Optimal. Leaf size=70 \[ -\frac{1}{2 \left (a+b x^2\right ) (b c-a d)}-\frac{d \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac{d \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]
[Out]
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Rubi [A] time = 0.120471, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{1}{2 \left (a+b x^2\right ) (b c-a d)}-\frac{d \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac{d \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[x/((a + b*x^2)^2*(c + d*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 21.8142, size = 56, normalized size = 0.8 \[ - \frac{d \log{\left (a + b x^{2} \right )}}{2 \left (a d - b c\right )^{2}} + \frac{d \log{\left (c + d x^{2} \right )}}{2 \left (a d - b c\right )^{2}} + \frac{1}{2 \left (a + b x^{2}\right ) \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b*x**2+a)**2/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.04828, size = 66, normalized size = 0.94 \[ \frac{d \left (a+b x^2\right ) \log \left (c+d x^2\right )-d \left (a+b x^2\right ) \log \left (a+b x^2\right )+a d-b c}{2 \left (a+b x^2\right ) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[x/((a + b*x^2)^2*(c + d*x^2)),x]
[Out]
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Maple [A] time = 0.017, size = 90, normalized size = 1.3 \[{\frac{d\ln \left ( d{x}^{2}+c \right ) }{2\, \left ( ad-bc \right ) ^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) d}{2\, \left ( ad-bc \right ) ^{2}}}+{\frac{ad}{2\, \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{bc}{2\, \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b*x^2+a)^2/(d*x^2+c),x)
[Out]
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Maxima [A] time = 1.35271, size = 134, normalized size = 1.91 \[ -\frac{d \log \left (b x^{2} + a\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac{d \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac{1}{2 \,{\left (a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^2 + a)^2*(d*x^2 + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231694, size = 139, normalized size = 1.99 \[ -\frac{b c - a d +{\left (b d x^{2} + a d\right )} \log \left (b x^{2} + a\right ) -{\left (b d x^{2} + a d\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^2 + a)^2*(d*x^2 + c)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.12281, size = 248, normalized size = 3.54 \[ \frac{d \log{\left (x^{2} + \frac{- \frac{a^{3} d^{4}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} + \frac{b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{2 \left (a d - b c\right )^{2}} - \frac{d \log{\left (x^{2} + \frac{\frac{a^{3} d^{4}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} - \frac{b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{2 \left (a d - b c\right )^{2}} + \frac{1}{2 a^{2} d - 2 a b c + x^{2} \left (2 a b d - 2 b^{2} c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x**2+a)**2/(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.265773, size = 115, normalized size = 1.64 \[ \frac{b d{\rm ln}\left ({\left | \frac{b c}{b x^{2} + a} - \frac{a d}{b x^{2} + a} + d \right |}\right )}{2 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}} - \frac{b}{2 \,{\left (b^{2} c - a b d\right )}{\left (b x^{2} + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^2 + a)^2*(d*x^2 + c)),x, algorithm="giac")
[Out]