Optimal. Leaf size=85 \[ \frac{b \log (x) (3 A b-2 a B)}{a^4}-\frac{b (3 A b-2 a B) \log (a+b x)}{a^4}+\frac{2 A b-a B}{a^3 x}+\frac{b (A b-a B)}{a^3 (a+b x)}-\frac{A}{2 a^2 x^2} \]
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Rubi [A] time = 0.163731, 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 \[ \frac{b \log (x) (3 A b-2 a B)}{a^4}-\frac{b (3 A b-2 a B) \log (a+b x)}{a^4}+\frac{2 A b-a B}{a^3 x}+\frac{b (A b-a B)}{a^3 (a+b x)}-\frac{A}{2 a^2 x^2} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^3*(a + b*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 30.1712, size = 80, normalized size = 0.94 \[ - \frac{A}{2 a^{2} x^{2}} + \frac{b \left (A b - B a\right )}{a^{3} \left (a + b x\right )} + \frac{2 A b - B a}{a^{3} x} + \frac{b \left (3 A b - 2 B a\right ) \log{\left (x \right )}}{a^{4}} - \frac{b \left (3 A b - 2 B a\right ) \log{\left (a + b x \right )}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**3/(b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.131025, size = 85, normalized size = 1. \[ \frac{-\frac{a \left (a^2 (A+2 B x)+a b x (4 B x-3 A)-6 A b^2 x^2\right )}{x^2 (a+b x)}+2 b \log (x) (3 A b-2 a B)+2 b (2 a B-3 A b) \log (a+b x)}{2 a^4} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^3*(a + b*x)^2),x]
[Out]
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Maple [A] time = 0.016, size = 107, normalized size = 1.3 \[ -{\frac{A}{2\,{a}^{2}{x}^{2}}}+2\,{\frac{Ab}{{a}^{3}x}}-{\frac{B}{{a}^{2}x}}+3\,{\frac{A\ln \left ( x \right ){b}^{2}}{{a}^{4}}}-2\,{\frac{bB\ln \left ( x \right ) }{{a}^{3}}}-3\,{\frac{{b}^{2}\ln \left ( bx+a \right ) A}{{a}^{4}}}+2\,{\frac{b\ln \left ( bx+a \right ) B}{{a}^{3}}}+{\frac{A{b}^{2}}{{a}^{3} \left ( bx+a \right ) }}-{\frac{Bb}{{a}^{2} \left ( bx+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^3/(b*x+a)^2,x)
[Out]
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Maxima [A] time = 1.33854, size = 134, normalized size = 1.58 \[ -\frac{A a^{2} + 2 \,{\left (2 \, B a b - 3 \, A b^{2}\right )} x^{2} +{\left (2 \, B a^{2} - 3 \, A a b\right )} x}{2 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} + \frac{{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (b x + a\right )}{a^{4}} - \frac{{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (x\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^2*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212768, size = 203, normalized size = 2.39 \[ -\frac{A a^{3} + 2 \,{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2} +{\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} x - 2 \,{\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{3} +{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) + 2 \,{\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{3} +{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^2*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.58925, size = 184, normalized size = 2.16 \[ - \frac{A a^{2} + x^{2} \left (- 6 A b^{2} + 4 B a b\right ) + x \left (- 3 A a b + 2 B a^{2}\right )}{2 a^{4} x^{2} + 2 a^{3} b x^{3}} - \frac{b \left (- 3 A b + 2 B a\right ) \log{\left (x + \frac{- 3 A a b^{2} + 2 B a^{2} b - a b \left (- 3 A b + 2 B a\right )}{- 6 A b^{3} + 4 B a b^{2}} \right )}}{a^{4}} + \frac{b \left (- 3 A b + 2 B a\right ) \log{\left (x + \frac{- 3 A a b^{2} + 2 B a^{2} b + a b \left (- 3 A b + 2 B a\right )}{- 6 A b^{3} + 4 B a b^{2}} \right )}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/x**3/(b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.424267, size = 176, normalized size = 2.07 \[ -\frac{{\left (2 \, B a b^{2} - 3 \, A b^{3}\right )}{\rm ln}\left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{4} b} - \frac{\frac{B a b^{4}}{b x + a} - \frac{A b^{5}}{b x + a}}{a^{3} b^{3}} - \frac{2 \, B a b - 5 \, A b^{2} - \frac{2 \,{\left (B a^{2} b^{2} - 3 \, A a b^{3}\right )}}{{\left (b x + a\right )} b}}{2 \, a^{4}{\left (\frac{a}{b x + a} - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^2*x^3),x, algorithm="giac")
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