3.3.48 \(\int \frac {x}{(d+e x^2) (a+c x^4)^2} \, dx\) [248]

Optimal. Leaf size=151 \[ \frac {a e+c d x^2}{4 a \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {\sqrt {c} d \left (c d^2+3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^2+a e^2\right )^2}+\frac {e^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}-\frac {e^3 \log \left (a+c x^4\right )}{4 \left (c d^2+a e^2\right )^2} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1262, 755, 815, 649, 211, 266} \begin {gather*} \frac {\sqrt {c} d \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) \left (3 a e^2+c d^2\right )}{4 a^{3/2} \left (a e^2+c d^2\right )^2}+\frac {a e+c d x^2}{4 a \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac {e^3 \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2}+\frac {e^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/((d + e*x^2)*(a + c*x^4)^2),x]

[Out]

(a*e + c*d*x^2)/(4*a*(c*d^2 + a*e^2)*(a + c*x^4)) + (Sqrt[c]*d*(c*d^2 + 3*a*e^2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]]
)/(4*a^(3/2)*(c*d^2 + a*e^2)^2) + (e^3*Log[d + e*x^2])/(2*(c*d^2 + a*e^2)^2) - (e^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 755

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

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 1262

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

Rubi steps

\begin {align*} \int \frac {x}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{(d+e x) \left (a+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {a e+c d x^2}{4 a \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {\text {Subst}\left (\int \frac {-c d^2-2 a e^2-c d e x}{(d+e x) \left (a+c x^2\right )} \, dx,x,x^2\right )}{4 a \left (c d^2+a e^2\right )}\\ &=\frac {a e+c d x^2}{4 a \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {\text {Subst}\left (\int \left (-\frac {2 a e^4}{\left (c d^2+a e^2\right ) (d+e x)}-\frac {c \left (c d^3+3 a d e^2-2 a e^3 x\right )}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx,x,x^2\right )}{4 a \left (c d^2+a e^2\right )}\\ &=\frac {a e+c d x^2}{4 a \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {e^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}+\frac {c \text {Subst}\left (\int \frac {c d^3+3 a d e^2-2 a e^3 x}{a+c x^2} \, dx,x,x^2\right )}{4 a \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x^2}{4 a \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {e^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}-\frac {\left (c e^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 (c d \left (c d^2+3 a e^2\right )\right ) \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 )^2}\\ &=\frac {a e+c d x^2}{4 a \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {\sqrt {c} d \left (c d^2+3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^2+a e^2\right )^2}+\frac {e^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}-\frac {e^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 = 0.09, size = 117, normalized size = 0.77 \begin {gather*} \frac {\frac {\left (c d^2+a e^2\right ) \left (a e+c d x^2\right )}{a \left (a+c x^4\right )}+\frac {\sqrt {c} d \left (c d^2+3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{a^{3/2}}+2 e^3 \log \left (d+e x^2\right )-e^3 \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/((d + e*x^2)*(a + c*x^4)^2),x]

[Out]

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

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Maple [A]
time = 0.20, size = 146, normalized size = 0.97

method	result	size
default	\(\frac {e^{3} \ln \left (e \,x^{2}+d \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}+\frac {c \left (\frac {\frac {d \left (a \,e^{2}+c \,d^{2}\right ) x^{2}}{2 a}+\frac {e \left (a \,e^{2}+c \,d^{2}\right )}{2 c}}{c \,x^{4}+a}+\frac {-\frac {a \,e^{3} \ln \left (c \,x^{4}+a \right )}{c}+\frac {\left (3 d \,e^{2} a +c \,d^{3}\right ) \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{\sqrt {a c}}}{2 a}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}\)	\(146\)
risch	\(\frac {\frac {c d \,x^{2}}{4 a \left (a \,e^{2}+c \,d^{2}\right )}+\frac {e}{4 a \,e^{2}+4 c \,d^{2}}}{c \,x^{4}+a}-\frac {\ln \left (\left (-18 a^{4} c d \,e^{7}+30 a^{3} c^{2} d^{3} e^{5}+18 a^{2} c^{3} d^{5} e^{3}+2 a \,c^{4} d^{7} e +15 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a^{2} c d \,e^{4}-2 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a \,c^{2} d^{3} e^{2}-\sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, c^{3} d^{5}\right ) x^{2}-45 a^{4} c \,d^{2} e^{6}-9 a^{3} c^{2} d^{4} e^{4}+5 a^{2} c^{3} d^{6} e^{2}+a \,c^{4} d^{8}-6 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a^{3} e^{5}+12 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a^{2} c \,d^{2} e^{3}+2 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a \,c^{2} d^{4} e \right ) e^{3}}{4 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}+\frac {\ln \left (\left (-18 a^{4} c d \,e^{7}+30 a^{3} c^{2} d^{3} e^{5}+18 a^{2} c^{3} d^{5} e^{3}+2 a \,c^{4} d^{7} e +15 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a^{2} c d \,e^{4}-2 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a \,c^{2} d^{3} e^{2}-\sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, c^{3} d^{5}\right ) x^{2}-45 a^{4} c \,d^{2} e^{6}-9 a^{3} c^{2} d^{4} e^{4}+5 a^{2} c^{3} d^{6} e^{2}+a \,c^{4} d^{8}-6 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a^{3} e^{5}+12 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a^{2} c \,d^{2} e^{3}+2 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a \,c^{2} d^{4} e \right ) \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}}{8 a^{2} \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}-\frac {\ln \left (\left (-18 a^{4} c d \,e^{7}+30 a^{3} c^{2} d^{3} e^{5}+18 a^{2} c^{3} d^{5} e^{3}+2 a \,c^{4} d^{7} e -15 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a^{2} c d \,e^{4}+2 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a \,c^{2} d^{3} e^{2}+\sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, c^{3} d^{5}\right ) x^{2}-45 a^{4} c \,d^{2} e^{6}-9 a^{3} c^{2} d^{4} e^{4}+5 a^{2} c^{3} d^{6} e^{2}+a \,c^{4} d^{8}+6 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a^{3} e^{5}-12 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a^{2} c \,d^{2} e^{3}-2 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a \,c^{2} d^{4} e \right ) e^{3}}{4 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}-\frac {\ln \left (\left (-18 a^{4} c d \,e^{7}+30 a^{3} c^{2} d^{3} e^{5}+18 a^{2} c^{3} d^{5} e^{3}+2 a \,c^{4} d^{7} e -15 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a^{2} c d \,e^{4}+2 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a \,c^{2} d^{3} e^{2}+\sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, c^{3} d^{5}\right ) x^{2}-45 a^{4} c \,d^{2} e^{6}-9 a^{3} c^{2} d^{4} e^{4}+5 a^{2} c^{3} d^{6} e^{2}+a \,c^{4} d^{8}+6 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a^{3} e^{5}-12 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a^{2} c \,d^{2} e^{3}-2 \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}\, a \,c^{2} d^{4} e \right ) \sqrt {-a c \,d^{2} \left (3 a \,e^{2}+c \,d^{2}\right )^{2}}}{8 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) a^{2}}+\frac {e^{3} \ln \left (e \,x^{2}+d \right )}{2 a^{2} e^{4}+4 a c \,d^{2} e^{2}+2 c^{2} d^{4}}\)	\(1468\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.51, size = 187, normalized size = 1.24 \begin {gather*} -\frac {e^{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 {e^{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 (c^{2} d^{3} + 3 \, a c d e^{2}\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 d x^{2} + a e}{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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 4.45, size = 433, normalized size = 2.87 \begin {gather*} \left [\frac {2 \, c^{2} d^{3} x^{2} + 2 \, a c d x^{2} e^{2} + 2 \, a c d^{2} e + 2 \, a^{2} e^{3} - 2 \, {\left (a c x^{4} + a^{2}\right )} e^{3} \log \left (c x^{4} + a\right ) + 4 \, {\left (a c x^{4} + a^{2}\right )} e^{3} \log \left (x^{2} e + d\right ) + {\left (c^{2} d^{3} x^{4} + a c d^{3} + 3 \, {\left (a c d x^{4} + a^{2} d\right )} e^{2}\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 )}{8 \, {\left (a c^{3} d^{4} x^{4} + a^{2} c^{2} d^{4} + {\left (a^{3} c x^{4} + a^{4}\right )} e^{4} + 2 \, {\left (a^{2} c^{2} d^{2} x^{4} + a^{3} c d^{2}\right )} e^{2}\right )}}, \frac {c^{2} d^{3} x^{2} + a c d x^{2} e^{2} + a c d^{2} e + a^{2} e^{3} - {\left (a c x^{4} + a^{2}\right )} e^{3} \log \left (c x^{4} + a\right ) + 2 \, {\left (a c x^{4} + a^{2}\right )} e^{3} \log \left (x^{2} e + d\right ) - {\left (c^{2} d^{3} x^{4} + a c d^{3} + 3 \, {\left (a c d x^{4} + a^{2} d\right )} e^{2}\right )} \sqrt {\frac {c}{a}} \arctan \left (\frac {a \sqrt {\frac {c}{a}}}{c x^{2}}\right )}{4 \, {\left (a c^{3} d^{4} x^{4} + a^{2} c^{2} d^{4} + {\left (a^{3} c x^{4} + a^{4}\right )} e^{4} + 2 \, {\left (a^{2} c^{2} d^{2} x^{4} + a^{3} c d^{2}\right )} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [A]
time = 3.77, size = 199, normalized size = 1.32 \begin {gather*} -\frac {e^{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 {e^{4} \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^{2} d^{3} + 3 \, a c d e^{2}\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 {a c d^{2} e + {\left (c^{2} d^{3} + a c d e^{2}\right )} x^{2} + a^{2} e^{3}}{4 \, {\left (c x^{4} + a\right )} {\left (c d^{2} + a e^{2}\right )}^{2} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.49, size = 649, normalized size = 4.30 \begin {gather*} \frac {\frac {e}{4\,\left (c\,d^2+a\,e^2\right )}+\frac {c\,d\,x^2}{4\,a\,\left (c\,d^2+a\,e^2\right )}}{c\,x^4+a}+\frac {e^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\,a^6\,e^{10}\,\sqrt {-a^3\,c}+36\,a^7\,c\,e^{10}\,x^2+a\,c^5\,d^{10}\,\sqrt {-a^3\,c}+a^2\,c^6\,d^{10}\,x^2-81\,a^2\,d^2\,e^8\,{\left (-a^3\,c\right )}^{3/2}-22\,c^2\,d^6\,e^4\,{\left (-a^3\,c\right )}^{3/2}+8\,a^3\,c^5\,d^8\,e^2\,x^2+22\,a^4\,c^4\,d^6\,e^4\,x^2+60\,a^5\,c^3\,d^4\,e^6\,x^2+81\,a^6\,c^2\,d^2\,e^8\,x^2+8\,a^2\,c^4\,d^8\,e^2\,\sqrt {-a^3\,c}-60\,a\,c\,d^4\,e^6\,{\left (-a^3\,c\right )}^{3/2}\right )\,\left (c\,d^3\,\sqrt {-a^3\,c}-2\,a^3\,e^3+3\,a\,d\,e^2\,\sqrt {-a^3\,c}\right )}{8\,\left (a^5\,e^4+2\,a^4\,c\,d^2\,e^2+a^3\,c^2\,d^4\right )}-\frac {\ln \left (36\,a^7\,c\,e^{10}\,x^2-36\,a^6\,e^{10}\,\sqrt {-a^3\,c}-a\,c^5\,d^{10}\,\sqrt {-a^3\,c}+a^2\,c^6\,d^{10}\,x^2+81\,a^2\,d^2\,e^8\,{\left (-a^3\,c\right )}^{3/2}+22\,c^2\,d^6\,e^4\,{\left (-a^3\,c\right )}^{3/2}+8\,a^3\,c^5\,d^8\,e^2\,x^2+22\,a^4\,c^4\,d^6\,e^4\,x^2+60\,a^5\,c^3\,d^4\,e^6\,x^2+81\,a^6\,c^2\,d^2\,e^8\,x^2-8\,a^2\,c^4\,d^8\,e^2\,\sqrt {-a^3\,c}+60\,a\,c\,d^4\,e^6\,{\left (-a^3\,c\right )}^{3/2}\right )\,\left (2\,a^3\,e^3+c\,d^3\,\sqrt {-a^3\,c}+3\,a\,d\,e^2\,\sqrt {-a^3\,c}\right )}{8\,\left (a^5\,e^4+2\,a^4\,c\,d^2\,e^2+a^3\,c^2\,d^4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((a + c*x^4)^2*(d + e*x^2)),x)

[Out]

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

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