Optimal. Leaf size=42 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b c^2}+\frac{1}{2 a b c^2 (a-b x)} \]
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Rubi [A] time = 0.0672751, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b c^2}+\frac{1}{2 a b c^2 (a-b x)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)*(a*c - b*c*x)^2),x]
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Rubi in Sympy [A] time = 21.1932, size = 31, normalized size = 0.74 \[ \frac{1}{2 a b c^{2} \left (a - b x\right )} + \frac{\operatorname{atanh}{\left (\frac{b x}{a} \right )}}{2 a^{2} b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)/(-b*c*x+a*c)**2,x)
[Out]
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Mathematica [A] time = 0.0237949, size = 53, normalized size = 1.26 \[ \frac{(b x-a) \log (a-b x)+(a-b x) \log (a+b x)+2 a}{4 a^2 b c^2 (a-b x)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)*(a*c - b*c*x)^2),x]
[Out]
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Maple [A] time = 0.014, size = 58, normalized size = 1.4 \[{\frac{\ln \left ( bx+a \right ) }{4\,{c}^{2}{a}^{2}b}}-{\frac{\ln \left ( bx-a \right ) }{4\,{c}^{2}{a}^{2}b}}-{\frac{1}{2\,{c}^{2}ba \left ( bx-a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)/(-b*c*x+a*c)^2,x)
[Out]
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Maxima [A] time = 1.35918, size = 81, normalized size = 1.93 \[ -\frac{1}{2 \,{\left (a b^{2} c^{2} x - a^{2} b c^{2}\right )}} + \frac{\log \left (b x + a\right )}{4 \, a^{2} b c^{2}} - \frac{\log \left (b x - a\right )}{4 \, a^{2} b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*c*x - a*c)^2*(b*x + a)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209273, size = 81, normalized size = 1.93 \[ \frac{{\left (b x - a\right )} \log \left (b x + a\right ) -{\left (b x - a\right )} \log \left (b x - a\right ) - 2 \, a}{4 \,{\left (a^{2} b^{2} c^{2} x - a^{3} b c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*c*x - a*c)^2*(b*x + a)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.63929, size = 48, normalized size = 1.14 \[ - \frac{1}{- 2 a^{2} b c^{2} + 2 a b^{2} c^{2} x} + \frac{- \frac{\log{\left (- \frac{a}{b} + x \right )}}{4} + \frac{\log{\left (\frac{a}{b} + x \right )}}{4}}{a^{2} b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)/(-b*c*x+a*c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.20643, size = 72, normalized size = 1.71 \[ -\frac{1}{2 \,{\left (b c x - a c\right )} a b c} + \frac{{\rm ln}\left ({\left | -\frac{2 \, a c}{b c x - a c} - 1 \right |}\right )}{4 \, a^{2} b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*c*x - a*c)^2*(b*x + a)),x, algorithm="giac")
[Out]