Optimal. Leaf size=55 \[ -\frac{A b-2 a B}{b^3 (a+b x)}+\frac{a (A b-a B)}{2 b^3 (a+b x)^2}+\frac{B \log (a+b x)}{b^3} \]
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Rubi [A] time = 0.0377139, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {77} \[ -\frac{A b-2 a B}{b^3 (a+b x)}+\frac{a (A b-a B)}{2 b^3 (a+b x)^2}+\frac{B \log (a+b x)}{b^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{x (A+B x)}{(a+b x)^3} \, dx &=\int \left (\frac{a (-A b+a B)}{b^2 (a+b x)^3}+\frac{A b-2 a B}{b^2 (a+b x)^2}+\frac{B}{b^2 (a+b x)}\right ) \, dx\\ &=\frac{a (A b-a B)}{2 b^3 (a+b x)^2}-\frac{A b-2 a B}{b^3 (a+b x)}+\frac{B \log (a+b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0163888, size = 54, normalized size = 0.98 \[ \frac{3 a^2 B-a b (A-4 B x)+2 B (a+b x)^2 \log (a+b x)-2 A b^2 x}{2 b^3 (a+b x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 70, normalized size = 1.3 \begin{align*} -{\frac{A}{{b}^{2} \left ( bx+a \right ) }}+2\,{\frac{Ba}{{b}^{3} \left ( bx+a \right ) }}+{\frac{Aa}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{B{a}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{B\ln \left ( bx+a \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04585, size = 88, normalized size = 1.6 \begin{align*} \frac{3 \, B a^{2} - A a b + 2 \,{\left (2 \, B a b - A b^{2}\right )} x}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac{B \log \left (b x + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94273, size = 173, normalized size = 3.15 \begin{align*} \frac{3 \, B a^{2} - A a b + 2 \,{\left (2 \, B a b - A b^{2}\right )} x + 2 \,{\left (B b^{2} x^{2} + 2 \, B a b x + B a^{2}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.628659, size = 63, normalized size = 1.15 \begin{align*} \frac{B \log{\left (a + b x \right )}}{b^{3}} + \frac{- A a b + 3 B a^{2} + x \left (- 2 A b^{2} + 4 B a b\right )}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18719, size = 73, normalized size = 1.33 \begin{align*} \frac{B \log \left ({\left | b x + a \right |}\right )}{b^{3}} + \frac{2 \,{\left (2 \, B a - A b\right )} x + \frac{3 \, B a^{2} - A a b}{b}}{2 \,{\left (b x + a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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