Optimal. Leaf size=54 \[ 3 a^2 A b x+a^3 A \log (x)+\frac{3}{2} a A b^2 x^2+\frac{B (a+b x)^4}{4 b}+\frac{1}{3} A b^3 x^3 \]
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Rubi [A] time = 0.0161688, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {80, 43} \[ 3 a^2 A b x+a^3 A \log (x)+\frac{3}{2} a A b^2 x^2+\frac{B (a+b x)^4}{4 b}+\frac{1}{3} A b^3 x^3 \]
Antiderivative was successfully verified.
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Rule 80
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^3 (A+B x)}{x} \, dx &=\frac{B (a+b x)^4}{4 b}+A \int \frac{(a+b x)^3}{x} \, dx\\ &=\frac{B (a+b x)^4}{4 b}+A \int \left (3 a^2 b+\frac{a^3}{x}+3 a b^2 x+b^3 x^2\right ) \, dx\\ &=3 a^2 A b x+\frac{3}{2} a A b^2 x^2+\frac{1}{3} A b^3 x^3+\frac{B (a+b x)^4}{4 b}+a^3 A \log (x)\\ \end{align*}
Mathematica [A] time = 0.0235164, size = 63, normalized size = 1.17 \[ \frac{1}{12} x \left (18 a^2 b (2 A+B x)+12 a^3 B+6 a b^2 x (3 A+2 B x)+b^3 x^2 (4 A+3 B x)\right )+a^3 A \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 70, normalized size = 1.3 \begin{align*}{\frac{B{b}^{3}{x}^{4}}{4}}+{\frac{A{b}^{3}{x}^{3}}{3}}+B{x}^{3}a{b}^{2}+{\frac{3\,aA{b}^{2}{x}^{2}}{2}}+{\frac{3\,B{x}^{2}{a}^{2}b}{2}}+3\,{a}^{2}Abx+{a}^{3}Bx+{a}^{3}A\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04244, size = 92, normalized size = 1.7 \begin{align*} \frac{1}{4} \, B b^{3} x^{4} + A a^{3} \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + \frac{3}{2} \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64857, size = 154, normalized size = 2.85 \begin{align*} \frac{1}{4} \, B b^{3} x^{4} + A a^{3} \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + \frac{3}{2} \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.36449, size = 73, normalized size = 1.35 \begin{align*} A a^{3} \log{\left (x \right )} + \frac{B b^{3} x^{4}}{4} + x^{3} \left (\frac{A b^{3}}{3} + B a b^{2}\right ) + x^{2} \left (\frac{3 A a b^{2}}{2} + \frac{3 B a^{2} b}{2}\right ) + x \left (3 A a^{2} b + B a^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28789, size = 95, normalized size = 1.76 \begin{align*} \frac{1}{4} \, B b^{3} x^{4} + B a b^{2} x^{3} + \frac{1}{3} \, A b^{3} x^{3} + \frac{3}{2} \, B a^{2} b x^{2} + \frac{3}{2} \, A a b^{2} x^{2} + B a^{3} x + 3 \, A a^{2} b x + A a^{3} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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