Optimal. Leaf size=81 \[ -\frac{b}{(a+b x) (b c-a d)^2}-\frac{d}{(c+d x) (b c-a d)^2}-\frac{2 b d \log (a+b x)}{(b c-a d)^3}+\frac{2 b d \log (c+d x)}{(b c-a d)^3} \]
[Out]
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Rubi [A] time = 0.106053, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{b}{(a+b x) (b c-a d)^2}-\frac{d}{(c+d x) (b c-a d)^2}-\frac{2 b d \log (a+b x)}{(b c-a d)^3}+\frac{2 b d \log (c+d x)}{(b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)^2*(c + d*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 20.4022, size = 70, normalized size = 0.86 \[ \frac{2 b d \log{\left (a + b x \right )}}{\left (a d - b c\right )^{3}} - \frac{2 b d \log{\left (c + d x \right )}}{\left (a d - b c\right )^{3}} - \frac{b}{\left (a + b x\right ) \left (a d - b c\right )^{2}} - \frac{d}{\left (c + d x\right ) \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**2/(d*x+c)**2,x)
[Out]
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Mathematica [A] time = 0.103596, size = 66, normalized size = 0.81 \[ \frac{\frac{b (a d-b c)}{a+b x}+\frac{d (a d-b c)}{c+d x}-2 b d \log (a+b x)+2 b d \log (c+d x)}{(b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^2*(c + d*x)^2),x]
[Out]
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Maple [A] time = 0.017, size = 82, normalized size = 1. \[ -{\frac{d}{ \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-2\,{\frac{db\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{3}}}-{\frac{b}{ \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }}+2\,{\frac{db\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^2/(d*x+c)^2,x)
[Out]
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Maxima [A] time = 1.36732, size = 281, normalized size = 3.47 \[ -\frac{2 \, b d \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac{2 \, b d \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{2 \, b d x + b c + a d}{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^2*(d*x + c)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.211902, size = 325, normalized size = 4.01 \[ -\frac{b^{2} c^{2} - a^{2} d^{2} + 2 \,{\left (b^{2} c d - a b d^{2}\right )} x + 2 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right ) - 2 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (d x + c\right )}{a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^2*(d*x + c)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.27631, size = 405, normalized size = 5. \[ - \frac{2 b d \log{\left (x + \frac{- \frac{2 a^{4} b d^{5}}{\left (a d - b c\right )^{3}} + \frac{8 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} - \frac{12 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} + \frac{8 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + 2 a b d^{2} - \frac{2 b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + 2 b^{2} c d}{4 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} + \frac{2 b d \log{\left (x + \frac{\frac{2 a^{4} b d^{5}}{\left (a d - b c\right )^{3}} - \frac{8 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} + \frac{12 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} - \frac{8 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + 2 a b d^{2} + \frac{2 b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + 2 b^{2} c d}{4 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} - \frac{a d + b c + 2 b d x}{a^{3} c d^{2} - 2 a^{2} b c^{2} d + a b^{2} c^{3} + x^{2} \left (a^{2} b d^{3} - 2 a b^{2} c d^{2} + b^{3} c^{2} d\right ) + x \left (a^{3} d^{3} - a^{2} b c d^{2} - a b^{2} c^{2} d + b^{3} c^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**2/(d*x+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.228461, size = 207, normalized size = 2.56 \[ \frac{2 \, b^{2} d{\rm ln}\left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac{b^{3}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}{\left (b x + a\right )}} + \frac{b d^{2}}{{\left (b c - a d\right )}^{3}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^2*(d*x + c)^2),x, algorithm="giac")
[Out]