Optimal. Leaf size=100 \[ -\frac{b^2 \log \left (a+b x^2\right )}{2 a (b c-a d)^2}+\frac{d (2 b c-a d) \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^2}-\frac{d}{2 c \left (c+d x^2\right ) (b c-a d)}+\frac{\log (x)}{a c^2} \]
[Out]
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Rubi [A] time = 0.229838, antiderivative size = 100, 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 \[ -\frac{b^2 \log \left (a+b x^2\right )}{2 a (b c-a d)^2}+\frac{d (2 b c-a d) \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^2}-\frac{d}{2 c \left (c+d x^2\right ) (b c-a d)}+\frac{\log (x)}{a c^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(a + b*x^2)*(c + d*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 38.8441, size = 85, normalized size = 0.85 \[ \frac{d}{2 c \left (c + d x^{2}\right ) \left (a d - b c\right )} - \frac{d \left (a d - 2 b c\right ) \log{\left (c + d x^{2} \right )}}{2 c^{2} \left (a d - b c\right )^{2}} - \frac{b^{2} \log{\left (a + b x^{2} \right )}}{2 a \left (a d - b c\right )^{2}} + \frac{\log{\left (x^{2} \right )}}{2 a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(b*x**2+a)/(d*x**2+c)**2,x)
[Out]
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Mathematica [A] time = 0.16313, size = 98, normalized size = 0.98 \[ \frac{1}{2} \left (-\frac{b^2 \log \left (a+b x^2\right )}{a (b c-a d)^2}+\frac{d (2 b c-a d) \log \left (c+d x^2\right )}{c^2 (b c-a d)^2}-\frac{d}{c \left (c+d x^2\right ) (b c-a d)}+\frac{2 \log (x)}{a c^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(a + b*x^2)*(c + d*x^2)^2),x]
[Out]
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Maple [A] time = 0.023, size = 139, normalized size = 1.4 \[{\frac{\ln \left ( x \right ) }{a{c}^{2}}}+{\frac{a{d}^{2}}{2\,c \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{bd}{2\, \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{{d}^{2}\ln \left ( d{x}^{2}+c \right ) a}{2\,{c}^{2} \left ( ad-bc \right ) ^{2}}}+{\frac{d\ln \left ( d{x}^{2}+c \right ) b}{c \left ( ad-bc \right ) ^{2}}}-{\frac{{b}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,a \left ( ad-bc \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(b*x^2+a)/(d*x^2+c)^2,x)
[Out]
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Maxima [A] time = 1.35437, size = 186, normalized size = 1.86 \[ -\frac{b^{2} \log \left (b x^{2} + a\right )}{2 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )}} + \frac{{\left (2 \, b c d - a d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}} - \frac{d}{2 \,{\left (b c^{3} - a c^{2} d +{\left (b c^{2} d - a c d^{2}\right )} x^{2}\right )}} + \frac{\log \left (x^{2}\right )}{2 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)^2*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.00962, size = 296, normalized size = 2.96 \[ -\frac{a b c^{2} d - a^{2} c d^{2} +{\left (b^{2} c^{2} d x^{2} + b^{2} c^{3}\right )} \log \left (b x^{2} + a\right ) -{\left (2 \, a b c^{2} d - a^{2} c d^{2} +{\left (2 \, a b c d^{2} - a^{2} d^{3}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) - 2 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a b^{2} c^{5} - 2 \, a^{2} b c^{4} d + a^{3} c^{3} d^{2} +{\left (a b^{2} c^{4} d - 2 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)^2*x),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/(b*x**2+a)/(d*x**2+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.282932, size = 250, normalized size = 2.5 \[ -\frac{b^{3}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )}} + \frac{{\left (2 \, b c d^{2} - a d^{3}\right )}{\rm ln}\left ({\left | d x^{2} + c \right |}\right )}{2 \,{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )}} - \frac{2 \, b c d^{2} x^{2} - a d^{3} x^{2} + 3 \, b c^{2} d - 2 \, a c d^{2}}{2 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}{\left (d x^{2} + c\right )}} + \frac{{\rm ln}\left (x^{2}\right )}{2 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)^2*x),x, algorithm="giac")
[Out]