3.5.89 \(\int \frac {x^{5/2}}{(a+b x^2)^2 (c+d x^2)^2} \, dx\) [489]

Optimal. Leaf size=609 \[ -\frac {d x^{3/2}}{(b c-a d)^2 \left (c+d x^2\right )}-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\sqrt [4]{b} (3 b c+5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}+\frac {\sqrt [4]{b} (3 b c+5 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}+\frac {\sqrt [4]{d} (5 b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{c} (b c-a d)^3}-\frac {\sqrt [4]{d} (5 b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{c} (b c-a d)^3}+\frac {\sqrt [4]{b} (3 b c+5 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}-\frac {\sqrt [4]{b} (3 b c+5 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}-\frac {\sqrt [4]{d} (5 b c+3 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} (b c-a d)^3}+\frac {\sqrt [4]{d} (5 b c+3 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} (b c-a d)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.52, antiderivative size = 609, 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 = {477, 482, 593, 598, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\sqrt [4]{b} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (5 a d+3 b c)}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}+\frac {\sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (5 a d+3 b c)}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}+\frac {\sqrt [4]{d} (3 a d+5 b c) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{c} (b c-a d)^3}-\frac {\sqrt [4]{d} (3 a d+5 b c) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} \sqrt [4]{c} (b c-a d)^3}-\frac {x^{3/2}}{2 \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {d x^{3/2}}{\left (c+d x^2\right ) (b c-a d)^2}+\frac {\sqrt [4]{b} (5 a d+3 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]{a} (b c-a d)^3}-\frac {\sqrt [4]{b} (5 a d+3 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]{a} (b c-a d)^3}-\frac {\sqrt [4]{d} (3 a d+5 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]{c} (b c-a d)^3}+\frac {\sqrt [4]{d} (3 a d+5 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]{c} (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 303

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 477

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 482

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 593

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 598

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 631

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 642

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 1176

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 1179

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 )^2 \left (c+d x^2\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {x^6}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {x^2 \left (3 c-5 d x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)}\\ &=-\frac {d x^{3/2}}{(b c-a d)^2 \left (c+d x^2\right )}-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {x^2 \left (12 c (b c+a d)-8 b c d x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{8 c (b c-a d)^2}\\ &=-\frac {d x^{3/2}}{(b c-a d)^2 \left (c+d x^2\right )}-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \left (\frac {4 b c (3 b c+5 a d) x^2}{(b c-a d) \left (a+b x^4\right )}-\frac {4 c d (5 b c+3 a d) x^2}{(b c-a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{8 c (b c-a d)^2}\\ &=-\frac {d x^{3/2}}{(b c-a d)^2 \left (c+d x^2\right )}-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {(d (5 b c+3 a d)) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)^3}+\frac {(b (3 b c+5 a d)) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)^3}\\ &=-\frac {d x^{3/2}}{(b c-a d)^2 \left (c+d x^2\right )}-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\left (\sqrt {d} (5 b c+3 a d)\right ) \text {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 {d} (5 b c+3 a d)\right ) \text {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 {b} (3 b c+5 a d)\right ) \text {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 {b} (3 b c+5 a d)\right ) \text {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 {d x^{3/2}}{(b c-a d)^2 \left (c+d x^2\right )}-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {(5 b c+3 a d) \text {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 (b c-a d)^3}-\frac {(5 b c+3 a d) \text {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 (b c-a d)^3}-\frac {\left (\sqrt [4]{d} (5 b c+3 a d)\right ) \text {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]{c} (b c-a d)^3}-\frac {\left (\sqrt [4]{d} (5 b c+3 a d)\right ) \text {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]{c} (b c-a d)^3}+\frac {(3 b c+5 a d) \text {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 (b c-a d)^3}+\frac {(3 b c+5 a d) \text {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 (b c-a d)^3}+\frac {\left (\sqrt [4]{b} (3 b c+5 a d)\right ) \text {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]{a} (b c-a d)^3}+\frac {\left (\sqrt [4]{b} (3 b c+5 a d)\right ) \text {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]{a} (b c-a d)^3}\\ &=-\frac {d x^{3/2}}{(b c-a d)^2 \left (c+d x^2\right )}-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\sqrt [4]{b} (3 b c+5 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]{a} (b c-a d)^3}-\frac {\sqrt [4]{b} (3 b c+5 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]{a} (b c-a d)^3}-\frac {\sqrt [4]{d} (5 b c+3 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]{c} (b c-a d)^3}+\frac {\sqrt [4]{d} (5 b c+3 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]{c} (b c-a d)^3}-\frac {\left (\sqrt [4]{d} (5 b c+3 a d)\right ) \text {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]{c} (b c-a d)^3}+\frac {\left (\sqrt [4]{d} (5 b c+3 a d)\right ) \text {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]{c} (b c-a d)^3}+\frac {\left (\sqrt [4]{b} (3 b c+5 a d)\right ) \text {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]{a} (b c-a d)^3}-\frac {\left (\sqrt [4]{b} (3 b c+5 a d)\right ) \text {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]{a} (b c-a d)^3}\\ &=-\frac {d x^{3/2}}{(b c-a d)^2 \left (c+d x^2\right )}-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\sqrt [4]{b} (3 b c+5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}+\frac {\sqrt [4]{b} (3 b c+5 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}+\frac {\sqrt [4]{d} (5 b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{c} (b c-a d)^3}-\frac {\sqrt [4]{d} (5 b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{c} (b c-a d)^3}+\frac {\sqrt [4]{b} (3 b c+5 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]{a} (b c-a d)^3}-\frac {\sqrt [4]{b} (3 b c+5 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]{a} (b c-a d)^3}-\frac {\sqrt [4]{d} (5 b c+3 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]{c} (b c-a d)^3}+\frac {\sqrt [4]{d} (5 b c+3 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]{c} (b c-a d)^3}\\ \end {align*}

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Mathematica [A]
time = 1.61, size = 340, normalized size = 0.56 \begin {gather*} \frac {1}{8} \left (-\frac {4 x^{3/2} \left (a d+b \left (c+2 d x^2\right )\right )}{(b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\sqrt {2} \sqrt [4]{b} (3 b c+5 a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{a} (-b c+a d)^3}+\frac {\sqrt {2} \sqrt [4]{d} (5 b c+3 a d) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{\sqrt [4]{c} (b c-a d)^3}+\frac {\sqrt {2} \sqrt [4]{b} (3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{a} (-b c+a d)^3}+\frac {\sqrt {2} \sqrt [4]{d} (5 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt [4]{c} (b c-a d)^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.12, size = 302, normalized size = 0.50

method	result	size
derivativedivides	\(-\frac {2 b \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {5 a d}{4}+\frac {3 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{3}}+\frac {2 d \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (\frac {5 b c}{4}+\frac {3 a d}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{3}}\)	\(302\)
default	\(-\frac {2 b \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {5 a d}{4}+\frac {3 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{3}}+\frac {2 d \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (\frac {5 b c}{4}+\frac {3 a d}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{3}}\)	\(302\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)/(b*x^2+a)^2/(d*x^2+c)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.51, size = 567, normalized size = 0.93 \begin {gather*} \frac {{\left (3 \, b^{2} c + 5 \, a b d\right )} {\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 )}}{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 (5 \, b c d + 3 \, a 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 )}}{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 {2 \, b d x^{\frac {7}{2}} + {\left (b c + a d\right )} x^{\frac {3}{2}}}{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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(3*b^2*c + 5*a*b*d)*(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*sqr
t(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)*sqr
t(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 + sqr
t(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) - 1/16*(5*b*c*d + 3*a*d^2)*(2*sqr
t(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)*sqr
t(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
^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) - 1/2*(2*b*d*x^(7/2) + (b*c + a*d)*x^(3/2))/(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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5814 vs. \(2 (459) = 918\).
time = 63.14, size = 5814, normalized size = 9.55 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(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)*(-(81*b^5*c^4 + 540*a*b^4*c^3*d + 1350*a^2*b^3*c^2*d^2 + 1500*a^3*b^2*c*d
^3 + 625*a^4*b*d^4)/(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b
^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*
b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12))^(1/4)*arctan(-((b^3*c^3 - 3*a*b^2*c^2*d +
3*a^2*b*c*d^2 - a^3*d^3)*sqrt((729*b^8*c^6 + 7290*a*b^7*c^5*d + 30375*a^2*b^6*c^4*d^2 + 67500*a^3*b^5*c^3*d^3
+ 84375*a^4*b^4*c^2*d^4 + 56250*a^5*b^3*c*d^5 + 15625*a^6*b^2*d^6)*x - (81*a*b^11*c^10 + 54*a^2*b^10*c^9*d - 6
75*a^3*b^9*c^8*d^2 - 120*a^4*b^8*c^7*d^3 + 2290*a^5*b^7*c^6*d^4 - 636*a^6*b^6*c^5*d^5 - 3534*a^7*b^5*c^4*d^6 +
2440*a^8*b^4*c^3*d^7 + 1725*a^9*b^3*c^2*d^8 - 2250*a^10*b^2*c*d^9 + 625*a^11*b*d^10)*sqrt(-(81*b^5*c^4 + 540*
a*b^4*c^3*d + 1350*a^2*b^3*c^2*d^2 + 1500*a^3*b^2*c*d^3 + 625*a^4*b*d^4)/(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 6
6*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 -
792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a
^13*d^12)))*(-(81*b^5*c^4 + 540*a*b^4*c^3*d + 1350*a^2*b^3*c^2*d^2 + 1500*a^3*b^2*c*d^3 + 625*a^4*b*d^4)/(a*b^
12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*
c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2
*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12))^(1/4) - (27*b^7*c^6 + 54*a*b^6*c^5*d - 99*a^2*b^5*c^4*d^2 - 172*a^3
*b^4*c^3*d^3 + 165*a^4*b^3*c^2*d^4 + 150*a^5*b^2*c*d^5 - 125*a^6*b*d^6)*sqrt(x)*(-(81*b^5*c^4 + 540*a*b^4*c^3*
d + 1350*a^2*b^3*c^2*d^2 + 1500*a^3*b^2*c*d^3 + 625*a^4*b*d^4)/(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10
*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^
5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12))
^(1/4))/(81*b^5*c^4 + 540*a*b^4*c^3*d + 1350*a^2*b^3*c^2*d^2 + 1500*a^3*b^2*c*d^3 + 625*a^4*b*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)*(-(625*b^4*c^4*d + 1500*a*b^3*c^3*d^2 + 1350*a^2*b^2*c^2*d^3 + 540*a^3*b*c*d^4 + 81*a^
4*d^5)/(b^12*c^13 - 12*a*b^11*c^12*d + 66*a^2*b^10*c^11*d^2 - 220*a^3*b^9*c^10*d^3 + 495*a^4*b^8*c^9*d^4 - 792
*a^5*b^7*c^8*d^5 + 924*a^6*b^6*c^7*d^6 - 792*a^7*b^5*c^6*d^7 + 495*a^8*b^4*c^5*d^8 - 220*a^9*b^3*c^4*d^9 + 66*
a^10*b^2*c^3*d^10 - 12*a^11*b*c^2*d^11 + a^12*c*d^12))^(1/4)*arctan(-((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2
- a^3*d^3)*sqrt((15625*b^6*c^6*d^2 + 56250*a*b^5*c^5*d^3 + 84375*a^2*b^4*c^4*d^4 + 67500*a^3*b^3*c^3*d^5 + 30
375*a^4*b^2*c^2*d^6 + 7290*a^5*b*c*d^7 + 729*a^6*d^8)*x - (625*b^10*c^11*d - 2250*a*b^9*c^10*d^2 + 1725*a^2*b^
8*c^9*d^3 + 2440*a^3*b^7*c^8*d^4 - 3534*a^4*b^6*c^7*d^5 - 636*a^5*b^5*c^6*d^6 + 2290*a^6*b^4*c^5*d^7 - 120*a^7
*b^3*c^4*d^8 - 675*a^8*b^2*c^3*d^9 + 54*a^9*b*c^2*d^10 + 81*a^10*c*d^11)*sqrt(-(625*b^4*c^4*d + 1500*a*b^3*c^3
*d^2 + 1350*a^2*b^2*c^2*d^3 + 540*a^3*b*c*d^4 + 81*a^4*d^5)/(b^12*c^13 - 12*a*b^11*c^12*d + 66*a^2*b^10*c^11*d
^2 - 220*a^3*b^9*c^10*d^3 + 495*a^4*b^8*c^9*d^4 - 792*a^5*b^7*c^8*d^5 + 924*a^6*b^6*c^7*d^6 - 792*a^7*b^5*c^6*
d^7 + 495*a^8*b^4*c^5*d^8 - 220*a^9*b^3*c^4*d^9 + 66*a^10*b^2*c^3*d^10 - 12*a^11*b*c^2*d^11 + a^12*c*d^12)))*(
-(625*b^4*c^4*d + 1500*a*b^3*c^3*d^2 + 1350*a^2*b^2*c^2*d^3 + 540*a^3*b*c*d^4 + 81*a^4*d^5)/(b^12*c^13 - 12*a*
b^11*c^12*d + 66*a^2*b^10*c^11*d^2 - 220*a^3*b^9*c^10*d^3 + 495*a^4*b^8*c^9*d^4 - 792*a^5*b^7*c^8*d^5 + 924*a^
6*b^6*c^7*d^6 - 792*a^7*b^5*c^6*d^7 + 495*a^8*b^4*c^5*d^8 - 220*a^9*b^3*c^4*d^9 + 66*a^10*b^2*c^3*d^10 - 12*a^
11*b*c^2*d^11 + a^12*c*d^12))^(1/4) - (125*b^6*c^6*d - 150*a*b^5*c^5*d^2 - 165*a^2*b^4*c^4*d^3 + 172*a^3*b^3*c
^3*d^4 + 99*a^4*b^2*c^2*d^5 - 54*a^5*b*c*d^6 - 27*a^6*d^7)*sqrt(x)*(-(625*b^4*c^4*d + 1500*a*b^3*c^3*d^2 + 135
0*a^2*b^2*c^2*d^3 + 540*a^3*b*c*d^4 + 81*a^4*d^5)/(b^12*c^13 - 12*a*b^11*c^12*d + 66*a^2*b^10*c^11*d^2 - 220*a
^3*b^9*c^10*d^3 + 495*a^4*b^8*c^9*d^4 - 792*a^5*b^7*c^8*d^5 + 924*a^6*b^6*c^7*d^6 - 792*a^7*b^5*c^6*d^7 + 495*
a^8*b^4*c^5*d^8 - 220*a^9*b^3*c^4*d^9 + 66*a^10*b^2*c^3*d^10 - 12*a^11*b*c^2*d^11 + a^12*c*d^12))^(1/4))/(625*
b^4*c^4*d + 1500*a*b^3*c^3*d^2 + 1350*a^2*b^2*c^2*d^3 + 540*a^3*b*c*d^4 + 81*a^4*d^5)) + (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^5*c^4 + 540*a*b^4*c^3*d + 1350*a^2*b^3*c^2*d^2 + 1500*a^3*b^2*c*d^3 + 625*a^4*b*d^4)/(a*b^12*
c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7
*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)/(b*x**2+a)**2/(d*x**2+c)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 952 vs. \(2 (459) = 918\).
time = 1.11, size = 952, normalized size = 1.56 \begin {gather*} \frac {{\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} b c + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} a d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} a b^{5} c^{3} - 3 \, \sqrt {2} a^{2} b^{4} c^{2} d + 3 \, \sqrt {2} a^{3} b^{3} c d^{2} - \sqrt {2} a^{4} b^{2} d^{3}\right )}} + \frac {{\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} b c + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} a d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} a b^{5} c^{3} - 3 \, \sqrt {2} a^{2} b^{4} c^{2} d + 3 \, \sqrt {2} a^{3} b^{3} c d^{2} - \sqrt {2} a^{4} b^{2} d^{3}\right )}} - \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} b^{3} c^{4} d^{2} - 3 \, \sqrt {2} a b^{2} c^{3} d^{3} + 3 \, \sqrt {2} a^{2} b c^{2} d^{4} - \sqrt {2} a^{3} c d^{5}\right )}} - \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} b^{3} c^{4} d^{2} - 3 \, \sqrt {2} a b^{2} c^{3} d^{3} + 3 \, \sqrt {2} a^{2} b c^{2} d^{4} - \sqrt {2} a^{3} c d^{5}\right )}} - \frac {{\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} b c + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{8 \, {\left (\sqrt {2} a b^{5} c^{3} - 3 \, \sqrt {2} a^{2} b^{4} c^{2} d + 3 \, \sqrt {2} a^{3} b^{3} c d^{2} - \sqrt {2} a^{4} b^{2} d^{3}\right )}} + \frac {{\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} b c + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{8 \, {\left (\sqrt {2} a b^{5} c^{3} - 3 \, \sqrt {2} a^{2} b^{4} c^{2} d + 3 \, \sqrt {2} a^{3} b^{3} c d^{2} - \sqrt {2} a^{4} b^{2} d^{3}\right )}} + \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{3} c^{4} d^{2} - 3 \, \sqrt {2} a b^{2} c^{3} d^{3} + 3 \, \sqrt {2} a^{2} b c^{2} d^{4} - \sqrt {2} a^{3} c d^{5}\right )}} - \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{3} c^{4} d^{2} - 3 \, \sqrt {2} a b^{2} c^{3} d^{3} + 3 \, \sqrt {2} a^{2} b c^{2} d^{4} - \sqrt {2} a^{3} c d^{5}\right )}} - \frac {2 \, b d x^{\frac {7}{2}} + b c x^{\frac {3}{2}} + a d x^{\frac {3}{2}}}{2 \, {\left (b d x^{4} + b c x^{2} + a d x^{2} + a c\right )} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(3*(a*b^3)^(3/4)*b*c + 5*(a*b^3)^(3/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/
4))/(sqrt(2)*a*b^5*c^3 - 3*sqrt(2)*a^2*b^4*c^2*d + 3*sqrt(2)*a^3*b^3*c*d^2 - sqrt(2)*a^4*b^2*d^3) + 1/4*(3*(a*
b^3)^(3/4)*b*c + 5*(a*b^3)^(3/4)*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt
(2)*a*b^5*c^3 - 3*sqrt(2)*a^2*b^4*c^2*d + 3*sqrt(2)*a^3*b^3*c*d^2 - sqrt(2)*a^4*b^2*d^3) - 1/4*(5*(c*d^3)^(3/4
)*b*c + 3*(c*d^3)^(3/4)*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^
4*d^2 - 3*sqrt(2)*a*b^2*c^3*d^3 + 3*sqrt(2)*a^2*b*c^2*d^4 - sqrt(2)*a^3*c*d^5) - 1/4*(5*(c*d^3)^(3/4)*b*c + 3*
(c*d^3)^(3/4)*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^4*d^2 - 3
*sqrt(2)*a*b^2*c^3*d^3 + 3*sqrt(2)*a^2*b*c^2*d^4 - sqrt(2)*a^3*c*d^5) - 1/8*(3*(a*b^3)^(3/4)*b*c + 5*(a*b^3)^(
3/4)*a*d)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a*b^5*c^3 - 3*sqrt(2)*a^2*b^4*c^2*d + 3*sq
rt(2)*a^3*b^3*c*d^2 - sqrt(2)*a^4*b^2*d^3) + 1/8*(3*(a*b^3)^(3/4)*b*c + 5*(a*b^3)^(3/4)*a*d)*log(-sqrt(2)*sqrt
(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a*b^5*c^3 - 3*sqrt(2)*a^2*b^4*c^2*d + 3*sqrt(2)*a^3*b^3*c*d^2 - sqrt
(2)*a^4*b^2*d^3) + 1/8*(5*(c*d^3)^(3/4)*b*c + 3*(c*d^3)^(3/4)*a*d)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(
c/d))/(sqrt(2)*b^3*c^4*d^2 - 3*sqrt(2)*a*b^2*c^3*d^3 + 3*sqrt(2)*a^2*b*c^2*d^4 - sqrt(2)*a^3*c*d^5) - 1/8*(5*(
c*d^3)^(3/4)*b*c + 3*(c*d^3)^(3/4)*a*d)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^3*c^4*d^2
- 3*sqrt(2)*a*b^2*c^3*d^3 + 3*sqrt(2)*a^2*b*c^2*d^4 - sqrt(2)*a^3*c*d^5) - 1/2*(2*b*d*x^(7/2) + b*c*x^(3/2) +
a*d*x^(3/2))/((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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Mupad [B]
time = 2.11, size = 2500, normalized size = 4.11 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atan((((-(81*b^5*c^4 + 625*a^4*b*d^4 + 1500*a^3*b^2*c*d^3 + 1350*a^2*b^3*c^2*d^2 + 540*a*b^4*c^3*d)/(4096*a^
13*d^12 + 4096*a*b^12*c^12 - 49152*a^2*b^11*c^11*d + 270336*a^3*b^10*c^10*d^2 - 901120*a^4*b^9*c^9*d^3 + 20275
20*a^5*b^8*c^8*d^4 - 3244032*a^6*b^7*c^7*d^5 + 3784704*a^7*b^6*c^6*d^6 - 3244032*a^8*b^5*c^5*d^7 + 2027520*a^9
*b^4*c^4*d^8 - 901120*a^10*b^3*c^3*d^9 + 270336*a^11*b^2*c^2*d^10 - 49152*a^12*b*c*d^11))^(3/4)*(((864*a*b^20*
c^17*d^4 + 864*a^17*b^4*c*d^20 - 5184*a^2*b^19*c^16*d^5 + 3200*a^3*b^18*c^15*d^6 + 56640*a^4*b^17*c^14*d^7 - 2
20800*a^5*b^16*c^13*d^8 + 369088*a^6*b^15*c^12*d^9 - 240768*a^7*b^14*c^11*d^10 - 158400*a^8*b^13*c^10*d^11 + 3
90720*a^9*b^12*c^9*d^12 - 158400*a^10*b^11*c^8*d^13 - 240768*a^11*b^10*c^7*d^14 + 369088*a^12*b^9*c^6*d^15 - 2
20800*a^13*b^8*c^5*d^16 + 56640*a^14*b^7*c^4*d^17 + 3200*a^15*b^6*c^3*d^18 - 5184*a^16*b^5*c^2*d^19)*1i)/(a^14
*d^14 + b^14*c^14 + 91*a^2*b^12*c^12*d^2 - 364*a^3*b^11*c^11*d^3 + 1001*a^4*b^10*c^10*d^4 - 2002*a^5*b^9*c^9*d
^5 + 3003*a^6*b^8*c^8*d^6 - 3432*a^7*b^7*c^7*d^7 + 3003*a^8*b^6*c^6*d^8 - 2002*a^9*b^5*c^5*d^9 + 1001*a^10*b^4
*c^4*d^10 - 364*a^11*b^3*c^3*d^11 + 91*a^12*b^2*c^2*d^12 - 14*a*b^13*c^13*d - 14*a^13*b*c*d^13) - (x^(1/2)*(-(
81*b^5*c^4 + 625*a^4*b*d^4 + 1500*a^3*b^2*c*d^3 + 1350*a^2*b^3*c^2*d^2 + 540*a*b^4*c^3*d)/(4096*a^13*d^12 + 40
96*a*b^12*c^12 - 49152*a^2*b^11*c^11*d + 270336*a^3*b^10*c^10*d^2 - 901120*a^4*b^9*c^9*d^3 + 2027520*a^5*b^8*c
^8*d^4 - 3244032*a^6*b^7*c^7*d^5 + 3784704*a^7*b^6*c^6*d^6 - 3244032*a^8*b^5*c^5*d^7 + 2027520*a^9*b^4*c^4*d^8
- 901120*a^10*b^3*c^3*d^9 + 270336*a^11*b^2*c^2*d^10 - 49152*a^12*b*c*d^11))^(1/4)*(36864*a*b^20*c^17*d^4 + 3
6864*a^17*b^4*c*d^20 - 319488*a^2*b^19*c^16*d^5 + 1163264*a^3*b^18*c^15*d^6 - 2334720*a^4*b^17*c^14*d^7 + 3293
184*a^5*b^16*c^13*d^8 - 5758976*a^6*b^15*c^12*d^9 + 13516800*a^7*b^14*c^11*d^10 - 25141248*a^8*b^13*c^10*d^11
+ 31088640*a^9*b^12*c^9*d^12 - 25141248*a^10*b^11*c^8*d^13 + 13516800*a^11*b^10*c^7*d^14 - 5758976*a^12*b^9*c^
6*d^15 + 3293184*a^13*b^8*c^5*d^16 - 2334720*a^14*b^7*c^4*d^17 + 1163264*a^15*b^6*c^3*d^18 - 319488*a^16*b^5*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))) - (x^(1/2)*(5625*a*b^12*c^8*d^5 + 5625*a^8*b^5*c*d^1
2 + 34275*a^2*b^11*c^7*d^6 + 88705*a^3*b^10*c^6*d^7 + 133539*a^4*b^9*c^5*d^8 + 133539*a^5*b^8*c^4*d^9 + 88705*
a^6*b^7*c^3*d^10 + 34275*a^7*b^6*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^5*c^4 + 62
5*a^4*b*d^4 + 1500*a^3*b^2*c*d^3 + 1350*a^2*b^3*c^2*d^2 + 540*a*b^4*c^3*d)/(4096*a^13*d^12 + 4096*a*b^12*c^12
- 49152*a^2*b^11*c^11*d + 270336*a^3*b^10*c^10*d^2 - 901120*a^4*b^9*c^9*d^3 + 2027520*a^5*b^8*c^8*d^4 - 324403
2*a^6*b^7*c^7*d^5 + 3784704*a^7*b^6*c^6*d^6 - 3244032*a^8*b^5*c^5*d^7 + 2027520*a^9*b^4*c^4*d^8 - 901120*a^10*
b^3*c^3*d^9 + 270336*a^11*b^2*c^2*d^10 - 49152*a^12*b*c*d^11))^(1/4) - ((-(81*b^5*c^4 + 625*a^4*b*d^4 + 1500*a
^3*b^2*c*d^3 + 1350*a^2*b^3*c^2*d^2 + 540*a*b^4*c^3*d)/(4096*a^13*d^12 + 4096*a*b^12*c^12 - 49152*a^2*b^11*c^1
1*d + 270336*a^3*b^10*c^10*d^2 - 901120*a^4*b^9*c^9*d^3 + 2027520*a^5*b^8*c^8*d^4 - 3244032*a^6*b^7*c^7*d^5 +
3784704*a^7*b^6*c^6*d^6 - 3244032*a^8*b^5*c^5*d^7 + 2027520*a^9*b^4*c^4*d^8 - 901120*a^10*b^3*c^3*d^9 + 270336
*a^11*b^2*c^2*d^10 - 49152*a^12*b*c*d^11))^(3/4)*(((864*a*b^20*c^17*d^4 + 864*a^17*b^4*c*d^20 - 5184*a^2*b^19*
c^16*d^5 + 3200*a^3*b^18*c^15*d^6 + 56640*a^4*b^17*c^14*d^7 - 220800*a^5*b^16*c^13*d^8 + 369088*a^6*b^15*c^12*
d^9 - 240768*a^7*b^14*c^11*d^10 - 158400*a^8*b^13*c^10*d^11 + 390720*a^9*b^12*c^9*d^12 - 158400*a^10*b^11*c^8*
d^13 - 240768*a^11*b^10*c^7*d^14 + 369088*a^12*b^9*c^6*d^15 - 220800*a^13*b^8*c^5*d^16 + 56640*a^14*b^7*c^4*d^
17 + 3200*a^15*b^6*c^3*d^18 - 5184*a^16*b^5*c^2*d^19)*1i)/(a^14*d^14 + b^14*c^14 + 91*a^2*b^12*c^12*d^2 - 364*
a^3*b^11*c^11*d^3 + 1001*a^4*b^10*c^10*d^4 - 2002*a^5*b^9*c^9*d^5 + 3003*a^6*b^8*c^8*d^6 - 3432*a^7*b^7*c^7*d^
7 + 3003*a^8*b^6*c^6*d^8 - 2002*a^9*b^5*c^5*d^9 + 1001*a^10*b^4*c^4*d^10 - 364*a^11*b^3*c^3*d^11 + 91*a^12*b^2
*c^2*d^12 - 14*a*b^13*c^13*d - 14*a^13*b*c*d^13) + (x^(1/2)*(-(81*b^5*c^4 + 625*a^4*b*d^4 + 1500*a^3*b^2*c*d^3
+ 1350*a^2*b^3*c^2*d^2 + 540*a*b^4*c^3*d)/(4096*a^13*d^12 + 4096*a*b^12*c^12 - 49152*a^2*b^11*c^11*d + 270336
*a^3*b^10*c^10*d^2 - 901120*a^4*b^9*c^9*d^3 + 2027520*a^5*b^8*c^8*d^4 - 3244032*a^6*b^7*c^7*d^5 + 3784704*a^7*
b^6*c^6*d^6 - 3244032*a^8*b^5*c^5*d^7 + 2027520*a^9*b^4*c^4*d^8 - 901120*a^10*b^3*c^3*d^9 + 270336*a^11*b^2*c^
2*d^10 - 49152*a^12*b*c*d^11))^(1/4)*(36864*a*b^20*c^17*d^4 + 36864*a^17*b^4*c*d^20 - 319488*a^2*b^19*c^16*d^5
+ 1163264*a^3*b^18*c^15*d^6 - 2334720*a^4*b^17...

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