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3.5.50 \(\int \frac {1}{x^{5/2} (a+b x^2) (c+d x^2)} \, dx\)

Optimal. Leaf size=478 \[ \frac {b^{7/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)}-\frac {b^{7/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)}+\frac {b^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)}-\frac {b^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{7/4} (b c-a d)}-\frac {d^{7/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{7/4} (b c-a d)}+\frac {d^{7/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{7/4} (b c-a d)}-\frac {d^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{7/4} (b c-a d)}+\frac {d^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} c^{7/4} (b c-a d)}-\frac {2}{3 a c x^{3/2}} \]

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Rubi [A]  time = 0.48, antiderivative size = 478, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {466, 480, 522, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {b^{7/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)}-\frac {b^{7/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)}+\frac {b^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)}-\frac {b^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{7/4} (b c-a d)}-\frac {d^{7/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{7/4} (b c-a d)}+\frac {d^{7/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{7/4} (b c-a d)}-\frac {d^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{7/4} (b c-a d)}+\frac {d^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} c^{7/4} (b c-a d)}-\frac {2}{3 a c x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)),x]

[Out]

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

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 480

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((e*x)^(m
 + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*e*(m + 1)), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[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 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 {1}{x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2}{3 a c x^{3/2}}+\frac {2 \operatorname {Subst}\left (\int \frac {-3 (b c+a d)-3 b d x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{3 a c}\\ &=-\frac {2}{3 a c x^{3/2}}-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a (b c-a d)}+\frac {\left (2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{c (b c-a d)}\\ &=-\frac {2}{3 a c x^{3/2}}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^{3/2} (b c-a d)}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^{3/2} (b c-a d)}+\frac {d^2 \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{c^{3/2} (b c-a d)}+\frac {d^2 \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{c^{3/2} (b c-a d)}\\ &=-\frac {2}{3 a c x^{3/2}}-\frac {b^{3/2} \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 a^{3/2} (b c-a d)}-\frac {b^{3/2} \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 a^{3/2} (b c-a d)}+\frac {b^{7/4} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} a^{7/4} (b c-a d)}+\frac {b^{7/4} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} a^{7/4} (b c-a d)}+\frac {d^{3/2} \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 )}{2 c^{3/2} (b c-a d)}+\frac {d^{3/2} \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 )}{2 c^{3/2} (b c-a d)}-\frac {d^{7/4} \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 )}{2 \sqrt {2} c^{7/4} (b c-a d)}-\frac {d^{7/4} \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 )}{2 \sqrt {2} c^{7/4} (b c-a d)}\\ &=-\frac {2}{3 a c x^{3/2}}+\frac {b^{7/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)}-\frac {b^{7/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)}-\frac {d^{7/4} \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{7/4} (b c-a d)}+\frac {d^{7/4} \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{7/4} (b c-a d)}-\frac {b^{7/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)}+\frac {b^{7/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)}+\frac {d^{7/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{7/4} (b c-a d)}-\frac {d^{7/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{7/4} (b c-a d)}\\ &=-\frac {2}{3 a c x^{3/2}}+\frac {b^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)}-\frac {b^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)}-\frac {d^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{7/4} (b c-a d)}+\frac {d^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{7/4} (b c-a d)}+\frac {b^{7/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)}-\frac {b^{7/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)}-\frac {d^{7/4} \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{7/4} (b c-a d)}+\frac {d^{7/4} \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{7/4} (b c-a d)}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 411, normalized size = 0.86 \begin {gather*} \frac {-\frac {3 \sqrt {2} b^{7/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{7/4}}+\frac {3 \sqrt {2} b^{7/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{7/4}}-\frac {6 \sqrt {2} b^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {6 \sqrt {2} b^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{a^{7/4}}+\frac {8 b}{a x^{3/2}}+\frac {3 \sqrt {2} d^{7/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{7/4}}-\frac {3 \sqrt {2} d^{7/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{7/4}}+\frac {6 \sqrt {2} d^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{c^{7/4}}-\frac {6 \sqrt {2} d^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{c^{7/4}}-\frac {8 d}{c x^{3/2}}}{12 a d-12 b c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)),x]

[Out]

((8*b)/(a*x^(3/2)) - (8*d)/(c*x^(3/2)) - (6*Sqrt[2]*b^(7/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(
7/4) + (6*Sqrt[2]*b^(7/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(7/4) + (6*Sqrt[2]*d^(7/4)*ArcTan[1
 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(7/4) - (6*Sqrt[2]*d^(7/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/
4)])/c^(7/4) - (3*Sqrt[2]*b^(7/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(7/4) + (3*Sqr
t[2]*b^(7/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(7/4) + (3*Sqrt[2]*d^(7/4)*Log[Sqrt
[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(7/4) - (3*Sqrt[2]*d^(7/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)
*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(7/4))/(-12*b*c + 12*a*d)

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IntegrateAlgebraic [A]  time = 0.64, size = 280, normalized size = 0.59 \begin {gather*} -\frac {b^{7/4} \tan ^{-1}\left (\frac {\frac {\sqrt [4]{a}}{\sqrt {2} \sqrt [4]{b}}-\frac {\sqrt [4]{b} x}{\sqrt {2} \sqrt [4]{a}}}{\sqrt {x}}\right )}{\sqrt {2} a^{7/4} (a d-b c)}+\frac {b^{7/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} a^{7/4} (a d-b c)}-\frac {d^{7/4} \tan ^{-1}\left (\frac {\frac {\sqrt [4]{c}}{\sqrt {2} \sqrt [4]{d}}-\frac {\sqrt [4]{d} x}{\sqrt {2} \sqrt [4]{c}}}{\sqrt {x}}\right )}{\sqrt {2} c^{7/4} (b c-a d)}+\frac {d^{7/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt {2} c^{7/4} (b c-a d)}-\frac {2}{3 a c x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)),x]

[Out]

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

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fricas [B]  time = 15.95, size = 1431, normalized size = 2.99

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="fricas")

[Out]

1/6*(12*(-b^7/(a^7*b^4*c^4 - 4*a^8*b^3*c^3*d + 6*a^9*b^2*c^2*d^2 - 4*a^10*b*c*d^3 + a^11*d^4))^(1/4)*a*c*x^2*a
rctan(-((a^5*b^3*c^3 - 3*a^6*b^2*c^2*d + 3*a^7*b*c*d^2 - a^8*d^3)*(-b^7/(a^7*b^4*c^4 - 4*a^8*b^3*c^3*d + 6*a^9
*b^2*c^2*d^2 - 4*a^10*b*c*d^3 + a^11*d^4))^(3/4)*sqrt(b^4*x + (a^4*b^2*c^2 - 2*a^5*b*c*d + a^6*d^2)*sqrt(-b^7/
(a^7*b^4*c^4 - 4*a^8*b^3*c^3*d + 6*a^9*b^2*c^2*d^2 - 4*a^10*b*c*d^3 + a^11*d^4))) - (a^5*b^5*c^3 - 3*a^6*b^4*c
^2*d + 3*a^7*b^3*c*d^2 - a^8*b^2*d^3)*(-b^7/(a^7*b^4*c^4 - 4*a^8*b^3*c^3*d + 6*a^9*b^2*c^2*d^2 - 4*a^10*b*c*d^
3 + a^11*d^4))^(3/4)*sqrt(x))/b^7) - 12*(-d^7/(b^4*c^11 - 4*a*b^3*c^10*d + 6*a^2*b^2*c^9*d^2 - 4*a^3*b*c^8*d^3
 + a^4*c^7*d^4))^(1/4)*a*c*x^2*arctan(-((b^3*c^8 - 3*a*b^2*c^7*d + 3*a^2*b*c^6*d^2 - a^3*c^5*d^3)*(-d^7/(b^4*c
^11 - 4*a*b^3*c^10*d + 6*a^2*b^2*c^9*d^2 - 4*a^3*b*c^8*d^3 + a^4*c^7*d^4))^(3/4)*sqrt(d^4*x + (b^2*c^6 - 2*a*b
*c^5*d + a^2*c^4*d^2)*sqrt(-d^7/(b^4*c^11 - 4*a*b^3*c^10*d + 6*a^2*b^2*c^9*d^2 - 4*a^3*b*c^8*d^3 + a^4*c^7*d^4
))) - (b^3*c^8*d^2 - 3*a*b^2*c^7*d^3 + 3*a^2*b*c^6*d^4 - a^3*c^5*d^5)*(-d^7/(b^4*c^11 - 4*a*b^3*c^10*d + 6*a^2
*b^2*c^9*d^2 - 4*a^3*b*c^8*d^3 + a^4*c^7*d^4))^(3/4)*sqrt(x))/d^7) - 3*(-b^7/(a^7*b^4*c^4 - 4*a^8*b^3*c^3*d +
6*a^9*b^2*c^2*d^2 - 4*a^10*b*c*d^3 + a^11*d^4))^(1/4)*a*c*x^2*log(b^2*sqrt(x) + (-b^7/(a^7*b^4*c^4 - 4*a^8*b^3
*c^3*d + 6*a^9*b^2*c^2*d^2 - 4*a^10*b*c*d^3 + a^11*d^4))^(1/4)*(a^2*b*c - a^3*d)) + 3*(-b^7/(a^7*b^4*c^4 - 4*a
^8*b^3*c^3*d + 6*a^9*b^2*c^2*d^2 - 4*a^10*b*c*d^3 + a^11*d^4))^(1/4)*a*c*x^2*log(b^2*sqrt(x) - (-b^7/(a^7*b^4*
c^4 - 4*a^8*b^3*c^3*d + 6*a^9*b^2*c^2*d^2 - 4*a^10*b*c*d^3 + a^11*d^4))^(1/4)*(a^2*b*c - a^3*d)) + 3*(-d^7/(b^
4*c^11 - 4*a*b^3*c^10*d + 6*a^2*b^2*c^9*d^2 - 4*a^3*b*c^8*d^3 + a^4*c^7*d^4))^(1/4)*a*c*x^2*log(d^2*sqrt(x) +
(-d^7/(b^4*c^11 - 4*a*b^3*c^10*d + 6*a^2*b^2*c^9*d^2 - 4*a^3*b*c^8*d^3 + a^4*c^7*d^4))^(1/4)*(b*c^3 - a*c^2*d)
) - 3*(-d^7/(b^4*c^11 - 4*a*b^3*c^10*d + 6*a^2*b^2*c^9*d^2 - 4*a^3*b*c^8*d^3 + a^4*c^7*d^4))^(1/4)*a*c*x^2*log
(d^2*sqrt(x) - (-d^7/(b^4*c^11 - 4*a*b^3*c^10*d + 6*a^2*b^2*c^9*d^2 - 4*a^3*b*c^8*d^3 + a^4*c^7*d^4))^(1/4)*(b
*c^3 - a*c^2*d)) - 4*sqrt(x))/(a*c*x^2)

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giac [A]  time = 0.70, size = 476, normalized size = 1.00 \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} a^{2} b c - \sqrt {2} a^{3} d} - \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} a^{2} b c - \sqrt {2} a^{3} d} + \frac {\left (c d^{3}\right )^{\frac {1}{4}} d \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 )}{\sqrt {2} b c^{3} - \sqrt {2} a c^{2} d} + \frac {\left (c d^{3}\right )^{\frac {1}{4}} d \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 )}{\sqrt {2} b c^{3} - \sqrt {2} a c^{2} d} - \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} a^{2} b c - \sqrt {2} a^{3} d\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} a^{2} b c - \sqrt {2} a^{3} d\right )}} + \frac {\left (c d^{3}\right )^{\frac {1}{4}} d \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{2 \, {\left (\sqrt {2} b c^{3} - \sqrt {2} a c^{2} d\right )}} - \frac {\left (c d^{3}\right )^{\frac {1}{4}} d \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{2 \, {\left (\sqrt {2} b c^{3} - \sqrt {2} a c^{2} d\right )}} - \frac {2}{3 \, a c x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^2+a)/(d*x^2+c),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)*a^2*b*c - sqrt(2)*
a^3*d) - (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)*a^2*b*c -
 sqrt(2)*a^3*d) + (c*d^3)^(1/4)*d*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b
*c^3 - sqrt(2)*a*c^2*d) + (c*d^3)^(1/4)*d*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(
sqrt(2)*b*c^3 - sqrt(2)*a*c^2*d) - 1/2*(a*b^3)^(1/4)*b*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(
2)*a^2*b*c - sqrt(2)*a^3*d) + 1/2*(a*b^3)^(1/4)*b*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a
^2*b*c - sqrt(2)*a^3*d) + 1/2*(c*d^3)^(1/4)*d*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b*c^3
- sqrt(2)*a*c^2*d) - 1/2*(c*d^3)^(1/4)*d*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b*c^3 - sq
rt(2)*a*c^2*d) - 2/3/(a*c*x^(3/2))

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maple [A]  time = 0.02, size = 351, normalized size = 0.73 \begin {gather*} \frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (a d -b c \right ) a^{2}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (a d -b c \right ) a^{2}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} \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 )}{4 \left (a d -b c \right ) a^{2}}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{2 \left (a d -b c \right ) c^{2}}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{2 \left (a d -b c \right ) c^{2}}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, d^{2} \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 )}{4 \left (a d -b c \right ) c^{2}}-\frac {2}{3 a c \,x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(5/2)/(b*x^2+a)/(d*x^2+c),x)

[Out]

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

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maxima [A]  time = 2.38, size = 396, normalized size = 0.83 \begin {gather*} -\frac {\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}}}}{4 \, {\left (a b c - a^{2} d\right )}} + \frac {\frac {2 \, \sqrt {2} d^{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 {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} d^{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 {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} d^{\frac {7}{4}} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}}} - \frac {\sqrt {2} d^{\frac {7}{4}} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}}}}{4 \, {\left (b c^{2} - a c d\right )}} - \frac {2}{3 \, a c x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^2+a)/(d*x^2+c),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*c - a^2*d) + 1/4*(2*sqrt(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)*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))) + sqrt(2)*d^(7/4)*log(sqrt(
2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/c^(3/4) - sqrt(2)*d^(7/4)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(
x) + sqrt(d)*x + sqrt(c))/c^(3/4))/(b*c^2 - a*c*d) - 2/3/(a*c*x^(3/2))

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mupad [B]  time = 2.40, size = 7540, normalized size = 15.77

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)),x)

[Out]

2*atan(((x^(1/2)*(256*a^9*b^11*c^11*d^9 + 256*a^11*b^9*c^9*d^11) - (-d^7/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^
3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d))^(1/4)*(((-d^7/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^
8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d))^(1/4)*(8192*a^13*b^12*c^21*d^4 - 40960*a^14*b^11*c^20*d^5 + 819
20*a^15*b^10*c^19*d^6 - 90112*a^16*b^9*c^18*d^7 + 81920*a^17*b^8*c^17*d^8 - 90112*a^18*b^7*c^16*d^9 + 81920*a^
19*b^6*c^15*d^10 - 40960*a^20*b^5*c^14*d^11 + 8192*a^21*b^4*c^13*d^12)*1i + x^(1/2)*(4096*a^11*b^13*c^20*d^4 -
 16384*a^12*b^12*c^19*d^5 + 24576*a^13*b^11*c^18*d^6 - 16384*a^14*b^10*c^17*d^7 + 4096*a^15*b^9*c^16*d^8 + 409
6*a^16*b^8*c^15*d^9 - 16384*a^17*b^7*c^14*d^10 + 24576*a^18*b^6*c^13*d^11 - 16384*a^19*b^5*c^12*d^12 + 4096*a^
20*b^4*c^11*d^13))*(-d^7/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10
*d))^(3/4)*1i + 512*a^9*b^12*c^14*d^7 - 512*a^10*b^11*c^13*d^8 - 512*a^13*b^8*c^10*d^11 + 512*a^14*b^7*c^9*d^1
2)*1i)*(-d^7/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d))^(1/4) +
 (x^(1/2)*(256*a^9*b^11*c^11*d^9 + 256*a^11*b^9*c^9*d^11) + (-d^7/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8
*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d))^(1/4)*(((-d^7/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 +
 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d))^(1/4)*(8192*a^13*b^12*c^21*d^4 - 40960*a^14*b^11*c^20*d^5 + 81920*a^15
*b^10*c^19*d^6 - 90112*a^16*b^9*c^18*d^7 + 81920*a^17*b^8*c^17*d^8 - 90112*a^18*b^7*c^16*d^9 + 81920*a^19*b^6*
c^15*d^10 - 40960*a^20*b^5*c^14*d^11 + 8192*a^21*b^4*c^13*d^12)*1i - x^(1/2)*(4096*a^11*b^13*c^20*d^4 - 16384*
a^12*b^12*c^19*d^5 + 24576*a^13*b^11*c^18*d^6 - 16384*a^14*b^10*c^17*d^7 + 4096*a^15*b^9*c^16*d^8 + 4096*a^16*
b^8*c^15*d^9 - 16384*a^17*b^7*c^14*d^10 + 24576*a^18*b^6*c^13*d^11 - 16384*a^19*b^5*c^12*d^12 + 4096*a^20*b^4*
c^11*d^13))*(-d^7/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d))^(3
/4)*1i + 512*a^9*b^12*c^14*d^7 - 512*a^10*b^11*c^13*d^8 - 512*a^13*b^8*c^10*d^11 + 512*a^14*b^7*c^9*d^12)*1i)*
(-d^7/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d))^(1/4))/((x^(1/
2)*(256*a^9*b^11*c^11*d^9 + 256*a^11*b^9*c^9*d^11) - (-d^7/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 +
96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d))^(1/4)*(((-d^7/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2
*b^2*c^9*d^2 - 64*a*b^3*c^10*d))^(1/4)*(8192*a^13*b^12*c^21*d^4 - 40960*a^14*b^11*c^20*d^5 + 81920*a^15*b^10*c
^19*d^6 - 90112*a^16*b^9*c^18*d^7 + 81920*a^17*b^8*c^17*d^8 - 90112*a^18*b^7*c^16*d^9 + 81920*a^19*b^6*c^15*d^
10 - 40960*a^20*b^5*c^14*d^11 + 8192*a^21*b^4*c^13*d^12)*1i + x^(1/2)*(4096*a^11*b^13*c^20*d^4 - 16384*a^12*b^
12*c^19*d^5 + 24576*a^13*b^11*c^18*d^6 - 16384*a^14*b^10*c^17*d^7 + 4096*a^15*b^9*c^16*d^8 + 4096*a^16*b^8*c^1
5*d^9 - 16384*a^17*b^7*c^14*d^10 + 24576*a^18*b^6*c^13*d^11 - 16384*a^19*b^5*c^12*d^12 + 4096*a^20*b^4*c^11*d^
13))*(-d^7/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d))^(3/4)*1i
+ 512*a^9*b^12*c^14*d^7 - 512*a^10*b^11*c^13*d^8 - 512*a^13*b^8*c^10*d^11 + 512*a^14*b^7*c^9*d^12)*1i)*(-d^7/(
16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d))^(1/4)*1i - (x^(1/2)*(
256*a^9*b^11*c^11*d^9 + 256*a^11*b^9*c^9*d^11) + (-d^7/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a
^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d))^(1/4)*(((-d^7/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2
*c^9*d^2 - 64*a*b^3*c^10*d))^(1/4)*(8192*a^13*b^12*c^21*d^4 - 40960*a^14*b^11*c^20*d^5 + 81920*a^15*b^10*c^19*
d^6 - 90112*a^16*b^9*c^18*d^7 + 81920*a^17*b^8*c^17*d^8 - 90112*a^18*b^7*c^16*d^9 + 81920*a^19*b^6*c^15*d^10 -
 40960*a^20*b^5*c^14*d^11 + 8192*a^21*b^4*c^13*d^12)*1i - x^(1/2)*(4096*a^11*b^13*c^20*d^4 - 16384*a^12*b^12*c
^19*d^5 + 24576*a^13*b^11*c^18*d^6 - 16384*a^14*b^10*c^17*d^7 + 4096*a^15*b^9*c^16*d^8 + 4096*a^16*b^8*c^15*d^
9 - 16384*a^17*b^7*c^14*d^10 + 24576*a^18*b^6*c^13*d^11 - 16384*a^19*b^5*c^12*d^12 + 4096*a^20*b^4*c^11*d^13))
*(-d^7/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d))^(3/4)*1i + 51
2*a^9*b^12*c^14*d^7 - 512*a^10*b^11*c^13*d^8 - 512*a^13*b^8*c^10*d^11 + 512*a^14*b^7*c^9*d^12)*1i)*(-d^7/(16*b
^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d))^(1/4)*1i))*(-d^7/(16*b^4*
c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d))^(1/4) - atan((a^2*b^5*d^7*x^
(1/2)*1i + b^7*c^2*d^5*x^(1/2)*1i - (a^2*b^16*c^11*x^(1/2)*16i)/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3
*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3) + (a^3*b^15*c^10*d*x^(1/2)*64i)/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*
a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3) - (a^4*b^14*c^9*d^2*x^(1/2)*96i)/(16*a^11*d^4 + 16*a^7*b
^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3) + (a^5*b^13*c^8*d^3*x^(1/2)*64i)/(16*a^11*d^
4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3) - (a^6*b^12*c^7*d^4*x^(1/2)*16i)
/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3) - (a^7*b^11*c^6*d^5*
x^(1/2)*16i)/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3) + (a^8*b
^10*c^5*d^6*x^(1/2)*64i)/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d
^3) - (a^9*b^9*c^4*d^7*x^(1/2)*96i)/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64
*a^10*b*c*d^3) + (a^10*b^8*c^3*d^8*x^(1/2)*64i)/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*
c^2*d^2 - 64*a^10*b*c*d^3) - (a^11*b^7*c^2*d^9*x^(1/2)*16i)/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d +
 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3))/((-b^7/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c
^2*d^2 - 64*a^10*b*c*d^3))^(1/4)*((b^7*(32*a^4*b^8*c^12 + 32*a^12*c^4*d^8 - 160*a^5*b^7*c^11*d - 160*a^11*b*c^
5*d^7 + 320*a^6*b^6*c^10*d^2 - 352*a^7*b^5*c^9*d^3 + 320*a^8*b^4*c^8*d^4 - 352*a^9*b^3*c^7*d^5 + 320*a^10*b^2*
c^6*d^6))/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3) + 2*a^5*b^3
*d^8 + 2*b^8*c^5*d^3 - 2*a*b^7*c^4*d^4 - 2*a^4*b^4*c*d^7)))*(-b^7/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c
^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3))^(1/4)*2i - atan((a^2*b^5*d^7*x^(1/2)*1i + b^7*c^2*d^5*x^(1/2)*1i
 - (a^11*c^2*d^16*x^(1/2)*16i)/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^
3*c^10*d) + (a^10*b*c^3*d^15*x^(1/2)*64i)/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^
2 - 64*a*b^3*c^10*d) - (a^2*b^9*c^11*d^7*x^(1/2)*16i)/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^
2*b^2*c^9*d^2 - 64*a*b^3*c^10*d) + (a^3*b^8*c^10*d^8*x^(1/2)*64i)/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8
*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d) - (a^4*b^7*c^9*d^9*x^(1/2)*96i)/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 6
4*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d) + (a^5*b^6*c^8*d^10*x^(1/2)*64i)/(16*b^4*c^11 + 16*a^4
*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d) - (a^6*b^5*c^7*d^11*x^(1/2)*16i)/(16*b^4*c
^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d) - (a^7*b^4*c^6*d^12*x^(1/2)*16
i)/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d) + (a^8*b^3*c^5*d^1
3*x^(1/2)*64i)/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d) - (a^9
*b^2*c^4*d^14*x^(1/2)*96i)/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^
10*d))/((-d^7/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d))^(1/4)*
((d^7*(32*a^4*b^8*c^12 + 32*a^12*c^4*d^8 - 160*a^5*b^7*c^11*d - 160*a^11*b*c^5*d^7 + 320*a^6*b^6*c^10*d^2 - 35
2*a^7*b^5*c^9*d^3 + 320*a^8*b^4*c^8*d^4 - 352*a^9*b^3*c^7*d^5 + 320*a^10*b^2*c^6*d^6))/(16*b^4*c^11 + 16*a^4*c
^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*b^3*c^10*d) + 2*a^5*b^3*d^8 + 2*b^8*c^5*d^3 - 2*a*b^7*c^
4*d^4 - 2*a^4*b^4*c*d^7)))*(-d^7/(16*b^4*c^11 + 16*a^4*c^7*d^4 - 64*a^3*b*c^8*d^3 + 96*a^2*b^2*c^9*d^2 - 64*a*
b^3*c^10*d))^(1/4)*2i + 2*atan(((x^(1/2)*(256*a^9*b^11*c^11*d^9 + 256*a^11*b^9*c^9*d^11) - (-b^7/(16*a^11*d^4
+ 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3))^(1/4)*(((-b^7/(16*a^11*d^4 + 16*a
^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3))^(1/4)*(8192*a^13*b^12*c^21*d^4 - 40960*
a^14*b^11*c^20*d^5 + 81920*a^15*b^10*c^19*d^6 - 90112*a^16*b^9*c^18*d^7 + 81920*a^17*b^8*c^17*d^8 - 90112*a^18
*b^7*c^16*d^9 + 81920*a^19*b^6*c^15*d^10 - 40960*a^20*b^5*c^14*d^11 + 8192*a^21*b^4*c^13*d^12)*1i + x^(1/2)*(4
096*a^11*b^13*c^20*d^4 - 16384*a^12*b^12*c^19*d^5 + 24576*a^13*b^11*c^18*d^6 - 16384*a^14*b^10*c^17*d^7 + 4096
*a^15*b^9*c^16*d^8 + 4096*a^16*b^8*c^15*d^9 - 16384*a^17*b^7*c^14*d^10 + 24576*a^18*b^6*c^13*d^11 - 16384*a^19
*b^5*c^12*d^12 + 4096*a^20*b^4*c^11*d^13))*(-b^7/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2
*c^2*d^2 - 64*a^10*b*c*d^3))^(3/4)*1i + 512*a^9*b^12*c^14*d^7 - 512*a^10*b^11*c^13*d^8 - 512*a^13*b^8*c^10*d^1
1 + 512*a^14*b^7*c^9*d^12)*1i)*(-b^7/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 6
4*a^10*b*c*d^3))^(1/4) + (x^(1/2)*(256*a^9*b^11*c^11*d^9 + 256*a^11*b^9*c^9*d^11) + (-b^7/(16*a^11*d^4 + 16*a^
7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3))^(1/4)*(((-b^7/(16*a^11*d^4 + 16*a^7*b^4*
c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3))^(1/4)*(8192*a^13*b^12*c^21*d^4 - 40960*a^14*b^
11*c^20*d^5 + 81920*a^15*b^10*c^19*d^6 - 90112*a^16*b^9*c^18*d^7 + 81920*a^17*b^8*c^17*d^8 - 90112*a^18*b^7*c^
16*d^9 + 81920*a^19*b^6*c^15*d^10 - 40960*a^20*b^5*c^14*d^11 + 8192*a^21*b^4*c^13*d^12)*1i - x^(1/2)*(4096*a^1
1*b^13*c^20*d^4 - 16384*a^12*b^12*c^19*d^5 + 24576*a^13*b^11*c^18*d^6 - 16384*a^14*b^10*c^17*d^7 + 4096*a^15*b
^9*c^16*d^8 + 4096*a^16*b^8*c^15*d^9 - 16384*a^17*b^7*c^14*d^10 + 24576*a^18*b^6*c^13*d^11 - 16384*a^19*b^5*c^
12*d^12 + 4096*a^20*b^4*c^11*d^13))*(-b^7/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^
2 - 64*a^10*b*c*d^3))^(3/4)*1i + 512*a^9*b^12*c^14*d^7 - 512*a^10*b^11*c^13*d^8 - 512*a^13*b^8*c^10*d^11 + 512
*a^14*b^7*c^9*d^12)*1i)*(-b^7/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*
b*c*d^3))^(1/4))/((x^(1/2)*(256*a^9*b^11*c^11*d^9 + 256*a^11*b^9*c^9*d^11) - (-b^7/(16*a^11*d^4 + 16*a^7*b^4*c
^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3))^(1/4)*(((-b^7/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 6
4*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3))^(1/4)*(8192*a^13*b^12*c^21*d^4 - 40960*a^14*b^11*c^20
*d^5 + 81920*a^15*b^10*c^19*d^6 - 90112*a^16*b^9*c^18*d^7 + 81920*a^17*b^8*c^17*d^8 - 90112*a^18*b^7*c^16*d^9
+ 81920*a^19*b^6*c^15*d^10 - 40960*a^20*b^5*c^14*d^11 + 8192*a^21*b^4*c^13*d^12)*1i + x^(1/2)*(4096*a^11*b^13*
c^20*d^4 - 16384*a^12*b^12*c^19*d^5 + 24576*a^13*b^11*c^18*d^6 - 16384*a^14*b^10*c^17*d^7 + 4096*a^15*b^9*c^16
*d^8 + 4096*a^16*b^8*c^15*d^9 - 16384*a^17*b^7*c^14*d^10 + 24576*a^18*b^6*c^13*d^11 - 16384*a^19*b^5*c^12*d^12
 + 4096*a^20*b^4*c^11*d^13))*(-b^7/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*
a^10*b*c*d^3))^(3/4)*1i + 512*a^9*b^12*c^14*d^7 - 512*a^10*b^11*c^13*d^8 - 512*a^13*b^8*c^10*d^11 + 512*a^14*b
^7*c^9*d^12)*1i)*(-b^7/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3
))^(1/4)*1i - (x^(1/2)*(256*a^9*b^11*c^11*d^9 + 256*a^11*b^9*c^9*d^11) + (-b^7/(16*a^11*d^4 + 16*a^7*b^4*c^4 -
 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3))^(1/4)*(((-b^7/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^
8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3))^(1/4)*(8192*a^13*b^12*c^21*d^4 - 40960*a^14*b^11*c^20*d^5
 + 81920*a^15*b^10*c^19*d^6 - 90112*a^16*b^9*c^18*d^7 + 81920*a^17*b^8*c^17*d^8 - 90112*a^18*b^7*c^16*d^9 + 81
920*a^19*b^6*c^15*d^10 - 40960*a^20*b^5*c^14*d^11 + 8192*a^21*b^4*c^13*d^12)*1i - x^(1/2)*(4096*a^11*b^13*c^20
*d^4 - 16384*a^12*b^12*c^19*d^5 + 24576*a^13*b^11*c^18*d^6 - 16384*a^14*b^10*c^17*d^7 + 4096*a^15*b^9*c^16*d^8
 + 4096*a^16*b^8*c^15*d^9 - 16384*a^17*b^7*c^14*d^10 + 24576*a^18*b^6*c^13*d^11 - 16384*a^19*b^5*c^12*d^12 + 4
096*a^20*b^4*c^11*d^13))*(-b^7/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10
*b*c*d^3))^(3/4)*1i + 512*a^9*b^12*c^14*d^7 - 512*a^10*b^11*c^13*d^8 - 512*a^13*b^8*c^10*d^11 + 512*a^14*b^7*c
^9*d^12)*1i)*(-b^7/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3))^(
1/4)*1i))*(-b^7/(16*a^11*d^4 + 16*a^7*b^4*c^4 - 64*a^8*b^3*c^3*d + 96*a^9*b^2*c^2*d^2 - 64*a^10*b*c*d^3))^(1/4
) - 2/(3*a*c*x^(3/2))

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sympy [F(-1)]  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(1/x**(5/2)/(b*x**2+a)/(d*x**2+c),x)

[Out]

Timed out

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