Optimal. Leaf size=85 \[ -\frac {b^4 n \log (b+c x)}{4 c^4}+\frac {b^3 n x}{4 c^3}-\frac {b^2 n x^2}{8 c^2}+\frac {1}{4} x^4 \log \left (d \left (b x+c x^2\right )^n\right )+\frac {b n x^3}{12 c}-\frac {n x^4}{8} \]
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Rubi [A] time = 0.06, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2525, 77} \[ -\frac {b^2 n x^2}{8 c^2}+\frac {b^3 n x}{4 c^3}-\frac {b^4 n \log (b+c x)}{4 c^4}+\frac {1}{4} x^4 \log \left (d \left (b x+c x^2\right )^n\right )+\frac {b n x^3}{12 c}-\frac {n x^4}{8} \]
Antiderivative was successfully verified.
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Rule 77
Rule 2525
Rubi steps
\begin {align*} \int x^3 \log \left (d \left (b x+c x^2\right )^n\right ) \, dx &=\frac {1}{4} x^4 \log \left (d \left (b x+c x^2\right )^n\right )-\frac {1}{4} n \int \frac {x^3 (b+2 c x)}{b+c x} \, dx\\ &=\frac {1}{4} x^4 \log \left (d \left (b x+c x^2\right )^n\right )-\frac {1}{4} n \int \left (-\frac {b^3}{c^3}+\frac {b^2 x}{c^2}-\frac {b x^2}{c}+2 x^3+\frac {b^4}{c^3 (b+c x)}\right ) \, dx\\ &=\frac {b^3 n x}{4 c^3}-\frac {b^2 n x^2}{8 c^2}+\frac {b n x^3}{12 c}-\frac {n x^4}{8}-\frac {b^4 n \log (b+c x)}{4 c^4}+\frac {1}{4} x^4 \log \left (d \left (b x+c x^2\right )^n\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 74, normalized size = 0.87 \[ \frac {-6 b^4 n \log (b+c x)+c n x \left (6 b^3-3 b^2 c x+2 b c^2 x^2-3 c^3 x^3\right )+6 c^4 x^4 \log \left (d (x (b+c x))^n\right )}{24 c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 86, normalized size = 1.01 \[ \frac {6 \, c^{4} n x^{4} \log \left (c x^{2} + b x\right ) - 3 \, c^{4} n x^{4} + 6 \, c^{4} x^{4} \log \relax (d) + 2 \, b c^{3} n x^{3} - 3 \, b^{2} c^{2} n x^{2} + 6 \, b^{3} c n x - 6 \, b^{4} n \log \left (c x + b\right )}{24 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 75, normalized size = 0.88 \[ \frac {1}{4} \, n x^{4} \log \left (c x^{2} + b x\right ) - \frac {1}{8} \, {\left (n - 2 \, \log \relax (d)\right )} x^{4} + \frac {b n x^{3}}{12 \, c} - \frac {b^{2} n x^{2}}{8 \, c^{2}} + \frac {b^{3} n x}{4 \, c^{3}} - \frac {b^{4} n \log \left (c x + b\right )}{4 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int x^{3} \ln \left (d \left (c \,x^{2}+b x \right )^{n}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 75, normalized size = 0.88 \[ \frac {1}{4} \, x^{4} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) - \frac {1}{24} \, n {\left (\frac {6 \, b^{4} \log \left (c x + b\right )}{c^{4}} + \frac {3 \, c^{3} x^{4} - 2 \, b c^{2} x^{3} + 3 \, b^{2} c x^{2} - 6 \, b^{3} x}{c^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 73, normalized size = 0.86 \[ \frac {x^4\,\ln \left (d\,{\left (c\,x^2+b\,x\right )}^n\right )}{4}-\frac {n\,x^4}{8}-\frac {b^2\,n\,x^2}{8\,c^2}-\frac {b^4\,n\,\ln \left (b+c\,x\right )}{4\,c^4}+\frac {b\,n\,x^3}{12\,c}+\frac {b^3\,n\,x}{4\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.09, size = 119, normalized size = 1.40 \[ \begin {cases} - \frac {b^{4} n \log {\left (b + c x \right )}}{4 c^{4}} + \frac {b^{3} n x}{4 c^{3}} - \frac {b^{2} n x^{2}}{8 c^{2}} + \frac {b n x^{3}}{12 c} + \frac {n x^{4} \log {\left (b x + c x^{2} \right )}}{4} - \frac {n x^{4}}{8} + \frac {x^{4} \log {\relax (d )}}{4} & \text {for}\: c \neq 0 \\\frac {n x^{4} \log {\relax (b )}}{4} + \frac {n x^{4} \log {\relax (x )}}{4} - \frac {n x^{4}}{16} + \frac {x^{4} \log {\relax (d )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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