Optimal. Leaf size=67 \[ -\frac {a^4 \log (x)}{4 b}-a^3 x-\frac {3}{4} a^2 b x^2-\frac {1}{3} a b^2 x^3+\frac {\log (x) (a+b x)^4}{4 b}-\frac {b^3 x^4}{16} \]
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Rubi [A] time = 0.03, antiderivative size = 67, 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} \[ -\frac {3}{4} a^2 b x^2-\frac {a^4 \log (x)}{4 b}-a^3 x-\frac {1}{3} a b^2 x^3+\frac {\log (x) (a+b x)^4}{4 b}-\frac {b^3 x^4}{16} \]
Antiderivative was successfully verified.
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Rule 12
Rule 32
Rule 43
Rule 2313
Rubi steps
\begin {align*} \int (a+b x)^3 \log (x) \, dx &=\frac {(a+b x)^4 \log (x)}{4 b}-\int \frac {(a+b x)^4}{4 b x} \, dx\\ &=\frac {(a+b x)^4 \log (x)}{4 b}-\frac {\int \frac {(a+b x)^4}{x} \, dx}{4 b}\\ &=\frac {(a+b x)^4 \log (x)}{4 b}-\frac {\int \left (4 a^3 b+\frac {a^4}{x}+6 a^2 b^2 x+4 a b^3 x^2+b^4 x^3\right ) \, dx}{4 b}\\ &=-a^3 x-\frac {3}{4} a^2 b x^2-\frac {1}{3} a b^2 x^3-\frac {b^3 x^4}{16}-\frac {a^4 \log (x)}{4 b}+\frac {(a+b x)^4 \log (x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 81, normalized size = 1.21 \[ -a^3 x+a^3 x \log (x)-\frac {3}{4} a^2 b x^2+\frac {3}{2} a^2 b x^2 \log (x)-\frac {1}{3} a b^2 x^3+a b^2 x^3 \log (x)-\frac {1}{16} b^3 x^4+\frac {1}{4} b^3 x^4 \log (x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int (a+b x)^3 \log (x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.69, size = 69, normalized size = 1.03 \[ -\frac {1}{16} \, b^{3} x^{4} - \frac {1}{3} \, a b^{2} x^{3} - \frac {3}{4} \, a^{2} b x^{2} - a^{3} x + \frac {1}{4} \, {\left (b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a^{3} x\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 71, normalized size = 1.06 \[ \frac {1}{4} \, b^{3} x^{4} \log \relax (x) - \frac {1}{16} \, b^{3} x^{4} + a b^{2} x^{3} \log \relax (x) - \frac {1}{3} \, a b^{2} x^{3} + \frac {3}{2} \, a^{2} b x^{2} \log \relax (x) - \frac {3}{4} \, a^{2} b x^{2} + a^{3} x \log \relax (x) - a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 58, normalized size = 0.87
method | result | size |
risch | \(-a^{3} x -\frac {3 a^{2} b \,x^{2}}{4}-\frac {a \,b^{2} x^{3}}{3}-\frac {b^{3} x^{4}}{16}-\frac {a^{4} \ln \relax (x )}{4 b}+\frac {\left (b x +a \right )^{4} \ln \relax (x )}{4 b}\) | \(58\) |
default | \(b^{3} \left (\frac {x^{4} \ln \relax (x )}{4}-\frac {x^{4}}{16}\right )+3 b^{2} a \left (\frac {x^{3} \ln \relax (x )}{3}-\frac {x^{3}}{9}\right )+3 a^{2} b \left (\frac {x^{2} \ln \relax (x )}{2}-\frac {x^{2}}{4}\right )+a^{3} \left (x \ln \relax (x )-x \right )\) | \(69\) |
norman | \(a^{3} x \ln \relax (x )+a \,b^{2} x^{3} \ln \relax (x )-a^{3} x -\frac {b^{3} x^{4}}{16}-\frac {a \,b^{2} x^{3}}{3}-\frac {3 a^{2} b \,x^{2}}{4}+\frac {b^{3} x^{4} \ln \relax (x )}{4}+\frac {3 a^{2} b \,x^{2} \ln \relax (x )}{2}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 69, normalized size = 1.03 \[ -\frac {1}{16} \, b^{3} x^{4} - \frac {1}{3} \, a b^{2} x^{3} - \frac {3}{4} \, a^{2} b x^{2} - a^{3} x + \frac {1}{4} \, {\left (b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a^{3} x\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 71, normalized size = 1.06 \[ a^3\,x\,\ln \relax (x)-\frac {b^3\,x^4}{16}-\frac {3\,a^2\,b\,x^2}{4}-\frac {a\,b^2\,x^3}{3}-a^3\,x+\frac {b^3\,x^4\,\ln \relax (x)}{4}+\frac {3\,a^2\,b\,x^2\,\ln \relax (x)}{2}+a\,b^2\,x^3\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 71, normalized size = 1.06 \[ - a^{3} x - \frac {3 a^{2} b x^{2}}{4} - \frac {a b^{2} x^{3}}{3} - \frac {b^{3} x^{4}}{16} + \left (a^{3} x + \frac {3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac {b^{3} x^{4}}{4}\right ) \log {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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