Optimal. Leaf size=54 \[ -\frac {a^2 x^3}{9}-\frac {b^3 \log (x)}{3 a}-\frac {1}{2} a b x^2+\frac {\log (x) (a x+b)^3}{3 a}-b^2 x \]
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Rubi [A] time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {32, 2313, 12, 43} \[ -\frac {a^2 x^3}{9}-\frac {b^3 \log (x)}{3 a}-\frac {1}{2} a b x^2+\frac {\log (x) (a x+b)^3}{3 a}-b^2 x \]
Antiderivative was successfully verified.
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Rule 12
Rule 32
Rule 43
Rule 2313
Rubi steps
\begin {align*} \int (b+a x)^2 \log (x) \, dx &=\frac {(b+a x)^3 \log (x)}{3 a}-\int \frac {(b+a x)^3}{3 a x} \, dx\\ &=\frac {(b+a x)^3 \log (x)}{3 a}-\frac {\int \frac {(b+a x)^3}{x} \, dx}{3 a}\\ &=\frac {(b+a x)^3 \log (x)}{3 a}-\frac {\int \left (3 a b^2+\frac {b^3}{x}+3 a^2 b x+a^3 x^2\right ) \, dx}{3 a}\\ &=-b^2 x-\frac {1}{2} a b x^2-\frac {a^2 x^3}{9}-\frac {b^3 \log (x)}{3 a}+\frac {(b+a x)^3 \log (x)}{3 a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 53, normalized size = 0.98 \[ -\frac {1}{9} a^2 x^3+\frac {1}{3} a^2 x^3 \log (x)-\frac {1}{2} a b x^2+a b x^2 \log (x)-b^2 x+b^2 x \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 47, normalized size = 0.87 \[ -\frac {1}{9} \, a^{2} x^{3} - \frac {1}{2} \, a b x^{2} - b^{2} x + \frac {1}{3} \, {\left (a^{2} x^{3} + 3 \, a b x^{2} + 3 \, b^{2} x\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 47, normalized size = 0.87 \[ \frac {1}{3} \, a^{2} x^{3} \log \relax (x) - \frac {1}{9} \, a^{2} x^{3} + a b x^{2} \log \relax (x) - \frac {1}{2} \, a b x^{2} + b^{2} x \log \relax (x) - b^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 48, normalized size = 0.89 \[ \frac {a^{2} x^{3} \ln \relax (x )}{3}-\frac {a^{2} x^{3}}{9}+a b \,x^{2} \ln \relax (x )-\frac {a b \,x^{2}}{2}+b^{2} x \ln \relax (x )-b^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 47, normalized size = 0.87 \[ -\frac {1}{9} \, a^{2} x^{3} - \frac {1}{2} \, a b x^{2} - b^{2} x + \frac {1}{3} \, {\left (a^{2} x^{3} + 3 \, a b x^{2} + 3 \, b^{2} x\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 47, normalized size = 0.87 \[ b^2\,x\,\ln \relax (x)-\frac {a^2\,x^3}{9}-b^2\,x+\frac {a^2\,x^3\,\ln \relax (x)}{3}-\frac {a\,b\,x^2}{2}+a\,b\,x^2\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 44, normalized size = 0.81 \[ - \frac {a^{2} x^{3}}{9} - \frac {a b x^{2}}{2} - b^{2} x + \left (\frac {a^{2} x^{3}}{3} + a b x^{2} + b^{2} x\right ) \log {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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