Optimal. Leaf size=62 \[ \frac{A b-a B}{a^2 x}+\frac{b \log (x) (A b-a B)}{a^3}-\frac{b (A b-a B) \log (a+b x)}{a^3}-\frac{A}{2 a x^2} \]
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Rubi [A] time = 0.0385763, antiderivative size = 62, 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} \[ \frac{A b-a B}{a^2 x}+\frac{b \log (x) (A b-a B)}{a^3}-\frac{b (A b-a B) \log (a+b x)}{a^3}-\frac{A}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{x^3 (a+b x)} \, dx &=\int \left (\frac{A}{a x^3}+\frac{-A b+a B}{a^2 x^2}-\frac{b (-A b+a B)}{a^3 x}+\frac{b^2 (-A b+a B)}{a^3 (a+b x)}\right ) \, dx\\ &=-\frac{A}{2 a x^2}+\frac{A b-a B}{a^2 x}+\frac{b (A b-a B) \log (x)}{a^3}-\frac{b (A b-a B) \log (a+b x)}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0326241, size = 58, normalized size = 0.94 \[ \frac{-\frac{a (a A+2 a B x-2 A b x)}{x^2}+2 b \log (x) (A b-a B)+2 b (a B-A b) \log (a+b x)}{2 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 75, normalized size = 1.2 \begin{align*} -{\frac{A}{2\,a{x}^{2}}}+{\frac{Ab}{{a}^{2}x}}-{\frac{B}{ax}}+{\frac{A\ln \left ( x \right ){b}^{2}}{{a}^{3}}}-{\frac{bB\ln \left ( x \right ) }{{a}^{2}}}-{\frac{{b}^{2}\ln \left ( bx+a \right ) A}{{a}^{3}}}+{\frac{b\ln \left ( bx+a \right ) B}{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01145, size = 85, normalized size = 1.37 \begin{align*} \frac{{\left (B a b - A b^{2}\right )} \log \left (b x + a\right )}{a^{3}} - \frac{{\left (B a b - A b^{2}\right )} \log \left (x\right )}{a^{3}} - \frac{A a + 2 \,{\left (B a - A b\right )} x}{2 \, a^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95602, size = 153, normalized size = 2.47 \begin{align*} \frac{2 \,{\left (B a b - A b^{2}\right )} x^{2} \log \left (b x + a\right ) - 2 \,{\left (B a b - A b^{2}\right )} x^{2} \log \left (x\right ) - A a^{2} - 2 \,{\left (B a^{2} - A a b\right )} x}{2 \, a^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.730744, size = 131, normalized size = 2.11 \begin{align*} - \frac{A a + x \left (- 2 A b + 2 B a\right )}{2 a^{2} x^{2}} - \frac{b \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{2} + B a^{2} b - a b \left (- A b + B a\right )}{- 2 A b^{3} + 2 B a b^{2}} \right )}}{a^{3}} + \frac{b \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{2} + B a^{2} b + a b \left (- A b + B a\right )}{- 2 A b^{3} + 2 B a b^{2}} \right )}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16324, size = 101, normalized size = 1.63 \begin{align*} -\frac{{\left (B a b - A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac{{\left (B a b^{2} - A b^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{3} b} - \frac{A a^{2} + 2 \,{\left (B a^{2} - A a b\right )} x}{2 \, a^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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