3.3.47 \(\int \frac {x^3}{(d+e x^2) (a+c x^4)^2} \, dx\) [247]

Optimal. Leaf size=149 \[ \frac {-d+e x^2}{4 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {e \left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 \sqrt {a} \sqrt {c} \left (c d^2+a e^2\right )^2}-\frac {d e^2 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}+\frac {d e^2 \log \left (a+c x^4\right )}{4 \left (c d^2+a e^2\right )^2} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 148, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1266, 837, 815, 649, 211, 266} \begin {gather*} -\frac {e \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) \left (c d^2-a e^2\right )}{4 \sqrt {a} \sqrt {c} \left (a e^2+c d^2\right )^2}+\frac {d e^2 \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2}-\frac {d e^2 \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )^2}-\frac {d-e x^2}{4 \left (a+c x^4\right ) \left (a e^2+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/((d + e*x^2)*(a + c*x^4)^2),x]

[Out]

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

Rule 211

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

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 815

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x)^m*((f + g*x)/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 1266

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m
- 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]

Rubi steps

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

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Mathematica [A]
time = 0.10, size = 114, normalized size = 0.77 \begin {gather*} \frac {\frac {\left (c d^2+a e^2\right ) \left (-d+e x^2\right )}{a+c x^4}+\frac {e \left (-c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}-2 d e^2 \log \left (d+e x^2\right )+d e^2 \log \left (a+c x^4\right )}{4 \left (c d^2+a e^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/((d + e*x^2)*(a + c*x^4)^2),x]

[Out]

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

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Maple [A]
time = 0.20, size = 132, normalized size = 0.89

method	result	size
default	\(-\frac {d \,e^{2} \ln \left (e \,x^{2}+d \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}+\frac {\frac {\left (\frac {1}{2} a \,e^{3}+\frac {1}{2} c \,d^{2} e \right ) x^{2}-\frac {d \left (a \,e^{2}+c \,d^{2}\right )}{2}}{c \,x^{4}+a}+\frac {e \left (d e \ln \left (c \,x^{4}+a \right )+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{\sqrt {a c}}\right )}{2}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}\)	\(132\)
risch	\(\frac {\frac {e \,x^{2}}{4 a \,e^{2}+4 c \,d^{2}}-\frac {d}{4 \left (a \,e^{2}+c \,d^{2}\right )}}{c \,x^{4}+a}+\frac {e^{2} \ln \left (\left (8 a^{3} c d \,e^{5}-16 a^{2} c^{2} d^{3} e^{3}+8 c^{3} d^{5} e a +\sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, a^{2} e^{4}-14 \sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, a c \,d^{2} e^{2}+\sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, c^{2} d^{4}\right ) x^{2}-e^{6} a^{4}+15 a^{3} c \,d^{2} e^{4}-15 a^{2} c^{2} d^{4} e^{2}+a \,c^{3} d^{6}+8 \sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, a^{2} d \,e^{3}-8 \sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, a c \,d^{3} e \right ) d}{4 a^{2} e^{4}+8 a c \,d^{2} e^{2}+4 c^{2} d^{4}}+\frac {e \ln \left (\left (8 a^{3} c d \,e^{5}-16 a^{2} c^{2} d^{3} e^{3}+8 c^{3} d^{5} e a +\sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, a^{2} e^{4}-14 \sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, a c \,d^{2} e^{2}+\sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, c^{2} d^{4}\right ) x^{2}-e^{6} a^{4}+15 a^{3} c \,d^{2} e^{4}-15 a^{2} c^{2} d^{4} e^{2}+a \,c^{3} d^{6}+8 \sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, a^{2} d \,e^{3}-8 \sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, a c \,d^{3} e \right ) \sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}}{8 a c \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}+\frac {e^{2} \ln \left (\left (8 a^{3} c d \,e^{5}-16 a^{2} c^{2} d^{3} e^{3}+8 c^{3} d^{5} e a -\sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, a^{2} e^{4}+14 \sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, a c \,d^{2} e^{2}-\sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, c^{2} d^{4}\right ) x^{2}-e^{6} a^{4}+15 a^{3} c \,d^{2} e^{4}-15 a^{2} c^{2} d^{4} e^{2}+a \,c^{3} d^{6}-8 \sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, a^{2} d \,e^{3}+8 \sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, a c \,d^{3} e \right ) d}{4 a^{2} e^{4}+8 a c \,d^{2} e^{2}+4 c^{2} d^{4}}-\frac {e \ln \left (\left (8 a^{3} c d \,e^{5}-16 a^{2} c^{2} d^{3} e^{3}+8 c^{3} d^{5} e a -\sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, a^{2} e^{4}+14 \sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, a c \,d^{2} e^{2}-\sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, c^{2} d^{4}\right ) x^{2}-e^{6} a^{4}+15 a^{3} c \,d^{2} e^{4}-15 a^{2} c^{2} d^{4} e^{2}+a \,c^{3} d^{6}-8 \sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, a^{2} d \,e^{3}+8 \sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}\, a c \,d^{3} e \right ) \sqrt {-a c \left (a \,e^{2}-c \,d^{2}\right )^{2}}}{8 a c \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}-\frac {d \,e^{2} \ln \left (e \,x^{2}+d \right )}{2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}\)	\(1167\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(e*x^2+d)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.51, size = 178, normalized size = 1.19 \begin {gather*} \frac {d e^{2} \log \left (c x^{4} + a\right )}{4 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} - \frac {d e^{2} \log \left (x^{2} e + d\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} - \frac {{\left (c d^{2} e - a e^{3}\right )} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {a c}} + \frac {x^{2} e - d}{4 \, {\left ({\left (c^{2} d^{2} + a c e^{2}\right )} x^{4} + a c d^{2} + a^{2} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.93, size = 475, normalized size = 3.19 \begin {gather*} \left [\frac {2 \, a c^{2} d^{2} x^{2} e - 2 \, a c^{2} d^{3} + 2 \, a^{2} c x^{2} e^{3} - 2 \, a^{2} c d e^{2} + 2 \, {\left (a c^{2} d x^{4} + a^{2} c d\right )} e^{2} \log \left (c x^{4} + a\right ) - 4 \, {\left (a c^{2} d x^{4} + a^{2} c d\right )} e^{2} \log \left (x^{2} e + d\right ) - \sqrt {-a c} {\left ({\left (a c x^{4} + a^{2}\right )} e^{3} - {\left (c^{2} d^{2} x^{4} + a c d^{2}\right )} e\right )} \log \left (\frac {c x^{4} - 2 \, \sqrt {-a c} x^{2} - a}{c x^{4} + a}\right )}{8 \, {\left (a c^{4} d^{4} x^{4} + a^{2} c^{3} d^{4} + {\left (a^{3} c^{2} x^{4} + a^{4} c\right )} e^{4} + 2 \, {\left (a^{2} c^{3} d^{2} x^{4} + a^{3} c^{2} d^{2}\right )} e^{2}\right )}}, \frac {a c^{2} d^{2} x^{2} e - a c^{2} d^{3} + a^{2} c x^{2} e^{3} - a^{2} c d e^{2} + {\left (a c^{2} d x^{4} + a^{2} c d\right )} e^{2} \log \left (c x^{4} + a\right ) - 2 \, {\left (a c^{2} d x^{4} + a^{2} c d\right )} e^{2} \log \left (x^{2} e + d\right ) - \sqrt {a c} {\left ({\left (a c x^{4} + a^{2}\right )} e^{3} - {\left (c^{2} d^{2} x^{4} + a c d^{2}\right )} e\right )} \arctan \left (\frac {\sqrt {a c}}{c x^{2}}\right )}{4 \, {\left (a c^{4} d^{4} x^{4} + a^{2} c^{3} d^{4} + {\left (a^{3} c^{2} x^{4} + a^{4} c\right )} e^{4} + 2 \, {\left (a^{2} c^{3} d^{2} x^{4} + a^{3} c^{2} d^{2}\right )} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="fricas")

[Out]

[1/8*(2*a*c^2*d^2*x^2*e - 2*a*c^2*d^3 + 2*a^2*c*x^2*e^3 - 2*a^2*c*d*e^2 + 2*(a*c^2*d*x^4 + a^2*c*d)*e^2*log(c*
x^4 + a) - 4*(a*c^2*d*x^4 + a^2*c*d)*e^2*log(x^2*e + d) - sqrt(-a*c)*((a*c*x^4 + a^2)*e^3 - (c^2*d^2*x^4 + a*c
*d^2)*e)*log((c*x^4 - 2*sqrt(-a*c)*x^2 - a)/(c*x^4 + a)))/(a*c^4*d^4*x^4 + a^2*c^3*d^4 + (a^3*c^2*x^4 + a^4*c)
*e^4 + 2*(a^2*c^3*d^2*x^4 + a^3*c^2*d^2)*e^2), 1/4*(a*c^2*d^2*x^2*e - a*c^2*d^3 + a^2*c*x^2*e^3 - a^2*c*d*e^2
+ (a*c^2*d*x^4 + a^2*c*d)*e^2*log(c*x^4 + a) - 2*(a*c^2*d*x^4 + a^2*c*d)*e^2*log(x^2*e + d) - sqrt(a*c)*((a*c*
x^4 + a^2)*e^3 - (c^2*d^2*x^4 + a*c*d^2)*e)*arctan(sqrt(a*c)/(c*x^2)))/(a*c^4*d^4*x^4 + a^2*c^3*d^4 + (a^3*c^2
*x^4 + a^4*c)*e^4 + 2*(a^2*c^3*d^2*x^4 + a^3*c^2*d^2)*e^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(e*x**2+d)/(c*x**4+a)**2,x)

[Out]

Timed out

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Giac [A]
time = 3.15, size = 188, normalized size = 1.26 \begin {gather*} \frac {d e^{2} \log \left (c x^{4} + a\right )}{4 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} - \frac {d e^{3} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}} - \frac {{\left (c d^{2} e - a e^{3}\right )} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {a c}} - \frac {c d^{3} - {\left (c d^{2} e + a e^{3}\right )} x^{2} + a d e^{2}}{4 \, {\left (c x^{4} + a\right )} {\left (c d^{2} + a e^{2}\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.41, size = 527, normalized size = 3.54 \begin {gather*} \frac {\ln \left (a^4\,e^8\,\sqrt {-a\,c}+c^4\,d^8\,\sqrt {-a\,c}+70\,d^4\,e^4\,{\left (-a\,c\right )}^{5/2}+c^5\,d^8\,x^2+a^4\,c\,e^8\,x^2-36\,a^2\,d^2\,e^6\,{\left (-a\,c\right )}^{3/2}-36\,c^2\,d^6\,e^2\,{\left (-a\,c\right )}^{3/2}+70\,a^2\,c^3\,d^4\,e^4\,x^2+36\,a^3\,c^2\,d^2\,e^6\,x^2+36\,a\,c^4\,d^6\,e^2\,x^2\right )\,\left (a\,\left (\frac {e^3\,\sqrt {-a\,c}}{8}+\frac {c\,d\,e^2}{4}\right )-\frac {c\,d^2\,e\,\sqrt {-a\,c}}{8}\right )}{a^3\,c\,e^4+2\,a^2\,c^2\,d^2\,e^2+a\,c^3\,d^4}-\frac {\frac {d}{4\,\left (c\,d^2+a\,e^2\right )}-\frac {e\,x^2}{4\,\left (c\,d^2+a\,e^2\right )}}{c\,x^4+a}-\frac {\ln \left (c^5\,d^8\,x^2-c^4\,d^8\,\sqrt {-a\,c}-70\,d^4\,e^4\,{\left (-a\,c\right )}^{5/2}-a^4\,e^8\,\sqrt {-a\,c}+a^4\,c\,e^8\,x^2+36\,a^2\,d^2\,e^6\,{\left (-a\,c\right )}^{3/2}+36\,c^2\,d^6\,e^2\,{\left (-a\,c\right )}^{3/2}+70\,a^2\,c^3\,d^4\,e^4\,x^2+36\,a^3\,c^2\,d^2\,e^6\,x^2+36\,a\,c^4\,d^6\,e^2\,x^2\right )\,\left (a\,\left (\frac {e^3\,\sqrt {-a\,c}}{8}-\frac {c\,d\,e^2}{4}\right )-\frac {c\,d^2\,e\,\sqrt {-a\,c}}{8}\right )}{a^3\,c\,e^4+2\,a^2\,c^2\,d^2\,e^2+a\,c^3\,d^4}-\frac {d\,e^2\,\ln \left (e\,x^2+d\right )}{2\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((a + c*x^4)^2*(d + e*x^2)),x)

[Out]

(log(a^4*e^8*(-a*c)^(1/2) + c^4*d^8*(-a*c)^(1/2) + 70*d^4*e^4*(-a*c)^(5/2) + c^5*d^8*x^2 + a^4*c*e^8*x^2 - 36*
a^2*d^2*e^6*(-a*c)^(3/2) - 36*c^2*d^6*e^2*(-a*c)^(3/2) + 70*a^2*c^3*d^4*e^4*x^2 + 36*a^3*c^2*d^2*e^6*x^2 + 36*
a*c^4*d^6*e^2*x^2)*(a*((e^3*(-a*c)^(1/2))/8 + (c*d*e^2)/4) - (c*d^2*e*(-a*c)^(1/2))/8))/(a*c^3*d^4 + a^3*c*e^4
+ 2*a^2*c^2*d^2*e^2) - (d/(4*(a*e^2 + c*d^2)) - (e*x^2)/(4*(a*e^2 + c*d^2)))/(a + c*x^4) - (log(c^5*d^8*x^2 -
c^4*d^8*(-a*c)^(1/2) - 70*d^4*e^4*(-a*c)^(5/2) - a^4*e^8*(-a*c)^(1/2) + a^4*c*e^8*x^2 + 36*a^2*d^2*e^6*(-a*c)
^(3/2) + 36*c^2*d^6*e^2*(-a*c)^(3/2) + 70*a^2*c^3*d^4*e^4*x^2 + 36*a^3*c^2*d^2*e^6*x^2 + 36*a*c^4*d^6*e^2*x^2)
*(a*((e^3*(-a*c)^(1/2))/8 - (c*d*e^2)/4) - (c*d^2*e*(-a*c)^(1/2))/8))/(a*c^3*d^4 + a^3*c*e^4 + 2*a^2*c^2*d^2*e
^2) - (d*e^2*log(d + e*x^2))/(2*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2))

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