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

Optimal. Leaf size=463 \[ \frac {\sqrt [4]{a} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{b} (b c-a d)}-\frac {\sqrt [4]{a} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{b} (b c-a d)}-\frac {\sqrt [4]{c} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{d} (b c-a d)}+\frac {\sqrt [4]{c} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{d} (b c-a d)}+\frac {\sqrt [4]{a} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{b} (b c-a d)}-\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{b} (b c-a d)}-\frac {\sqrt [4]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{d} (b c-a d)}+\frac {\sqrt [4]{c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{d} (b c-a d)} \]

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Rubi [A]  time = 0.36, antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {466, 481, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {\sqrt [4]{a} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{b} (b c-a d)}-\frac {\sqrt [4]{a} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{b} (b c-a d)}-\frac {\sqrt [4]{c} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{d} (b c-a d)}+\frac {\sqrt [4]{c} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{d} (b c-a d)}+\frac {\sqrt [4]{a} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{b} (b c-a d)}-\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{b} (b c-a d)}-\frac {\sqrt [4]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{d} (b c-a d)}+\frac {\sqrt [4]{c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{d} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(1/4)*(b*c - a*d)) - (a^(1/4)*ArcTan[1 + (S
qrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(1/4)*(b*c - a*d)) - (c^(1/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x]
)/c^(1/4)])/(Sqrt[2]*d^(1/4)*(b*c - a*d)) + (c^(1/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*d
^(1/4)*(b*c - a*d)) + (a^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(1/4)*
(b*c - a*d)) - (a^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(1/4)*(b*c -
a*d)) - (c^(1/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*d^(1/4)*(b*c - a*d)) +
(c^(1/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*d^(1/4)*(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 481

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> -Dist[(a*e^n)/(b*c -
a*d), Int[(e*x)^(m - n)/(a + b*x^n), x], x] + Dist[(c*e^n)/(b*c - a*d), Int[(e*x)^(m - n)/(c + d*x^n), x], x]
/; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 364, normalized size = 0.79 \begin {gather*} \frac {\sqrt [4]{a} \sqrt [4]{d} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-\sqrt [4]{a} \sqrt [4]{d} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+2 \sqrt [4]{a} \sqrt [4]{d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )-2 \sqrt [4]{a} \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )-\sqrt [4]{b} \sqrt [4]{c} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )+\sqrt [4]{b} \sqrt [4]{c} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )-2 \sqrt [4]{b} \sqrt [4]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )+2 \sqrt [4]{b} \sqrt [4]{c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{d} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^(1/4)*d^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*a^(1/4)*d^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/
4)*Sqrt[x])/a^(1/4)] - 2*b^(1/4)*c^(1/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 2*b^(1/4)*c^(1/4)*Arc
Tan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + a^(1/4)*d^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + S
qrt[b]*x] - a^(1/4)*d^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - b^(1/4)*c^(1/4)*Log[S
qrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] + b^(1/4)*c^(1/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*
Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*b^(1/4)*d^(1/4)*(b*c - a*d))

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IntegrateAlgebraic [A]  time = 0.46, size = 265, normalized size = 0.57 \begin {gather*} \frac {\sqrt [4]{a} \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} \sqrt [4]{b} (b c-a d)}+\frac {\sqrt [4]{c} \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} \sqrt [4]{d} (a d-b c)}-\frac {\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} \sqrt [4]{b} (b c-a d)}-\frac {\sqrt [4]{c} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt {2} \sqrt [4]{d} (a d-b c)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.15, size = 1249, normalized size = 2.70

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.64, size = 441, normalized size = 0.95 \begin {gather*} -\frac {\left (a b^{3}\right )^{\frac {1}{4}} \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^{2} c - \sqrt {2} a b d} - \frac {\left (a b^{3}\right )^{\frac {1}{4}} \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^{2} c - \sqrt {2} a b d} + \frac {\left (c d^{3}\right )^{\frac {1}{4}} \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 d - \sqrt {2} a d^{2}} + \frac {\left (c d^{3}\right )^{\frac {1}{4}} \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 d - \sqrt {2} a d^{2}} - \frac {\left (a b^{3}\right )^{\frac {1}{4}} \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^{2} c - \sqrt {2} a b d\right )}} + \frac {\left (a b^{3}\right )^{\frac {1}{4}} \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^{2} c - \sqrt {2} a b d\right )}} + \frac {\left (c d^{3}\right )^{\frac {1}{4}} \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 d - \sqrt {2} a d^{2}\right )}} - \frac {\left (c d^{3}\right )^{\frac {1}{4}} \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 d - \sqrt {2} a d^{2}\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),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 304, normalized size = 0.66 \begin {gather*} \frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 a d -2 b c}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 a d -2 b c}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {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 )}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {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 )}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {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 a d -4 b c}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.10, size = 367, normalized size = 0.79 \begin {gather*} -\frac {\frac {2 \, \sqrt {2} \sqrt {a} \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}}} + \frac {2 \, \sqrt {2} \sqrt {a} \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}}} + \frac {\sqrt {2} a^{\frac {1}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} a^{\frac {1}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}}}{4 \, {\left (b c - a d\right )}} + \frac {\frac {2 \, \sqrt {2} \sqrt {c} \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}}} + \frac {2 \, \sqrt {2} \sqrt {c} \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}}} + \frac {\sqrt {2} c^{\frac {1}{4}} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{d^{\frac {1}{4}}} - \frac {\sqrt {2} c^{\frac {1}{4}} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{d^{\frac {1}{4}}}}{4 \, {\left (b c - a d\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),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.42, size = 5963, normalized size = 12.88

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atan(-((-a/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*d))^(1/4)*((-a/
(16*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*d))^(1/4)*((x^(1/2)*(4096*a^
2*b^9*c^7*d^4 - 12288*a^3*b^8*c^6*d^5 + 8192*a^4*b^7*c^5*d^6 + 8192*a^5*b^6*c^4*d^7 - 12288*a^6*b^5*c^3*d^8 +
4096*a^7*b^4*c^2*d^9) - (-a/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*
d))^(1/4)*(8192*a^2*b^10*c^8*d^4 - 49152*a^3*b^9*c^7*d^5 + 122880*a^4*b^8*c^6*d^6 - 163840*a^5*b^7*c^5*d^7 + 1
22880*a^6*b^6*c^4*d^8 - 49152*a^7*b^5*c^3*d^9 + 8192*a^8*b^4*c^2*d^10)*1i)*(-a/(16*b^5*c^4 + 16*a^4*b*d^4 - 64
*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*d))^(3/4)*1i + 512*a^2*b^6*c^5*d^3 - 512*a^3*b^5*c^4*d^4 -
512*a^4*b^4*c^3*d^5 + 512*a^5*b^3*c^2*d^6)*1i - x^(1/2)*(256*a^2*b^5*c^4*d^3 + 256*a^4*b^3*c^2*d^5)) + (-a/(16
*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*d))^(1/4)*((-a/(16*b^5*c^4 + 16
*a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*d))^(1/4)*((x^(1/2)*(4096*a^2*b^9*c^7*d^4 -
12288*a^3*b^8*c^6*d^5 + 8192*a^4*b^7*c^5*d^6 + 8192*a^5*b^6*c^4*d^7 - 12288*a^6*b^5*c^3*d^8 + 4096*a^7*b^4*c^2
*d^9) + (-a/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*d))^(1/4)*(8192*
a^2*b^10*c^8*d^4 - 49152*a^3*b^9*c^7*d^5 + 122880*a^4*b^8*c^6*d^6 - 163840*a^5*b^7*c^5*d^7 + 122880*a^6*b^6*c^
4*d^8 - 49152*a^7*b^5*c^3*d^9 + 8192*a^8*b^4*c^2*d^10)*1i)*(-a/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3 +
96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*d))^(3/4)*1i - 512*a^2*b^6*c^5*d^3 + 512*a^3*b^5*c^4*d^4 + 512*a^4*b^4*c^3*
d^5 - 512*a^5*b^3*c^2*d^6)*1i - x^(1/2)*(256*a^2*b^5*c^4*d^3 + 256*a^4*b^3*c^2*d^5)))/((-a/(16*b^5*c^4 + 16*a^
4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*d))^(1/4)*((-a/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*
a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*d))^(1/4)*((x^(1/2)*(4096*a^2*b^9*c^7*d^4 - 12288*a^3*b^8*c^
6*d^5 + 8192*a^4*b^7*c^5*d^6 + 8192*a^5*b^6*c^4*d^7 - 12288*a^6*b^5*c^3*d^8 + 4096*a^7*b^4*c^2*d^9) - (-a/(16*
b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*d))^(1/4)*(8192*a^2*b^10*c^8*d^4
- 49152*a^3*b^9*c^7*d^5 + 122880*a^4*b^8*c^6*d^6 - 163840*a^5*b^7*c^5*d^7 + 122880*a^6*b^6*c^4*d^8 - 49152*a^
7*b^5*c^3*d^9 + 8192*a^8*b^4*c^2*d^10)*1i)*(-a/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*
d^2 - 64*a*b^4*c^3*d))^(3/4)*1i + 512*a^2*b^6*c^5*d^3 - 512*a^3*b^5*c^4*d^4 - 512*a^4*b^4*c^3*d^5 + 512*a^5*b^
3*c^2*d^6)*1i - x^(1/2)*(256*a^2*b^5*c^4*d^3 + 256*a^4*b^3*c^2*d^5))*1i - (-a/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*
a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*d))^(1/4)*((-a/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3
+ 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*d))^(1/4)*((x^(1/2)*(4096*a^2*b^9*c^7*d^4 - 12288*a^3*b^8*c^6*d^5 + 8192*
a^4*b^7*c^5*d^6 + 8192*a^5*b^6*c^4*d^7 - 12288*a^6*b^5*c^3*d^8 + 4096*a^7*b^4*c^2*d^9) + (-a/(16*b^5*c^4 + 16*
a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*d))^(1/4)*(8192*a^2*b^10*c^8*d^4 - 49152*a^3*
b^9*c^7*d^5 + 122880*a^4*b^8*c^6*d^6 - 163840*a^5*b^7*c^5*d^7 + 122880*a^6*b^6*c^4*d^8 - 49152*a^7*b^5*c^3*d^9
+ 8192*a^8*b^4*c^2*d^10)*1i)*(-a/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b^
4*c^3*d))^(3/4)*1i - 512*a^2*b^6*c^5*d^3 + 512*a^3*b^5*c^4*d^4 + 512*a^4*b^4*c^3*d^5 - 512*a^5*b^3*c^2*d^6)*1i
- x^(1/2)*(256*a^2*b^5*c^4*d^3 + 256*a^4*b^3*c^2*d^5))*1i))*(-a/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3
+ 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*d))^(1/4) - atan((a^2*d^2*x^(1/2)*1i + b^2*c^2*x^(1/2)*1i - (b^6*c^6*d*x^
(1/2)*16i)/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a^3*b*c*d^4) - (a^2*b^4*c^4
*d^3*x^(1/2)*32i)/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a^3*b*c*d^4) - (a^3*
b^3*c^3*d^4*x^(1/2)*32i)/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a^3*b*c*d^4)
+ (a^4*b^2*c^2*d^5*x^(1/2)*48i)/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a^3*b*
c*d^4) - (a^5*b*c*d^6*x^(1/2)*16i)/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a^3
*b*c*d^4) + (a*b^5*c^5*d^2*x^(1/2)*48i)/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 6
4*a^3*b*c*d^4))/((-c/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a^3*b*c*d^4))^(1/
4)*((c*(32*a^6*b*d^7 + 32*b^7*c^6*d - 192*a*b^6*c^5*d^2 - 192*a^5*b^2*c*d^6 + 480*a^2*b^5*c^4*d^3 - 640*a^3*b^
4*c^3*d^4 + 480*a^4*b^3*c^2*d^5))/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a^3*
b*c*d^4) - 2*b^3*c^3 - 2*a^3*d^3 + 2*a*b^2*c^2*d + 2*a^2*b*c*d^2)))*(-c/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*
c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a^3*b*c*d^4))^(1/4)*2i - atan((a^2*d^2*x^(1/2)*1i + b^2*c^2*x^(1/2)*1i - (a^
6*b*d^6*x^(1/2)*16i)/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*d) + (a
^2*b^5*c^4*d^2*x^(1/2)*48i)/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*
d) - (a^3*b^4*c^3*d^3*x^(1/2)*32i)/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b
^4*c^3*d) - (a^4*b^3*c^2*d^4*x^(1/2)*32i)/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 -
64*a*b^4*c^3*d) - (a*b^6*c^5*d*x^(1/2)*16i)/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^
2 - 64*a*b^4*c^3*d) + (a^5*b^2*c*d^5*x^(1/2)*48i)/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c
^2*d^2 - 64*a*b^4*c^3*d))/((-a/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c
^3*d))^(1/4)*((a*(32*a^6*b*d^7 + 32*b^7*c^6*d - 192*a*b^6*c^5*d^2 - 192*a^5*b^2*c*d^6 + 480*a^2*b^5*c^4*d^3 -
640*a^3*b^4*c^3*d^4 + 480*a^4*b^3*c^2*d^5))/(16*b^5*c^4 + 16*a^4*b*d^4 - 64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2
- 64*a*b^4*c^3*d) - 2*b^3*c^3 - 2*a^3*d^3 + 2*a*b^2*c^2*d + 2*a^2*b*c*d^2)))*(-a/(16*b^5*c^4 + 16*a^4*b*d^4 -
64*a^3*b^2*c*d^3 + 96*a^2*b^3*c^2*d^2 - 64*a*b^4*c^3*d))^(1/4)*2i + 2*atan(-((-c/(16*a^4*d^5 + 16*b^4*c^4*d -
64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a^3*b*c*d^4))^(1/4)*((-c/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3
*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a^3*b*c*d^4))^(1/4)*((x^(1/2)*(4096*a^2*b^9*c^7*d^4 - 12288*a^3*b^8*c^6*d^5 + 8
192*a^4*b^7*c^5*d^6 + 8192*a^5*b^6*c^4*d^7 - 12288*a^6*b^5*c^3*d^8 + 4096*a^7*b^4*c^2*d^9) - (-c/(16*a^4*d^5 +
16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a^3*b*c*d^4))^(1/4)*(8192*a^2*b^10*c^8*d^4 - 49152*
a^3*b^9*c^7*d^5 + 122880*a^4*b^8*c^6*d^6 - 163840*a^5*b^7*c^5*d^7 + 122880*a^6*b^6*c^4*d^8 - 49152*a^7*b^5*c^3
*d^9 + 8192*a^8*b^4*c^2*d^10)*1i)*(-c/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*
a^3*b*c*d^4))^(3/4)*1i + 512*a^2*b^6*c^5*d^3 - 512*a^3*b^5*c^4*d^4 - 512*a^4*b^4*c^3*d^5 + 512*a^5*b^3*c^2*d^6
)*1i - x^(1/2)*(256*a^2*b^5*c^4*d^3 + 256*a^4*b^3*c^2*d^5)) + (-c/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^
2 + 96*a^2*b^2*c^2*d^3 - 64*a^3*b*c*d^4))^(1/4)*((-c/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^
2*c^2*d^3 - 64*a^3*b*c*d^4))^(1/4)*((x^(1/2)*(4096*a^2*b^9*c^7*d^4 - 12288*a^3*b^8*c^6*d^5 + 8192*a^4*b^7*c^5*
d^6 + 8192*a^5*b^6*c^4*d^7 - 12288*a^6*b^5*c^3*d^8 + 4096*a^7*b^4*c^2*d^9) + (-c/(16*a^4*d^5 + 16*b^4*c^4*d -
64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a^3*b*c*d^4))^(1/4)*(8192*a^2*b^10*c^8*d^4 - 49152*a^3*b^9*c^7*d^5
+ 122880*a^4*b^8*c^6*d^6 - 163840*a^5*b^7*c^5*d^7 + 122880*a^6*b^6*c^4*d^8 - 49152*a^7*b^5*c^3*d^9 + 8192*a^8*
b^4*c^2*d^10)*1i)*(-c/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a^3*b*c*d^4))^(3
/4)*1i - 512*a^2*b^6*c^5*d^3 + 512*a^3*b^5*c^4*d^4 + 512*a^4*b^4*c^3*d^5 - 512*a^5*b^3*c^2*d^6)*1i - x^(1/2)*(
256*a^2*b^5*c^4*d^3 + 256*a^4*b^3*c^2*d^5)))/((-c/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c
^2*d^3 - 64*a^3*b*c*d^4))^(1/4)*((-c/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a
^3*b*c*d^4))^(1/4)*((x^(1/2)*(4096*a^2*b^9*c^7*d^4 - 12288*a^3*b^8*c^6*d^5 + 8192*a^4*b^7*c^5*d^6 + 8192*a^5*b
^6*c^4*d^7 - 12288*a^6*b^5*c^3*d^8 + 4096*a^7*b^4*c^2*d^9) - (-c/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2
+ 96*a^2*b^2*c^2*d^3 - 64*a^3*b*c*d^4))^(1/4)*(8192*a^2*b^10*c^8*d^4 - 49152*a^3*b^9*c^7*d^5 + 122880*a^4*b^8
*c^6*d^6 - 163840*a^5*b^7*c^5*d^7 + 122880*a^6*b^6*c^4*d^8 - 49152*a^7*b^5*c^3*d^9 + 8192*a^8*b^4*c^2*d^10)*1i
)*(-c/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a^3*b*c*d^4))^(3/4)*1i + 512*a^2
*b^6*c^5*d^3 - 512*a^3*b^5*c^4*d^4 - 512*a^4*b^4*c^3*d^5 + 512*a^5*b^3*c^2*d^6)*1i - x^(1/2)*(256*a^2*b^5*c^4*
d^3 + 256*a^4*b^3*c^2*d^5))*1i - (-c/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a
^3*b*c*d^4))^(1/4)*((-c/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a^3*b*c*d^4))^
(1/4)*((x^(1/2)*(4096*a^2*b^9*c^7*d^4 - 12288*a^3*b^8*c^6*d^5 + 8192*a^4*b^7*c^5*d^6 + 8192*a^5*b^6*c^4*d^7 -
12288*a^6*b^5*c^3*d^8 + 4096*a^7*b^4*c^2*d^9) + (-c/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2
*c^2*d^3 - 64*a^3*b*c*d^4))^(1/4)*(8192*a^2*b^10*c^8*d^4 - 49152*a^3*b^9*c^7*d^5 + 122880*a^4*b^8*c^6*d^6 - 16
3840*a^5*b^7*c^5*d^7 + 122880*a^6*b^6*c^4*d^8 - 49152*a^7*b^5*c^3*d^9 + 8192*a^8*b^4*c^2*d^10)*1i)*(-c/(16*a^4
*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a^3*b*c*d^4))^(3/4)*1i - 512*a^2*b^6*c^5*d^3
+ 512*a^3*b^5*c^4*d^4 + 512*a^4*b^4*c^3*d^5 - 512*a^5*b^3*c^2*d^6)*1i - x^(1/2)*(256*a^2*b^5*c^4*d^3 + 256*a^4
*b^3*c^2*d^5))*1i))*(-c/(16*a^4*d^5 + 16*b^4*c^4*d - 64*a*b^3*c^3*d^2 + 96*a^2*b^2*c^2*d^3 - 64*a^3*b*c*d^4))^
(1/4)

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

[Out]

Timed out

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