - 2002*a^5*b^9*c^11*d^5 + 3003*a^6*b^8*c^10*d^6 - 3432*a^7*b^7*c^9*d^7 + 3003*a^8*b^6*c^8*d^8 - 2002*a^9*b^5

*c^7*d^9 + 1001*a^10*b^4*c^6*d^10 - 364*a^11*b^3*c^5*d^11 + 91*a^12*b^2*c^4*d^12 - 14*a*b^13*c^15*d) - (x^(1/2

)*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*

d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*

b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*(147456*a^19*b^4*c*d^20 +

17186816*a^3*b^20*c^17*d^4 - 201326592*a^4*b^19*c^16*d^5 + 1089601536*a^5*b^18*c^15*d^6 - 3630694400*a^6*b^17

*c^14*d^7 + 8402436096*a^7*b^16*c^13*d^8 - 14511243264*a^8*b^15*c^12*d^9 + 19702087680*a^9*b^14*c^11*d^10 - 21

851799552*a^10*b^13*c^10*d^11 + 20194099200*a^11*b^12*c^9*d^12 - 15479078912*a^12*b^11*c^8*d^13 + 9580707840*a

^13*b^10*c^7*d^14 - 4594335744*a^14*b^9*c^6*d^15 + 1620770816*a^15*b^8*c^5*d^16 - 393216000*a^16*b^7*c^4*d^17

+ 59375616*a^17*b^6*c^3*d^18 - 4718592*a^18*b^5*c^2*d^19))/(4096*(b^12*c^14 + a^12*c^2*d^12 - 12*a^11*b*c^3*d^

11 + 66*a^2*b^10*c^12*d^2 - 220*a^3*b^9*c^11*d^3 + 495*a^4*b^8*c^10*d^4 - 792*a^5*b^7*c^9*d^5 + 924*a^6*b^6*c^

8*d^6 - 792*a^7*b^5*c^7*d^7 + 495*a^8*b^4*c^6*d^8 - 220*a^9*b^3*c^5*d^9 + 66*a^10*b^2*c^4*d^10 - 12*a*b^11*c^1

3*d)))*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8

*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520

*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(3/4)*1i - (x^(1/2)*(81*a^

11*b^8*d^9 + 625*a^3*b^16*c^8*d + 5976*a^10*b^9*c*d^8 + 15000*a^4*b^15*c^7*d^2 + 133500*a^5*b^14*c^6*d^3 + 538

600*a^6*b^13*c^5*d^4 + 956550*a^7*b^12*c^4*d^5 + 583080*a^8*b^11*c^3*d^6 - 136260*a^9*b^10*c^2*d^7)*1i)/(4096*

(b^12*c^14 + a^12*c^2*d^12 - 12*a^11*b*c^3*d^11 + 66*a^2*b^10*c^12*d^2 - 220*a^3*b^9*c^11*d^3 + 495*a^4*b^8*c^

10*d^4 - 792*a^5*b^7*c^9*d^5 + 924*a^6*b^6*c^8*d^6 - 792*a^7*b^5*c^7*d^7 + 495*a^8*b^4*c^6*d^8 - 220*a^9*b^3*c

^5*d^9 + 66*a^10*b^2*c^4*d^10 - 12*a*b^11*c^13*d)))*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c

^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*

a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 1

92*a^11*b*c*d^11))^(1/4) + (((((27*a^20*b^4*d^20)/16 - (1107*a^19*b^5*c*d^19)/16 + (125*a^3*b^21*c^17*d^3)/16

- (31893*a^4*b^20*c^16*d^4)/16 + (44481*a^5*b^19*c^15*d^5)/2 - (227605*a^6*b^18*c^14*d^6)/2 + (1382895*a^7*b^1

7*c^13*d^7)/4 - (2723535*a^8*b^16*c^12*d^8)/4 + (1760163*a^9*b^15*c^11*d^9)/2 - (1361943*a^10*b^14*c^10*d^10)/

2 + (1117215*a^11*b^13*c^9*d^11)/8 + (2877545*a^12*b^12*c^8*d^12)/8 - (1026465*a^13*b^11*c^7*d^13)/2 + (744837

*a^14*b^10*c^6*d^14)/2 - (688489*a^15*b^9*c^5*d^15)/4 + (208665*a^16*b^8*c^4*d^16)/4 - (20115*a^17*b^7*c^3*d^1

7)/2 + (2295*a^18*b^6*c^2*d^18)/2)*1i)/(b^14*c^16 + a^14*c^2*d^14 - 14*a^13*b*c^3*d^13 + 91*a^2*b^12*c^14*d^2

- 364*a^3*b^11*c^13*d^3 + 1001*a^4*b^10*c^12*d^4 - 2002*a^5*b^9*c^11*d^5 + 3003*a^6*b^8*c^10*d^6 - 3432*a^7*b^

7*c^9*d^7 + 3003*a^8*b^6*c^8*d^8 - 2002*a^9*b^5*c^7*d^9 + 1001*a^10*b^4*c^6*d^10 - 364*a^11*b^3*c^5*d^11 + 91*

a^12*b^2*c^4*d^12 - 14*a*b^13*c^15*d) + (x^(1/2)*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10

*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7

*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*

a^11*b*c*d^11))^(1/4)*(147456*a^19*b^4*c*d^20 + 17186816*a^3*b^20*c^17*d^4 - 201326592*a^4*b^19*c^16*d^5 + 108

9601536*a^5*b^18*c^15*d^6 - 3630694400*a^6*b^17*c^14*d^7 + 8402436096*a^7*b^16*c^13*d^8 - 14511243264*a^8*b^15

*c^12*d^9 + 19702087680*a^9*b^14*c^11*d^10 - 21851799552*a^10*b^13*c^10*d^11 + 20194099200*a^11*b^12*c^9*d^12

- 15479078912*a^12*b^11*c^8*d^13 + 9580707840*a^13*b^10*c^7*d^14 - 4594335744*a^14*b^9*c^6*d^15 + 1620770816*a

^15*b^8*c^5*d^16 - 393216000*a^16*b^7*c^4*d^17 + 59375616*a^17*b^6*c^3*d^18 - 4718592*a^18*b^5*c^2*d^19))/(409

6*(b^12*c^14 + a^12*c^2*d^12 - 12*a^11*b*c^3*d^11 + 66*a^2*b^10*c^12*d^2 - 220*a^3*b^9*c^11*d^3 + 495*a^4*b^8*

c^10*d^4 - 792*a^5*b^7*c^9*d^5 + 924*a^6*b^6*c^8*d^6 - 792*a^7*b^5*c^7*d^7 + 495*a^8*b^4*c^6*d^8 - 220*a^9*b^3

*c^5*d^9 + 66*a^10*b^2*c^4*d^10 - 12*a*b^11*c^13*d)))*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10

*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 1267

2*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d -

192*a^11*b*c*d^11))^(3/4)*1i + (x^(1/2)*(81*a^11*b^8*d^9 + 625*a^3*b^16*c^8*d + 5976*a^10*b^9*c*d^8 + 15000*a

^4*b^15*c^7*d^2 + 133500*a^5*b^14*c^6*d^3 + 538600*a^6*b^13*c^5*d^4 + 956550*a^7*b^12*c^4*d^5 + 583080*a^8*b^1

1*c^3*d^6 - 136260*a^9*b^10*c^2*d^7)*1i)/(4096*(b^12*c^14 + a^12*c^2*d^12 - 12*a^11*b*c^3*d^11 + 66*a^2*b^10*c

^12*d^2 - 220*a^3*b^9*c^11*d^3 + 495*a^4*b^8*c^10*d^4 - 792*a^5*b^7*c^9*d^5 + 924*a^6*b^6*c^8*d^6 - 792*a^7*b^

5*c^7*d^7 + 495*a^8*b^4*c^6*d^8 - 220*a^9*b^3*c^5*d^9 + 66*a^10*b^2*c^4*d^10 - 12*a*b^11*c^13*d)))*(-(a^3*b^5)

/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a

^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 +

1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4) + ((625*a^4*b^16*c^7*d)/4096 - (945*a^

11*b^9*d^8)/4096 + (28215*a^10*b^10*c*d^7)/4096 + (15625*a^5*b^15*c^6*d^2)/4096 + (145125*a^6*b^14*c^5*d^3)/40

96 + (586125*a^7*b^13*c^4*d^4)/4096 + (810675*a^8*b^12*c^3*d^5)/4096 - (274725*a^9*b^11*c^2*d^6)/4096)/(b^14*c

^16 + a^14*c^2*d^14 - 14*a^13*b*c^3*d^13 + 91*a^2*b^12*c^14*d^2 - 364*a^3*b^11*c^13*d^3 + 1001*a^4*b^10*c^12*d

^4 - 2002*a^5*b^9*c^11*d^5 + 3003*a^6*b^8*c^10*d^6 - 3432*a^7*b^7*c^9*d^7 + 3003*a^8*b^6*c^8*d^8 - 2002*a^9*b^

5*c^7*d^9 + 1001*a^10*b^4*c^6*d^10 - 364*a^11*b^3*c^5*d^11 + 91*a^12*b^2*c^4*d^12 - 14*a*b^13*c^15*d)))*(-(a^3

*b^5)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12

672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*

d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4) - atan((((((27*a^20*b^4*d^20)/16

- (1107*a^19*b^5*c*d^19)/16 + (125*a^3*b^21*c^17*d^3)/16 - (31893*a^4*b^20*c^16*d^4)/16 + (44481*a^5*b^19*c^15

*d^5)/2 - (227605*a^6*b^18*c^14*d^6)/2 + (1382895*a^7*b^17*c^13*d^7)/4 - (2723535*a^8*b^16*c^12*d^8)/4 + (1760

163*a^9*b^15*c^11*d^9)/2 - (1361943*a^10*b^14*c^10*d^10)/2 + (1117215*a^11*b^13*c^9*d^11)/8 + (2877545*a^12*b^

12*c^8*d^12)/8 - (1026465*a^13*b^11*c^7*d^13)/2 + (744837*a^14*b^10*c^6*d^14)/2 - (688489*a^15*b^9*c^5*d^15)/4

+ (208665*a^16*b^8*c^4*d^16)/4 - (20115*a^17*b^7*c^3*d^17)/2 + (2295*a^18*b^6*c^2*d^18)/2)/(b^14*c^16 + a^14*

c^2*d^14 - 14*a^13*b*c^3*d^13 + 91*a^2*b^12*c^14*d^2 - 364*a^3*b^11*c^13*d^3 + 1001*a^4*b^10*c^12*d^4 - 2002*a

^5*b^9*c^11*d^5 + 3003*a^6*b^8*c^10*d^6 - 3432*a^7*b^7*c^9*d^7 + 3003*a^8*b^6*c^8*d^8 - 2002*a^9*b^5*c^7*d^9 +

1001*a^10*b^4*c^6*d^10 - 364*a^11*b^3*c^5*d^11 + 91*a^12*b^2*c^4*d^12 - 14*a*b^13*c^15*d) - (x^(1/2)*(-(a^3*b

^5)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 1267

2*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^

9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*(147456*a^19*b^4*c*d^20 + 17186816*

a^3*b^20*c^17*d^4 - 201326592*a^4*b^19*c^16*d^5 + 1089601536*a^5*b^18*c^15*d^6 - 3630694400*a^6*b^17*c^14*d^7

+ 8402436096*a^7*b^16*c^13*d^8 - 14511243264*a^8*b^15*c^12*d^9 + 19702087680*a^9*b^14*c^11*d^10 - 21851799552*

a^10*b^13*c^10*d^11 + 20194099200*a^11*b^12*c^9*d^12 - 15479078912*a^12*b^11*c^8*d^13 + 9580707840*a^13*b^10*c

^7*d^14 - 4594335744*a^14*b^9*c^6*d^15 + 1620770816*a^15*b^8*c^5*d^16 - 393216000*a^16*b^7*c^4*d^17 + 59375616

*a^17*b^6*c^3*d^18 - 4718592*a^18*b^5*c^2*d^19))/(4096*(b^12*c^14 + a^12*c^2*d^12 - 12*a^11*b*c^3*d^11 + 66*a^

2*b^10*c^12*d^2 - 220*a^3*b^9*c^11*d^3 + 495*a^4*b^8*c^10*d^4 - 792*a^5*b^7*c^9*d^5 + 924*a^6*b^6*c^8*d^6 - 79

2*a^7*b^5*c^7*d^7 + 495*a^8*b^4*c^6*d^8 - 220*a^9*b^3*c^5*d^9 + 66*a^10*b^2*c^4*d^10 - 12*a*b^11*c^13*d)))*(-(

a^3*b^5)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 -

12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c

^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(3/4)*1i - (x^(1/2)*(81*a^11*b^8*d^9

+ 625*a^3*b^16*c^8*d + 5976*a^10*b^9*c*d^8 + 15000*a^4*b^15*c^7*d^2 + 133500*a^5*b^14*c^6*d^3 + 538600*a^6*b^

13*c^5*d^4 + 956550*a^7*b^12*c^4*d^5 + 583080*a^8*b^11*c^3*d^6 - 136260*a^9*b^10*c^2*d^7)*1i)/(4096*(b^12*c^14

+ a^12*c^2*d^12 - 12*a^11*b*c^3*d^11 + 66*a^2*b^10*c^12*d^2 - 220*a^3*b^9*c^11*d^3 + 495*a^4*b^8*c^10*d^4 - 7

92*a^5*b^7*c^9*d^5 + 924*a^6*b^6*c^8*d^6 - 792*a^7*b^5*c^7*d^7 + 495*a^8*b^4*c^6*d^8 - 220*a^9*b^3*c^5*d^9 + 6

6*a^10*b^2*c^4*d^10 - 12*a*b^11*c^13*d)))*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 -

3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^

5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*

c*d^11))^(1/4) - ((((27*a^20*b^4*d^20)/16 - (1107*a^19*b^5*c*d^19)/16 + (125*a^3*b^21*c^17*d^3)/16 - (31893*a^

4*b^20*c^16*d^4)/16 + (44481*a^5*b^19*c^15*d^5)/2 - (227605*a^6*b^18*c^14*d^6)/2 + (1382895*a^7*b^17*c^13*d^7)

/4 - (2723535*a^8*b^16*c^12*d^8)/4 + (1760163*a^9*b^15*c^11*d^9)/2 - (1361943*a^10*b^14*c^10*d^10)/2 + (111721

5*a^11*b^13*c^9*d^11)/8 + (2877545*a^12*b^12*c^8*d^12)/8 - (1026465*a^13*b^11*c^7*d^13)/2 + (744837*a^14*b^10*

c^6*d^14)/2 - (688489*a^15*b^9*c^5*d^15)/4 + (208665*a^16*b^8*c^4*d^16)/4 - (20115*a^17*b^7*c^3*d^17)/2 + (229

5*a^18*b^6*c^2*d^18)/2)/(b^14*c^16 + a^14*c^2*d^14 - 14*a^13*b*c^3*d^13 + 91*a^2*b^12*c^14*d^2 - 364*a^3*b^11*

c^13*d^3 + 1001*a^4*b^10*c^12*d^4 - 2002*a^5*b^9*c^11*d^5 + 3003*a^6*b^8*c^10*d^6 - 3432*a^7*b^7*c^9*d^7 + 300

3*a^8*b^6*c^8*d^8 - 2002*a^9*b^5*c^7*d^9 + 1001*a^10*b^4*c^6*d^10 - 364*a^11*b^3*c^5*d^11 + 91*a^12*b^2*c^4*d^

12 - 14*a*b^13*c^15*d) + (x^(1/2)*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3

*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 +

7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))

^(1/4)*(147456*a^19*b^4*c*d^20 + 17186816*a^3*b^20*c^17*d^4 - 201326592*a^4*b^19*c^16*d^5 + 1089601536*a^5*b^1

8*c^15*d^6 - 3630694400*a^6*b^17*c^14*d^7 + 8402436096*a^7*b^16*c^13*d^8 - 14511243264*a^8*b^15*c^12*d^9 + 197

02087680*a^9*b^14*c^11*d^10 - 21851799552*a^10*b^13*c^10*d^11 + 20194099200*a^11*b^12*c^9*d^12 - 15479078912*a

^12*b^11*c^8*d^13 + 9580707840*a^13*b^10*c^7*d^14 - 4594335744*a^14*b^9*c^6*d^15 + 1620770816*a^15*b^8*c^5*d^1

6 - 393216000*a^16*b^7*c^4*d^17 + 59375616*a^17*b^6*c^3*d^18 - 4718592*a^18*b^5*c^2*d^19))/(4096*(b^12*c^14 +

a^12*c^2*d^12 - 12*a^11*b*c^3*d^11 + 66*a^2*b^10*c^12*d^2 - 220*a^3*b^9*c^11*d^3 + 495*a^4*b^8*c^10*d^4 - 792*

a^5*b^7*c^9*d^5 + 924*a^6*b^6*c^8*d^6 - 792*a^7*b^5*c^7*d^7 + 495*a^8*b^4*c^6*d^8 - 220*a^9*b^3*c^5*d^9 + 66*a

^10*b^2*c^4*d^10 - 12*a*b^11*c^13*d)))*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 352

0*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d

^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d

^11))^(3/4)*1i + (x^(1/2)*(81*a^11*b^8*d^9 + 625*a^3*b^16*c^8*d + 5976*a^10*b^9*c*d^8 + 15000*a^4*b^15*c^7*d^2

+ 133500*a^5*b^14*c^6*d^3 + 538600*a^6*b^13*c^5*d^4 + 956550*a^7*b^12*c^4*d^5 + 583080*a^8*b^11*c^3*d^6 - 136

260*a^9*b^10*c^2*d^7)*1i)/(4096*(b^12*c^14 + a^12*c^2*d^12 - 12*a^11*b*c^3*d^11 + 66*a^2*b^10*c^12*d^2 - 220*a

^3*b^9*c^11*d^3 + 495*a^4*b^8*c^10*d^4 - 792*a^5*b^7*c^9*d^5 + 924*a^6*b^6*c^8*d^6 - 792*a^7*b^5*c^7*d^7 + 495

*a^8*b^4*c^6*d^8 - 220*a^9*b^3*c^5*d^9 + 66*a^10*b^2*c^4*d^10 - 12*a*b^11*c^13*d)))*(-(a^3*b^5)/(16*a^12*d^12

+ 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5

+ 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*

c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4))/(((((27*a^20*b^4*d^20)/16 - (1107*a^19*b^5*c*d^19)/1

6 + (125*a^3*b^21*c^17*d^3)/16 - (31893*a^4*b^20*c^16*d^4)/16 + (44481*a^5*b^19*c^15*d^5)/2 - (227605*a^6*b^18

*c^14*d^6)/2 + (1382895*a^7*b^17*c^13*d^7)/4 - (2723535*a^8*b^16*c^12*d^8)/4 + (1760163*a^9*b^15*c^11*d^9)/2 -

(1361943*a^10*b^14*c^10*d^10)/2 + (1117215*a^11*b^13*c^9*d^11)/8 + (2877545*a^12*b^12*c^8*d^12)/8 - (1026465*

a^13*b^11*c^7*d^13)/2 + (744837*a^14*b^10*c^6*d^14)/2 - (688489*a^15*b^9*c^5*d^15)/4 + (208665*a^16*b^8*c^4*d^

16)/4 - (20115*a^17*b^7*c^3*d^17)/2 + (2295*a^18*b^6*c^2*d^18)/2)/(b^14*c^16 + a^14*c^2*d^14 - 14*a^13*b*c^3*d

^13 + 91*a^2*b^12*c^14*d^2 - 364*a^3*b^11*c^13*d^3 + 1001*a^4*b^10*c^12*d^4 - 2002*a^5*b^9*c^11*d^5 + 3003*a^6

*b^8*c^10*d^6 - 3432*a^7*b^7*c^9*d^7 + 3003*a^8*b^6*c^8*d^8 - 2002*a^9*b^5*c^7*d^9 + 1001*a^10*b^4*c^6*d^10 -

364*a^11*b^3*c^5*d^11 + 91*a^12*b^2*c^4*d^12 - 14*a*b^13*c^15*d) - (x^(1/2)*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^1

2*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*

a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10

- 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*(147456*a^19*b^4*c*d^20 + 17186816*a^3*b^20*c^17*d^4 - 201326

592*a^4*b^19*c^16*d^5 + 1089601536*a^5*b^18*c^15*d^6 - 3630694400*a^6*b^17*c^14*d^7 + 8402436096*a^7*b^16*c^13

*d^8 - 14511243264*a^8*b^15*c^12*d^9 + 19702087680*a^9*b^14*c^11*d^10 - 21851799552*a^10*b^13*c^10*d^11 + 2019

4099200*a^11*b^12*c^9*d^12 - 15479078912*a^12*b^11*c^8*d^13 + 9580707840*a^13*b^10*c^7*d^14 - 4594335744*a^14*

b^9*c^6*d^15 + 1620770816*a^15*b^8*c^5*d^16 - 393216000*a^16*b^7*c^4*d^17 + 59375616*a^17*b^6*c^3*d^18 - 47185

92*a^18*b^5*c^2*d^19))/(4096*(b^12*c^14 + a^12*c^2*d^12 - 12*a^11*b*c^3*d^11 + 66*a^2*b^10*c^12*d^2 - 220*a^3*

b^9*c^11*d^3 + 495*a^4*b^8*c^10*d^4 - 792*a^5*b^7*c^9*d^5 + 924*a^6*b^6*c^8*d^6 - 792*a^7*b^5*c^7*d^7 + 495*a^

8*b^4*c^6*d^8 - 220*a^9*b^3*c^5*d^9 + 66*a^10*b^2*c^4*d^10 - 12*a*b^11*c^13*d)))*(-(a^3*b^5)/(16*a^12*d^12 + 1

6*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 1

4784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2

*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(3/4) - (x^(1/2)*(81*a^11*b^8*d^9 + 625*a^3*b^16*c^8*d + 5976*

a^10*b^9*c*d^8 + 15000*a^4*b^15*c^7*d^2 + 133500*a^5*b^14*c^6*d^3 + 538600*a^6*b^13*c^5*d^4 + 956550*a^7*b^12*

c^4*d^5 + 583080*a^8*b^11*c^3*d^6 - 136260*a^9*b^10*c^2*d^7))/(4096*(b^12*c^14 + a^12*c^2*d^12 - 12*a^11*b*c^3

*d^11 + 66*a^2*b^10*c^12*d^2 - 220*a^3*b^9*c^11*d^3 + 495*a^4*b^8*c^10*d^4 - 792*a^5*b^7*c^9*d^5 + 924*a^6*b^6

*c^8*d^6 - 792*a^7*b^5*c^7*d^7 + 495*a^8*b^4*c^6*d^8 - 220*a^9*b^3*c^5*d^9 + 66*a^10*b^2*c^4*d^10 - 12*a*b^11*

c^13*d)))*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*

b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3

520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4) + ((((27*a^20*b^4

*d^20)/16 - (1107*a^19*b^5*c*d^19)/16 + (125*a^3*b^21*c^17*d^3)/16 - (31893*a^4*b^20*c^16*d^4)/16 + (44481*a^5

*b^19*c^15*d^5)/2 - (227605*a^6*b^18*c^14*d^6)/2 + (1382895*a^7*b^17*c^13*d^7)/4 - (2723535*a^8*b^16*c^12*d^8)

/4 + (1760163*a^9*b^15*c^11*d^9)/2 - (1361943*a^10*b^14*c^10*d^10)/2 + (1117215*a^11*b^13*c^9*d^11)/8 + (28775

45*a^12*b^12*c^8*d^12)/8 - (1026465*a^13*b^11*c^7*d^13)/2 + (744837*a^14*b^10*c^6*d^14)/2 - (688489*a^15*b^9*c

^5*d^15)/4 + (208665*a^16*b^8*c^4*d^16)/4 - (20115*a^17*b^7*c^3*d^17)/2 + (2295*a^18*b^6*c^2*d^18)/2)/(b^14*c^

16 + a^14*c^2*d^14 - 14*a^13*b*c^3*d^13 + 91*a^2*b^12*c^14*d^2 - 364*a^3*b^11*c^13*d^3 + 1001*a^4*b^10*c^12*d^

4 - 2002*a^5*b^9*c^11*d^5 + 3003*a^6*b^8*c^10*d^6 - 3432*a^7*b^7*c^9*d^7 + 3003*a^8*b^6*c^8*d^8 - 2002*a^9*b^5

*c^7*d^9 + 1001*a^10*b^4*c^6*d^10 - 364*a^11*b^3*c^5*d^11 + 91*a^12*b^2*c^4*d^12 - 14*a*b^13*c^15*d) + (x^(1/2