Optimal. Leaf size=82 \[ \frac{d^2 \log (a+b x)}{(b c-a d)^3}-\frac{d^2 \log (c+d x)}{(b c-a d)^3}+\frac{d}{(a+b x) (b c-a d)^2}-\frac{1}{2 (a+b x)^2 (b c-a d)} \]
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Rubi [A] time = 0.135, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{d^2 \log (a+b x)}{(b c-a d)^3}-\frac{d^2 \log (c+d x)}{(b c-a d)^3}+\frac{d}{(a+b x) (b c-a d)^2}-\frac{1}{2 (a+b x)^2 (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)^2*(a*c + (b*c + a*d)*x + b*d*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 31.6386, size = 68, normalized size = 0.83 \[ - \frac{d^{2} \log{\left (a + b x \right )}}{\left (a d - b c\right )^{3}} + \frac{d^{2} \log{\left (c + d x \right )}}{\left (a d - b c\right )^{3}} + \frac{d}{\left (a + b x\right ) \left (a d - b c\right )^{2}} + \frac{1}{2 \left (a + b x\right )^{2} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**2/(a*c+(a*d+b*c)*x+b*d*x**2),x)
[Out]
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Mathematica [A] time = 0.116028, size = 67, normalized size = 0.82 \[ \frac{\frac{(b c-a d) (3 a d-b c+2 b d x)}{(a+b x)^2}+2 d^2 \log (a+b x)-2 d^2 \log (c+d x)}{2 (b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^2*(a*c + (b*c + a*d)*x + b*d*x^2)),x]
[Out]
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Maple [A] time = 0.013, size = 81, normalized size = 1. \[{\frac{1}{ \left ( 2\,ad-2\,bc \right ) \left ( bx+a \right ) ^{2}}}+{\frac{d}{ \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }}-{\frac{{d}^{2}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{3}}}+{\frac{{d}^{2}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^2/(a*c+(a*d+b*c)*x+x^2*b*d),x)
[Out]
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Maxima [A] time = 0.766232, size = 273, normalized size = 3.33 \[ \frac{d^{2} \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{d^{2} \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac{2 \, b d x - b c + 3 \, a d}{2 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210188, size = 327, normalized size = 3.99 \[ -\frac{b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2} - 2 \,{\left (b^{2} c d - a b d^{2}\right )} x - 2 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right ) + 2 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (d x + c\right )}{2 \,{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3} +{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{2} + 2 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.61831, size = 381, normalized size = 4.65 \[ \frac{d^{2} \log{\left (x + \frac{- \frac{a^{4} d^{6}}{\left (a d - b c\right )^{3}} + \frac{4 a^{3} b c d^{5}}{\left (a d - b c\right )^{3}} - \frac{6 a^{2} b^{2} c^{2} d^{4}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{3} c^{3} d^{3}}{\left (a d - b c\right )^{3}} + a d^{3} - \frac{b^{4} c^{4} d^{2}}{\left (a d - b c\right )^{3}} + b c d^{2}}{2 b d^{3}} \right )}}{\left (a d - b c\right )^{3}} - \frac{d^{2} \log{\left (x + \frac{\frac{a^{4} d^{6}}{\left (a d - b c\right )^{3}} - \frac{4 a^{3} b c d^{5}}{\left (a d - b c\right )^{3}} + \frac{6 a^{2} b^{2} c^{2} d^{4}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{3} c^{3} d^{3}}{\left (a d - b c\right )^{3}} + a d^{3} + \frac{b^{4} c^{4} d^{2}}{\left (a d - b c\right )^{3}} + b c d^{2}}{2 b d^{3}} \right )}}{\left (a d - b c\right )^{3}} + \frac{3 a d - b c + 2 b d x}{2 a^{4} d^{2} - 4 a^{3} b c d + 2 a^{2} b^{2} c^{2} + x^{2} \left (2 a^{2} b^{2} d^{2} - 4 a b^{3} c d + 2 b^{4} c^{2}\right ) + x \left (4 a^{3} b d^{2} - 8 a^{2} b^{2} c d + 4 a b^{3} c^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**2/(a*c+(a*d+b*c)*x+b*d*x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.215471, size = 196, normalized size = 2.39 \[ -\frac{b d^{2}{\rm ln}\left ({\left | -\frac{b c}{b x + a} + \frac{a d}{b x + a} - d \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac{\frac{b^{3} c}{{\left (b x + a\right )}^{2}} - \frac{2 \, b^{2} d}{b x + a} - \frac{a b^{2} d}{{\left (b x + a\right )}^{2}}}{2 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)^2),x, algorithm="giac")
[Out]