Optimal. Leaf size=65 \[ a^2 \log (x) (a B+3 A b)-\frac{a^3 A}{x}+\frac{1}{2} b^2 x^2 (3 a B+A b)+3 a b x (a B+A b)+\frac{1}{3} b^3 B x^3 \]
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Rubi [A] time = 0.0385159, antiderivative size = 65, 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 = {76} \[ a^2 \log (x) (a B+3 A b)-\frac{a^3 A}{x}+\frac{1}{2} b^2 x^2 (3 a B+A b)+3 a b x (a B+A b)+\frac{1}{3} b^3 B x^3 \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int \frac{(a+b x)^3 (A+B x)}{x^2} \, dx &=\int \left (3 a b (A b+a B)+\frac{a^3 A}{x^2}+\frac{a^2 (3 A b+a B)}{x}+b^2 (A b+3 a B) x+b^3 B x^2\right ) \, dx\\ &=-\frac{a^3 A}{x}+3 a b (A b+a B) x+\frac{1}{2} b^2 (A b+3 a B) x^2+\frac{1}{3} b^3 B x^3+a^2 (3 A b+a B) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0256477, size = 67, normalized size = 1.03 \[ \log (x) \left (3 a^2 A b+a^3 B\right )-\frac{a^3 A}{x}+\frac{1}{2} b^2 x^2 (3 a B+A b)+3 a b x (a B+A b)+\frac{1}{3} b^3 B x^3 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 71, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}B{x}^{3}}{3}}+{\frac{A{x}^{2}{b}^{3}}{2}}+{\frac{3\,B{x}^{2}a{b}^{2}}{2}}+3\,a{b}^{2}Ax+3\,{a}^{2}bBx+3\,A\ln \left ( x \right ){a}^{2}b+B\ln \left ( x \right ){a}^{3}-{\frac{A{a}^{3}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02371, size = 93, normalized size = 1.43 \begin{align*} \frac{1}{3} \, B b^{3} x^{3} - \frac{A a^{3}}{x} + \frac{1}{2} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} x +{\left (B a^{3} + 3 \, A a^{2} b\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66924, size = 163, normalized size = 2.51 \begin{align*} \frac{2 \, B b^{3} x^{4} - 6 \, A a^{3} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 18 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 6 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x \log \left (x\right )}{6 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.406116, size = 70, normalized size = 1.08 \begin{align*} - \frac{A a^{3}}{x} + \frac{B b^{3} x^{3}}{3} + a^{2} \left (3 A b + B a\right ) \log{\left (x \right )} + x^{2} \left (\frac{A b^{3}}{2} + \frac{3 B a b^{2}}{2}\right ) + x \left (3 A a b^{2} + 3 B a^{2} b\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20815, size = 96, normalized size = 1.48 \begin{align*} \frac{1}{3} \, B b^{3} x^{3} + \frac{3}{2} \, B a b^{2} x^{2} + \frac{1}{2} \, A b^{3} x^{2} + 3 \, B a^{2} b x + 3 \, A a b^{2} x - \frac{A a^{3}}{x} +{\left (B a^{3} + 3 \, A a^{2} b\right )} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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