Optimal. Leaf size=88 \[ \frac{a}{(a+b x) (b c-a d)^2}+\frac{c}{(c+d x) (b c-a d)^2}+\frac{(a d+b c) \log (a+b x)}{(b c-a d)^3}-\frac{(a d+b c) \log (c+d x)}{(b c-a d)^3} \]
[Out]
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Rubi [A] time = 0.146465, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{a}{(a+b x) (b c-a d)^2}+\frac{c}{(c+d x) (b c-a d)^2}+\frac{(a d+b c) \log (a+b x)}{(b c-a d)^3}-\frac{(a d+b c) \log (c+d x)}{(b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Int[x/((a + b*x)^2*(c + d*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 24.5747, size = 73, normalized size = 0.83 \[ \frac{a}{\left (a + b x\right ) \left (a d - b c\right )^{2}} + \frac{c}{\left (c + d x\right ) \left (a d - b c\right )^{2}} - \frac{\left (a d + b c\right ) \log{\left (a + b x \right )}}{\left (a d - b c\right )^{3}} + \frac{\left (a d + b c\right ) \log{\left (c + d x \right )}}{\left (a d - b c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b*x+a)**2/(d*x+c)**2,x)
[Out]
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Mathematica [A] time = 0.0964646, size = 75, normalized size = 0.85 \[ \frac{\frac{a (b c-a d)}{a+b x}+\frac{c (b c-a d)}{c+d x}+(a d+b c) \log (a+b x)-(a d+b c) \log (c+d x)}{(b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Integrate[x/((a + b*x)^2*(c + d*x)^2),x]
[Out]
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Maple [A] time = 0.017, size = 118, normalized size = 1.3 \[{\frac{\ln \left ( dx+c \right ) ad}{ \left ( ad-bc \right ) ^{3}}}+{\frac{\ln \left ( dx+c \right ) bc}{ \left ( ad-bc \right ) ^{3}}}+{\frac{c}{ \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}+{\frac{a}{ \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }}-{\frac{\ln \left ( bx+a \right ) ad}{ \left ( ad-bc \right ) ^{3}}}-{\frac{\ln \left ( bx+a \right ) bc}{ \left ( ad-bc \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b*x+a)^2/(d*x+c)^2,x)
[Out]
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Maxima [A] time = 1.35921, size = 294, normalized size = 3.34 \[ \frac{{\left (b c + a d\right )} \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{{\left (b c + a d\right )} \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac{2 \, a c +{\left (b c + a d\right )} x}{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x + a)^2*(d*x + c)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210712, size = 382, normalized size = 4.34 \[ \frac{2 \, a b c^{2} - 2 \, a^{2} c d +{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x +{\left (a b c^{2} + a^{2} c d +{\left (b^{2} c d + a b d^{2}\right )} x^{2} +{\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x\right )} \log \left (b x + a\right ) -{\left (a b c^{2} + a^{2} c d +{\left (b^{2} c d + a b d^{2}\right )} x^{2} +{\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x\right )} \log \left (d x + c\right )}{a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x + a)^2*(d*x + c)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.67756, size = 483, normalized size = 5.49 \[ \frac{2 a c + x \left (a d + b c\right )}{a^{3} c d^{2} - 2 a^{2} b c^{2} d + a b^{2} c^{3} + x^{2} \left (a^{2} b d^{3} - 2 a b^{2} c d^{2} + b^{3} c^{2} d\right ) + x \left (a^{3} d^{3} - a^{2} b c d^{2} - a b^{2} c^{2} d + b^{3} c^{3}\right )} + \frac{\left (a d + b c\right ) \log{\left (x + \frac{- \frac{a^{4} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + \frac{4 a^{3} b c d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} - \frac{6 a^{2} b^{2} c^{2} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + a^{2} d^{2} + \frac{4 a b^{3} c^{3} d \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + 2 a b c d - \frac{b^{4} c^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + b^{2} c^{2}}{2 a b d^{2} + 2 b^{2} c d} \right )}}{\left (a d - b c\right )^{3}} - \frac{\left (a d + b c\right ) \log{\left (x + \frac{\frac{a^{4} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} - \frac{4 a^{3} b c d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + \frac{6 a^{2} b^{2} c^{2} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + a^{2} d^{2} - \frac{4 a b^{3} c^{3} d \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + 2 a b c d + \frac{b^{4} c^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + b^{2} c^{2}}{2 a b d^{2} + 2 b^{2} c d} \right )}}{\left (a d - b c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x+a)**2/(d*x+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.323034, size = 225, normalized size = 2.56 \[ \frac{\frac{a b^{3}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}{\left (b x + a\right )}} - \frac{{\left (b^{3} c + a b^{2} d\right )}{\rm ln}\left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac{b^{2} c d}{{\left (b c - a d\right )}^{3}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x + a)^2*(d*x + c)^2),x, algorithm="giac")
[Out]