Optimal. Leaf size=57 \[ \frac {b n x}{2 c}-\frac {n x^2}{2}-\frac {b^2 n \log (b+c x)}{2 c^2}+\frac {1}{2} x^2 \log \left (d \left (b x+c x^2\right )^n\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 57, 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 = {2605, 78}
\begin {gather*} -\frac {b^2 n \log (b+c x)}{2 c^2}+\frac {1}{2} x^2 \log \left (d \left (b x+c x^2\right )^n\right )+\frac {b n x}{2 c}-\frac {n x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 2605
Rubi steps
\begin {align*} \int x \log \left (d \left (b x+c x^2\right )^n\right ) \, dx &=\frac {1}{2} x^2 \log \left (d \left (b x+c x^2\right )^n\right )-\frac {1}{2} n \int \frac {x (b+2 c x)}{b+c x} \, dx\\ &=\frac {1}{2} x^2 \log \left (d \left (b x+c x^2\right )^n\right )-\frac {1}{2} n \int \left (-\frac {b}{c}+2 x+\frac {b^2}{c (b+c x)}\right ) \, dx\\ &=\frac {b n x}{2 c}-\frac {n x^2}{2}-\frac {b^2 n \log (b+c x)}{2 c^2}+\frac {1}{2} x^2 \log \left (d \left (b x+c x^2\right )^n\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 49, normalized size = 0.86 \begin {gather*} -\frac {1}{2} n \left (-\frac {b x}{c}+x^2+\frac {b^2 \log (b+c x)}{c^2}\right )+\frac {1}{2} x^2 \log \left (d (x (b+c x))^n\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x \ln \left (d \left (c \,x^{2}+b x \right )^{n}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 51, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, x^{2} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) - \frac {1}{2} \, n {\left (\frac {b^{2} \log \left (c x + b\right )}{c^{2}} + \frac {c x^{2} - b x}{c}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 59, normalized size = 1.04 \begin {gather*} \frac {c^{2} n x^{2} \log \left (c x^{2} + b x\right ) - c^{2} n x^{2} + c^{2} x^{2} \log \left (d\right ) + b c n x - b^{2} n \log \left (c x + b\right )}{2 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.73, size = 70, normalized size = 1.23 \begin {gather*} \begin {cases} - \frac {b^{2} n \log {\left (b + c x \right )}}{2 c^{2}} + \frac {b n x}{2 c} - \frac {n x^{2}}{2} + \frac {x^{2} \log {\left (d \left (b x + c x^{2}\right )^{n} \right )}}{2} & \text {for}\: c \neq 0 \\- \frac {n x^{2}}{4} + \frac {x^{2} \log {\left (d \left (b x\right )^{n} \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.44, size = 51, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, n x^{2} \log \left (c x^{2} + b x\right ) - \frac {1}{2} \, {\left (n - \log \left (d\right )\right )} x^{2} + \frac {b n x}{2 \, c} - \frac {b^{2} n \log \left (c x + b\right )}{2 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 49, normalized size = 0.86 \begin {gather*} \frac {x^2\,\ln \left (d\,{\left (c\,x^2+b\,x\right )}^n\right )}{2}-\frac {n\,x^2}{2}+\frac {b\,n\,x}{2\,c}-\frac {b^2\,n\,\ln \left (b+c\,x\right )}{2\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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