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

Optimal. Leaf size=463 \[ -\frac {a^{3/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{3/4} (b c-a d)}+\frac {a^{3/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{3/4} (b c-a d)}+\frac {a^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{3/4} (b c-a d)}-\frac {a^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} b^{3/4} (b c-a d)}+\frac {c^{3/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{3/4} (b c-a d)}-\frac {c^{3/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{3/4} (b c-a d)}-\frac {c^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{3/4} (b c-a d)}+\frac {c^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} d^{3/4} (b c-a d)} \]

[Out]

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

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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, 297, 1162, 617, 204, 1165, 628} \[ -\frac {a^{3/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{3/4} (b c-a d)}+\frac {a^{3/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{3/4} (b c-a d)}+\frac {a^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{3/4} (b c-a d)}-\frac {a^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} b^{3/4} (b c-a d)}+\frac {c^{3/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{3/4} (b c-a d)}-\frac {c^{3/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{3/4} (b c-a d)}-\frac {c^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{3/4} (b c-a d)}+\frac {c^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} d^{3/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(3/4)*(b*c - a*d)) - (a^(3/4)*ArcTan[1 + (S
qrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(3/4)*(b*c - a*d)) - (c^(3/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x]
)/c^(1/4)])/(Sqrt[2]*d^(3/4)*(b*c - a*d)) + (c^(3/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*d
^(3/4)*(b*c - a*d)) - (a^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(3/4)*
(b*c - a*d)) + (a^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(3/4)*(b*c -
a*d)) + (c^(3/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*d^(3/4)*(b*c - a*d)) -
(c^(3/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*d^(3/4)*(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 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

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

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Mathematica [A]  time = 0.11, size = 142, normalized size = 0.31 \[ \frac {\frac {(-a)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{-a}}\right )}{b^{3/4}}-\frac {(-a)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{-a}}\right )}{b^{3/4}}-\frac {(-c)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{-c}}\right )}{d^{3/4}}+\frac {(-c)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{-c}}\right )}{d^{3/4}}}{b c-a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.61, size = 1385, normalized size = 2.99 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(-a^3/(b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4))^(1/4)*arctan(-(sqrt(a^
4*x - (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*sqrt(-a^3/(b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3
*b^4*c*d^3 + a^4*b^3*d^4)))*(b^2*c - a*b*d)*(-a^3/(b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d
^3 + a^4*b^3*d^4))^(1/4) - (a^2*b^2*c - a^3*b*d)*(-a^3/(b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^
4*c*d^3 + a^4*b^3*d^4))^(1/4)*sqrt(x))/a^3) + 2*(-c^3/(b^4*c^4*d^3 - 4*a*b^3*c^3*d^4 + 6*a^2*b^2*c^2*d^5 - 4*a
^3*b*c*d^6 + a^4*d^7))^(1/4)*arctan(-(sqrt(c^4*x - (b^2*c^5*d - 2*a*b*c^4*d^2 + a^2*c^3*d^3)*sqrt(-c^3/(b^4*c^
4*d^3 - 4*a*b^3*c^3*d^4 + 6*a^2*b^2*c^2*d^5 - 4*a^3*b*c*d^6 + a^4*d^7)))*(b*c*d - a*d^2)*(-c^3/(b^4*c^4*d^3 -
4*a*b^3*c^3*d^4 + 6*a^2*b^2*c^2*d^5 - 4*a^3*b*c*d^6 + a^4*d^7))^(1/4) - (b*c^3*d - a*c^2*d^2)*(-c^3/(b^4*c^4*d
^3 - 4*a*b^3*c^3*d^4 + 6*a^2*b^2*c^2*d^5 - 4*a^3*b*c*d^6 + a^4*d^7))^(1/4)*sqrt(x))/c^3) - 1/2*(-a^3/(b^7*c^4
- 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4))^(1/4)*log(a^2*sqrt(x) + (b^5*c^3 - 3*a*b
^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*(-a^3/(b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3
+ a^4*b^3*d^4))^(3/4)) + 1/2*(-a^3/(b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d
^4))^(1/4)*log(a^2*sqrt(x) - (b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*(-a^3/(b^7*c^4 - 4*a*b^
6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4))^(3/4)) + 1/2*(-c^3/(b^4*c^4*d^3 - 4*a*b^3*c^3*d^
4 + 6*a^2*b^2*c^2*d^5 - 4*a^3*b*c*d^6 + a^4*d^7))^(1/4)*log(c^2*sqrt(x) + (b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a
^2*b*c*d^4 - a^3*d^5)*(-c^3/(b^4*c^4*d^3 - 4*a*b^3*c^3*d^4 + 6*a^2*b^2*c^2*d^5 - 4*a^3*b*c*d^6 + a^4*d^7))^(3/
4)) - 1/2*(-c^3/(b^4*c^4*d^3 - 4*a*b^3*c^3*d^4 + 6*a^2*b^2*c^2*d^5 - 4*a^3*b*c*d^6 + a^4*d^7))^(1/4)*log(c^2*s
qrt(x) - (b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*(-c^3/(b^4*c^4*d^3 - 4*a*b^3*c^3*d^4 + 6*a^
2*b^2*c^2*d^5 - 4*a^3*b*c*d^6 + a^4*d^7))^(3/4))

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giac [A]  time = 0.69, size = 457, normalized size = 0.99 \[ -\frac {\left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b^{4} c - \sqrt {2} a b^{3} d} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b^{4} c - \sqrt {2} a b^{3} d} + \frac {\left (c d^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b c d^{3} - \sqrt {2} a d^{4}} + \frac {\left (c d^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b c d^{3} - \sqrt {2} a d^{4}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} b^{4} c - \sqrt {2} a b^{3} d\right )}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} b^{4} c - \sqrt {2} a b^{3} d\right )}} - \frac {\left (c d^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{2 \, {\left (\sqrt {2} b c d^{3} - \sqrt {2} a d^{4}\right )}} + \frac {\left (c d^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{2 \, {\left (\sqrt {2} b c d^{3} - \sqrt {2} a d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 328, normalized size = 0.71 \[ \frac {\sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} b}+\frac {\sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} b}+\frac {\sqrt {2}\, a \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} b}-\frac {\sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{2 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} d}-\frac {\sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{2 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} d}-\frac {\sqrt {2}\, c \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{4 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a/(a*d-b*c)/b/(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/(a*d-b*c)/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+1/2*a/(a*d-b*c
)/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)-1/4*c/(a*d-b*c)/d/(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/(a*d-b*c)/d/(c/d)^(1/4
)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-1/2*c/(a*d-b*c)/d/(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 = 2.60, size = 369, normalized size = 0.80 \[ -\frac {a {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, {\left (b c - a d\right )}} + \frac {c {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{4 \, {\left (b c - a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.27, size = 2609, normalized size = 5.63 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- 2*atan((2*b^4*c^3*x^(1/2)*(-a^3/(16*b^7*c^4 + 16*a^4*b^3*d^4 - 64*a^3*b^4*c*d^3 + 96*a^2*b^5*c^2*d^2 - 64*a*
b^6*c^3*d))^(1/4) + 64*a^4*b^4*d^7*x^(1/2)*(-a^3/(16*b^7*c^4 + 16*a^4*b^3*d^4 - 64*a^3*b^4*c*d^3 + 96*a^2*b^5*
c^2*d^2 - 64*a*b^6*c^3*d))^(5/4) + 64*b^8*c^4*d^3*x^(1/2)*(-a^3/(16*b^7*c^4 + 16*a^4*b^3*d^4 - 64*a^3*b^4*c*d^
3 + 96*a^2*b^5*c^2*d^2 - 64*a*b^6*c^3*d))^(5/4) + 2*a^3*b*d^3*x^(1/2)*(-a^3/(16*b^7*c^4 + 16*a^4*b^3*d^4 - 64*
a^3*b^4*c*d^3 + 96*a^2*b^5*c^2*d^2 - 64*a*b^6*c^3*d))^(1/4) + 384*a^2*b^6*c^2*d^5*x^(1/2)*(-a^3/(16*b^7*c^4 +
16*a^4*b^3*d^4 - 64*a^3*b^4*c*d^3 + 96*a^2*b^5*c^2*d^2 - 64*a*b^6*c^3*d))^(5/4) - 256*a*b^7*c^3*d^4*x^(1/2)*(-
a^3/(16*b^7*c^4 + 16*a^4*b^3*d^4 - 64*a^3*b^4*c*d^3 + 96*a^2*b^5*c^2*d^2 - 64*a*b^6*c^3*d))^(5/4) - 256*a^3*b^
5*c*d^6*x^(1/2)*(-a^3/(16*b^7*c^4 + 16*a^4*b^3*d^4 - 64*a^3*b^4*c*d^3 + 96*a^2*b^5*c^2*d^2 - 64*a*b^6*c^3*d))^
(5/4))/(a^3*d^2 + a*b^2*c^2 + a^2*b*c*d))*(-a^3/(16*b^7*c^4 + 16*a^4*b^3*d^4 - 64*a^3*b^4*c*d^3 + 96*a^2*b^5*c
^2*d^2 - 64*a*b^6*c^3*d))^(1/4) - atan((b^4*c^3*x^(1/2)*(-a^3/(16*b^7*c^4 + 16*a^4*b^3*d^4 - 64*a^3*b^4*c*d^3
+ 96*a^2*b^5*c^2*d^2 - 64*a*b^6*c^3*d))^(1/4)*2i + a^4*b^4*d^7*x^(1/2)*(-a^3/(16*b^7*c^4 + 16*a^4*b^3*d^4 - 64
*a^3*b^4*c*d^3 + 96*a^2*b^5*c^2*d^2 - 64*a*b^6*c^3*d))^(5/4)*64i + b^8*c^4*d^3*x^(1/2)*(-a^3/(16*b^7*c^4 + 16*
a^4*b^3*d^4 - 64*a^3*b^4*c*d^3 + 96*a^2*b^5*c^2*d^2 - 64*a*b^6*c^3*d))^(5/4)*64i + a^3*b*d^3*x^(1/2)*(-a^3/(16
*b^7*c^4 + 16*a^4*b^3*d^4 - 64*a^3*b^4*c*d^3 + 96*a^2*b^5*c^2*d^2 - 64*a*b^6*c^3*d))^(1/4)*2i + a^2*b^6*c^2*d^
5*x^(1/2)*(-a^3/(16*b^7*c^4 + 16*a^4*b^3*d^4 - 64*a^3*b^4*c*d^3 + 96*a^2*b^5*c^2*d^2 - 64*a*b^6*c^3*d))^(5/4)*
384i - a*b^7*c^3*d^4*x^(1/2)*(-a^3/(16*b^7*c^4 + 16*a^4*b^3*d^4 - 64*a^3*b^4*c*d^3 + 96*a^2*b^5*c^2*d^2 - 64*a
*b^6*c^3*d))^(5/4)*256i - a^3*b^5*c*d^6*x^(1/2)*(-a^3/(16*b^7*c^4 + 16*a^4*b^3*d^4 - 64*a^3*b^4*c*d^3 + 96*a^2
*b^5*c^2*d^2 - 64*a*b^6*c^3*d))^(5/4)*256i)/(a^3*d^2 + a*b^2*c^2 + a^2*b*c*d))*(-a^3/(16*b^7*c^4 + 16*a^4*b^3*
d^4 - 64*a^3*b^4*c*d^3 + 96*a^2*b^5*c^2*d^2 - 64*a*b^6*c^3*d))^(1/4)*2i - 2*atan((2*a^3*d^4*x^(1/2)*(-c^3/(16*
a^4*d^7 + 16*b^4*c^4*d^3 - 64*a*b^3*c^3*d^4 + 96*a^2*b^2*c^2*d^5 - 64*a^3*b*c*d^6))^(1/4) + 2*b^3*c^3*d*x^(1/2
)*(-c^3/(16*a^4*d^7 + 16*b^4*c^4*d^3 - 64*a*b^3*c^3*d^4 + 96*a^2*b^2*c^2*d^5 - 64*a^3*b*c*d^6))^(1/4) + 64*a^4
*b^3*d^8*x^(1/2)*(-c^3/(16*a^4*d^7 + 16*b^4*c^4*d^3 - 64*a*b^3*c^3*d^4 + 96*a^2*b^2*c^2*d^5 - 64*a^3*b*c*d^6))
^(5/4) + 64*b^7*c^4*d^4*x^(1/2)*(-c^3/(16*a^4*d^7 + 16*b^4*c^4*d^3 - 64*a*b^3*c^3*d^4 + 96*a^2*b^2*c^2*d^5 - 6
4*a^3*b*c*d^6))^(5/4) + 384*a^2*b^5*c^2*d^6*x^(1/2)*(-c^3/(16*a^4*d^7 + 16*b^4*c^4*d^3 - 64*a*b^3*c^3*d^4 + 96
*a^2*b^2*c^2*d^5 - 64*a^3*b*c*d^6))^(5/4) - 256*a*b^6*c^3*d^5*x^(1/2)*(-c^3/(16*a^4*d^7 + 16*b^4*c^4*d^3 - 64*
a*b^3*c^3*d^4 + 96*a^2*b^2*c^2*d^5 - 64*a^3*b*c*d^6))^(5/4) - 256*a^3*b^4*c*d^7*x^(1/2)*(-c^3/(16*a^4*d^7 + 16
*b^4*c^4*d^3 - 64*a*b^3*c^3*d^4 + 96*a^2*b^2*c^2*d^5 - 64*a^3*b*c*d^6))^(5/4))/(b^2*c^3 + a^2*c*d^2 + a*b*c^2*
d))*(-c^3/(16*a^4*d^7 + 16*b^4*c^4*d^3 - 64*a*b^3*c^3*d^4 + 96*a^2*b^2*c^2*d^5 - 64*a^3*b*c*d^6))^(1/4) - atan
((a^3*d^4*x^(1/2)*(-c^3/(16*a^4*d^7 + 16*b^4*c^4*d^3 - 64*a*b^3*c^3*d^4 + 96*a^2*b^2*c^2*d^5 - 64*a^3*b*c*d^6)
)^(1/4)*2i + b^3*c^3*d*x^(1/2)*(-c^3/(16*a^4*d^7 + 16*b^4*c^4*d^3 - 64*a*b^3*c^3*d^4 + 96*a^2*b^2*c^2*d^5 - 64
*a^3*b*c*d^6))^(1/4)*2i + a^4*b^3*d^8*x^(1/2)*(-c^3/(16*a^4*d^7 + 16*b^4*c^4*d^3 - 64*a*b^3*c^3*d^4 + 96*a^2*b
^2*c^2*d^5 - 64*a^3*b*c*d^6))^(5/4)*64i + b^7*c^4*d^4*x^(1/2)*(-c^3/(16*a^4*d^7 + 16*b^4*c^4*d^3 - 64*a*b^3*c^
3*d^4 + 96*a^2*b^2*c^2*d^5 - 64*a^3*b*c*d^6))^(5/4)*64i + a^2*b^5*c^2*d^6*x^(1/2)*(-c^3/(16*a^4*d^7 + 16*b^4*c
^4*d^3 - 64*a*b^3*c^3*d^4 + 96*a^2*b^2*c^2*d^5 - 64*a^3*b*c*d^6))^(5/4)*384i - a*b^6*c^3*d^5*x^(1/2)*(-c^3/(16
*a^4*d^7 + 16*b^4*c^4*d^3 - 64*a*b^3*c^3*d^4 + 96*a^2*b^2*c^2*d^5 - 64*a^3*b*c*d^6))^(5/4)*256i - a^3*b^4*c*d^
7*x^(1/2)*(-c^3/(16*a^4*d^7 + 16*b^4*c^4*d^3 - 64*a*b^3*c^3*d^4 + 96*a^2*b^2*c^2*d^5 - 64*a^3*b*c*d^6))^(5/4)*
256i)/(b^2*c^3 + a^2*c*d^2 + a*b*c^2*d))*(-c^3/(16*a^4*d^7 + 16*b^4*c^4*d^3 - 64*a*b^3*c^3*d^4 + 96*a^2*b^2*c^
2*d^5 - 64*a^3*b*c*d^6))^(1/4)*2i

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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