Optimal. Leaf size=61 \[ -\frac{c}{d (c+d x) (b c-a d)}-\frac{a \log (a+b x)}{(b c-a d)^2}+\frac{a \log (c+d x)}{(b c-a d)^2} \]
[Out]
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Rubi [A] time = 0.0872763, antiderivative size = 61, 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 \[ -\frac{c}{d (c+d x) (b c-a d)}-\frac{a \log (a+b x)}{(b c-a d)^2}+\frac{a \log (c+d x)}{(b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[x/((a + b*x)*(c + d*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 25.2652, size = 48, normalized size = 0.79 \[ - \frac{a \log{\left (a + b x \right )}}{\left (a d - b c\right )^{2}} + \frac{a \log{\left (c + d x \right )}}{\left (a d - b c\right )^{2}} + \frac{c}{d \left (c + d x\right ) \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b*x+a)/(d*x+c)**2,x)
[Out]
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Mathematica [A] time = 0.0515013, size = 60, normalized size = 0.98 \[ \frac{c}{d (c+d x) (a d-b c)}-\frac{a \log (a+b x)}{(b c-a d)^2}+\frac{a \log (c+d x)}{(b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[x/((a + b*x)*(c + d*x)^2),x]
[Out]
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Maple [A] time = 0.013, size = 61, normalized size = 1. \[{\frac{a\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{2}}}+{\frac{c}{d \left ( ad-bc \right ) \left ( dx+c \right ) }}-{\frac{a\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b*x+a)/(d*x+c)^2,x)
[Out]
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Maxima [A] time = 1.34932, size = 132, normalized size = 2.16 \[ -\frac{a \log \left (b x + a\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} + \frac{a \log \left (d x + c\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} - \frac{c}{b c^{2} d - a c d^{2} +{\left (b c d^{2} - a d^{3}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x + a)*(d*x + c)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224019, size = 144, normalized size = 2.36 \[ -\frac{b c^{2} - a c d +{\left (a d^{2} x + a c d\right )} \log \left (b x + a\right ) -{\left (a d^{2} x + a c d\right )} \log \left (d x + c\right )}{b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x + a)*(d*x + c)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.06708, size = 238, normalized size = 3.9 \[ \frac{a \log{\left (x + \frac{- \frac{a^{4} d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a^{3} b c d^{2}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b^{2} c^{2} d}{\left (a d - b c\right )^{2}} + a^{2} d + \frac{a b^{3} c^{3}}{\left (a d - b c\right )^{2}} + a b c}{2 a b d} \right )}}{\left (a d - b c\right )^{2}} - \frac{a \log{\left (x + \frac{\frac{a^{4} d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a^{3} b c d^{2}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b^{2} c^{2} d}{\left (a d - b c\right )^{2}} + a^{2} d - \frac{a b^{3} c^{3}}{\left (a d - b c\right )^{2}} + a b c}{2 a b d} \right )}}{\left (a d - b c\right )^{2}} + \frac{c}{a c d^{2} - b c^{2} d + x \left (a d^{3} - b c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x+a)/(d*x+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.30503, size = 115, normalized size = 1.89 \[ -\frac{\frac{a d^{2}{\rm ln}\left ({\left | b - \frac{b c}{d x + c} + \frac{a d}{d x + c} \right |}\right )}{b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}} + \frac{c d}{{\left (b c d - a d^{2}\right )}{\left (d x + c\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x + a)*(d*x + c)^2),x, algorithm="giac")
[Out]