Optimal. Leaf size=85 \[ -\frac{a}{(c+d x) (b c-a d)^2}-\frac{c}{2 d (c+d x)^2 (b c-a d)}-\frac{a b \log (a+b x)}{(b c-a d)^3}+\frac{a b \log (c+d x)}{(b c-a d)^3} \]
[Out]
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Rubi [A] time = 0.131087, antiderivative size = 85, 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 \[ -\frac{a}{(c+d x) (b c-a d)^2}-\frac{c}{2 d (c+d x)^2 (b c-a d)}-\frac{a b \log (a+b x)}{(b c-a d)^3}+\frac{a b \log (c+d x)}{(b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Int[x/((a + b*x)*(c + d*x)^3),x]
[Out]
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Rubi in Sympy [A] time = 23.9009, size = 70, normalized size = 0.82 \[ \frac{a b \log{\left (a + b x \right )}}{\left (a d - b c\right )^{3}} - \frac{a b \log{\left (c + d x \right )}}{\left (a d - b c\right )^{3}} - \frac{a}{\left (c + d x\right ) \left (a d - b c\right )^{2}} + \frac{c}{2 d \left (c + d x\right )^{2} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b*x+a)/(d*x+c)**3,x)
[Out]
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Mathematica [A] time = 0.0785235, size = 85, normalized size = 1. \[ -\frac{a}{(c+d x) (b c-a d)^2}+\frac{c}{2 d (c+d x)^2 (a d-b c)}-\frac{a b \log (a+b x)}{(b c-a d)^3}+\frac{a b \log (c+d x)}{(b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Integrate[x/((a + b*x)*(c + d*x)^3),x]
[Out]
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Maple [A] time = 0.013, size = 84, normalized size = 1. \[ -{\frac{a}{ \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}+{\frac{c}{ \left ( 2\,ad-2\,bc \right ) d \left ( dx+c \right ) ^{2}}}-{\frac{ab\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{3}}}+{\frac{ab\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b*x+a)/(d*x+c)^3,x)
[Out]
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Maxima [A] time = 1.35348, size = 281, normalized size = 3.31 \[ -\frac{a b \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac{a b \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{2 \, a d^{2} x + b c^{2} + a c d}{2 \,{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} +{\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x + a)*(d*x + c)^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209268, size = 329, normalized size = 3.87 \[ -\frac{b^{2} c^{3} - a^{2} c d^{2} + 2 \,{\left (a b c d^{2} - a^{2} d^{3}\right )} x + 2 \,{\left (a b d^{3} x^{2} + 2 \, a b c d^{2} x + a b c^{2} d\right )} \log \left (b x + a\right ) - 2 \,{\left (a b d^{3} x^{2} + 2 \, a b c d^{2} x + a b c^{2} d\right )} \log \left (d x + c\right )}{2 \,{\left (b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{3} d^{3} - a^{3} c^{2} d^{4} +{\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )} x^{2} + 2 \,{\left (b^{3} c^{4} d^{2} - 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c^{2} d^{4} - a^{3} c d^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x + a)*(d*x + c)^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.3869, size = 400, normalized size = 4.71 \[ - \frac{a b \log{\left (x + \frac{- \frac{a^{5} b d^{4}}{\left (a d - b c\right )^{3}} + \frac{4 a^{4} b^{2} c d^{3}}{\left (a d - b c\right )^{3}} - \frac{6 a^{3} b^{3} c^{2} d^{2}}{\left (a d - b c\right )^{3}} + \frac{4 a^{2} b^{4} c^{3} d}{\left (a d - b c\right )^{3}} + a^{2} b d - \frac{a b^{5} c^{4}}{\left (a d - b c\right )^{3}} + a b^{2} c}{2 a b^{2} d} \right )}}{\left (a d - b c\right )^{3}} + \frac{a b \log{\left (x + \frac{\frac{a^{5} b d^{4}}{\left (a d - b c\right )^{3}} - \frac{4 a^{4} b^{2} c d^{3}}{\left (a d - b c\right )^{3}} + \frac{6 a^{3} b^{3} c^{2} d^{2}}{\left (a d - b c\right )^{3}} - \frac{4 a^{2} b^{4} c^{3} d}{\left (a d - b c\right )^{3}} + a^{2} b d + \frac{a b^{5} c^{4}}{\left (a d - b c\right )^{3}} + a b^{2} c}{2 a b^{2} d} \right )}}{\left (a d - b c\right )^{3}} - \frac{a c d + 2 a d^{2} x + b c^{2}}{2 a^{2} c^{2} d^{3} - 4 a b c^{3} d^{2} + 2 b^{2} c^{4} d + x^{2} \left (2 a^{2} d^{5} - 4 a b c d^{4} + 2 b^{2} c^{2} d^{3}\right ) + x \left (4 a^{2} c d^{4} - 8 a b c^{2} d^{3} + 4 b^{2} c^{3} d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x+a)/(d*x+c)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.265168, size = 223, normalized size = 2.62 \[ -\frac{a b^{2}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} + \frac{a b d{\rm ln}\left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}} - \frac{b^{2} c^{3} - a^{2} c d^{2} + 2 \,{\left (a b c d^{2} - a^{2} d^{3}\right )} x}{2 \,{\left (b c - a d\right )}^{3}{\left (d x + c\right )}^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x + a)*(d*x + c)^3),x, algorithm="giac")
[Out]