Optimal. Leaf size=126 \[ \frac{b^3 \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^2}-\frac{\log (x) (2 a d+b c)}{a^2 c^3}-\frac{d^2 (3 b c-2 a d) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^2}+\frac{d^2}{2 c^2 \left (c+d x^2\right ) (b c-a d)}-\frac{1}{2 a c^2 x^2} \]
[Out]
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Rubi [A] time = 0.329082, antiderivative size = 126, 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 \[ \frac{b^3 \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^2}-\frac{\log (x) (2 a d+b c)}{a^2 c^3}-\frac{d^2 (3 b c-2 a d) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^2}+\frac{d^2}{2 c^2 \left (c+d x^2\right ) (b c-a d)}-\frac{1}{2 a c^2 x^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a + b*x^2)*(c + d*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 51.5114, size = 116, normalized size = 0.92 \[ - \frac{d^{2}}{2 c^{2} \left (c + d x^{2}\right ) \left (a d - b c\right )} + \frac{d^{2} \left (2 a d - 3 b c\right ) \log{\left (c + d x^{2} \right )}}{2 c^{3} \left (a d - b c\right )^{2}} - \frac{1}{2 a c^{2} x^{2}} + \frac{b^{3} \log{\left (a + b x^{2} \right )}}{2 a^{2} \left (a d - b c\right )^{2}} - \frac{\left (2 a d + b c\right ) \log{\left (x^{2} \right )}}{2 a^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(b*x**2+a)/(d*x**2+c)**2,x)
[Out]
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Mathematica [A] time = 0.549605, size = 117, normalized size = 0.93 \[ \frac{1}{2} \left (\frac{b^3 \log \left (a+b x^2\right )}{a^2 (b c-a d)^2}-\frac{2 \log (x) (2 a d+b c)}{a^2 c^3}+\frac{\frac{c d^2}{\left (c+d x^2\right ) (b c-a d)}+\frac{d^2 (2 a d-3 b c) \log \left (c+d x^2\right )}{(b c-a d)^2}-\frac{c}{a x^2}}{c^3}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a + b*x^2)*(c + d*x^2)^2),x]
[Out]
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Maple [A] time = 0.026, size = 170, normalized size = 1.4 \[ -{\frac{1}{2\,a{c}^{2}{x}^{2}}}-2\,{\frac{\ln \left ( x \right ) d}{a{c}^{3}}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}{c}^{2}}}-{\frac{{d}^{3}a}{2\,{c}^{2} \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{b{d}^{2}}{2\,c \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{{d}^{3}\ln \left ( d{x}^{2}+c \right ) a}{{c}^{3} \left ( ad-bc \right ) ^{2}}}-{\frac{3\,{d}^{2}\ln \left ( d{x}^{2}+c \right ) b}{2\,{c}^{2} \left ( ad-bc \right ) ^{2}}}+{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{2} \left ( ad-bc \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(b*x^2+a)/(d*x^2+c)^2,x)
[Out]
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Maxima [A] time = 1.36354, size = 254, normalized size = 2.02 \[ \frac{b^{3} \log \left (b x^{2} + a\right )}{2 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}} - \frac{{\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )}} - \frac{b c^{2} - a c d +{\left (b c d - 2 \, a d^{2}\right )} x^{2}}{2 \,{\left ({\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{4} +{\left (a b c^{4} - a^{2} c^{3} d\right )} x^{2}\right )}} - \frac{{\left (b c + 2 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)^2*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 2.36857, size = 408, normalized size = 3.24 \[ -\frac{a b^{2} c^{4} - 2 \, a^{2} b c^{3} d + a^{3} c^{2} d^{2} +{\left (a b^{2} c^{3} d - 3 \, a^{2} b c^{2} d^{2} + 2 \, a^{3} c d^{3}\right )} x^{2} -{\left (b^{3} c^{3} d x^{4} + b^{3} c^{4} x^{2}\right )} \log \left (b x^{2} + a\right ) +{\left ({\left (3 \, a^{2} b c d^{3} - 2 \, a^{3} d^{4}\right )} x^{4} +{\left (3 \, a^{2} b c^{2} d^{2} - 2 \, a^{3} c d^{3}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 2 \,{\left ({\left (b^{3} c^{3} d - 3 \, a^{2} b c d^{3} + 2 \, a^{3} d^{4}\right )} x^{4} +{\left (b^{3} c^{4} - 3 \, a^{2} b c^{2} d^{2} + 2 \, a^{3} c d^{3}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left ({\left (a^{2} b^{2} c^{5} d - 2 \, a^{3} b c^{4} d^{2} + a^{4} c^{3} d^{3}\right )} x^{4} +{\left (a^{2} b^{2} c^{6} - 2 \, a^{3} b c^{5} d + a^{4} c^{4} d^{2}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)^2*x^3),x, algorithm="fricas")
[Out]
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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**3/(b*x**2+a)/(d*x**2+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.291994, size = 347, normalized size = 2.75 \[ \frac{b^{4}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \,{\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )}} - \frac{{\left (3 \, b c d^{3} - 2 \, a d^{4}\right )}{\rm ln}\left ({\left | d x^{2} + c \right |}\right )}{2 \,{\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )}} + \frac{b^{3} c^{2} d x^{4} + b^{3} c^{3} x^{2} - 2 \, a b^{2} c^{2} d x^{2} + 6 \, a^{2} b c d^{2} x^{2} - 4 \, a^{3} d^{3} x^{2} - 2 \, a b^{2} c^{3} + 4 \, a^{2} b c^{2} d - 2 \, a^{3} c d^{2}}{4 \,{\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )}{\left (d x^{4} + c x^{2}\right )}} - \frac{{\left (b c + 2 \, a d\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)^2*x^3),x, algorithm="giac")
[Out]