Optimal. Leaf size=59 \[ \frac {b^3 \log (a x+b)}{3 a^3}-\frac {b^2 x}{3 a^2}+\frac {1}{3} x^3 \log (a x+b)+\frac {b x^2}{6 a}-\frac {x^3}{9} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2395, 43} \[ -\frac {b^2 x}{3 a^2}+\frac {b^3 \log (a x+b)}{3 a^3}+\frac {b x^2}{6 a}+\frac {1}{3} x^3 \log (a x+b)-\frac {x^3}{9} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2395
Rubi steps
\begin {align*} \int x^2 \log (b+a x) \, dx &=\frac {1}{3} x^3 \log (b+a x)-\frac {1}{3} a \int \frac {x^3}{b+a x} \, dx\\ &=\frac {1}{3} x^3 \log (b+a x)-\frac {1}{3} a \int \left (\frac {b^2}{a^3}-\frac {b x}{a^2}+\frac {x^2}{a}-\frac {b^3}{a^3 (b+a x)}\right ) \, dx\\ &=-\frac {b^2 x}{3 a^2}+\frac {b x^2}{6 a}-\frac {x^3}{9}+\frac {b^3 \log (b+a x)}{3 a^3}+\frac {1}{3} x^3 \log (b+a x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 59, normalized size = 1.00 \[ \frac {b^3 \log (a x+b)}{3 a^3}-\frac {b^2 x}{3 a^2}+\frac {1}{3} x^3 \log (a x+b)+\frac {b x^2}{6 a}-\frac {x^3}{9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 49, normalized size = 0.83 \[ -\frac {2 \, a^{3} x^{3} - 3 \, a^{2} b x^{2} + 6 \, a b^{2} x - 6 \, {\left (a^{3} x^{3} + b^{3}\right )} \log \left (a x + b\right )}{18 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.27, size = 94, normalized size = 1.59 \[ \frac {{\left (a x + b\right )}^{3} \log \left (a x + b\right )}{3 \, a^{3}} - \frac {{\left (a x + b\right )}^{2} b \log \left (a x + b\right )}{a^{3}} + \frac {{\left (a x + b\right )} b^{2} \log \left (a x + b\right )}{a^{3}} - \frac {{\left (a x + b\right )}^{3}}{9 \, a^{3}} + \frac {{\left (a x + b\right )}^{2} b}{2 \, a^{3}} - \frac {{\left (a x + b\right )} b^{2}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 58, normalized size = 0.98 \[ \frac {x^{3} \ln \left (a x +b \right )}{3}-\frac {x^{3}}{9}+\frac {b \,x^{2}}{6 a}-\frac {b^{2} x}{3 a^{2}}+\frac {b^{3} \ln \left (a x +b \right )}{3 a^{3}}-\frac {11 b^{3}}{18 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 57, normalized size = 0.97 \[ \frac {1}{3} \, x^{3} \log \left (a x + b\right ) + \frac {1}{18} \, a {\left (\frac {6 \, b^{3} \log \left (a x + b\right )}{a^{4}} - \frac {2 \, a^{2} x^{3} - 3 \, a b x^{2} + 6 \, b^{2} x}{a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 75, normalized size = 1.27 \[ \left \{\begin {array}{cl} \frac {x^3\,\left (\ln \left (a\,x\right )-\frac {1}{3}\right )}{3} & \text {\ if\ \ }b=0\\ \frac {\ln \left (b+a\,x\right )\,\left (x^3+\frac {b^3}{a^3}\right )}{3}-\frac {b^3\,\left (\frac {a^3\,x^3}{3\,b^3}-\frac {a^2\,x^2}{2\,b^2}+\frac {a\,x}{b}\right )}{3\,a^3} & \text {\ if\ \ }b\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 54, normalized size = 0.92 \[ - a \left (\frac {x^{3}}{9 a} - \frac {b x^{2}}{6 a^{2}} + \frac {b^{2} x}{3 a^{3}} - \frac {b^{3} \log {\left (a x + b \right )}}{3 a^{4}}\right ) + \frac {x^{3} \log {\left (a x + b \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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