Optimal. Leaf size=54 \[ -b^2 x-\frac {1}{2} a b x^2-\frac {a^2 x^3}{9}-\frac {b^3 \log (x)}{3 a}+\frac {(b+a x)^3 \log (x)}{3 a} \]
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Rubi [A]
time = 0.02, 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, 2350, 12,
45} \begin {gather*} -\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 \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 32
Rule 45
Rule 2350
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 \begin {gather*} -b^2 x-\frac {1}{2} a b x^2-\frac {a^2 x^3}{9}+b^2 x \log (x)+a b x^2 \log (x)+\frac {1}{3} a^2 x^3 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.08, size = 44, normalized size = 0.81 \begin {gather*} \frac {x \left (-2 a^2 x^2-9 a b x-18 b^2+6 \text {Log}\left [x\right ] \left (a^2 x^2+3 a b x+3 b^2\right )\right )}{18} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 48, normalized size = 0.89
method | result | size |
risch | \(-b^{2} x -\frac {a b \,x^{2}}{2}-\frac {a^{2} x^{3}}{9}-\frac {b^{3} \ln \left (x \right )}{3 a}+\frac {\left (a x +b \right )^{3} \ln \left (x \right )}{3 a}\) | \(47\) |
default | \(a^{2} \left (-\frac {x^{3}}{9}+\frac {x^{3} \ln \left (x \right )}{3}\right )+2 a b \left (-\frac {x^{2}}{4}+\frac {x^{2} \ln \left (x \right )}{2}\right )+b^{2} \left (-x +x \ln \left (x \right )\right )\) | \(48\) |
norman | \(b^{2} x \ln \left (x \right )+a b \,x^{2} \ln \left (x \right )-\frac {a^{2} x^{3}}{9}-b^{2} x -\frac {a b \,x^{2}}{2}+\frac {a^{2} x^{3} \ln \left (x \right )}{3}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 47, normalized size = 0.87 \begin {gather*} -\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 \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 47, normalized size = 0.87 \begin {gather*} -\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 \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 44, normalized size = 0.81 \begin {gather*} - \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 {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 52, normalized size = 0.96 \begin {gather*} -\frac {1}{9} a^{2} x^{3}-\frac {1}{2} a b x^{2}-b^{2} x+\frac {1}{3} a^{2} x^{3} \ln x+b^{2} x \ln x+a b x^{2} \ln x \end {gather*}
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 \begin {gather*} b^2\,x\,\ln \left (x\right )-\frac {a^2\,x^3}{9}-b^2\,x+\frac {a^2\,x^3\,\ln \left (x\right )}{3}-\frac {a\,b\,x^2}{2}+a\,b\,x^2\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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