Optimal. Leaf size=193 \[ \frac {6 \text {Li}_5\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^4 c^4 n^4 \log ^4(f)}-\frac {6 x \text {Li}_4\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac {3 x^2 \text {Li}_3\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {x^3 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {1}{4} x^4 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac {1}{4} x^4 \log \left (\frac {e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2532, 2531, 6609, 2282, 6589} \[ \frac {3 x^2 \text {PolyLog}\left (3,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {6 x \text {PolyLog}\left (4,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac {6 \text {PolyLog}\left (5,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^4 c^4 n^4 \log ^4(f)}-\frac {x^3 \text {PolyLog}\left (2,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {1}{4} x^4 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac {1}{4} x^4 \log \left (\frac {e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 2532
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right ) \, dx &=\frac {1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{4} x^4 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )+\int x^3 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx\\ &=\frac {1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{4} x^4 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x^3 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {3 \int x^2 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx}{b c n \log (f)}\\ &=\frac {1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{4} x^4 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x^3 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {3 x^2 \text {Li}_3\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {6 \int x \text {Li}_3\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx}{b^2 c^2 n^2 \log ^2(f)}\\ &=\frac {1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{4} x^4 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x^3 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {3 x^2 \text {Li}_3\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {6 x \text {Li}_4\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac {6 \int \text {Li}_4\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx}{b^3 c^3 n^3 \log ^3(f)}\\ &=\frac {1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{4} x^4 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x^3 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {3 x^2 \text {Li}_3\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {6 x \text {Li}_4\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac {6 \operatorname {Subst}\left (\int \frac {\text {Li}_4\left (-\frac {e x^n}{d}\right )}{x} \, dx,x,f^{c (a+b x)}\right )}{b^4 c^4 n^3 \log ^4(f)}\\ &=\frac {1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{4} x^4 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x^3 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {3 x^2 \text {Li}_3\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {6 x \text {Li}_4\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac {6 \text {Li}_5\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^4 c^4 n^4 \log ^4(f)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 193, normalized size = 1.00 \[ \frac {6 \text {Li}_5\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^4 c^4 n^4 \log ^4(f)}-\frac {6 x \text {Li}_4\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac {3 x^2 \text {Li}_3\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {x^3 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {1}{4} x^4 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac {1}{4} x^4 \log \left (\frac {e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.76, size = 245, normalized size = 1.27 \[ -\frac {4 \, b^{3} c^{3} n^{3} x^{3} {\rm Li}_2\left (-\frac {e f^{b c n x + a c n} + d}{d} + 1\right ) \log \relax (f)^{3} - 12 \, b^{2} c^{2} n^{2} x^{2} \log \relax (f)^{2} {\rm polylog}\left (3, -\frac {e f^{b c n x + a c n}}{d}\right ) - {\left (b^{4} c^{4} n^{4} x^{4} - a^{4} c^{4} n^{4}\right )} \log \left (e f^{b c n x + a c n} + d\right ) \log \relax (f)^{4} + {\left (b^{4} c^{4} n^{4} x^{4} - a^{4} c^{4} n^{4}\right )} \log \relax (f)^{4} \log \left (\frac {e f^{b c n x + a c n} + d}{d}\right ) + 24 \, b c n x \log \relax (f) {\rm polylog}\left (4, -\frac {e f^{b c n x + a c n}}{d}\right ) - 24 \, {\rm polylog}\left (5, -\frac {e f^{b c n x + a c n}}{d}\right )}{4 \, b^{4} c^{4} n^{4} \log \relax (f)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \log \left (e {\left (f^{{\left (b x + a\right )} c}\right )}^{n} + d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.36, size = 1276, normalized size = 6.61 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 204, normalized size = 1.06 \[ \frac {1}{4} \, x^{4} \log \left (e f^{{\left (b x + a\right )} c n} + d\right ) - \frac {b^{4} c^{4} n^{4} x^{4} \log \left (\frac {e f^{b c n x} f^{a c n}}{d} + 1\right ) \log \relax (f)^{4} + 4 \, b^{3} c^{3} n^{3} x^{3} {\rm Li}_2\left (-\frac {e f^{b c n x} f^{a c n}}{d}\right ) \log \relax (f)^{3} - 12 \, b^{2} c^{2} n^{2} x^{2} \log \relax (f)^{2} {\rm Li}_{3}(-\frac {e f^{b c n x} f^{a c n}}{d}) + 24 \, b c n x \log \relax (f) {\rm Li}_{4}(-\frac {e f^{b c n x} f^{a c n}}{d}) - 24 \, {\rm Li}_{5}(-\frac {e f^{b c n x} f^{a c n}}{d})}{4 \, b^{4} c^{4} n^{4} \log \relax (f)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,\ln \left (d+e\,{\left (f^{c\,\left (a+b\,x\right )}\right )}^n\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {b c e n e^{a c n \log {\relax (f )}} \log {\relax (f )} \int \frac {x^{4} e^{b c n x \log {\relax (f )}}}{d + e e^{a c n \log {\relax (f )}} e^{b c n x \log {\relax (f )}}}\, dx}{4} + \frac {x^{4} \log {\left (d + e \left (f^{c \left (a + b x\right )}\right )^{n} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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