Optimal. Leaf size=110 \[ \frac{3 A b-a B}{a^4 x}+\frac{b (3 A b-2 a B)}{a^4 (a+b x)}+\frac{b (A b-a B)}{2 a^3 (a+b x)^2}+\frac{3 b \log (x) (2 A b-a B)}{a^5}-\frac{3 b (2 A b-a B) \log (a+b x)}{a^5}-\frac{A}{2 a^3 x^2} \]
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Rubi [A] time = 0.0889311, antiderivative size = 110, 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, Rules used = {77} \[ \frac{3 A b-a B}{a^4 x}+\frac{b (3 A b-2 a B)}{a^4 (a+b x)}+\frac{b (A b-a B)}{2 a^3 (a+b x)^2}+\frac{3 b \log (x) (2 A b-a B)}{a^5}-\frac{3 b (2 A b-a B) \log (a+b x)}{a^5}-\frac{A}{2 a^3 x^2} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{x^3 (a+b x)^3} \, dx &=\int \left (\frac{A}{a^3 x^3}+\frac{-3 A b+a B}{a^4 x^2}-\frac{3 b (-2 A b+a B)}{a^5 x}+\frac{b^2 (-A b+a B)}{a^3 (a+b x)^3}+\frac{b^2 (-3 A b+2 a B)}{a^4 (a+b x)^2}+\frac{3 b^2 (-2 A b+a B)}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac{A}{2 a^3 x^2}+\frac{3 A b-a B}{a^4 x}+\frac{b (A b-a B)}{2 a^3 (a+b x)^2}+\frac{b (3 A b-2 a B)}{a^4 (a+b x)}+\frac{3 b (2 A b-a B) \log (x)}{a^5}-\frac{3 b (2 A b-a B) \log (a+b x)}{a^5}\\ \end{align*}
Mathematica [A] time = 0.0795557, size = 102, normalized size = 0.93 \[ \frac{-\frac{a \left (a^2 b x (9 B x-4 A)+a^3 (A+2 B x)+6 a b^2 x^2 (B x-3 A)-12 A b^3 x^3\right )}{x^2 (a+b x)^2}+6 b \log (x) (2 A b-a B)+6 b (a B-2 A b) \log (a+b x)}{2 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 138, normalized size = 1.3 \begin{align*} -{\frac{A}{2\,{a}^{3}{x}^{2}}}+3\,{\frac{Ab}{{a}^{4}x}}-{\frac{B}{{a}^{3}x}}+6\,{\frac{A\ln \left ( x \right ){b}^{2}}{{a}^{5}}}-3\,{\frac{bB\ln \left ( x \right ) }{{a}^{4}}}-6\,{\frac{{b}^{2}\ln \left ( bx+a \right ) A}{{a}^{5}}}+3\,{\frac{b\ln \left ( bx+a \right ) B}{{a}^{4}}}+3\,{\frac{A{b}^{2}}{{a}^{4} \left ( bx+a \right ) }}-2\,{\frac{Bb}{{a}^{3} \left ( bx+a \right ) }}+{\frac{A{b}^{2}}{2\,{a}^{3} \left ( bx+a \right ) ^{2}}}-{\frac{Bb}{2\,{a}^{2} \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05811, size = 177, normalized size = 1.61 \begin{align*} -\frac{A a^{3} + 6 \,{\left (B a b^{2} - 2 \, A b^{3}\right )} x^{3} + 9 \,{\left (B a^{2} b - 2 \, A a b^{2}\right )} x^{2} + 2 \,{\left (B a^{3} - 2 \, A a^{2} b\right )} x}{2 \,{\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}} + \frac{3 \,{\left (B a b - 2 \, A b^{2}\right )} \log \left (b x + a\right )}{a^{5}} - \frac{3 \,{\left (B a b - 2 \, A b^{2}\right )} \log \left (x\right )}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.99737, size = 467, normalized size = 4.25 \begin{align*} -\frac{A a^{4} + 6 \,{\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + 9 \,{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 2 \,{\left (B a^{4} - 2 \, A a^{3} b\right )} x - 6 \,{\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{4} + 2 \,{\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} +{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) + 6 \,{\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{4} + 2 \,{\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} +{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.19129, size = 219, normalized size = 1.99 \begin{align*} - \frac{A a^{3} + x^{3} \left (- 12 A b^{3} + 6 B a b^{2}\right ) + x^{2} \left (- 18 A a b^{2} + 9 B a^{2} b\right ) + x \left (- 4 A a^{2} b + 2 B a^{3}\right )}{2 a^{6} x^{2} + 4 a^{5} b x^{3} + 2 a^{4} b^{2} x^{4}} - \frac{3 b \left (- 2 A b + B a\right ) \log{\left (x + \frac{- 6 A a b^{2} + 3 B a^{2} b - 3 a b \left (- 2 A b + B a\right )}{- 12 A b^{3} + 6 B a b^{2}} \right )}}{a^{5}} + \frac{3 b \left (- 2 A b + B a\right ) \log{\left (x + \frac{- 6 A a b^{2} + 3 B a^{2} b + 3 a b \left (- 2 A b + B a\right )}{- 12 A b^{3} + 6 B a b^{2}} \right )}}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20415, size = 167, normalized size = 1.52 \begin{align*} -\frac{3 \,{\left (B a b - 2 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac{3 \,{\left (B a b^{2} - 2 \, A b^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac{6 \, B a b^{2} x^{3} - 12 \, A b^{3} x^{3} + 9 \, B a^{2} b x^{2} - 18 \, A a b^{2} x^{2} + 2 \, B a^{3} x - 4 \, A a^{2} b x + A a^{3}}{2 \,{\left (b x^{2} + a x\right )}^{2} a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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