3.5.82 \(\int \frac {x^{3/2}}{(a+b x^2) (c+d x^2)^3} \, dx\) [482]

Optimal. Leaf size=627 \[ \frac {\sqrt {x}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(7 b c+a d) \sqrt {x}}{16 c (b c-a d)^2 \left (c+d x^2\right )}+\frac {\sqrt [4]{a} b^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {\sqrt [4]{a} b^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{a} b^{7/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}-\frac {\sqrt [4]{a} b^{7/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}-\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3} \]

[Out]

1/2*a^(1/4)*b^(7/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/(-a*d+b*c)^3*2^(1/2)-1/2*a^(1/4)*b^(7/4)*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+14*a*b*c*d+21*b^2*c^2)*arctan(1-d^(1/
4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(7/4)/d^(1/4)/(-a*d+b*c)^3*2^(1/2)+1/64*(-3*a^2*d^2+14*a*b*c*d+21*b^2*c^2)*arcta
n(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(7/4)/d^(1/4)/(-a*d+b*c)^3*2^(1/2)+1/4*a^(1/4)*b^(7/4)*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^(1/4)*b^(7/4)*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+14*a*b*c*d+21*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)-c^(1
/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(7/4)/d^(1/4)/(-a*d+b*c)^3*2^(1/2)+1/128*(-3*a^2*d^2+14*a*b*c*d+21*b^2*c^2)*ln(
c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(7/4)/d^(1/4)/(-a*d+b*c)^3*2^(1/2)+1/4*x^(1/2)/(-a*d+b*c)
/(d*x^2+c)^2+1/16*(a*d+7*b*c)*x^(1/2)/c/(-a*d+b*c)^2/(d*x^2+c)

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Rubi [A]
time = 0.48, antiderivative size = 627, 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 = {477, 482, 541, 536, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}-\frac {\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{a} b^{7/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {\sqrt [4]{a} b^{7/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} (b c-a d)^3}+\frac {\sqrt [4]{a} b^{7/4} \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 {\sqrt [4]{a} b^{7/4} \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 {\sqrt {x} (a d+7 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^2}+\frac {\sqrt {x}}{4 \left (c+d x^2\right )^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[x]/(4*(b*c - a*d)*(c + d*x^2)^2) + ((7*b*c + a*d)*Sqrt[x])/(16*c*(b*c - a*d)^2*(c + d*x^2)) + (a^(1/4)*b^
(7/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*(b*c - a*d)^3) - (a^(1/4)*b^(7/4)*ArcTan[1 + (Sq
rt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*(b*c - a*d)^3) - ((21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (
Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4)*d^(1/4)*(b*c - a*d)^3) + ((21*b^2*c^2 + 14*a*b*c*d - 3*
a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4)*d^(1/4)*(b*c - a*d)^3) + (a^(1/4)*
b^(7/4)*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^(1/4)*b^(7/
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) - ((21*b^2*c^2 + 14*a
*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(7/4)*d^(1/4)*(b
*c - a*d)^3) + ((21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*
x])/(64*Sqrt[2]*c^(7/4)*d^(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 217

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 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 536

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 541

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 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^{3/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=2 \text {Subst}\left (\int \frac {x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=\frac {\sqrt {x}}{4 (b c-a d) \left (c+d x^2\right )^2}-\frac {\text {Subst}\left (\int \frac {a-7 b x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)}\\ &=\frac {\sqrt {x}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(7 b c+a d) \sqrt {x}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {a (11 b c-3 a d)-3 b (7 b c+a d) x^4}{\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 {\sqrt {x}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(7 b c+a d) \sqrt {x}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\left (2 a b^2\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c (b c-a d)^3}\\ &=\frac {\sqrt {x}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(7 b c+a d) \sqrt {x}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\left (\sqrt {a} b^2\right ) \text {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 (\sqrt {a} b^2\right ) \text {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 (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{3/2} (b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{3/2} (b c-a d)^3}\\ &=\frac {\sqrt {x}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(7 b c+a d) \sqrt {x}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\left (\sqrt {a} b^{3/2}\right ) \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 )}{2 (b c-a d)^3}-\frac {\left (\sqrt {a} b^{3/2}\right ) \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 )}{2 (b c-a d)^3}+\frac {\left (\sqrt [4]{a} b^{7/4}\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 )}{2 \sqrt {2} (b c-a d)^3}+\frac {\left (\sqrt [4]{a} b^{7/4}\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 )}{2 \sqrt {2} (b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \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 )}{64 c^{3/2} \sqrt {d} (b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \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 )}{64 c^{3/2} \sqrt {d} (b c-a d)^3}-\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\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 )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}-\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\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 )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}\\ &=\frac {\sqrt {x}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(7 b c+a d) \sqrt {x}}{16 c (b c-a d)^2 \left (c+d x^2\right )}+\frac {\sqrt [4]{a} b^{7/4} \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 {\sqrt [4]{a} b^{7/4} \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 (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}-\frac {\left (\sqrt [4]{a} b^{7/4}\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 )}{\sqrt {2} (b c-a d)^3}+\frac {\left (\sqrt [4]{a} b^{7/4}\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 )}{\sqrt {2} (b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\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 )}{32 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}-\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\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 )}{32 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}\\ &=\frac {\sqrt {x}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(7 b c+a d) \sqrt {x}}{16 c (b c-a d)^2 \left (c+d x^2\right )}+\frac {\sqrt [4]{a} b^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {\sqrt [4]{a} b^{7/4} \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 (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{a} b^{7/4} \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 {\sqrt [4]{a} b^{7/4} \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 (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}\\ \end {align*}

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Mathematica [A]
time = 1.18, size = 327, normalized size = 0.52 \begin {gather*} \frac {\frac {4 (b c-a d) \sqrt {x} \left (a d \left (-3 c+d x^2\right )+b c \left (11 c+7 d x^2\right )\right )}{c \left (c+d x^2\right )^2}+32 \sqrt {2} \sqrt [4]{a} b^{7/4} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-\frac {\sqrt {2} \left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{7/4} \sqrt [4]{d}}-32 \sqrt {2} \sqrt [4]{a} b^{7/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )+\frac {\sqrt {2} \left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{7/4} \sqrt [4]{d}}}{64 (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*(b*c - a*d)*Sqrt[x]*(a*d*(-3*c + d*x^2) + b*c*(11*c + 7*d*x^2)))/(c*(c + d*x^2)^2) + 32*Sqrt[2]*a^(1/4)*b^
(7/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - (Sqrt[2]*(21*b^2*c^2 + 14*a*b*c*d - 3*
a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(c^(7/4)*d^(1/4)) - 32*Sqrt[2]*a^(1/
4)*b^(7/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + (Sqrt[2]*(21*b^2*c^2 + 14*a*b*c*
d - 3*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c^(7/4)*d^(1/4)))/(64*(b*c -
a*d)^3)

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Maple [A]
time = 0.09, size = 327, normalized size = 0.52

method	result	size
derivativedivides	\(\frac {b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{3}}+\frac {\frac {2 \left (\frac {d \left (a^{2} d^{2}+6 a b c d -7 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{32 c}+\left (\frac {7}{16} a b c d -\frac {11}{32} b^{2} c^{2}-\frac {3}{32} a^{2} d^{2}\right ) \sqrt {x}\right )}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (3 a^{2} d^{2}-14 a b c d -21 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c^{2}}}{\left (a d -b c \right )^{3}}\)	\(327\)
default	\(\frac {b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{3}}+\frac {\frac {2 \left (\frac {d \left (a^{2} d^{2}+6 a b c d -7 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{32 c}+\left (\frac {7}{16} a b c d -\frac {11}{32} b^{2} c^{2}-\frac {3}{32} a^{2} d^{2}\right ) \sqrt {x}\right )}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (3 a^{2} d^{2}-14 a b c d -21 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c^{2}}}{\left (a d -b c \right )^{3}}\)	\(327\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^2/(a*d-b*c)^3*(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/(a*d
-b*c)^3*((1/32*d*(a^2*d^2+6*a*b*c*d-7*b^2*c^2)/c*x^(5/2)+(7/16*a*b*c*d-11/32*b^2*c^2-3/32*a^2*d^2)*x^(1/2))/(d
*x^2+c)^2+1/256*(3*a^2*d^2-14*a*b*c*d-21*b^2*c^2)/c^2*(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 = 654, normalized size = 1.04 \begin {gather*} -\frac {{\left (\frac {2 \, \sqrt {2} b^{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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} b^{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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} b^{\frac {7}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {7}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}}\right )} a}{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 (7 \, b c d + a d^{2}\right )} x^{\frac {5}{2}} + {\left (11 \, b c^{2} - 3 \, a c d\right )} \sqrt {x}}{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 )}} + \frac {\frac {2 \, \sqrt {2} {\left (21 \, b^{2} c^{2} + 14 \, a b c d - 3 \, a^{2} d^{2}\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 (21 \, b^{2} c^{2} + 14 \, a b c d - 3 \, a^{2} d^{2}\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 (21 \, b^{2} c^{2} + 14 \, a b c d - 3 \, a^{2} d^{2}\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 (21 \, b^{2} c^{2} + 14 \, a b c d - 3 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{128 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*sqrt(2)*b^2*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(s
qrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*b^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(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*b^(7/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x)
+ sqrt(b)*x + sqrt(a))/a^(3/4) - sqrt(2)*b^(7/4)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/
a^(3/4))*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*((7*b*c*d + a*d^2)*x^(5/2) + (11*b*c^2 -
3*a*c*d)*sqrt(x))/(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) + 1/128*(2*sqrt(2)*(21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^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(c)*sqrt(sqrt(c)*sqrt(d))
) + 2*sqrt(2)*(21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*s
qrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)
*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(21*b^2*c^2 + 14*a*b*c
*d - 3*a^2*d^2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/(b^3*c^4 - 3*a*
b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5450 vs. \(2 (480) = 960\).
time = 88.78, size = 5450, normalized size = 8.69 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(4*(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)*(-(194481*b^8*c^8 + 518616*a*b^7*c^7*d + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b
^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 - 1512*a^7*b*c*d^7 + 81*a^8*d
^8)/(b^12*c^19*d - 12*a*b^11*c^18*d^2 + 66*a^2*b^10*c^17*d^3 - 220*a^3*b^9*c^16*d^4 + 495*a^4*b^8*c^15*d^5 - 7
92*a^5*b^7*c^14*d^6 + 924*a^6*b^6*c^13*d^7 - 792*a^7*b^5*c^12*d^8 + 495*a^8*b^4*c^11*d^9 - 220*a^9*b^3*c^10*d^
10 + 66*a^10*b^2*c^9*d^11 - 12*a^11*b*c^8*d^12 + a^12*c^7*d^13))^(1/4)*arctan(-((b^9*c^14*d - 9*a*b^8*c^13*d^2
+ 36*a^2*b^7*c^12*d^3 - 84*a^3*b^6*c^11*d^4 + 126*a^4*b^5*c^10*d^5 - 126*a^5*b^4*c^9*d^6 + 84*a^6*b^3*c^8*d^7
- 36*a^7*b^2*c^7*d^8 + 9*a^8*b*c^6*d^9 - a^9*c^5*d^10)*sqrt((441*b^4*c^4 + 588*a*b^3*c^3*d + 70*a^2*b^2*c^2*d
^2 - 84*a^3*b*c*d^3 + 9*a^4*d^4)*x + (b^6*c^10 - 6*a*b^5*c^9*d + 15*a^2*b^4*c^8*d^2 - 20*a^3*b^3*c^7*d^3 + 15*
a^4*b^2*c^6*d^4 - 6*a^5*b*c^5*d^5 + a^6*c^4*d^6)*sqrt(-(194481*b^8*c^8 + 518616*a*b^7*c^7*d + 407484*a^2*b^6*c
^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 - 1512*a^7
*b*c*d^7 + 81*a^8*d^8)/(b^12*c^19*d - 12*a*b^11*c^18*d^2 + 66*a^2*b^10*c^17*d^3 - 220*a^3*b^9*c^16*d^4 + 495*a
^4*b^8*c^15*d^5 - 792*a^5*b^7*c^14*d^6 + 924*a^6*b^6*c^13*d^7 - 792*a^7*b^5*c^12*d^8 + 495*a^8*b^4*c^11*d^9 -
220*a^9*b^3*c^10*d^10 + 66*a^10*b^2*c^9*d^11 - 12*a^11*b*c^8*d^12 + a^12*c^7*d^13)))*(-(194481*b^8*c^8 + 51861
6*a*b^7*c^7*d + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 +
8316*a^6*b^2*c^2*d^6 - 1512*a^7*b*c*d^7 + 81*a^8*d^8)/(b^12*c^19*d - 12*a*b^11*c^18*d^2 + 66*a^2*b^10*c^17*d^
3 - 220*a^3*b^9*c^16*d^4 + 495*a^4*b^8*c^15*d^5 - 792*a^5*b^7*c^14*d^6 + 924*a^6*b^6*c^13*d^7 - 792*a^7*b^5*c^
12*d^8 + 495*a^8*b^4*c^11*d^9 - 220*a^9*b^3*c^10*d^10 + 66*a^10*b^2*c^9*d^11 - 12*a^11*b*c^8*d^12 + a^12*c^7*d
^13))^(3/4) + (21*b^11*c^16*d - 175*a*b^10*c^15*d^2 + 627*a^2*b^9*c^14*d^3 - 1233*a^3*b^8*c^13*d^4 + 1362*a^4*
b^7*c^12*d^5 - 630*a^5*b^6*c^11*d^6 - 378*a^6*b^5*c^10*d^7 + 798*a^7*b^4*c^9*d^8 - 567*a^8*b^3*c^8*d^9 + 213*a
^9*b^2*c^7*d^10 - 41*a^10*b*c^6*d^11 + 3*a^11*c^5*d^12)*sqrt(x)*(-(194481*b^8*c^8 + 518616*a*b^7*c^7*d + 40748
4*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6
- 1512*a^7*b*c*d^7 + 81*a^8*d^8)/(b^12*c^19*d - 12*a*b^11*c^18*d^2 + 66*a^2*b^10*c^17*d^3 - 220*a^3*b^9*c^16*
d^4 + 495*a^4*b^8*c^15*d^5 - 792*a^5*b^7*c^14*d^6 + 924*a^6*b^6*c^13*d^7 - 792*a^7*b^5*c^12*d^8 + 495*a^8*b^4*
c^11*d^9 - 220*a^9*b^3*c^10*d^10 + 66*a^10*b^2*c^9*d^11 - 12*a^11*b*c^8*d^12 + a^12*c^7*d^13))^(3/4))/(194481*
b^8*c^8 + 518616*a*b^7*c^7*d + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^
5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 - 1512*a^7*b*c*d^7 + 81*a^8*d^8)) + 128*(-a*b^7/(b^12*c^12 - 12*a*b^11*c^
11*d + 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^11*b*c*d
^11 + a^12*d^12))^(1/4)*(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)*arctan(-((b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84*
a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8*
b*c*d^8 - a^9*d^9)*(-a*b^7/(b^12*c^12 - 12*a*b^11*c^11*d + 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^11*b*c*d^11 + a^12*d^12))^(3/4)*sqrt(b^4*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(-a*b^7/(b^1
2*c^12 - 12*a*b^11*c^11*d + 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^11*b*c*d^11 + a^12*d^12))) - (b^11*c^9 - 9*a*b^10*c^8*d + 36*a^2*b^9*c^7*d^2 - 84*a^3*b^8*c^6*d^3
+ 126*a^4*b^7*c^5*d^4 - 126*a^5*b^6*c^4*d^5 + 84*a^6*b^5*c^3*d^6 - 36*a^7*b^4*c^2*d^7 + 9*a^8*b^3*c*d^8 - a^9
*b^2*d^9)*(-a*b^7/(b^12*c^12 - 12*a*b^11*c^11*d + 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^11*b*c*d^11 + a^12*d^12))^(3/4)*sqrt(x))/(a*b^7)) - 32*(-a*b^7/(b^12*c^12 -
12*a*b^11*c^11*d + 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 + 9
24*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^11*b*c*d^11 + a^12*d^12))^(1/4)*(b^2*c^5 -...

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

[Out]

Timed out

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Giac [A]
time = 1.81, size = 946, normalized size = 1.51 \begin {gather*} -\frac {\left (a b^{3}\right )^{\frac {1}{4}} b \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 )}{\sqrt {2} b^{3} c^{3} - 3 \, \sqrt {2} a b^{2} c^{2} d + 3 \, \sqrt {2} a^{2} b c d^{2} - \sqrt {2} a^{3} d^{3}} - \frac {\left (a b^{3}\right )^{\frac {1}{4}} b \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 )}{\sqrt {2} b^{3} c^{3} - 3 \, \sqrt {2} a b^{2} c^{2} d + 3 \, \sqrt {2} a^{2} b c d^{2} - \sqrt {2} a^{3} d^{3}} - \frac {\left (a b^{3}\right )^{\frac {1}{4}} b \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} b^{3} c^{3} - 3 \, \sqrt {2} a b^{2} c^{2} d + 3 \, \sqrt {2} a^{2} b c d^{2} - \sqrt {2} a^{3} d^{3}\right )}} + \frac {\left (a b^{3}\right )^{\frac {1}{4}} b \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} b^{3} c^{3} - 3 \, \sqrt {2} a b^{2} c^{2} d + 3 \, \sqrt {2} a^{2} b c d^{2} - \sqrt {2} a^{3} d^{3}\right )}} + \frac {{\left (21 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 14 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\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 )}{32 \, {\left (\sqrt {2} b^{3} c^{5} d - 3 \, \sqrt {2} a b^{2} c^{4} d^{2} + 3 \, \sqrt {2} a^{2} b c^{3} d^{3} - \sqrt {2} a^{3} c^{2} d^{4}\right )}} + \frac {{\left (21 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 14 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\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 )}{32 \, {\left (\sqrt {2} b^{3} c^{5} d - 3 \, \sqrt {2} a b^{2} c^{4} d^{2} + 3 \, \sqrt {2} a^{2} b c^{3} d^{3} - \sqrt {2} a^{3} c^{2} d^{4}\right )}} + \frac {{\left (21 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 14 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{64 \, {\left (\sqrt {2} b^{3} c^{5} d - 3 \, \sqrt {2} a b^{2} c^{4} d^{2} + 3 \, \sqrt {2} a^{2} b c^{3} d^{3} - \sqrt {2} a^{3} c^{2} d^{4}\right )}} - \frac {{\left (21 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 14 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{64 \, {\left (\sqrt {2} b^{3} c^{5} d - 3 \, \sqrt {2} a b^{2} c^{4} d^{2} + 3 \, \sqrt {2} a^{2} b c^{3} d^{3} - \sqrt {2} a^{3} c^{2} d^{4}\right )}} + \frac {7 \, b c d x^{\frac {5}{2}} + a d^{2} x^{\frac {5}{2}} + 11 \, b c^{2} \sqrt {x} - 3 \, a c d \sqrt {x}}{16 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} {\left (d x^{2} + c\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*b^3)^(1/4)*b*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^3*c^3 - 3*sqrt(2
)*a*b^2*c^2*d + 3*sqrt(2)*a^2*b*c*d^2 - sqrt(2)*a^3*d^3) - (a*b^3)^(1/4)*b*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^
(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^3*c^3 - 3*sqrt(2)*a*b^2*c^2*d + 3*sqrt(2)*a^2*b*c*d^2 - sqrt(2)*a^3
*d^3) - 1/2*(a*b^3)^(1/4)*b*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*b^3*c^3 - 3*sqrt(2)*a*b^
2*c^2*d + 3*sqrt(2)*a^2*b*c*d^2 - sqrt(2)*a^3*d^3) + 1/2*(a*b^3)^(1/4)*b*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x
+ sqrt(a/b))/(sqrt(2)*b^3*c^3 - 3*sqrt(2)*a*b^2*c^2*d + 3*sqrt(2)*a^2*b*c*d^2 - sqrt(2)*a^3*d^3) + 1/32*(21*(c
*d^3)^(1/4)*b^2*c^2 + 14*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3)^(1/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1
/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^5*d - 3*sqrt(2)*a*b^2*c^4*d^2 + 3*sqrt(2)*a^2*b*c^3*d^3 - sqrt(2)
*a^3*c^2*d^4) + 1/32*(21*(c*d^3)^(1/4)*b^2*c^2 + 14*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3)^(1/4)*a^2*d^2)*arctan(-1
/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^5*d - 3*sqrt(2)*a*b^2*c^4*d^2 + 3*sqr
t(2)*a^2*b*c^3*d^3 - sqrt(2)*a^3*c^2*d^4) + 1/64*(21*(c*d^3)^(1/4)*b^2*c^2 + 14*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d
^3)^(1/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^3*c^5*d - 3*sqrt(2)*a*b^2*c^4*d
^2 + 3*sqrt(2)*a^2*b*c^3*d^3 - sqrt(2)*a^3*c^2*d^4) - 1/64*(21*(c*d^3)^(1/4)*b^2*c^2 + 14*(c*d^3)^(1/4)*a*b*c*
d - 3*(c*d^3)^(1/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^3*c^5*d - 3*sqrt(2)*
a*b^2*c^4*d^2 + 3*sqrt(2)*a^2*b*c^3*d^3 - sqrt(2)*a^3*c^2*d^4) + 1/16*(7*b*c*d*x^(5/2) + a*d^2*x^(5/2) + 11*b*
c^2*sqrt(x) - 3*a*c*d*sqrt(x))/((b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*(d*x^2 + c)^2)

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Mupad [B]
time = 2.35, size = 2500, normalized size = 3.99 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atan(((((((81*a^9*b^7*d^10)/2048 - (1431*a^8*b^8*c*d^9)/2048 - (194481*a^2*b^14*c^7*d^3)/2048 - (713097*a^3*
b^13*c^6*d^4)/2048 - (432453*a^4*b^12*c^5*d^5)/2048 + (18067*a^5*b^11*c^4*d^6)/2048 + (5709*a^6*b^10*c^3*d^7)/
2048 + (6885*a^7*b^9*c^2*d^8)/2048)*1i)/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a
^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) - (((-(a*b^7)/
(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)*(8192*a^2*b^18*c^18*d^4 - 95488*a^3*b^1
7*c^17*d^5 + 506112*a^4*b^16*c^16*d^6 - 1607168*a^5*b^15*c^15*d^7 + 3384832*a^6*b^14*c^14*d^8 - 4925184*a^7*b^
13*c^13*d^9 + 4958976*a^8*b^12*c^12*d^10 - 3277824*a^9*b^11*c^11*d^11 + 1115136*a^10*b^10*c^10*d^12 + 199936*a
^11*b^9*c^9*d^13 - 459008*a^12*b^8*c^8*d^14 + 256512*a^13*b^7*c^7*d^15 - 76288*a^14*b^6*c^6*d^16 + 12032*a^15*
b^5*c^5*d^17 - 768*a^16*b^4*c^4*d^18))/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^
3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) - (x^(1/2)*(167
77216*a^2*b^21*c^19*d^4 - 194101248*a^3*b^20*c^18*d^5 + 1030225920*a^4*b^19*c^17*d^6 - 3328573440*a^5*b^18*c^1
6*d^7 + 7335837696*a^6*b^17*c^15*d^8 - 11738087424*a^7*b^16*c^14*d^9 + 14203486208*a^8*b^15*c^13*d^10 - 133610
86464*a^9*b^14*c^12*d^11 + 9861857280*a^10*b^13*c^11*d^12 - 5521702912*a^11*b^12*c^10*d^13 + 1989672960*a^12*b
^11*c^9*d^14 - 49938432*a^13*b^10*c^8*d^15 - 484442112*a^14*b^9*c^7*d^16 + 343080960*a^15*b^8*c^6*d^17 - 12740
1984*a^16*b^7*c^5*d^18 + 27394048*a^17*b^6*c^4*d^19 - 3145728*a^18*b^5*c^3*d^20 + 147456*a^19*b^4*c^2*d^21)*1i
)/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b^10*c^14*d^2 - 220*a^3*b^9*c^13*d^3 + 495*a^
4*b^8*c^12*d^4 - 792*a^5*b^7*c^11*d^5 + 924*a^6*b^6*c^10*d^6 - 792*a^7*b^5*c^9*d^7 + 495*a^8*b^4*c^8*d^8 - 220
*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d)))*(-(a*b^7)/(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)*(-(a*b^7)/(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) - (x^(1/2)*(81*a^10*b^9*d^11 - 1512*a^9*b^10*c*d^10 + 194481*a^2*b^17*c^8*d^3 + 518616*a^3*b^16*c^7*d^
4 + 859068*a^4*b^15*c^6*d^5 + 610344*a^5*b^14*c^5*d^6 - 14266*a^6*b^13*c^4*d^7 - 87192*a^7*b^12*c^3*d^8 + 1753
2*a^8*b^11*c^2*d^9))/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b^10*c^14*d^2 - 220*a^3*b^
9*c^13*d^3 + 495*a^4*b^8*c^12*d^4 - 792*a^5*b^7*c^11*d^5 + 924*a^6*b^6*c^10*d^6 - 792*a^7*b^5*c^9*d^7 + 495*a^
8*b^4*c^8*d^8 - 220*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d)))*(-(a*b^7)/(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 + 147
84*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) - (((((81*a^9*b^7*d^10)/2048 - (1431*a^8*b^8*c*d^9)/2048 -
(194481*a^2*b^14*c^7*d^3)/2048 - (713097*a^3*b^13*c^6*d^4)/2048 - (432453*a^4*b^12*c^5*d^5)/2048 + (18067*a^5
*b^11*c^4*d^6)/2048 + (5709*a^6*b^10*c^3*d^7)/2048 + (6885*a^7*b^9*c^2*d^8)/2048)*1i)/(b^8*c^12 + a^8*c^4*d^8
- 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^
6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) - (((-(a*b^7)/(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)*(8192*a^2*b^18*c^18*d^4 - 95488*a^3*b^17*c^17*d^5 + 506112*a^4*b^16*c^16*d^6 - 1607168*a^5*b^15*c^15*d^
7 + 3384832*a^6*b^14*c^14*d^8 - 4925184*a^7*b^13*c^13*d^9 + 4958976*a^8*b^12*c^12*d^10 - 3277824*a^9*b^11*c^11
*d^11 + 1115136*a^10*b^10*c^10*d^12 + 199936*a^11*b^9*c^9*d^13 - 459008*a^12*b^8*c^8*d^14 + 256512*a^13*b^7*c^
7*d^15 - 76288*a^14*b^6*c^6*d^16 + 12032*a^15*b^5*c^5*d^17 - 768*a^16*b^4*c^4*d^18))/(b^8*c^12 + a^8*c^4*d^8 -
8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6
*b^2*c^6*d^6 - 8*a*b^7*c^11*d) + (x^(1/2)*(16777216*a^2*b^21*c^19*d^4 - 194101248*a^3*b^20*c^18*d^5 + 10302259
20*a^4*b^19*c^17*d^6 - 3328573440*a^5*b^18*c^16...

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