3.3.45 \(\int \frac {x^7}{(d+e x^2) (a+c x^4)^2} \, dx\) [245]

Optimal. Leaf size=150 \[ \frac {a \left (d-e x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {\sqrt {a} e \left (3 c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 c^{3/2} \left (c d^2+a e^2\right )^2}-\frac {d^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}+\frac {d^3 \log \left (a+c x^4\right )}{4 \left (c d^2+a e^2\right )^2} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 150, normalized size of antiderivative = 1.00, 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, 1661, 815, 649, 211, 266} \begin {gather*} \frac {\sqrt {a} e \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) \left (a e^2+3 c d^2\right )}{4 c^{3/2} \left (a e^2+c d^2\right )^2}+\frac {a \left (d-e x^2\right )}{4 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac {d^3 \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2}-\frac {d^3 \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7/((d + e*x^2)*(a + c*x^4)^2),x]

[Out]

(a*(d - e*x^2))/(4*c*(c*d^2 + a*e^2)*(a + c*x^4)) + (Sqrt[a]*e*(3*c*d^2 + a*e^2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]]
)/(4*c^(3/2)*(c*d^2 + a*e^2)^2) - (d^3*Log[d + e*x^2])/(2*(c*d^2 + a*e^2)^2) + (d^3*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 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]

Rule 1661

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[(a*g - c*f*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {x^7}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3}{(d+e x) \left (a+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {a \left (d-e x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {\text {Subst}\left (\int \frac {-\frac {a^2 d e}{c d^2+a e^2}-\frac {a \left (2 c d^2+a e^2\right ) x}{c d^2+a e^2}}{(d+e x) \left (a+c x^2\right )} \, dx,x,x^2\right )}{4 a c}\\ &=\frac {a \left (d-e x^2\right )}{4 c \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^3 e}{\left (c d^2+a e^2\right )^2 (d+e x)}-\frac {a \left (3 a c d^2 e+a^2 e^3+2 c^2 d^3 x\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx,x,x^2\right )}{4 a c}\\ &=\frac {a \left (d-e x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {d^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}+\frac {\text {Subst}\left (\int \frac {3 a c d^2 e+a^2 e^3+2 c^2 d^3 x}{a+c x^2} \, dx,x,x^2\right )}{4 c \left (c d^2+a e^2\right )^2}\\ &=\frac {a \left (d-e x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {d^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}+\frac {\left (c d^3\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 (a e \left (3 c d^2+a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{4 c \left (c d^2+a e^2\right )^2}\\ &=\frac {a \left (d-e x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {\sqrt {a} e \left (3 c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 c^{3/2} \left (c d^2+a e^2\right )^2}-\frac {d^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}+\frac {d^3 \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 = 0.08, size = 142, normalized size = 0.95 \begin {gather*} \frac {\sqrt {a} e \left (3 c d^2+a e^2\right ) \left (a+c x^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )+\sqrt {c} \left (a \left (c d^2+a e^2\right ) \left (d-e x^2\right )-2 c d^3 \left (a+c x^4\right ) \log \left (d+e x^2\right )+c d^3 \left (a+c x^4\right ) \log \left (a+c x^4\right )\right )}{4 c^{3/2} \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/((d + e*x^2)*(a + c*x^4)^2),x]

[Out]

(Sqrt[a]*e*(3*c*d^2 + a*e^2)*(a + c*x^4)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]] + Sqrt[c]*(a*(c*d^2 + a*e^2)*(d - e*x^2
) - 2*c*d^3*(a + c*x^4)*Log[d + e*x^2] + c*d^3*(a + c*x^4)*Log[a + c*x^4]))/(4*c^(3/2)*(c*d^2 + a*e^2)^2*(a +
c*x^4))

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Maple [A]
time = 0.19, size = 146, normalized size = 0.97

method	result	size
default	\(-\frac {d^{3} \ln \left (e \,x^{2}+d \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}+\frac {\frac {-\frac {a e \left (a \,e^{2}+c \,d^{2}\right ) x^{2}}{2 c}+\frac {a d \left (a \,e^{2}+c \,d^{2}\right )}{2 c}}{c \,x^{4}+a}+\frac {c \,d^{3} \ln \left (c \,x^{4}+a \right )+\frac {\left (a^{2} e^{3}+3 a \,d^{2} e c \right ) \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{\sqrt {a c}}}{2 c}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}\)	\(146\)
risch	\(\frac {-\frac {a e \,x^{2}}{4 c \left (a \,e^{2}+c \,d^{2}\right )}+\frac {a d}{4 c \left (a \,e^{2}+c \,d^{2}\right )}}{c \,x^{4}+a}-\frac {d^{3} \ln \left (e \,x^{2}+d \right )}{2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a^{2} c^{3} e^{4}+2 a \,c^{4} d^{2} e^{2}+c^{5} d^{4}\right ) \textit {\_Z}^{2}-4 c^{3} d^{3} \textit {\_Z} +a \,e^{2}+4 c \,d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (2 a^{3} c^{3} e^{7}+2 a^{2} c^{4} d^{2} e^{5}-2 a \,c^{5} d^{4} e^{3}-2 c^{6} d^{6} e \right ) \textit {\_R}^{2}+\left (a^{2} c^{2} d \,e^{5}-6 a \,c^{3} d^{3} e^{3}-7 c^{4} d^{5} e \right ) \textit {\_R} +2 a^{2} e^{5}+8 a c \,d^{2} e^{3}+4 c^{2} d^{4} e \right ) x^{2}+\left (3 a^{3} c^{3} d \,e^{6}+5 a^{2} c^{4} d^{3} e^{4}+a \,c^{5} d^{5} e^{2}-c^{6} d^{7}\right ) \textit {\_R}^{2}+\left (2 a^{2} c^{2} d^{2} e^{4}-2 c^{4} d^{6}\right ) \textit {\_R} +2 a^{2} d \,e^{4}+8 a c \,d^{3} e^{2}+8 c^{2} d^{5}\right )\right )}{8}\)	\(381\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(e*x^2+d)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.49, size = 191, normalized size = 1.27 \begin {gather*} \frac {d^{3} \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^{3} \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 (3 \, a c d^{2} e + a^{2} e^{3}\right )} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} \sqrt {a c}} - \frac {a x^{2} e - a d}{4 \, {\left (a c^{2} d^{2} + {\left (c^{3} d^{2} + a c^{2} e^{2}\right )} x^{4} + a^{2} c e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 4.63, size = 460, normalized size = 3.07 \begin {gather*} \left [-\frac {2 \, a c d^{2} x^{2} e - 2 \, a c d^{3} + 2 \, a^{2} x^{2} e^{3} - 2 \, a^{2} d e^{2} - {\left ({\left (a c x^{4} + a^{2}\right )} e^{3} + 3 \, {\left (c^{2} d^{2} x^{4} + a c d^{2}\right )} e\right )} \sqrt {-\frac {a}{c}} \log \left (\frac {c x^{4} + 2 \, c x^{2} \sqrt {-\frac {a}{c}} - a}{c x^{4} + a}\right ) - 2 \, {\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (c x^{4} + a\right ) + 4 \, {\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (x^{2} e + d\right )}{8 \, {\left (c^{4} d^{4} x^{4} + a c^{3} d^{4} + {\left (a^{2} c^{2} x^{4} + a^{3} c\right )} e^{4} + 2 \, {\left (a c^{3} d^{2} x^{4} + a^{2} c^{2} d^{2}\right )} e^{2}\right )}}, -\frac {a c d^{2} x^{2} e - a c d^{3} + a^{2} x^{2} e^{3} - a^{2} d e^{2} - {\left ({\left (a c x^{4} + a^{2}\right )} e^{3} + 3 \, {\left (c^{2} d^{2} x^{4} + a c d^{2}\right )} e\right )} \sqrt {\frac {a}{c}} \arctan \left (\frac {c x^{2} \sqrt {\frac {a}{c}}}{a}\right ) - {\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (c x^{4} + a\right ) + 2 \, {\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (x^{2} e + d\right )}{4 \, {\left (c^{4} d^{4} x^{4} + a c^{3} d^{4} + {\left (a^{2} c^{2} x^{4} + a^{3} c\right )} e^{4} + 2 \, {\left (a c^{3} d^{2} x^{4} + a^{2} c^{2} d^{2}\right )} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="fricas")

[Out]

[-1/8*(2*a*c*d^2*x^2*e - 2*a*c*d^3 + 2*a^2*x^2*e^3 - 2*a^2*d*e^2 - ((a*c*x^4 + a^2)*e^3 + 3*(c^2*d^2*x^4 + a*c
*d^2)*e)*sqrt(-a/c)*log((c*x^4 + 2*c*x^2*sqrt(-a/c) - a)/(c*x^4 + a)) - 2*(c^2*d^3*x^4 + a*c*d^3)*log(c*x^4 +
a) + 4*(c^2*d^3*x^4 + a*c*d^3)*log(x^2*e + d))/(c^4*d^4*x^4 + a*c^3*d^4 + (a^2*c^2*x^4 + a^3*c)*e^4 + 2*(a*c^3
*d^2*x^4 + a^2*c^2*d^2)*e^2), -1/4*(a*c*d^2*x^2*e - a*c*d^3 + a^2*x^2*e^3 - a^2*d*e^2 - ((a*c*x^4 + a^2)*e^3 +
3*(c^2*d^2*x^4 + a*c*d^2)*e)*sqrt(a/c)*arctan(c*x^2*sqrt(a/c)/a) - (c^2*d^3*x^4 + a*c*d^3)*log(c*x^4 + a) + 2
*(c^2*d^3*x^4 + a*c*d^3)*log(x^2*e + d))/(c^4*d^4*x^4 + a*c^3*d^4 + (a^2*c^2*x^4 + a^3*c)*e^4 + 2*(a*c^3*d^2*x
^4 + a^2*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**7/(e*x**2+d)/(c*x**4+a)**2,x)

[Out]

Timed out

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Giac [A]
time = 4.38, size = 223, normalized size = 1.49 \begin {gather*} -\frac {d^{3} e \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 {d^{3} \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 {{\left (3 \, a c d^{2} e + a^{2} e^{3}\right )} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} \sqrt {a c}} - \frac {c^{2} d^{3} x^{4} + a c d^{2} x^{2} e + a^{2} x^{2} e^{3} - a^{2} d e^{2}}{4 \, {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} {\left (c x^{4} + a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.49, size = 647, normalized size = 4.31 \begin {gather*} \frac {\frac {a\,d}{4\,c\,\left (c\,d^2+a\,e^2\right )}-\frac {a\,e\,x^2}{4\,c\,\left (c\,d^2+a\,e^2\right )}}{c\,x^4+a}-\frac {d^3\,\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 )}+\frac {\ln \left (36\,c^8\,d^{10}\,x^2+36\,c^6\,d^{10}\,\sqrt {-a\,c^3}+a^5\,c\,e^{10}\,\sqrt {-a\,c^3}+a^5\,c^3\,e^{10}\,x^2-22\,a^2\,d^4\,e^6\,{\left (-a\,c^3\right )}^{3/2}-81\,c^2\,d^8\,e^2\,{\left (-a\,c^3\right )}^{3/2}+60\,a^2\,c^6\,d^6\,e^4\,x^2+22\,a^3\,c^5\,d^4\,e^6\,x^2+8\,a^4\,c^4\,d^2\,e^8\,x^2+8\,a^4\,c^2\,d^2\,e^8\,\sqrt {-a\,c^3}-60\,a\,c\,d^6\,e^4\,{\left (-a\,c^3\right )}^{3/2}+81\,a\,c^7\,d^8\,e^2\,x^2\right )\,\left (2\,c^3\,d^3+a\,e^3\,\sqrt {-a\,c^3}+3\,c\,d^2\,e\,\sqrt {-a\,c^3}\right )}{8\,\left (a^2\,c^3\,e^4+2\,a\,c^4\,d^2\,e^2+c^5\,d^4\right )}-\frac {\ln \left (36\,c^8\,d^{10}\,x^2-36\,c^6\,d^{10}\,\sqrt {-a\,c^3}-a^5\,c\,e^{10}\,\sqrt {-a\,c^3}+a^5\,c^3\,e^{10}\,x^2+22\,a^2\,d^4\,e^6\,{\left (-a\,c^3\right )}^{3/2}+81\,c^2\,d^8\,e^2\,{\left (-a\,c^3\right )}^{3/2}+60\,a^2\,c^6\,d^6\,e^4\,x^2+22\,a^3\,c^5\,d^4\,e^6\,x^2+8\,a^4\,c^4\,d^2\,e^8\,x^2-8\,a^4\,c^2\,d^2\,e^8\,\sqrt {-a\,c^3}+60\,a\,c\,d^6\,e^4\,{\left (-a\,c^3\right )}^{3/2}+81\,a\,c^7\,d^8\,e^2\,x^2\right )\,\left (a\,e^3\,\sqrt {-a\,c^3}-2\,c^3\,d^3+3\,c\,d^2\,e\,\sqrt {-a\,c^3}\right )}{8\,\left (a^2\,c^3\,e^4+2\,a\,c^4\,d^2\,e^2+c^5\,d^4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/((a + c*x^4)^2*(d + e*x^2)),x)

[Out]

((a*d)/(4*c*(a*e^2 + c*d^2)) - (a*e*x^2)/(4*c*(a*e^2 + c*d^2)))/(a + c*x^4) - (d^3*log(d + e*x^2))/(2*(a^2*e^4
+ c^2*d^4 + 2*a*c*d^2*e^2)) + (log(36*c^8*d^10*x^2 + 36*c^6*d^10*(-a*c^3)^(1/2) + a^5*c*e^10*(-a*c^3)^(1/2) +
a^5*c^3*e^10*x^2 - 22*a^2*d^4*e^6*(-a*c^3)^(3/2) - 81*c^2*d^8*e^2*(-a*c^3)^(3/2) + 60*a^2*c^6*d^6*e^4*x^2 + 2
2*a^3*c^5*d^4*e^6*x^2 + 8*a^4*c^4*d^2*e^8*x^2 + 8*a^4*c^2*d^2*e^8*(-a*c^3)^(1/2) - 60*a*c*d^6*e^4*(-a*c^3)^(3/
2) + 81*a*c^7*d^8*e^2*x^2)*(2*c^3*d^3 + a*e^3*(-a*c^3)^(1/2) + 3*c*d^2*e*(-a*c^3)^(1/2)))/(8*(c^5*d^4 + a^2*c^
3*e^4 + 2*a*c^4*d^2*e^2)) - (log(36*c^8*d^10*x^2 - 36*c^6*d^10*(-a*c^3)^(1/2) - a^5*c*e^10*(-a*c^3)^(1/2) + a^
5*c^3*e^10*x^2 + 22*a^2*d^4*e^6*(-a*c^3)^(3/2) + 81*c^2*d^8*e^2*(-a*c^3)^(3/2) + 60*a^2*c^6*d^6*e^4*x^2 + 22*a
^3*c^5*d^4*e^6*x^2 + 8*a^4*c^4*d^2*e^8*x^2 - 8*a^4*c^2*d^2*e^8*(-a*c^3)^(1/2) + 60*a*c*d^6*e^4*(-a*c^3)^(3/2)
+ 81*a*c^7*d^8*e^2*x^2)*(a*e^3*(-a*c^3)^(1/2) - 2*c^3*d^3 + 3*c*d^2*e*(-a*c^3)^(1/2)))/(8*(c^5*d^4 + a^2*c^3*e
^4 + 2*a*c^4*d^2*e^2))

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