3.3.50 \(\int \frac {1}{x^3 (d+e x^2) (a+c x^4)^2} \, dx\) [250]

Optimal. Leaf size=236 \[ -\frac {1}{2 a^2 d x^2}-\frac {c \left (a e+c d x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {c^{3/2} d \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{5/2} \left (c d^2+a e^2\right )}-\frac {c^{3/2} d \left (c d^2+2 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{5/2} \left (c d^2+a e^2\right )^2}-\frac {e \log (x)}{a^2 d^2}+\frac {e^5 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2+a e^2\right )^2}+\frac {c e \left (c d^2+2 a e^2\right ) \log \left (a+c x^4\right )}{4 a^2 \left (c d^2+a e^2\right )^2} \]

[Out]

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

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Rubi [A]
time = 0.18, antiderivative size = 236, 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 {c^{3/2} d \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) \left (2 a e^2+c d^2\right )}{2 a^{5/2} \left (a e^2+c d^2\right )^2}-\frac {c^{3/2} d \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{5/2} \left (a e^2+c d^2\right )}+\frac {c e \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 {c \left (a e+c d x^2\right )}{4 a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac {e \log (x)}{a^2 d^2}-\frac {1}{2 a^2 d x^2}+\frac {e^5 \log \left (d+e x^2\right )}{2 d^2 \left (a e^2+c d^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(d + e*x^2)*(a + c*x^4)^2),x]

[Out]

-1/2*1/(a^2*d*x^2) - (c*(a*e + c*d*x^2))/(4*a^2*(c*d^2 + a*e^2)*(a + c*x^4)) - (c^(3/2)*d*ArcTan[(Sqrt[c]*x^2)
/Sqrt[a]])/(4*a^(5/2)*(c*d^2 + a*e^2)) - (c^(3/2)*d*(c*d^2 + 2*a*e^2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*a^(5/2
)*(c*d^2 + a*e^2)^2) - (e*Log[x])/(a^2*d^2) + (e^5*Log[d + e*x^2])/(2*d^2*(c*d^2 + a*e^2)^2) + (c*e*(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^3 \left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (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^2}-\frac {e}{a^2 d^2 x}+\frac {e^6}{d^2 \left (c d^2+a e^2\right )^2 (d+e x)}-\frac {c^2 (d-e x)}{a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}-\frac {c^2 \left (c d^2+2 a e^2\right ) (d-e x)}{a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 a^2 d x^2}-\frac {e \log (x)}{a^2 d^2}+\frac {e^5 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2+a e^2\right )^2}-\frac {c^2 \text {Subst}\left (\int \frac {d-e x}{\left (a+c x^2\right )^2} \, dx,x,x^2\right )}{2 a \left (c d^2+a e^2\right )}-\frac {\left (c^2 \left (c d^2+2 a e^2\right )\right ) \text {Subst}\left (\int \frac {d-e x}{a+c x^2} \, dx,x,x^2\right )}{2 a^2 \left (c d^2+a e^2\right )^2}\\ &=-\frac {1}{2 a^2 d x^2}-\frac {c \left (a e+c d x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {e \log (x)}{a^2 d^2}+\frac {e^5 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2+a e^2\right )^2}-\frac {\left (c^2 d\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{4 a^2 \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 {1}{a+c x^2} \, dx,x,x^2\right )}{2 a^2 \left (c d^2+a e^2\right )^2}+\frac {\left (c^2 e \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 {1}{2 a^2 d x^2}-\frac {c \left (a e+c d x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {c^{3/2} d \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{5/2} \left (c d^2+a e^2\right )}-\frac {c^{3/2} d \left (c d^2+2 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{5/2} \left (c d^2+a e^2\right )^2}-\frac {e \log (x)}{a^2 d^2}+\frac {e^5 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2+a e^2\right )^2}+\frac {c e \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 = 0.29, size = 248, normalized size = 1.05 \begin {gather*} \frac {1}{4} \left (-\frac {2}{a^2 d x^2}-\frac {c \left (a e+c d x^2\right )}{a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {c^{3/2} d \left (3 c d^2+5 a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{5/2} \left (c d^2+a e^2\right )^2}+\frac {c^{3/2} d \left (3 c d^2+5 a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{5/2} \left (c d^2+a e^2\right )^2}-\frac {4 e \log (x)}{a^2 d^2}+\frac {2 e^5 \log \left (d+e x^2\right )}{\left (c d^3+a d e^2\right )^2}+\frac {c \left (c d^2 e+2 a e^3\right ) \log \left (a+c x^4\right )}{a^2 \left (c d^2+a e^2\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(d + e*x^2)*(a + c*x^4)^2),x]

[Out]

(-2/(a^2*d*x^2) - (c*(a*e + c*d*x^2))/(a^2*(c*d^2 + a*e^2)*(a + c*x^4)) + (c^(3/2)*d*(3*c*d^2 + 5*a*e^2)*ArcTa
n[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(5/2)*(c*d^2 + a*e^2)^2) + (c^(3/2)*d*(3*c*d^2 + 5*a*e^2)*ArcTan[1 + (S
qrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(5/2)*(c*d^2 + a*e^2)^2) - (4*e*Log[x])/(a^2*d^2) + (2*e^5*Log[d + e*x^2])/(c*d
^3 + a*d*e^2)^2 + (c*(c*d^2*e + 2*a*e^3)*Log[a + c*x^4])/(a^2*(c*d^2 + a*e^2)^2))/4

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Maple [A]
time = 0.20, size = 181, normalized size = 0.77

method	result	size
default	\(\frac {e^{5} \ln \left (e \,x^{2}+d \right )}{2 d^{2} \left (a \,e^{2}+c \,d^{2}\right )^{2}}-\frac {c^{2} \left (\frac {\left (\frac {1}{2} d \,e^{2} a +\frac {1}{2} c \,d^{3}\right ) x^{2}+\frac {a e \left (a \,e^{2}+c \,d^{2}\right )}{2 c}}{c \,x^{4}+a}+\frac {\left (-4 a \,e^{3}-2 c \,d^{2} e \right ) \ln \left (c \,x^{4}+a \right )}{4 c}+\frac {\left (5 d \,e^{2} a +3 c \,d^{3}\right ) \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{2 \sqrt {a c}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} a^{2}}-\frac {1}{2 a^{2} d \,x^{2}}-\frac {e \ln \left (x \right )}{a^{2} d^{2}}\)	\(181\)
risch	\(\frac {-\frac {c \left (2 a \,e^{2}+3 c \,d^{2}\right ) x^{4}}{4 d \,a^{2} \left (a \,e^{2}+c \,d^{2}\right )}-\frac {c e \,x^{2}}{4 a \left (a \,e^{2}+c \,d^{2}\right )}-\frac {1}{2 d a}}{x^{2} \left (c \,x^{4}+a \right )}-\frac {e \ln \left (x \right )}{a^{2} d^{2}}+\frac {e^{5} \ln \left (-e \,x^{2}-d \right )}{2 d^{2} \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (e^{4} a^{7}+2 a^{6} c \,d^{2} e^{2}+a^{5} c^{2} d^{4}\right ) \textit {\_Z}^{2}+\left (-8 a^{4} c \,e^{3}-4 a^{3} c^{2} d^{2} e \right ) \textit {\_Z} +16 a \,c^{2} e^{2}+9 c^{3} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-6 a^{9} d^{2} e^{8}-19 a^{8} c \,d^{4} e^{6}-25 a^{7} c^{2} d^{6} e^{4}-17 a^{6} c^{3} d^{8} e^{2}-5 a^{5} c^{4} d^{10}\right ) \textit {\_R}^{3}+\left (52 a^{6} c \,d^{2} e^{7}+96 a^{5} c^{2} d^{4} e^{5}+48 a^{4} c^{3} d^{6} e^{3}+4 a^{3} c^{4} d^{8} e \right ) \textit {\_R}^{2}+\left (-32 a^{4} e^{8} c -80 a^{3} c^{2} d^{2} e^{6}-246 a^{2} c^{3} d^{4} e^{4}-168 a \,c^{4} d^{6} e^{2}-36 c^{5} d^{8}\right ) \textit {\_R} +128 e^{7} a \,c^{2}\right ) x^{2}+\left (-2 a^{9} d^{3} e^{7}-2 a^{8} c \,d^{5} e^{5}+2 a^{7} c^{2} d^{7} e^{3}+2 a^{6} c^{3} d^{9} e \right ) \textit {\_R}^{3}+\left (-16 a^{7} d \,e^{8}-8 a^{6} c \,d^{3} e^{6}+21 a^{5} c^{2} d^{5} e^{4}+10 a^{4} c^{3} d^{7} e^{2}-3 a^{3} c^{4} d^{9}\right ) \textit {\_R}^{2}+\left (64 a^{4} c d \,e^{7}+20 a^{3} c^{2} d^{3} e^{5}-66 a^{2} c^{3} d^{5} e^{3}-24 a \,c^{4} d^{7} e \right ) \textit {\_R} -160 a \,c^{2} d \,e^{6}-144 c^{3} d^{3} e^{4}\right )\right )}{8}\)	\(617\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(e*x^2+d)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.51, size = 269, normalized size = 1.14 \begin {gather*} \frac {{\left (c^{2} d^{2} e + 2 \, a c e^{3}\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 (x^{2} e + d\right )}{2 \, {\left (c^{2} d^{6} + 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4}\right )}} - \frac {{\left (3 \, c^{3} d^{3} + 5 \, a c^{2} d e^{2}\right )} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} \sqrt {a c}} - \frac {a c d x^{2} e + {\left (3 \, c^{2} d^{2} + 2 \, a c e^{2}\right )} x^{4} + 2 \, a c d^{2} + 2 \, a^{2} e^{2}}{4 \, {\left ({\left (a^{2} c^{2} d^{3} + a^{3} c d e^{2}\right )} x^{6} + {\left (a^{3} c d^{3} + a^{4} d e^{2}\right )} x^{2}\right )}} - \frac {e \log \left (x^{2}\right )}{2 \, a^{2} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (204) = 408\).
time = 210.30, size = 863, normalized size = 3.66 \begin {gather*} \left [-\frac {6 \, c^{3} d^{5} x^{4} + 2 \, a c^{2} d^{4} x^{2} e + 4 \, a c^{2} d^{5} + 2 \, a^{2} c d^{2} x^{2} e^{3} - 4 \, {\left (a^{2} c x^{6} + a^{3} x^{2}\right )} e^{5} \log \left (x^{2} e + d\right ) - {\left (3 \, c^{3} d^{5} x^{6} + 3 \, a c^{2} d^{5} x^{2} + 5 \, {\left (a c^{2} d^{3} x^{6} + a^{2} c d^{3} x^{2}\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 ) + 4 \, {\left (a^{2} c d x^{4} + a^{3} d\right )} e^{4} + 2 \, {\left (5 \, a c^{2} d^{3} x^{4} + 4 \, a^{2} c d^{3}\right )} e^{2} - 2 \, {\left (2 \, {\left (a c^{2} d^{2} x^{6} + a^{2} c d^{2} x^{2}\right )} e^{3} + {\left (c^{3} d^{4} x^{6} + a c^{2} d^{4} x^{2}\right )} e\right )} \log \left (c x^{4} + a\right ) + 8 \, {\left ({\left (a^{2} c x^{6} + a^{3} x^{2}\right )} e^{5} + 2 \, {\left (a c^{2} d^{2} x^{6} + a^{2} c d^{2} x^{2}\right )} e^{3} + {\left (c^{3} d^{4} x^{6} + a c^{2} d^{4} x^{2}\right )} e\right )} \log \left (x\right )}{8 \, {\left (a^{2} c^{3} d^{6} x^{6} + a^{3} c^{2} d^{6} x^{2} + {\left (a^{4} c d^{2} x^{6} + a^{5} d^{2} x^{2}\right )} e^{4} + 2 \, {\left (a^{3} c^{2} d^{4} x^{6} + a^{4} c d^{4} x^{2}\right )} e^{2}\right )}}, -\frac {3 \, c^{3} d^{5} x^{4} + a c^{2} d^{4} x^{2} e + 2 \, a c^{2} d^{5} + a^{2} c d^{2} x^{2} e^{3} - 2 \, {\left (a^{2} c x^{6} + a^{3} x^{2}\right )} e^{5} \log \left (x^{2} e + d\right ) - {\left (3 \, c^{3} d^{5} x^{6} + 3 \, a c^{2} d^{5} x^{2} + 5 \, {\left (a c^{2} d^{3} x^{6} + a^{2} c d^{3} x^{2}\right )} e^{2}\right )} \sqrt {\frac {c}{a}} \arctan \left (\frac {a \sqrt {\frac {c}{a}}}{c x^{2}}\right ) + 2 \, {\left (a^{2} c d x^{4} + a^{3} d\right )} e^{4} + {\left (5 \, a c^{2} d^{3} x^{4} + 4 \, a^{2} c d^{3}\right )} e^{2} - {\left (2 \, {\left (a c^{2} d^{2} x^{6} + a^{2} c d^{2} x^{2}\right )} e^{3} + {\left (c^{3} d^{4} x^{6} + a c^{2} d^{4} x^{2}\right )} e\right )} \log \left (c x^{4} + a\right ) + 4 \, {\left ({\left (a^{2} c x^{6} + a^{3} x^{2}\right )} e^{5} + 2 \, {\left (a c^{2} d^{2} x^{6} + a^{2} c d^{2} x^{2}\right )} e^{3} + {\left (c^{3} d^{4} x^{6} + a c^{2} d^{4} x^{2}\right )} e\right )} \log \left (x\right )}{4 \, {\left (a^{2} c^{3} d^{6} x^{6} + a^{3} c^{2} d^{6} x^{2} + {\left (a^{4} c d^{2} x^{6} + a^{5} d^{2} x^{2}\right )} e^{4} + 2 \, {\left (a^{3} c^{2} d^{4} x^{6} + a^{4} c d^{4} x^{2}\right )} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="fricas")

[Out]

[-1/8*(6*c^3*d^5*x^4 + 2*a*c^2*d^4*x^2*e + 4*a*c^2*d^5 + 2*a^2*c*d^2*x^2*e^3 - 4*(a^2*c*x^6 + a^3*x^2)*e^5*log
(x^2*e + d) - (3*c^3*d^5*x^6 + 3*a*c^2*d^5*x^2 + 5*(a*c^2*d^3*x^6 + a^2*c*d^3*x^2)*e^2)*sqrt(-c/a)*log((c*x^4
- 2*a*x^2*sqrt(-c/a) - a)/(c*x^4 + a)) + 4*(a^2*c*d*x^4 + a^3*d)*e^4 + 2*(5*a*c^2*d^3*x^4 + 4*a^2*c*d^3)*e^2 -
2*(2*(a*c^2*d^2*x^6 + a^2*c*d^2*x^2)*e^3 + (c^3*d^4*x^6 + a*c^2*d^4*x^2)*e)*log(c*x^4 + a) + 8*((a^2*c*x^6 +
a^3*x^2)*e^5 + 2*(a*c^2*d^2*x^6 + a^2*c*d^2*x^2)*e^3 + (c^3*d^4*x^6 + a*c^2*d^4*x^2)*e)*log(x))/(a^2*c^3*d^6*x
^6 + a^3*c^2*d^6*x^2 + (a^4*c*d^2*x^6 + a^5*d^2*x^2)*e^4 + 2*(a^3*c^2*d^4*x^6 + a^4*c*d^4*x^2)*e^2), -1/4*(3*c
^3*d^5*x^4 + a*c^2*d^4*x^2*e + 2*a*c^2*d^5 + a^2*c*d^2*x^2*e^3 - 2*(a^2*c*x^6 + a^3*x^2)*e^5*log(x^2*e + d) -
(3*c^3*d^5*x^6 + 3*a*c^2*d^5*x^2 + 5*(a*c^2*d^3*x^6 + a^2*c*d^3*x^2)*e^2)*sqrt(c/a)*arctan(a*sqrt(c/a)/(c*x^2)
) + 2*(a^2*c*d*x^4 + a^3*d)*e^4 + (5*a*c^2*d^3*x^4 + 4*a^2*c*d^3)*e^2 - (2*(a*c^2*d^2*x^6 + a^2*c*d^2*x^2)*e^3
+ (c^3*d^4*x^6 + a*c^2*d^4*x^2)*e)*log(c*x^4 + a) + 4*((a^2*c*x^6 + a^3*x^2)*e^5 + 2*(a*c^2*d^2*x^6 + a^2*c*d
^2*x^2)*e^3 + (c^3*d^4*x^6 + a*c^2*d^4*x^2)*e)*log(x))/(a^2*c^3*d^6*x^6 + a^3*c^2*d^6*x^2 + (a^4*c*d^2*x^6 + a
^5*d^2*x^2)*e^4 + 2*(a^3*c^2*d^4*x^6 + a^4*c*d^4*x^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(1/x**3/(e*x**2+d)/(c*x**4+a)**2,x)

[Out]

Timed out

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Giac [A]
time = 3.10, size = 344, normalized size = 1.46 \begin {gather*} \frac {{\left (c^{2} d^{2} e + 2 \, a c e^{3}\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^{6} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \, {\left (c^{2} d^{6} e + 2 \, a c d^{4} e^{3} + a^{2} d^{2} e^{5}\right )}} - \frac {{\left (3 \, c^{3} d^{3} + 5 \, a c^{2} d e^{2}\right )} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} \sqrt {a c}} - \frac {9 \, c^{3} d^{5} x^{4} + 15 \, a c^{2} d^{3} x^{4} e^{2} - 2 \, a^{2} c x^{6} e^{5} + 3 \, a c^{2} d^{4} x^{2} e + 6 \, a^{2} c d x^{4} e^{4} + 6 \, a c^{2} d^{5} + 3 \, a^{2} c d^{2} x^{2} e^{3} + 12 \, a^{2} c d^{3} e^{2} - 2 \, a^{3} x^{2} e^{5} + 6 \, a^{3} d e^{4}}{12 \, {\left (a^{2} c^{2} d^{6} + 2 \, a^{3} c d^{4} e^{2} + a^{4} d^{2} e^{4}\right )} {\left (c x^{6} + a x^{2}\right )}} - \frac {e \log \left (x^{2}\right )}{2 \, a^{2} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="giac")

[Out]

1/4*(c^2*d^2*e + 2*a*c*e^3)*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^6*log(abs(x^2*e +
d))/(c^2*d^6*e + 2*a*c*d^4*e^3 + a^2*d^2*e^5) - 1/4*(3*c^3*d^3 + 5*a*c^2*d*e^2)*arctan(c*x^2/sqrt(a*c))/((a^2
*c^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e^4)*sqrt(a*c)) - 1/12*(9*c^3*d^5*x^4 + 15*a*c^2*d^3*x^4*e^2 - 2*a^2*c*x^6*e^
5 + 3*a*c^2*d^4*x^2*e + 6*a^2*c*d*x^4*e^4 + 6*a*c^2*d^5 + 3*a^2*c*d^2*x^2*e^3 + 12*a^2*c*d^3*e^2 - 2*a^3*x^2*e
^5 + 6*a^3*d*e^4)/((a^2*c^2*d^6 + 2*a^3*c*d^4*e^2 + a^4*d^2*e^4)*(c*x^6 + a*x^2)) - 1/2*e*log(x^2)/(a^2*d^2)

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Mupad [B]
time = 2.94, size = 1337, normalized size = 5.67 \begin {gather*} \frac {\ln \left (81\,a^{10}\,c^{16}\,d^{24}\,x^2+1024\,a^{22}\,c^4\,e^{24}\,x^2-81\,a^3\,c^{11}\,d^{24}\,{\left (-a^5\,c^3\right )}^{3/2}+1024\,a^{20}\,c^2\,e^{24}\,\sqrt {-a^5\,c^3}-14496\,a^6\,d^8\,e^{16}\,{\left (-a^5\,c^3\right )}^{5/2}-5120\,a^{14}\,d^2\,e^{22}\,{\left (-a^5\,c^3\right )}^{3/2}+11647\,c^6\,d^{20}\,e^4\,{\left (-a^5\,c^3\right )}^{5/2}+1638\,a^{11}\,c^{15}\,d^{22}\,e^2\,x^2+11647\,a^{12}\,c^{14}\,d^{20}\,e^4\,x^2+43524\,a^{13}\,c^{13}\,d^{18}\,e^6\,x^2+97311\,a^{14}\,c^{12}\,d^{16}\,e^8\,x^2+133334\,a^{15}\,c^{11}\,d^{14}\,e^{10}\,x^2+103633\,a^{16}\,c^{10}\,d^{12}\,e^{12}\,x^2+29456\,a^{17}\,c^9\,d^{10}\,e^{14}\,x^2-14496\,a^{18}\,c^8\,d^8\,e^{16}\,x^2-7984\,a^{19}\,c^7\,d^6\,e^{18}\,x^2+5888\,a^{20}\,c^6\,d^4\,e^{20}\,x^2+5120\,a^{21}\,c^5\,d^2\,e^{22}\,x^2+43524\,a\,c^5\,d^{18}\,e^6\,{\left (-a^5\,c^3\right )}^{5/2}+29456\,a^5\,c\,d^{10}\,e^{14}\,{\left (-a^5\,c^3\right )}^{5/2}-5888\,a^{13}\,c\,d^4\,e^{20}\,{\left (-a^5\,c^3\right )}^{3/2}+97311\,a^2\,c^4\,d^{16}\,e^8\,{\left (-a^5\,c^3\right )}^{5/2}+133334\,a^3\,c^3\,d^{14}\,e^{10}\,{\left (-a^5\,c^3\right )}^{5/2}+103633\,a^4\,c^2\,d^{12}\,e^{12}\,{\left (-a^5\,c^3\right )}^{5/2}-1638\,a^4\,c^{10}\,d^{22}\,e^2\,{\left (-a^5\,c^3\right )}^{3/2}+7984\,a^{12}\,c^2\,d^6\,e^{18}\,{\left (-a^5\,c^3\right )}^{3/2}\right )\,\left (4\,a^4\,c\,e^3-3\,c\,d^3\,\sqrt {-a^5\,c^3}+2\,a^3\,c^2\,d^2\,e-5\,a\,d\,e^2\,\sqrt {-a^5\,c^3}\right )}{8\,\left (a^7\,e^4+2\,a^6\,c\,d^2\,e^2+a^5\,c^2\,d^4\right )}-\frac {\frac {1}{2\,a\,d}+\frac {c\,e\,x^2}{4\,a\,\left (c\,d^2+a\,e^2\right )}+\frac {c\,x^4\,\left (3\,c\,d^2+2\,a\,e^2\right )}{4\,a^2\,d\,\left (c\,d^2+a\,e^2\right )}}{c\,x^6+a\,x^2}+\frac {\ln \left (81\,a^{10}\,c^{16}\,d^{24}\,x^2+1024\,a^{22}\,c^4\,e^{24}\,x^2+81\,a^3\,c^{11}\,d^{24}\,{\left (-a^5\,c^3\right )}^{3/2}-1024\,a^{20}\,c^2\,e^{24}\,\sqrt {-a^5\,c^3}+14496\,a^6\,d^8\,e^{16}\,{\left (-a^5\,c^3\right )}^{5/2}+5120\,a^{14}\,d^2\,e^{22}\,{\left (-a^5\,c^3\right )}^{3/2}-11647\,c^6\,d^{20}\,e^4\,{\left (-a^5\,c^3\right )}^{5/2}+1638\,a^{11}\,c^{15}\,d^{22}\,e^2\,x^2+11647\,a^{12}\,c^{14}\,d^{20}\,e^4\,x^2+43524\,a^{13}\,c^{13}\,d^{18}\,e^6\,x^2+97311\,a^{14}\,c^{12}\,d^{16}\,e^8\,x^2+133334\,a^{15}\,c^{11}\,d^{14}\,e^{10}\,x^2+103633\,a^{16}\,c^{10}\,d^{12}\,e^{12}\,x^2+29456\,a^{17}\,c^9\,d^{10}\,e^{14}\,x^2-14496\,a^{18}\,c^8\,d^8\,e^{16}\,x^2-7984\,a^{19}\,c^7\,d^6\,e^{18}\,x^2+5888\,a^{20}\,c^6\,d^4\,e^{20}\,x^2+5120\,a^{21}\,c^5\,d^2\,e^{22}\,x^2-43524\,a\,c^5\,d^{18}\,e^6\,{\left (-a^5\,c^3\right )}^{5/2}-29456\,a^5\,c\,d^{10}\,e^{14}\,{\left (-a^5\,c^3\right )}^{5/2}+5888\,a^{13}\,c\,d^4\,e^{20}\,{\left (-a^5\,c^3\right )}^{3/2}-97311\,a^2\,c^4\,d^{16}\,e^8\,{\left (-a^5\,c^3\right )}^{5/2}-133334\,a^3\,c^3\,d^{14}\,e^{10}\,{\left (-a^5\,c^3\right )}^{5/2}-103633\,a^4\,c^2\,d^{12}\,e^{12}\,{\left (-a^5\,c^3\right )}^{5/2}+1638\,a^4\,c^{10}\,d^{22}\,e^2\,{\left (-a^5\,c^3\right )}^{3/2}-7984\,a^{12}\,c^2\,d^6\,e^{18}\,{\left (-a^5\,c^3\right )}^{3/2}\right )\,\left (4\,a^4\,c\,e^3+3\,c\,d^3\,\sqrt {-a^5\,c^3}+2\,a^3\,c^2\,d^2\,e+5\,a\,d\,e^2\,\sqrt {-a^5\,c^3}\right )}{8\,\left (a^7\,e^4+2\,a^6\,c\,d^2\,e^2+a^5\,c^2\,d^4\right )}+\frac {e^5\,\ln \left (e\,x^2+d\right )}{2\,a^2\,d^2\,e^4+4\,a\,c\,d^4\,e^2+2\,c^2\,d^6}-\frac {e\,\ln \left (x\right )}{a^2\,d^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + c*x^4)^2*(d + e*x^2)),x)

[Out]

(log(81*a^10*c^16*d^24*x^2 + 1024*a^22*c^4*e^24*x^2 - 81*a^3*c^11*d^24*(-a^5*c^3)^(3/2) + 1024*a^20*c^2*e^24*(
-a^5*c^3)^(1/2) - 14496*a^6*d^8*e^16*(-a^5*c^3)^(5/2) - 5120*a^14*d^2*e^22*(-a^5*c^3)^(3/2) + 11647*c^6*d^20*e
^4*(-a^5*c^3)^(5/2) + 1638*a^11*c^15*d^22*e^2*x^2 + 11647*a^12*c^14*d^20*e^4*x^2 + 43524*a^13*c^13*d^18*e^6*x^
2 + 97311*a^14*c^12*d^16*e^8*x^2 + 133334*a^15*c^11*d^14*e^10*x^2 + 103633*a^16*c^10*d^12*e^12*x^2 + 29456*a^1
7*c^9*d^10*e^14*x^2 - 14496*a^18*c^8*d^8*e^16*x^2 - 7984*a^19*c^7*d^6*e^18*x^2 + 5888*a^20*c^6*d^4*e^20*x^2 +
5120*a^21*c^5*d^2*e^22*x^2 + 43524*a*c^5*d^18*e^6*(-a^5*c^3)^(5/2) + 29456*a^5*c*d^10*e^14*(-a^5*c^3)^(5/2) -
5888*a^13*c*d^4*e^20*(-a^5*c^3)^(3/2) + 97311*a^2*c^4*d^16*e^8*(-a^5*c^3)^(5/2) + 133334*a^3*c^3*d^14*e^10*(-a
^5*c^3)^(5/2) + 103633*a^4*c^2*d^12*e^12*(-a^5*c^3)^(5/2) - 1638*a^4*c^10*d^22*e^2*(-a^5*c^3)^(3/2) + 7984*a^1
2*c^2*d^6*e^18*(-a^5*c^3)^(3/2))*(4*a^4*c*e^3 - 3*c*d^3*(-a^5*c^3)^(1/2) + 2*a^3*c^2*d^2*e - 5*a*d*e^2*(-a^5*c
^3)^(1/2)))/(8*(a^7*e^4 + a^5*c^2*d^4 + 2*a^6*c*d^2*e^2)) - (1/(2*a*d) + (c*e*x^2)/(4*a*(a*e^2 + c*d^2)) + (c*
x^4*(2*a*e^2 + 3*c*d^2))/(4*a^2*d*(a*e^2 + c*d^2)))/(a*x^2 + c*x^6) + (log(81*a^10*c^16*d^24*x^2 + 1024*a^22*c
^4*e^24*x^2 + 81*a^3*c^11*d^24*(-a^5*c^3)^(3/2) - 1024*a^20*c^2*e^24*(-a^5*c^3)^(1/2) + 14496*a^6*d^8*e^16*(-a
^5*c^3)^(5/2) + 5120*a^14*d^2*e^22*(-a^5*c^3)^(3/2) - 11647*c^6*d^20*e^4*(-a^5*c^3)^(5/2) + 1638*a^11*c^15*d^2
2*e^2*x^2 + 11647*a^12*c^14*d^20*e^4*x^2 + 43524*a^13*c^13*d^18*e^6*x^2 + 97311*a^14*c^12*d^16*e^8*x^2 + 13333
4*a^15*c^11*d^14*e^10*x^2 + 103633*a^16*c^10*d^12*e^12*x^2 + 29456*a^17*c^9*d^10*e^14*x^2 - 14496*a^18*c^8*d^8
*e^16*x^2 - 7984*a^19*c^7*d^6*e^18*x^2 + 5888*a^20*c^6*d^4*e^20*x^2 + 5120*a^21*c^5*d^2*e^22*x^2 - 43524*a*c^5
*d^18*e^6*(-a^5*c^3)^(5/2) - 29456*a^5*c*d^10*e^14*(-a^5*c^3)^(5/2) + 5888*a^13*c*d^4*e^20*(-a^5*c^3)^(3/2) -
97311*a^2*c^4*d^16*e^8*(-a^5*c^3)^(5/2) - 133334*a^3*c^3*d^14*e^10*(-a^5*c^3)^(5/2) - 103633*a^4*c^2*d^12*e^12
*(-a^5*c^3)^(5/2) + 1638*a^4*c^10*d^22*e^2*(-a^5*c^3)^(3/2) - 7984*a^12*c^2*d^6*e^18*(-a^5*c^3)^(3/2))*(4*a^4*
c*e^3 + 3*c*d^3*(-a^5*c^3)^(1/2) + 2*a^3*c^2*d^2*e + 5*a*d*e^2*(-a^5*c^3)^(1/2)))/(8*(a^7*e^4 + a^5*c^2*d^4 +
2*a^6*c*d^2*e^2)) + (e^5*log(d + e*x^2))/(2*c^2*d^6 + 2*a^2*d^2*e^4 + 4*a*c*d^4*e^2) - (e*log(x))/(a^2*d^2)

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