Optimal. Leaf size=40 \[ a^2 A \log (x)+2 a A b x+\frac {B (a+b x)^3}{3 b}+\frac {1}{2} A b^2 x^2 \]
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Rubi [A] time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, 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} \[ a^2 A \log (x)+2 a A b x+\frac {B (a+b x)^3}{3 b}+\frac {1}{2} A b^2 x^2 \]
Antiderivative was successfully verified.
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Rule 43
Rule 80
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 (A+B x)}{x} \, dx &=\frac {B (a+b x)^3}{3 b}+A \int \frac {(a+b x)^2}{x} \, dx\\ &=\frac {B (a+b x)^3}{3 b}+A \int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx\\ &=2 a A b x+\frac {1}{2} A b^2 x^2+\frac {B (a+b x)^3}{3 b}+a^2 A \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 1.08 \[ a^2 A \log (x)+a^2 B x+a b x (2 A+B x)+\frac {1}{6} b^2 x^2 (3 A+2 B x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 46, normalized size = 1.15 \[ \frac {1}{3} \, B b^{2} x^{3} + A a^{2} \log \relax (x) + \frac {1}{2} \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + {\left (B a^{2} + 2 \, A a b\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.35, size = 46, normalized size = 1.15 \[ \frac {1}{3} \, B b^{2} x^{3} + B a b x^{2} + \frac {1}{2} \, A b^{2} x^{2} + B a^{2} x + 2 \, A a b x + A a^{2} \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 46, normalized size = 1.15 \[ \frac {B \,b^{2} x^{3}}{3}+\frac {A \,b^{2} x^{2}}{2}+B a b \,x^{2}+A \,a^{2} \ln \relax (x )+2 A a b x +B \,a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 46, normalized size = 1.15 \[ \frac {1}{3} \, B b^{2} x^{3} + A a^{2} \log \relax (x) + \frac {1}{2} \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + {\left (B a^{2} + 2 \, A a b\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 45, normalized size = 1.12 \[ x^2\,\left (\frac {A\,b^2}{2}+B\,a\,b\right )+x\,\left (B\,a^2+2\,A\,b\,a\right )+\frac {B\,b^2\,x^3}{3}+A\,a^2\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 46, normalized size = 1.15 \[ A a^{2} \log {\relax (x )} + \frac {B b^{2} x^{3}}{3} + x^{2} \left (\frac {A b^{2}}{2} + B a b\right ) + x \left (2 A a b + B a^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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