3.3.49 \(\int \frac {1}{x (d+e x^2) (a+c x^4)^2} \, dx\) [249]

Optimal. Leaf size=209 \[ \frac {c \left (d-e x^2\right )}{4 a \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {\sqrt {c} e^3 \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^2+a e^2\right )^2}-\frac {\sqrt {c} e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^2+a e^2\right )}+\frac {\log (x)}{a^2 d}-\frac {e^4 \log \left (d+e x^2\right )}{2 d \left (c d^2+a e^2\right )^2}-\frac {c d \left (c d^2+2 a e^2\right ) \log \left (a+c x^4\right )}{4 a^2 \left (c d^2+a e^2\right )^2} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1266, 908, 653, 211, 649, 266} \begin {gather*} -\frac {\sqrt {c} e \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (a e^2+c d^2\right )}-\frac {c d \left (2 a e^2+c d^2\right ) \log \left (a+c x^4\right )}{4 a^2 \left (a e^2+c d^2\right )^2}+\frac {\log (x)}{a^2 d}-\frac {\sqrt {c} e^3 \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (a e^2+c d^2\right )^2}+\frac {c \left (d-e x^2\right )}{4 a \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac {e^4 \log \left (d+e x^2\right )}{2 d \left (a e^2+c d^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(d + e*x^2)*(a + c*x^4)^2),x]

[Out]

(c*(d - e*x^2))/(4*a*(c*d^2 + a*e^2)*(a + c*x^4)) - (Sqrt[c]*e^3*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*Sqrt[a]*(c*
d^2 + a*e^2)^2) - (Sqrt[c]*e*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(4*a^(3/2)*(c*d^2 + a*e^2)) + Log[x]/(a^2*d) - (e^
4*Log[d + e*x^2])/(2*d*(c*d^2 + a*e^2)^2) - (c*d*(c*d^2 + 2*a*e^2)*Log[a + c*x^4])/(4*a^2*(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 653

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)/(2*a*c*(p + 1)))*(a + c*x
^2)^(p + 1), x] + Dist[d*((2*p + 3)/(2*a*(p + 1))), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 908

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

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

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Mathematica [A]
time = 0.12, size = 241, normalized size = 1.15 \begin {gather*} \frac {a c d \left (c d^2+a e^2\right ) \left (d-e x^2\right )+\sqrt {a} \sqrt {c} d e \left (c d^2+3 a e^2\right ) \left (a+c x^4\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+\sqrt {a} \sqrt {c} d e \left (c d^2+3 a e^2\right ) \left (a+c x^4\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+4 \left (c d^2+a e^2\right )^2 \left (a+c x^4\right ) \log (x)-2 a^2 e^4 \left (a+c x^4\right ) \log \left (d+e x^2\right )-c d^2 \left (c d^2+2 a e^2\right ) \left (a+c x^4\right ) \log \left (a+c x^4\right )}{4 a^2 d \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(d + e*x^2)*(a + c*x^4)^2),x]

[Out]

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

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Maple [A]
time = 0.20, size = 171, normalized size = 0.82

method	result	size
default	\(-\frac {e^{4} \ln \left (e \,x^{2}+d \right )}{2 d \left (a \,e^{2}+c \,d^{2}\right )^{2}}-\frac {c \left (\frac {\left (\frac {1}{2} a^{2} e^{3}+\frac {1}{2} a \,d^{2} e c \right ) x^{2}-\frac {a d \left (a \,e^{2}+c \,d^{2}\right )}{2}}{c \,x^{4}+a}+\frac {\left (4 a c d \,e^{2}+2 c^{2} d^{3}\right ) \ln \left (c \,x^{4}+a \right )}{4 c}+\frac {\left (3 a^{2} e^{3}+a \,d^{2} e c \right ) \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{2 \sqrt {a c}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} a^{2}}+\frac {\ln \left (x \right )}{d \,a^{2}}\)	\(171\)
risch	\(\frac {-\frac {c e \,x^{2}}{4 a \left (a \,e^{2}+c \,d^{2}\right )}+\frac {c d}{4 a \left (a \,e^{2}+c \,d^{2}\right )}}{c \,x^{4}+a}+\frac {\ln \left (x \right )}{d \,a^{2}}-\frac {e^{4} \ln \left (e \,x^{2}+d \right )}{2 d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a^{6} e^{4}+2 d^{2} a^{5} c \,e^{2}+a^{4} d^{4} c^{2}\right ) \textit {\_Z}^{2}+\left (8 a^{3} c d \,e^{2}+4 a^{2} d^{3} c^{2}\right ) \textit {\_Z} +9 a c \,e^{2}+4 c^{2} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-6 a^{8} e^{8}-19 a^{7} c \,d^{2} e^{6}-25 a^{6} c^{2} d^{4} e^{4}-17 a^{5} c^{3} d^{6} e^{2}-5 a^{4} c^{4} d^{8}\right ) \textit {\_R}^{3}+\left (-52 a^{5} c d \,e^{6}-102 a^{4} c^{2} d^{3} e^{4}-60 a^{3} c^{3} d^{5} e^{2}-10 a^{2} c^{4} d^{7}\right ) \textit {\_R}^{2}+\left (-86 a^{3} c \,e^{6}-56 a^{2} c^{2} d^{2} e^{4}-36 a \,c^{3} d^{4} e^{2}\right ) \textit {\_R} +64 c^{2} d \,e^{4}\right ) x^{2}+\left (-2 a^{8} d \,e^{7}-2 a^{7} c \,d^{3} e^{5}+2 a^{6} c^{2} d^{5} e^{3}+2 d^{7} a^{5} c^{3} e \right ) \textit {\_R}^{3}+\left (16 a^{6} e^{7}+11 a^{5} c \,d^{2} e^{5}-18 a^{4} c^{2} d^{4} e^{3}-13 d^{6} a^{3} c^{3} e \right ) \textit {\_R}^{2}+\left (34 a^{3} c d \,e^{5}+8 a^{2} c^{2} d^{3} e^{3}-24 c^{3} d^{5} e a \right ) \textit {\_R} +48 a c \,e^{5}+64 c^{2} d^{2} e^{3}\right )\right )}{8}\)	\(520\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(e*x^2+d)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.50, size = 220, normalized size = 1.05 \begin {gather*} -\frac {{\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} \log \left (c x^{4} + a\right )}{4 \, {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )}} - \frac {e^{4} \log \left (x^{2} e + d\right )}{2 \, {\left (c^{2} d^{5} + 2 \, a c d^{3} e^{2} + a^{2} d e^{4}\right )}} - \frac {{\left (c^{2} d^{2} e + 3 \, a c e^{3}\right )} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt {a c}} - \frac {c x^{2} e - c d}{4 \, {\left (a^{2} c d^{2} + {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{4} + a^{3} e^{2}\right )}} + \frac {\log \left (x^{2}\right )}{2 \, a^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 72.45, size = 671, normalized size = 3.21 \begin {gather*} \left [-\frac {2 \, a c^{2} d^{3} x^{2} e - 2 \, a c^{2} d^{4} + 2 \, a^{2} c d x^{2} e^{3} - 2 \, a^{2} c d^{2} e^{2} + 4 \, {\left (a^{2} c x^{4} + a^{3}\right )} e^{4} \log \left (x^{2} e + d\right ) - {\left (3 \, {\left (a^{2} c d x^{4} + a^{3} d\right )} e^{3} + {\left (a c^{2} d^{3} x^{4} + a^{2} c d^{3}\right )} e\right )} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{4} - 2 \, a x^{2} \sqrt {-\frac {c}{a}} - a}{c x^{4} + a}\right ) + 2 \, {\left (c^{3} d^{4} x^{4} + a c^{2} d^{4} + 2 \, {\left (a c^{2} d^{2} x^{4} + a^{2} c d^{2}\right )} e^{2}\right )} \log \left (c x^{4} + a\right ) - 8 \, {\left (c^{3} d^{4} x^{4} + a c^{2} d^{4} + {\left (a^{2} c x^{4} + a^{3}\right )} e^{4} + 2 \, {\left (a c^{2} d^{2} x^{4} + a^{2} c d^{2}\right )} e^{2}\right )} \log \left (x\right )}{8 \, {\left (a^{2} c^{3} d^{5} x^{4} + a^{3} c^{2} d^{5} + {\left (a^{4} c d x^{4} + a^{5} d\right )} e^{4} + 2 \, {\left (a^{3} c^{2} d^{3} x^{4} + a^{4} c d^{3}\right )} e^{2}\right )}}, -\frac {a c^{2} d^{3} x^{2} e - a c^{2} d^{4} + a^{2} c d x^{2} e^{3} - a^{2} c d^{2} e^{2} + 2 \, {\left (a^{2} c x^{4} + a^{3}\right )} e^{4} \log \left (x^{2} e + d\right ) - {\left (3 \, {\left (a^{2} c d x^{4} + a^{3} d\right )} e^{3} + {\left (a c^{2} d^{3} x^{4} + a^{2} c d^{3}\right )} e\right )} \sqrt {\frac {c}{a}} \arctan \left (\frac {a \sqrt {\frac {c}{a}}}{c x^{2}}\right ) + {\left (c^{3} d^{4} x^{4} + a c^{2} d^{4} + 2 \, {\left (a c^{2} d^{2} x^{4} + a^{2} c d^{2}\right )} e^{2}\right )} \log \left (c x^{4} + a\right ) - 4 \, {\left (c^{3} d^{4} x^{4} + a c^{2} d^{4} + {\left (a^{2} c x^{4} + a^{3}\right )} e^{4} + 2 \, {\left (a c^{2} d^{2} x^{4} + a^{2} c d^{2}\right )} e^{2}\right )} \log \left (x\right )}{4 \, {\left (a^{2} c^{3} d^{5} x^{4} + a^{3} c^{2} d^{5} + {\left (a^{4} c d x^{4} + a^{5} d\right )} e^{4} + 2 \, {\left (a^{3} c^{2} d^{3} x^{4} + a^{4} c d^{3}\right )} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="fricas")

[Out]

[-1/8*(2*a*c^2*d^3*x^2*e - 2*a*c^2*d^4 + 2*a^2*c*d*x^2*e^3 - 2*a^2*c*d^2*e^2 + 4*(a^2*c*x^4 + a^3)*e^4*log(x^2
*e + d) - (3*(a^2*c*d*x^4 + a^3*d)*e^3 + (a*c^2*d^3*x^4 + a^2*c*d^3)*e)*sqrt(-c/a)*log((c*x^4 - 2*a*x^2*sqrt(-
c/a) - a)/(c*x^4 + a)) + 2*(c^3*d^4*x^4 + a*c^2*d^4 + 2*(a*c^2*d^2*x^4 + a^2*c*d^2)*e^2)*log(c*x^4 + a) - 8*(c
^3*d^4*x^4 + a*c^2*d^4 + (a^2*c*x^4 + a^3)*e^4 + 2*(a*c^2*d^2*x^4 + a^2*c*d^2)*e^2)*log(x))/(a^2*c^3*d^5*x^4 +
a^3*c^2*d^5 + (a^4*c*d*x^4 + a^5*d)*e^4 + 2*(a^3*c^2*d^3*x^4 + a^4*c*d^3)*e^2), -1/4*(a*c^2*d^3*x^2*e - a*c^2
*d^4 + a^2*c*d*x^2*e^3 - a^2*c*d^2*e^2 + 2*(a^2*c*x^4 + a^3)*e^4*log(x^2*e + d) - (3*(a^2*c*d*x^4 + a^3*d)*e^3
+ (a*c^2*d^3*x^4 + a^2*c*d^3)*e)*sqrt(c/a)*arctan(a*sqrt(c/a)/(c*x^2)) + (c^3*d^4*x^4 + a*c^2*d^4 + 2*(a*c^2*
d^2*x^4 + a^2*c*d^2)*e^2)*log(c*x^4 + a) - 4*(c^3*d^4*x^4 + a*c^2*d^4 + (a^2*c*x^4 + a^3)*e^4 + 2*(a*c^2*d^2*x
^4 + a^2*c*d^2)*e^2)*log(x))/(a^2*c^3*d^5*x^4 + a^3*c^2*d^5 + (a^4*c*d*x^4 + a^5*d)*e^4 + 2*(a^3*c^2*d^3*x^4 +
a^4*c*d^3)*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(1/x/(e*x**2+d)/(c*x**4+a)**2,x)

[Out]

Timed out

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Giac [A]
time = 5.19, size = 279, normalized size = 1.33 \begin {gather*} -\frac {{\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} \log \left (c x^{4} + a\right )}{4 \, {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )}} - \frac {e^{5} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \, {\left (c^{2} d^{5} e + 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )}} - \frac {{\left (c^{2} d^{2} e + 3 \, a c e^{3}\right )} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt {a c}} + \frac {c^{3} d^{3} x^{4} + 2 \, a c^{2} d x^{4} e^{2} - a c^{2} d^{2} x^{2} e + 2 \, a c^{2} d^{3} - a^{2} c x^{2} e^{3} + 3 \, a^{2} c d e^{2}}{4 \, {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} {\left (c x^{4} + a\right )}} + \frac {\log \left (x^{2}\right )}{2 \, a^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 2.58, size = 1082, normalized size = 5.18 \begin {gather*} \frac {\frac {c\,d}{4\,a\,\left (c\,d^2+a\,e^2\right )}-\frac {c\,e\,x^2}{4\,a\,\left (c\,d^2+a\,e^2\right )}}{c\,x^4+a}-\frac {\ln \left (400\,a^9\,c^{12}\,d^{20}\,x^2-10481\,d^4\,e^{16}\,{\left (-a^5\,c\right )}^{7/2}-1024\,a^{12}\,e^{20}\,{\left (-a^5\,c\right )}^{3/2}+1024\,a^{19}\,c^2\,e^{20}\,x^2-400\,a^2\,c^{10}\,d^{20}\,{\left (-a^5\,c\right )}^{3/2}+5840\,a^6\,d^2\,e^{18}\,{\left (-a^5\,c\right )}^{5/2}+33710\,c^6\,d^{14}\,e^6\,{\left (-a^5\,c\right )}^{5/2}+4104\,a^{10}\,c^{11}\,d^{18}\,e^2\,x^2+16689\,a^{11}\,c^{10}\,d^{16}\,e^4\,x^2+33710\,a^{12}\,c^9\,d^{14}\,e^6\,x^2+33391\,a^{13}\,c^8\,d^{12}\,e^8\,x^2+10748\,a^{14}\,c^7\,d^{10}\,e^{10}\,x^2-3585\,a^{15}\,c^6\,d^8\,e^{12}\,x^2+3998\,a^{16}\,c^5\,d^6\,e^{14}\,x^2+10481\,a^{17}\,c^4\,d^4\,e^{16}\,x^2+5840\,a^{18}\,c^3\,d^2\,e^{18}\,x^2+10748\,a^2\,c^4\,d^{10}\,e^{10}\,{\left (-a^5\,c\right )}^{5/2}-3585\,a^3\,c^3\,d^8\,e^{12}\,{\left (-a^5\,c\right )}^{5/2}+3998\,a^4\,c^2\,d^6\,e^{14}\,{\left (-a^5\,c\right )}^{5/2}-4104\,a^3\,c^9\,d^{18}\,e^2\,{\left (-a^5\,c\right )}^{3/2}-16689\,a^4\,c^8\,d^{16}\,e^4\,{\left (-a^5\,c\right )}^{3/2}+33391\,a\,c^5\,d^{12}\,e^8\,{\left (-a^5\,c\right )}^{5/2}\right )\,\left (3\,a\,e^3\,\sqrt {-a^5\,c}+2\,a^2\,c^2\,d^3+4\,a^3\,c\,d\,e^2+c\,d^2\,e\,\sqrt {-a^5\,c}\right )}{8\,\left (a^6\,e^4+2\,a^5\,c\,d^2\,e^2+a^4\,c^2\,d^4\right )}+\frac {\ln \left (1024\,a^{12}\,e^{20}\,{\left (-a^5\,c\right )}^{3/2}+10481\,d^4\,e^{16}\,{\left (-a^5\,c\right )}^{7/2}+400\,a^9\,c^{12}\,d^{20}\,x^2+1024\,a^{19}\,c^2\,e^{20}\,x^2+400\,a^2\,c^{10}\,d^{20}\,{\left (-a^5\,c\right )}^{3/2}-5840\,a^6\,d^2\,e^{18}\,{\left (-a^5\,c\right )}^{5/2}-33710\,c^6\,d^{14}\,e^6\,{\left (-a^5\,c\right )}^{5/2}+4104\,a^{10}\,c^{11}\,d^{18}\,e^2\,x^2+16689\,a^{11}\,c^{10}\,d^{16}\,e^4\,x^2+33710\,a^{12}\,c^9\,d^{14}\,e^6\,x^2+33391\,a^{13}\,c^8\,d^{12}\,e^8\,x^2+10748\,a^{14}\,c^7\,d^{10}\,e^{10}\,x^2-3585\,a^{15}\,c^6\,d^8\,e^{12}\,x^2+3998\,a^{16}\,c^5\,d^6\,e^{14}\,x^2+10481\,a^{17}\,c^4\,d^4\,e^{16}\,x^2+5840\,a^{18}\,c^3\,d^2\,e^{18}\,x^2-10748\,a^2\,c^4\,d^{10}\,e^{10}\,{\left (-a^5\,c\right )}^{5/2}+3585\,a^3\,c^3\,d^8\,e^{12}\,{\left (-a^5\,c\right )}^{5/2}-3998\,a^4\,c^2\,d^6\,e^{14}\,{\left (-a^5\,c\right )}^{5/2}+4104\,a^3\,c^9\,d^{18}\,e^2\,{\left (-a^5\,c\right )}^{3/2}+16689\,a^4\,c^8\,d^{16}\,e^4\,{\left (-a^5\,c\right )}^{3/2}-33391\,a\,c^5\,d^{12}\,e^8\,{\left (-a^5\,c\right )}^{5/2}\right )\,\left (3\,a\,e^3\,\sqrt {-a^5\,c}-2\,a^2\,c^2\,d^3-4\,a^3\,c\,d\,e^2+c\,d^2\,e\,\sqrt {-a^5\,c}\right )}{8\,\left (a^6\,e^4+2\,a^5\,c\,d^2\,e^2+a^4\,c^2\,d^4\right )}-\frac {e^4\,\ln \left (e\,x^2+d\right )}{2\,a^2\,d\,e^4+4\,a\,c\,d^3\,e^2+2\,c^2\,d^5}+\frac {\ln \left (x\right )}{a^2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + c*x^4)^2*(d + e*x^2)),x)

[Out]

((c*d)/(4*a*(a*e^2 + c*d^2)) - (c*e*x^2)/(4*a*(a*e^2 + c*d^2)))/(a + c*x^4) - (log(400*a^9*c^12*d^20*x^2 - 104
81*d^4*e^16*(-a^5*c)^(7/2) - 1024*a^12*e^20*(-a^5*c)^(3/2) + 1024*a^19*c^2*e^20*x^2 - 400*a^2*c^10*d^20*(-a^5*
c)^(3/2) + 5840*a^6*d^2*e^18*(-a^5*c)^(5/2) + 33710*c^6*d^14*e^6*(-a^5*c)^(5/2) + 4104*a^10*c^11*d^18*e^2*x^2
+ 16689*a^11*c^10*d^16*e^4*x^2 + 33710*a^12*c^9*d^14*e^6*x^2 + 33391*a^13*c^8*d^12*e^8*x^2 + 10748*a^14*c^7*d^
10*e^10*x^2 - 3585*a^15*c^6*d^8*e^12*x^2 + 3998*a^16*c^5*d^6*e^14*x^2 + 10481*a^17*c^4*d^4*e^16*x^2 + 5840*a^1
8*c^3*d^2*e^18*x^2 + 10748*a^2*c^4*d^10*e^10*(-a^5*c)^(5/2) - 3585*a^3*c^3*d^8*e^12*(-a^5*c)^(5/2) + 3998*a^4*
c^2*d^6*e^14*(-a^5*c)^(5/2) - 4104*a^3*c^9*d^18*e^2*(-a^5*c)^(3/2) - 16689*a^4*c^8*d^16*e^4*(-a^5*c)^(3/2) + 3
3391*a*c^5*d^12*e^8*(-a^5*c)^(5/2))*(3*a*e^3*(-a^5*c)^(1/2) + 2*a^2*c^2*d^3 + 4*a^3*c*d*e^2 + c*d^2*e*(-a^5*c)
^(1/2)))/(8*(a^6*e^4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)) + (log(1024*a^12*e^20*(-a^5*c)^(3/2) + 10481*d^4*e^16*(
-a^5*c)^(7/2) + 400*a^9*c^12*d^20*x^2 + 1024*a^19*c^2*e^20*x^2 + 400*a^2*c^10*d^20*(-a^5*c)^(3/2) - 5840*a^6*d
^2*e^18*(-a^5*c)^(5/2) - 33710*c^6*d^14*e^6*(-a^5*c)^(5/2) + 4104*a^10*c^11*d^18*e^2*x^2 + 16689*a^11*c^10*d^1
6*e^4*x^2 + 33710*a^12*c^9*d^14*e^6*x^2 + 33391*a^13*c^8*d^12*e^8*x^2 + 10748*a^14*c^7*d^10*e^10*x^2 - 3585*a^
15*c^6*d^8*e^12*x^2 + 3998*a^16*c^5*d^6*e^14*x^2 + 10481*a^17*c^4*d^4*e^16*x^2 + 5840*a^18*c^3*d^2*e^18*x^2 -
10748*a^2*c^4*d^10*e^10*(-a^5*c)^(5/2) + 3585*a^3*c^3*d^8*e^12*(-a^5*c)^(5/2) - 3998*a^4*c^2*d^6*e^14*(-a^5*c)
^(5/2) + 4104*a^3*c^9*d^18*e^2*(-a^5*c)^(3/2) + 16689*a^4*c^8*d^16*e^4*(-a^5*c)^(3/2) - 33391*a*c^5*d^12*e^8*(
-a^5*c)^(5/2))*(3*a*e^3*(-a^5*c)^(1/2) - 2*a^2*c^2*d^3 - 4*a^3*c*d*e^2 + c*d^2*e*(-a^5*c)^(1/2)))/(8*(a^6*e^4
+ a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)) - (e^4*log(d + e*x^2))/(2*c^2*d^5 + 2*a^2*d*e^4 + 4*a*c*d^3*e^2) + log(x)/(a
^2*d)

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