Optimal. Leaf size=126 \[ \frac{b^2 (2 b c-3 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^2}-\frac{\log (x) (a d+2 b c)}{a^3 c^2}-\frac{b^2}{2 a^2 \left (a+b x^2\right ) (b c-a d)}-\frac{1}{2 a^2 c x^2}+\frac{d^3 \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^2} \]
[Out]
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Rubi [A] time = 0.324945, 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 \[ \frac{b^2 (2 b c-3 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^2}-\frac{\log (x) (a d+2 b c)}{a^3 c^2}-\frac{b^2}{2 a^2 \left (a+b x^2\right ) (b c-a d)}-\frac{1}{2 a^2 c x^2}+\frac{d^3 \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a + b*x^2)^2*(c + d*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 48.3724, size = 116, normalized size = 0.92 \[ \frac{d^{3} \log{\left (c + d x^{2} \right )}}{2 c^{2} \left (a d - b c\right )^{2}} + \frac{b^{2}}{2 a^{2} \left (a + b x^{2}\right ) \left (a d - b c\right )} - \frac{1}{2 a^{2} c x^{2}} - \frac{b^{2} \left (3 a d - 2 b c\right ) \log{\left (a + b x^{2} \right )}}{2 a^{3} \left (a d - b c\right )^{2}} - \frac{\left (a d + 2 b c\right ) \log{\left (x^{2} \right )}}{2 a^{3} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(b*x**2+a)**2/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.27241, size = 119, normalized size = 0.94 \[ \frac{1}{2} \left (\frac{b^2 (2 b c-3 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^2}-\frac{2 \log (x) (a d+2 b c)}{a^3 c^2}+\frac{b^2}{a^2 \left (a+b x^2\right ) (a d-b c)}-\frac{1}{a^2 c x^2}+\frac{d^3 \log \left (c+d x^2\right )}{c^2 (b c-a d)^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a + b*x^2)^2*(c + d*x^2)),x]
[Out]
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Maple [A] time = 0.027, size = 170, normalized size = 1.4 \[ -{\frac{1}{2\,{a}^{2}c{x}^{2}}}-{\frac{\ln \left ( x \right ) d}{{a}^{2}{c}^{2}}}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}c}}+{\frac{{d}^{3}\ln \left ( d{x}^{2}+c \right ) }{2\,{c}^{2} \left ( ad-bc \right ) ^{2}}}-{\frac{3\,{b}^{2}\ln \left ( b{x}^{2}+a \right ) d}{2\,{a}^{2} \left ( ad-bc \right ) ^{2}}}+{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) c}{{a}^{3} \left ( ad-bc \right ) ^{2}}}+{\frac{{b}^{2}d}{2\,a \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{3}c}{2\,{a}^{2} \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(b*x^2+a)^2/(d*x^2+c),x)
[Out]
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Maxima [A] time = 1.36836, size = 255, normalized size = 2.02 \[ \frac{d^{3} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}} + \frac{{\left (2 \, b^{3} c - 3 \, a b^{2} d\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )}} - \frac{a b c - a^{2} d +{\left (2 \, b^{2} c - a b d\right )} x^{2}}{2 \,{\left ({\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{4} +{\left (a^{3} b c^{2} - a^{4} c d\right )} x^{2}\right )}} - \frac{{\left (2 \, b c + a d\right )} \log \left (x^{2}\right )}{2 \, a^{3} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*(d*x^2 + c)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 2.46711, size = 409, normalized size = 3.25 \[ -\frac{a^{2} b^{2} c^{3} - 2 \, a^{3} b c^{2} d + a^{4} c d^{2} +{\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + a^{3} b c d^{2}\right )} x^{2} -{\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d\right )} x^{4} +{\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) -{\left (a^{3} b d^{3} x^{4} + a^{4} d^{3} x^{2}\right )} \log \left (d x^{2} + c\right ) + 2 \,{\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d + a^{3} b d^{3}\right )} x^{4} +{\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + a^{4} d^{3}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left ({\left (a^{3} b^{3} c^{4} - 2 \, a^{4} b^{2} c^{3} d + a^{5} b c^{2} d^{2}\right )} x^{4} +{\left (a^{4} b^{2} c^{4} - 2 \, a^{5} b c^{3} d + a^{6} c^{2} d^{2}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*(d*x^2 + c)*x^3),x, algorithm="fricas")
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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)**2/(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.239393, size = 347, normalized size = 2.75 \[ \frac{d^{4}{\rm ln}\left ({\left | d x^{2} + c \right |}\right )}{2 \,{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )}} + \frac{{\left (2 \, b^{4} c - 3 \, a b^{3} d\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \,{\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )}} + \frac{a^{2} b d^{3} x^{4} - 4 \, b^{3} c^{3} x^{2} + 6 \, a b^{2} c^{2} d x^{2} - 2 \, a^{2} b c d^{2} x^{2} + a^{3} d^{3} x^{2} - 2 \, a b^{2} c^{3} + 4 \, a^{2} b c^{2} d - 2 \, a^{3} c d^{2}}{4 \,{\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )}{\left (b x^{4} + a x^{2}\right )}} - \frac{{\left (2 \, b c + a d\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{3} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*(d*x^2 + c)*x^3),x, algorithm="giac")
[Out]