Optimal. Leaf size=65 \[ -\frac{a^2 (a B+3 A b)}{x}-\frac{a^3 A}{2 x^2}+b^2 x (3 a B+A b)+3 a b \log (x) (a B+A b)+\frac{1}{2} b^3 B x^2 \]
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Rubi [A] time = 0.040332, 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} \[ -\frac{a^2 (a B+3 A b)}{x}-\frac{a^3 A}{2 x^2}+b^2 x (3 a B+A b)+3 a b \log (x) (a B+A b)+\frac{1}{2} b^3 B x^2 \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int \frac{(a+b x)^3 (A+B x)}{x^3} \, dx &=\int \left (b^2 (A b+3 a B)+\frac{a^3 A}{x^3}+\frac{a^2 (3 A b+a B)}{x^2}+\frac{3 a b (A b+a B)}{x}+b^3 B x\right ) \, dx\\ &=-\frac{a^3 A}{2 x^2}-\frac{a^2 (3 A b+a B)}{x}+b^2 (A b+3 a B) x+\frac{1}{2} b^3 B x^2+3 a b (A b+a B) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0256161, size = 62, normalized size = 0.95 \[ \frac{1}{2} \left (-\frac{6 a^2 A b}{x}-\frac{a^3 (A+2 B x)}{x^2}+6 a b \log (x) (a B+A b)+6 a b^2 B x+b^3 x (2 A+B x)\right ) \]
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}^{2}}{2}}+{b}^{3}Ax+3\,a{b}^{2}Bx+3\,A\ln \left ( x \right ) a{b}^{2}+3\,B\ln \left ( x \right ){a}^{2}b-{\frac{A{a}^{3}}{2\,{x}^{2}}}-3\,{\frac{{a}^{2}bA}{x}}-{\frac{{a}^{3}B}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987261, size = 93, normalized size = 1.43 \begin{align*} \frac{1}{2} \, B b^{3} x^{2} +{\left (3 \, B a b^{2} + A b^{3}\right )} x + 3 \,{\left (B a^{2} b + A a b^{2}\right )} \log \left (x\right ) - \frac{A a^{3} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55146, size = 159, normalized size = 2.45 \begin{align*} \frac{B b^{3} x^{4} - A a^{3} + 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} \log \left (x\right ) - 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.593348, size = 66, normalized size = 1.02 \begin{align*} \frac{B b^{3} x^{2}}{2} + 3 a b \left (A b + B a\right ) \log{\left (x \right )} + x \left (A b^{3} + 3 B a b^{2}\right ) - \frac{A a^{3} + x \left (6 A a^{2} b + 2 B a^{3}\right )}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22381, size = 93, normalized size = 1.43 \begin{align*} \frac{1}{2} \, B b^{3} x^{2} + 3 \, B a b^{2} x + A b^{3} x + 3 \,{\left (B a^{2} b + A a b^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac{A a^{3} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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