Optimal. Leaf size=70 \[ \frac{1}{2 \left (c+d x^2\right ) (b c-a d)}+\frac{b \log \left (a+b x^2\right )}{2 (b c-a d)^2}-\frac{b \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]
[Out]
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Rubi [A] time = 0.121814, 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 \[ \frac{1}{2 \left (c+d x^2\right ) (b c-a d)}+\frac{b \log \left (a+b x^2\right )}{2 (b c-a d)^2}-\frac{b \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[x/((a + b*x^2)*(c + d*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 21.9376, size = 56, normalized size = 0.8 \[ \frac{b \log{\left (a + b x^{2} \right )}}{2 \left (a d - b c\right )^{2}} - \frac{b \log{\left (c + d x^{2} \right )}}{2 \left (a d - b c\right )^{2}} - \frac{1}{2 \left (c + d x^{2}\right ) \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b*x**2+a)/(d*x**2+c)**2,x)
[Out]
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Mathematica [A] time = 0.0477443, size = 66, normalized size = 0.94 \[ \frac{b \left (c+d x^2\right ) \log \left (a+b x^2\right )-a d-b \left (c+d x^2\right ) \log \left (c+d x^2\right )+b c}{2 \left (c+d x^2\right ) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[x/((a + b*x^2)*(c + d*x^2)^2),x]
[Out]
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Maple [A] time = 0.016, size = 90, normalized size = 1.3 \[ -{\frac{ad}{2\, \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{bc}{2\, \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{b\ln \left ( d{x}^{2}+c \right ) }{2\, \left ( ad-bc \right ) ^{2}}}+{\frac{b\ln \left ( b{x}^{2}+a \right ) }{2\, \left ( ad-bc \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b*x^2+a)/(d*x^2+c)^2,x)
[Out]
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Maxima [A] time = 1.35508, size = 134, normalized size = 1.91 \[ \frac{b \log \left (b x^{2} + a\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac{b \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac{1}{2 \,{\left (b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^2 + a)*(d*x^2 + c)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.235126, size = 139, normalized size = 1.99 \[ \frac{b c - a d +{\left (b d x^{2} + b c\right )} \log \left (b x^{2} + a\right ) -{\left (b d x^{2} + b c\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^2 + a)*(d*x^2 + c)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.39844, size = 248, normalized size = 3.54 \[ - \frac{b \log{\left (x^{2} + \frac{- \frac{a^{3} b d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} - \frac{3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d + \frac{b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{2 \left (a d - b c\right )^{2}} + \frac{b \log{\left (x^{2} + \frac{\frac{a^{3} b d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} + \frac{3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d - \frac{b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{2 \left (a d - b c\right )^{2}} - \frac{1}{2 a c d - 2 b c^{2} + x^{2} \left (2 a d^{2} - 2 b c d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x**2+a)/(d*x**2+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.270448, size = 115, normalized size = 1.64 \[ \frac{b d{\rm ln}\left ({\left | b - \frac{b c}{d x^{2} + c} + \frac{a d}{d x^{2} + c} \right |}\right )}{2 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}} + \frac{d}{2 \,{\left (b c d - a d^{2}\right )}{\left (d x^{2} + c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^2 + a)*(d*x^2 + c)^2),x, algorithm="giac")
[Out]