Optimal. Leaf size=119 \[ -\frac{b^3 \log \left (a+b x^2\right )}{2 a^3 (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}+\frac{\log (x) \left (a^2 d^2+a b c d+b^2 c^2\right )}{a^3 c^3}+\frac{d^3 \log \left (c+d x^2\right )}{2 c^3 (b c-a d)}-\frac{1}{4 a c x^4} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.293266, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{b^3 \log \left (a+b x^2\right )}{2 a^3 (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}+\frac{\log (x) \left (a^2 d^2+a b c d+b^2 c^2\right )}{a^3 c^3}+\frac{d^3 \log \left (c+d x^2\right )}{2 c^3 (b c-a d)}-\frac{1}{4 a c x^4} \]
Antiderivative was successfully verified.
[In] Int[1/(x^5*(a + b*x^2)*(c + d*x^2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 43.685, size = 109, normalized size = 0.92 \[ - \frac{d^{3} \log{\left (c + d x^{2} \right )}}{2 c^{3} \left (a d - b c\right )} - \frac{1}{4 a c x^{4}} + \frac{a d + b c}{2 a^{2} c^{2} x^{2}} + \frac{b^{3} \log{\left (a + b x^{2} \right )}}{2 a^{3} \left (a d - b c\right )} + \frac{\left (a^{2} d^{2} + a b c d + b^{2} c^{2}\right ) \log{\left (x^{2} \right )}}{2 a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**5/(b*x**2+a)/(d*x**2+c),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0964288, size = 119, normalized size = 1. \[ \frac{b^3 \log \left (a+b x^2\right )}{2 a^3 (a d-b c)}+\frac{a d+b c}{2 a^2 c^2 x^2}+\frac{\log (x) \left (a^2 d^2+a b c d+b^2 c^2\right )}{a^3 c^3}+\frac{d^3 \log \left (c+d x^2\right )}{2 c^3 (b c-a d)}-\frac{1}{4 a c x^4} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^5*(a + b*x^2)*(c + d*x^2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.02, size = 124, normalized size = 1. \[ -{\frac{1}{4\,ac{x}^{4}}}+{\frac{d}{2\,a{x}^{2}{c}^{2}}}+{\frac{b}{2\,{a}^{2}c{x}^{2}}}+{\frac{\ln \left ( x \right ){d}^{2}}{a{c}^{3}}}+{\frac{b\ln \left ( x \right ) d}{{a}^{2}{c}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{3}c}}-{\frac{{d}^{3}\ln \left ( d{x}^{2}+c \right ) }{2\,{c}^{3} \left ( ad-bc \right ) }}+{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{3} \left ( ad-bc \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^5/(b*x^2+a)/(d*x^2+c),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.35002, size = 158, normalized size = 1.33 \[ -\frac{b^{3} \log \left (b x^{2} + a\right )}{2 \,{\left (a^{3} b c - a^{4} d\right )}} + \frac{d^{3} \log \left (d x^{2} + c\right )}{2 \,{\left (b c^{4} - a c^{3} d\right )}} + \frac{{\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{3} c^{3}} + \frac{2 \,{\left (b c + a d\right )} x^{2} - a c}{4 \, a^{2} c^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)*x^5),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 1.25235, size = 171, normalized size = 1.44 \[ -\frac{2 \, b^{3} c^{3} x^{4} \log \left (b x^{2} + a\right ) - 2 \, a^{3} d^{3} x^{4} \log \left (d x^{2} + c\right ) + a^{2} b c^{3} - a^{3} c^{2} d - 4 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{4} \log \left (x\right ) - 2 \,{\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x^{2}}{4 \,{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)*x^5),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**5/(b*x**2+a)/(d*x**2+c),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)*x^5),x, algorithm="giac")
[Out]