∫ x7∕2 ___________________________

3.488 \(\int \frac {x^{7/2}}{(a+b x^2)^2 (c+d x^2)^2} \, dx\)

Optimal. Leaf size=624 \[ \frac {a \sqrt {x}}{2 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac {\sqrt {x} (a d+b c)}{2 b \left (c+d x^2\right ) (b c-a d)^2}+\frac {\sqrt [4]{a} (3 a d+5 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{a} (3 a d+5 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{c} (5 a d+3 b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{c} (5 a d+3 b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{a} (3 a d+5 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{a} (3 a d+5 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{c} (5 a d+3 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{c} (5 a d+3 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} \sqrt [4]{d} (b c-a d)^3} \]

[Out]

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

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

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

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

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

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

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

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

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

________________________________________________________________________________________

Rubi [A] time = 0.76, antiderivative size = 624, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {466, 470, 527, 522, 211, 1165, 628, 1162, 617, 204} \[ \frac {a \sqrt {x}}{2 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac {\sqrt {x} (a d+b c)}{2 b \left (c+d x^2\right ) (b c-a d)^2}+\frac {\sqrt [4]{a} (3 a d+5 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{a} (3 a d+5 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{c} (5 a d+3 b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{c} (5 a d+3 b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{a} (3 a d+5 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{a} (3 a d+5 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{c} (5 a d+3 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{c} (5 a d+3 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} \sqrt [4]{d} (b c-a d)^3} \]

Antiderivative was successfully veriﬁed.

[In]

Int[x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^2),x]

[Out]

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

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

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

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

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

4)*(5*b*c + 3*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*b^(1/4)*(b*c - a*d)^

3) - (a^(1/4)*(5*b*c + 3*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*b^(1/4)*(

b*c - a*d)^3) - (c^(1/4)*(3*b*c + 5*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2

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

)/(8*Sqrt[2]*d^(1/4)*(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 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 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2

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

*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +

(a*d*(m - n + n*q + 1) + b*c*n*(p + 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] && GtQ[m - n + 1, n] && 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^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^8}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {a c+(-4 b c-3 a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 b (b c-a d)}\\ &=\frac {(b c+a d) \sqrt {x}}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {8 a b c^2-12 b c (b c+a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{8 b c (b c-a d)^2}\\ &=\frac {(b c+a d) \sqrt {x}}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {(a (5 b c+3 a d)) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)^3}+\frac {(c (3 b c+5 a d)) \operatorname {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)^3}\\ &=\frac {(b c+a d) \sqrt {x}}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\left (\sqrt {a} (5 b c+3 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)^3}-\frac {\left (\sqrt {a} (5 b c+3 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)^3}+\frac {\left (\sqrt {c} (3 b c+5 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)^3}+\frac {\left (\sqrt {c} (3 b c+5 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)^3}\\ &=\frac {(b c+a d) \sqrt {x}}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\left (\sqrt {a} (5 b c+3 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 {b} (b c-a d)^3}-\frac {\left (\sqrt {a} (5 b c+3 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 {b} (b c-a d)^3}+\frac {\left (\sqrt [4]{a} (5 b c+3 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} \sqrt [4]{b} (b c-a d)^3}+\frac {\left (\sqrt [4]{a} (5 b c+3 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} \sqrt [4]{b} (b c-a d)^3}+\frac {\left (\sqrt {c} (3 b c+5 a d)\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 )}{8 \sqrt {d} (b c-a d)^3}+\frac {\left (\sqrt {c} (3 b c+5 a d)\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 )}{8 \sqrt {d} (b c-a d)^3}-\frac {\left (\sqrt [4]{c} (3 b c+5 a d)\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 )}{8 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}-\frac {\left (\sqrt [4]{c} (3 b c+5 a d)\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 )}{8 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}\\ &=\frac {(b c+a d) \sqrt {x}}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\sqrt [4]{a} (5 b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{a} (5 b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{c} (3 b c+5 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{c} (3 b c+5 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}-\frac {\left (\sqrt [4]{a} (5 b c+3 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} \sqrt [4]{b} (b c-a d)^3}+\frac {\left (\sqrt [4]{a} (5 b c+3 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} \sqrt [4]{b} (b c-a d)^3}+\frac {\left (\sqrt [4]{c} (3 b c+5 a d)\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 )}{4 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}-\frac {\left (\sqrt [4]{c} (3 b c+5 a d)\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 )}{4 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}\\ &=\frac {(b c+a d) \sqrt {x}}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\sqrt [4]{a} (5 b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{a} (5 b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{c} (3 b c+5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{c} (3 b c+5 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{a} (5 b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{a} (5 b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{c} (3 b c+5 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{c} (3 b c+5 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}\\ \end {align*}

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Mathematica [A] time = 0.92, size = 585, normalized size = 0.94 \[ \frac {1}{16} \left (\frac {8 a \sqrt {x}}{\left (a+b x^2\right ) (b c-a d)^2}+\frac {8 c \sqrt {x}}{\left (c+d x^2\right ) (b c-a d)^2}+\frac {\sqrt {2} \sqrt [4]{a} (3 a d+5 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt {2} \sqrt [4]{a} (3 a d+5 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{b} (b c-a d)^3}+\frac {\sqrt {2} \sqrt [4]{c} (5 a d+3 b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{\sqrt [4]{d} (a d-b c)^3}-\frac {\sqrt {2} \sqrt [4]{c} (5 a d+3 b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{\sqrt [4]{d} (a d-b c)^3}+\frac {2 \sqrt {2} \sqrt [4]{a} (3 a d+5 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{b} (b c-a d)^3}-\frac {2 \sqrt {2} \sqrt [4]{a} (3 a d+5 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{b} (b c-a d)^3}+\frac {2 \sqrt {2} \sqrt [4]{c} (5 a d+3 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt [4]{d} (a d-b c)^3}+\frac {2 \sqrt {2} \sqrt [4]{c} (5 a d+3 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt [4]{d} (b c-a d)^3}\right ) \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^2),x]

[Out]

((8*a*Sqrt[x])/((b*c - a*d)^2*(a + b*x^2)) + (8*c*Sqrt[x])/((b*c - a*d)^2*(c + d*x^2)) + (2*Sqrt[2]*a^(1/4)*(5

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

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

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

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

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

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

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

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

*c) + a*d)^3))/16

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fricas [B] time = 89.94, size = 5375, normalized size = 8.61 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="fricas")

[Out]

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

*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(625*a*b^4*c^4 + 1500*a^2*b^3*c^3*d + 1350*a^3*b^2*c^2*d^2 + 540*a^4*b*

c*d^3 + 81*a^5*d^4)/(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 220*a^3*b^10*c^9*d^3 + 495*a^4*b^9*

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

c^3*d^9 + 66*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12*b*d^12))^(1/4)*arctan(-((b^10*c^9 - 9*a*b^9*c^8*d +

36*a^2*b^8*c^7*d^2 - 84*a^3*b^7*c^6*d^3 + 126*a^4*b^6*c^5*d^4 - 126*a^5*b^5*c^4*d^5 + 84*a^6*b^4*c^3*d^6 - 36

*a^7*b^3*c^2*d^7 + 9*a^8*b^2*c*d^8 - a^9*b*d^9)*sqrt((25*b^2*c^2 + 30*a*b*c*d + 9*a^2*d^2)*x + (b^6*c^6 - 6*a*

b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*sqrt(-(625

*a*b^4*c^4 + 1500*a^2*b^3*c^3*d + 1350*a^3*b^2*c^2*d^2 + 540*a^4*b*c*d^3 + 81*a^5*d^4)/(b^13*c^12 - 12*a*b^12*

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

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

2*c*d^11 + a^12*b*d^12)))*(-(625*a*b^4*c^4 + 1500*a^2*b^3*c^3*d + 1350*a^3*b^2*c^2*d^2 + 540*a^4*b*c*d^3 + 81*

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

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

6*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12*b*d^12))^(3/4) - (5*b^11*c^10 - 42*a*b^10*c^9*d + 153*a^2*b^9*

c^8*d^2 - 312*a^3*b^8*c^7*d^3 + 378*a^4*b^7*c^6*d^4 - 252*a^5*b^6*c^5*d^5 + 42*a^6*b^5*c^4*d^6 + 72*a^7*b^4*c^

3*d^7 - 63*a^8*b^3*c^2*d^8 + 22*a^9*b^2*c*d^9 - 3*a^10*b*d^10)*sqrt(x)*(-(625*a*b^4*c^4 + 1500*a^2*b^3*c^3*d +

1350*a^3*b^2*c^2*d^2 + 540*a^4*b*c*d^3 + 81*a^5*d^4)/(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 2

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

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

625*a*b^4*c^4 + 1500*a^2*b^3*c^3*d + 1350*a^3*b^2*c^2*d^2 + 540*a^4*b*c*d^3 + 81*a^5*d^4)) - 4*(a*b^2*c^3 - 2*

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

a^3*d^3)*x^2)*(-(81*b^4*c^5 + 540*a*b^3*c^4*d + 1350*a^2*b^2*c^3*d^2 + 1500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b

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

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

b^2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^13))^(1/4)*arctan(-((b^9*c^9*d - 9*a*b^8*c^8*d^2 + 36*a^2*b^7*c^7*d^3

- 84*a^3*b^6*c^6*d^4 + 126*a^4*b^5*c^5*d^5 - 126*a^5*b^4*c^4*d^6 + 84*a^6*b^3*c^3*d^7 - 36*a^7*b^2*c^2*d^8 +

9*a^8*b*c*d^9 - a^9*d^10)*sqrt((9*b^2*c^2 + 30*a*b*c*d + 25*a^2*d^2)*x + (b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4

*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*sqrt(-(81*b^4*c^5 + 540*a*b^3*c^

4*d + 1350*a^2*b^2*c^3*d^2 + 1500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^

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

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

)))*(-(81*b^4*c^5 + 540*a*b^3*c^4*d + 1350*a^2*b^2*c^3*d^2 + 1500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^12*c^12*d

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

+ 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^1

1 - 12*a^11*b*c*d^12 + a^12*d^13))^(3/4) - (3*b^10*c^10*d - 22*a*b^9*c^9*d^2 + 63*a^2*b^8*c^8*d^3 - 72*a^3*b^7

*c^7*d^4 - 42*a^4*b^6*c^6*d^5 + 252*a^5*b^5*c^5*d^6 - 378*a^6*b^4*c^4*d^7 + 312*a^7*b^3*c^3*d^8 - 153*a^8*b^2*

c^2*d^9 + 42*a^9*b*c*d^10 - 5*a^10*d^11)*sqrt(x)*(-(81*b^4*c^5 + 540*a*b^3*c^4*d + 1350*a^2*b^2*c^3*d^2 + 1500

*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4

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

- 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^13))^(3/4))/(81*b^4*c^5 + 540*a*b^3

*c^4*d + 1350*a^2*b^2*c^3*d^2 + 1500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)) - (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2

+ (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(625*a*

b^4*c^4 + 1500*a^2*b^3*c^3*d + 1350*a^3*b^2*c^2*d^2 + 540*a^4*b*c*d^3 + 81*a^5*d^4)/(b^13*c^12 - 12*a*b^12*c^1

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

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

*d^11 + a^12*b*d^12))^(1/4)*log((5*b*c + 3*a*d)*sqrt(x) + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*

(-(625*a*b^4*c^4 + 1500*a^2*b^3*c^3*d + 1350*a^3*b^2*c^2*d^2 + 540*a^4*b*c*d^3 + 81*a^5*d^4)/(b^13*c^12 - 12*a

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

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

^11*b^2*c*d^11 + a^12*b*d^12))^(1/4)) + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 +

a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(625*a*b^4*c^4 + 1500*a^2*b^3*c^3*d +

1350*a^3*b^2*c^2*d^2 + 540*a^4*b*c*d^3 + 81*a^5*d^4)/(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 22

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

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

((5*b*c + 3*a*d)*sqrt(x) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(-(625*a*b^4*c^4 + 1500*a^2*b^3

*c^3*d + 1350*a^3*b^2*c^2*d^2 + 540*a^4*b*c*d^3 + 81*a^5*d^4)/(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10

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

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

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

^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(81*b^4*c^5 + 540*a*b^3*c^4*d + 1350*a^2*b^2*c^3*d^2 + 1500*a^3*b*c^2

*d^3 + 625*a^4*c*d^4)/(b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4

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

*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^13))^(1/4)*log((3*b*c + 5*a*d)*sqrt(x) + (b^3

*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(-(81*b^4*c^5 + 540*a*b^3*c^4*d + 1350*a^2*b^2*c^3*d^2 + 1500*

a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4

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

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

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

3)*x^2)*(-(81*b^4*c^5 + 540*a*b^3*c^4*d + 1350*a^2*b^2*c^3*d^2 + 1500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^12*c^1

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

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

*d^11 - 12*a^11*b*c*d^12 + a^12*d^13))^(1/4)*log((3*b*c + 5*a*d)*sqrt(x) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*

c*d^2 - a^3*d^3)*(-(81*b^4*c^5 + 540*a*b^3*c^4*d + 1350*a^2*b^2*c^3*d^2 + 1500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/

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

*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^1

0*b^2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^13))^(1/4)) + 4*((b*c + a*d)*x^2 + 2*a*c)*sqrt(x))/(a*b^2*c^3 - 2*a

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

a^3*d^3)*x^2)

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giac [A] time = 1.30, size = 912, normalized size = 1.46 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="giac")

[Out]

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

(1/4)*b*c + 3*(a*b^3)^(1/4)*a*d)*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) + 1/4*(3*(c*d^3)^(1/4)*b*c + 5*(

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

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

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

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

qrt(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) + 1/8*(5*(a*b^3)^(1/4)*b*c + 3*(a*b^3)^(1/4)*a*d)*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) + 1/8*(3*(c*d^3)^(

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

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

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

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

*c)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2))

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maple [A] time = 0.02, size = 740, normalized size = 1.19 \[ \frac {a^{2} d \sqrt {x}}{2 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right )}-\frac {a b c \sqrt {x}}{2 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right )}+\frac {a c d \sqrt {x}}{2 \left (a d -b c \right )^{3} \left (d \,x^{2}+c \right )}-\frac {b \,c^{2} \sqrt {x}}{2 \left (a d -b c \right )^{3} \left (d \,x^{2}+c \right )}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{3}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{3}}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{3}}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{3}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a d \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (a d -b c \right )^{3}}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a d \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{16 \left (a d -b c \right )^{3}}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{3}}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{3}}-\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{3}}-\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{3}}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (a d -b c \right )^{3}}-\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b c \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{16 \left (a d -b c \right )^{3}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)/(b*x^2+a)^2/(d*x^2+c)^2,x)

[Out]

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

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

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

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

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

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

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

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maxima [A] time = 2.52, size = 617, normalized size = 0.99 \[ -\frac {{\left (\frac {2 \, \sqrt {2} {\left (5 \, b c + 3 \, 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 (5 \, b c + 3 \, 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 (5 \, b c + 3 \, 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 (5 \, b c + 3 \, 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 )} a}{16 \, {\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 (\frac {2 \, \sqrt {2} {\left (3 \, b c + 5 \, a d\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 (3 \, b c + 5 \, a d\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 (3 \, b c + 5 \, a d\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 (3 \, b c + 5 \, a d\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}}}\right )} c}{16 \, {\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 (b c + a d\right )} x^{\frac {5}{2}} + 2 \, a c \sqrt {x}}{2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="maxima")

[Out]

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

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

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

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

t(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)*(3*

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

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

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

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

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mupad [B] time = 3.98, size = 34921, normalized size = 55.96 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^2),x)

[Out]

atan(((-(81*b^4*c^5 + 625*a^4*c*d^4 + 1500*a^3*b*c^2*d^3 + 1350*a^2*b^2*c^3*d^2 + 540*a*b^3*c^4*d)/(4096*a^12*

d^13 + 4096*b^12*c^12*d - 49152*a*b^11*c^11*d^2 + 270336*a^2*b^10*c^10*d^3 - 901120*a^3*b^9*c^9*d^4 + 2027520*

a^4*b^8*c^8*d^5 - 3244032*a^5*b^7*c^7*d^6 + 3784704*a^6*b^6*c^6*d^7 - 3244032*a^7*b^5*c^5*d^8 + 2027520*a^8*b^

4*c^4*d^9 - 901120*a^9*b^3*c^3*d^10 + 270336*a^10*b^2*c^2*d^11 - 49152*a^11*b*c*d^12))^(1/4)*((-(81*b^4*c^5 +

625*a^4*c*d^4 + 1500*a^3*b*c^2*d^3 + 1350*a^2*b^2*c^3*d^2 + 540*a*b^3*c^4*d)/(4096*a^12*d^13 + 4096*b^12*c^12*

d - 49152*a*b^11*c^11*d^2 + 270336*a^2*b^10*c^10*d^3 - 901120*a^3*b^9*c^9*d^4 + 2027520*a^4*b^8*c^8*d^5 - 3244

032*a^5*b^7*c^7*d^6 + 3784704*a^6*b^6*c^6*d^7 - 3244032*a^7*b^5*c^5*d^8 + 2027520*a^8*b^4*c^4*d^9 - 901120*a^9

*b^3*c^3*d^10 + 270336*a^10*b^2*c^2*d^11 - 49152*a^11*b*c*d^12))^(1/4)*(((405*a^2*b^9*c^8*d^3)/2 + 1674*a^3*b^

8*c^7*d^4 + (9843*a^4*b^7*c^6*d^5)/2 + 6884*a^5*b^6*c^5*d^6 + (9843*a^6*b^5*c^4*d^7)/2 + 1674*a^7*b^4*c^3*d^8

+ (405*a^8*b^3*c^2*d^9)/2)/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 -

56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7) + (-(81*b^4*c^5 + 625*a^4*c*d^4 + 15

00*a^3*b*c^2*d^3 + 1350*a^2*b^2*c^3*d^2 + 540*a*b^3*c^4*d)/(4096*a^12*d^13 + 4096*b^12*c^12*d - 49152*a*b^11*c

^11*d^2 + 270336*a^2*b^10*c^10*d^3 - 901120*a^3*b^9*c^9*d^4 + 2027520*a^4*b^8*c^8*d^5 - 3244032*a^5*b^7*c^7*d^

6 + 3784704*a^6*b^6*c^6*d^7 - 3244032*a^7*b^5*c^5*d^8 + 2027520*a^8*b^4*c^4*d^9 - 901120*a^9*b^3*c^3*d^10 + 27

0336*a^10*b^2*c^2*d^11 - 49152*a^11*b*c*d^12))^(3/4)*((x^(1/2)*(102400*a^2*b^19*c^17*d^4 - 1069056*a^3*b^18*c^

16*d^5 + 5001216*a^4*b^17*c^15*d^6 - 13799424*a^5*b^16*c^14*d^7 + 24858624*a^6*b^15*c^13*d^8 - 30412800*a^7*b^

14*c^12*d^9 + 24645632*a^8*b^13*c^11*d^10 - 9326592*a^9*b^12*c^10*d^11 - 9326592*a^10*b^11*c^9*d^12 + 24645632

*a^11*b^10*c^8*d^13 - 30412800*a^12*b^9*c^7*d^14 + 24858624*a^13*b^8*c^6*d^15 - 13799424*a^14*b^7*c^5*d^16 + 5

001216*a^15*b^6*c^4*d^17 - 1069056*a^16*b^5*c^3*d^18 + 102400*a^17*b^4*c^2*d^19))/(16*(a^12*d^12 + b^12*c^12 +

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

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

12*a^11*b*c*d^11)) + ((-(81*b^4*c^5 + 625*a^4*c*d^4 + 1500*a^3*b*c^2*d^3 + 1350*a^2*b^2*c^3*d^2 + 540*a*b^3*c^

4*d)/(4096*a^12*d^13 + 4096*b^12*c^12*d - 49152*a*b^11*c^11*d^2 + 270336*a^2*b^10*c^10*d^3 - 901120*a^3*b^9*c^

9*d^4 + 2027520*a^4*b^8*c^8*d^5 - 3244032*a^5*b^7*c^7*d^6 + 3784704*a^6*b^6*c^6*d^7 - 3244032*a^7*b^5*c^5*d^8

+ 2027520*a^8*b^4*c^4*d^9 - 901120*a^9*b^3*c^3*d^10 + 270336*a^10*b^2*c^2*d^11 - 49152*a^11*b*c*d^12))^(1/4)*(

10240*a^2*b^17*c^15*d^4 - 112640*a^3*b^16*c^14*d^5 + 552960*a^4*b^15*c^13*d^6 - 1576960*a^5*b^14*c^12*d^7 + 28

16000*a^6*b^13*c^11*d^8 - 3041280*a^7*b^12*c^10*d^9 + 1351680*a^8*b^11*c^9*d^10 + 1351680*a^9*b^10*c^8*d^11 -

3041280*a^10*b^9*c^7*d^12 + 2816000*a^11*b^8*c^6*d^13 - 1576960*a^12*b^7*c^5*d^14 + 552960*a^13*b^6*c^4*d^15 -

112640*a^14*b^5*c^3*d^16 + 10240*a^15*b^4*c^2*d^17))/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5

*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7)))*1i + (x

^(1/2)*(2025*a^2*b^11*c^10*d^3 + 15930*a^3*b^10*c^9*d^4 + 56304*a^4*b^9*c^8*d^5 + 115110*a^5*b^8*c^7*d^6 + 145

550*a^6*b^7*c^6*d^7 + 115110*a^7*b^6*c^5*d^8 + 56304*a^8*b^5*c^4*d^9 + 15930*a^9*b^4*c^3*d^10 + 2025*a^10*b^3*

c^2*d^11)*1i)/(16*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 -

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

66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11))) - (-(81*b^4*c^5 + 625*a^4*c*d^4 + 1500*a^3*b*c^2

*d^3 + 1350*a^2*b^2*c^3*d^2 + 540*a*b^3*c^4*d)/(4096*a^12*d^13 + 4096*b^12*c^12*d - 49152*a*b^11*c^11*d^2 + 27

0336*a^2*b^10*c^10*d^3 - 901120*a^3*b^9*c^9*d^4 + 2027520*a^4*b^8*c^8*d^5 - 3244032*a^5*b^7*c^7*d^6 + 3784704*

a^6*b^6*c^6*d^7 - 3244032*a^7*b^5*c^5*d^8 + 2027520*a^8*b^4*c^4*d^9 - 901120*a^9*b^3*c^3*d^10 + 270336*a^10*b^

2*c^2*d^11 - 49152*a^11*b*c*d^12))^(1/4)*((-(81*b^4*c^5 + 625*a^4*c*d^4 + 1500*a^3*b*c^2*d^3 + 1350*a^2*b^2*c^

3*d^2 + 540*a*b^3*c^4*d)/(4096*a^12*d^13 + 4096*b^12*c^12*d - 49152*a*b^11*c^11*d^2 + 270336*a^2*b^10*c^10*d^3

- 901120*a^3*b^9*c^9*d^4 + 2027520*a^4*b^8*c^8*d^5 - 3244032*a^5*b^7*c^7*d^6 + 3784704*a^6*b^6*c^6*d^7 - 3244

032*a^7*b^5*c^5*d^8 + 2027520*a^8*b^4*c^4*d^9 - 901120*a^9*b^3*c^3*d^10 + 270336*a^10*b^2*c^2*d^11 - 49152*a^1

1*b*c*d^12))^(1/4)*(((405*a^2*b^9*c^8*d^3)/2 + 1674*a^3*b^8*c^7*d^4 + (9843*a^4*b^7*c^6*d^5)/2 + 6884*a^5*b^6*

c^5*d^6 + (9843*a^6*b^5*c^4*d^7)/2 + 1674*a^7*b^4*c^3*d^8 + (405*a^8*b^3*c^2*d^9)/2)/(a^8*d^8 + b^8*c^8 + 28*a

^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c

^7*d - 8*a^7*b*c*d^7) - (-(81*b^4*c^5 + 625*a^4*c*d^4 + 1500*a^3*b*c^2*d^3 + 1350*a^2*b^2*c^3*d^2 + 540*a*b^3*

c^4*d)/(4096*a^12*d^13 + 4096*b^12*c^12*d - 49152*a*b^11*c^11*d^2 + 270336*a^2*b^10*c^10*d^3 - 901120*a^3*b^9*

c^9*d^4 + 2027520*a^4*b^8*c^8*d^5 - 3244032*a^5*b^7*c^7*d^6 + 3784704*a^6*b^6*c^6*d^7 - 3244032*a^7*b^5*c^5*d^

8 + 2027520*a^8*b^4*c^4*d^9 - 901120*a^9*b^3*c^3*d^10 + 270336*a^10*b^2*c^2*d^11 - 49152*a^11*b*c*d^12))^(3/4)

*((x^(1/2)*(102400*a^2*b^19*c^17*d^4 - 1069056*a^3*b^18*c^16*d^5 + 5001216*a^4*b^17*c^15*d^6 - 13799424*a^5*b^

16*c^14*d^7 + 24858624*a^6*b^15*c^13*d^8 - 30412800*a^7*b^14*c^12*d^9 + 24645632*a^8*b^13*c^11*d^10 - 9326592*

a^9*b^12*c^10*d^11 - 9326592*a^10*b^11*c^9*d^12 + 24645632*a^11*b^10*c^8*d^13 - 30412800*a^12*b^9*c^7*d^14 + 2

4858624*a^13*b^8*c^6*d^15 - 13799424*a^14*b^7*c^5*d^16 + 5001216*a^15*b^6*c^4*d^17 - 1069056*a^16*b^5*c^3*d^18

+ 102400*a^17*b^4*c^2*d^19))/(16*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^

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

9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)) - ((-(81*b^4*c^5 + 625*a^4*c*d^4

+ 1500*a^3*b*c^2*d^3 + 1350*a^2*b^2*c^3*d^2 + 540*a*b^3*c^4*d)/(4096*a^12*d^13 + 4096*b^12*c^12*d - 49152*a*b^

11*c^11*d^2 + 270336*a^2*b^10*c^10*d^3 - 901120*a^3*b^9*c^9*d^4 + 2027520*a^4*b^8*c^8*d^5 - 3244032*a^5*b^7*c^

7*d^6 + 3784704*a^6*b^6*c^6*d^7 - 3244032*a^7*b^5*c^5*d^8 + 2027520*a^8*b^4*c^4*d^9 - 901120*a^9*b^3*c^3*d^10

+ 270336*a^10*b^2*c^2*d^11 - 49152*a^11*b*c*d^12))^(1/4)*(10240*a^2*b^17*c^15*d^4 - 112640*a^3*b^16*c^14*d^5 +

552960*a^4*b^15*c^13*d^6 - 1576960*a^5*b^14*c^12*d^7 + 2816000*a^6*b^13*c^11*d^8 - 3041280*a^7*b^12*c^10*d^9

+ 1351680*a^8*b^11*c^9*d^10 + 1351680*a^9*b^10*c^8*d^11 - 3041280*a^10*b^9*c^7*d^12 + 2816000*a^11*b^8*c^6*d^1

3 - 1576960*a^12*b^7*c^5*d^14 + 552960*a^13*b^6*c^4*d^15 - 112640*a^14*b^5*c^3*d^16 + 10240*a^15*b^4*c^2*d^17)

)/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*

a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7)))*1i - (x^(1/2)*(2025*a^2*b^11*c^10*d^3 + 15930*a^3*b^10*c^9*

d^4 + 56304*a^4*b^9*c^8*d^5 + 115110*a^5*b^8*c^7*d^6 + 145550*a^6*b^7*c^6*d^7 + 115110*a^7*b^6*c^5*d^8 + 56304

*a^8*b^5*c^4*d^9 + 15930*a^9*b^4*c^3*d^10 + 2025*a^10*b^3*c^2*d^11)*1i)/(16*(a^12*d^12 + b^12*c^12 + 66*a^2*b^

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

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

c*d^11))))/((-(81*b^4*c^5 + 625*a^4*c*d^4 + 1500*a^3*b*c^2*d^3 + 1350*a^2*b^2*c^3*d^2 + 540*a*b^3*c^4*d)/(4096

*a^12*d^13 + 4096*b^12*c^12*d - 49152*a*b^11*c^11*d^2 + 270336*a^2*b^10*c^10*d^3 - 901120*a^3*b^9*c^9*d^4 + 20

27520*a^4*b^8*c^8*d^5 - 3244032*a^5*b^7*c^7*d^6 + 3784704*a^6*b^6*c^6*d^7 - 3244032*a^7*b^5*c^5*d^8 + 2027520*

a^8*b^4*c^4*d^9 - 901120*a^9*b^3*c^3*d^10 + 270336*a^10*b^2*c^2*d^11 - 49152*a^11*b*c*d^12))^(1/4)*((-(81*b^4*

c^5 + 625*a^4*c*d^4 + 1500*a^3*b*c^2*d^3 + 1350*a^2*b^2*c^3*d^2 + 540*a*b^3*c^4*d)/(4096*a^12*d^13 + 4096*b^12

*c^12*d - 49152*a*b^11*c^11*d^2 + 270336*a^2*b^10*c^10*d^3 - 901120*a^3*b^9*c^9*d^4 + 2027520*a^4*b^8*c^8*d^5

- 3244032*a^5*b^7*c^7*d^6 + 3784704*a^6*b^6*c^6*d^7 - 3244032*a^7*b^5*c^5*d^8 + 2027520*a^8*b^4*c^4*d^9 - 9011

20*a^9*b^3*c^3*d^10 + 270336*a^10*b^2*c^2*d^11 - 49152*a^11*b*c*d^12))^(1/4)*(((405*a^2*b^9*c^8*d^3)/2 + 1674*

a^3*b^8*c^7*d^4 + (9843*a^4*b^7*c^6*d^5)/2 + 6884*a^5*b^6*c^5*d^6 + (9843*a^6*b^5*c^4*d^7)/2 + 1674*a^7*b^4*c^

3*d^8 + (405*a^8*b^3*c^2*d^9)/2)/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4

*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7) + (-(81*b^4*c^5 + 625*a^4*c*d^

4 + 1500*a^3*b*c^2*d^3 + 1350*a^2*b^2*c^3*d^2 + 540*a*b^3*c^4*d)/(4096*a^12*d^13 + 4096*b^12*c^12*d - 49152*a*

b^11*c^11*d^2 + 270336*a^2*b^10*c^10*d^3 - 901120*a^3*b^9*c^9*d^4 + 2027520*a^4*b^8*c^8*d^5 - 3244032*a^5*b^7*

c^7*d^6 + 3784704*a^6*b^6*c^6*d^7 - 3244032*a^7*b^5*c^5*d^8 + 2027520*a^8*b^4*c^4*d^9 - 901120*a^9*b^3*c^3*d^1

0 + 270336*a^10*b^2*c^2*d^11 - 49152*a^11*b*c*d^12))^(3/4)*((x^(1/2)*(102400*a^2*b^19*c^17*d^4 - 1069056*a^3*b

^18*c^16*d^5 + 5001216*a^4*b^17*c^15*d^6 - 13799424*a^5*b^16*c^14*d^7 + 24858624*a^6*b^15*c^13*d^8 - 30412800*

a^7*b^14*c^12*d^9 + 24645632*a^8*b^13*c^11*d^10 - 9326592*a^9*b^12*c^10*d^11 - 9326592*a^10*b^11*c^9*d^12 + 24

645632*a^11*b^10*c^8*d^13 - 30412800*a^12*b^9*c^7*d^14 + 24858624*a^13*b^8*c^6*d^15 - 13799424*a^14*b^7*c^5*d^

16 + 5001216*a^15*b^6*c^4*d^17 - 1069056*a^16*b^5*c^3*d^18 + 102400*a^17*b^4*c^2*d^19))/(16*(a^12*d^12 + b^12*

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

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

1*d - 12*a^11*b*c*d^11)) + ((-(81*b^4*c^5 + 625*a^4*c*d^4 + 1500*a^3*b*c^2*d^3 + 1350*a^2*b^2*c^3*d^2 + 540*a*

b^3*c^4*d)/(4096*a^12*d^13 + 4096*b^12*c^12*d - 49152*a*b^11*c^11*d^2 + 270336*a^2*b^10*c^10*d^3 - 901120*a^3*

b^9*c^9*d^4 + 2027520*a^4*b^8*c^8*d^5 - 3244032*a^5*b^7*c^7*d^6 + 3784704*a^6*b^6*c^6*d^7 - 3244032*a^7*b^5*c^

5*d^8 + 2027520*a^8*b^4*c^4*d^9 - 901120*a^9*b^3*c^3*d^10 + 270336*a^10*b^2*c^2*d^11 - 49152*a^11*b*c*d^12))^(

1/4)*(10240*a^2*b^17*c^15*d^4 - 112640*a^3*b^16*c^14*d^5 + 552960*a^4*b^15*c^13*d^6 - 1576960*a^5*b^14*c^12*d^

7 + 2816000*a^6*b^13*c^11*d^8 - 3041280*a^7*b^12*c^10*d^9 + 1351680*a^8*b^11*c^9*d^10 + 1351680*a^9*b^10*c^8*d

^11 - 3041280*a^10*b^9*c^7*d^12 + 2816000*a^11*b^8*c^6*d^13 - 1576960*a^12*b^7*c^5*d^14 + 552960*a^13*b^6*c^4*

d^15 - 112640*a^14*b^5*c^3*d^16 + 10240*a^15*b^4*c^2*d^17))/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b

^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))) +

(x^(1/2)*(2025*a^2*b^11*c^10*d^3 + 15930*a^3*b^10*c^9*d^4 + 56304*a^4*b^9*c^8*d^5 + 115110*a^5*b^8*c^7*d^6 +

145550*a^6*b^7*c^6*d^7 + 115110*a^7*b^6*c^5*d^8 + 56304*a^8*b^5*c^4*d^9 + 15930*a^9*b^4*c^3*d^10 + 2025*a^10*b

^3*c^2*d^11))/(16*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 -

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

66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11))) + (-(81*b^4*c^5 + 625*a^4*c*d^4 + 1500*a^3*b*c^2