Optimal. Leaf size=156 \[ -\frac {2 \text {Li}_4\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac {2 x \text {Li}_3\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {x^2 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {1}{3} x^3 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac {1}{3} x^3 \log \left (\frac {e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ \frac {2 x \text {PolyLog}\left (3,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {2 \text {PolyLog}\left (4,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}-\frac {x^2 \text {PolyLog}\left (2,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {1}{3} x^3 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac {1}{3} x^3 \log \left (\frac {e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 2531
Rule 2532
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int x^2 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right ) \, dx &=\frac {1}{3} x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{3} x^3 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )+\int x^2 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx\\ &=\frac {1}{3} x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{3} x^3 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x^2 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {2 \int x \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx}{b c n \log (f)}\\ &=\frac {1}{3} x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{3} x^3 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x^2 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {2 x \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 {2 \int \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}{3} x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{3} x^3 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x^2 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {2 x \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 {2 \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {e x^n}{d}\right )}{x} \, dx,x,f^{c (a+b x)}\right )}{b^3 c^3 n^2 \log ^3(f)}\\ &=\frac {1}{3} x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{3} x^3 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x^2 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {2 x \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 {2 \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)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 156, normalized size = 1.00 \[ -\frac {2 \text {Li}_4\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac {2 x \text {Li}_3\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {x^2 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {1}{3} x^3 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac {1}{3} x^3 \log \left (\frac {e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.59, size = 205, normalized size = 1.31 \[ -\frac {3 \, b^{2} c^{2} n^{2} x^{2} {\rm Li}_2\left (-\frac {e f^{b c n x + a c n} + d}{d} + 1\right ) \log \relax (f)^{2} - 6 \, b c n x \log \relax (f) {\rm polylog}\left (3, -\frac {e f^{b c n x + a c n}}{d}\right ) - {\left (b^{3} c^{3} n^{3} x^{3} + a^{3} c^{3} n^{3}\right )} \log \left (e f^{b c n x + a c n} + d\right ) \log \relax (f)^{3} + {\left (b^{3} c^{3} n^{3} x^{3} + a^{3} c^{3} n^{3}\right )} \log \relax (f)^{3} \log \left (\frac {e f^{b c n x + a c n} + d}{d}\right ) + 6 \, {\rm polylog}\left (4, -\frac {e f^{b c n x + a c n}}{d}\right )}{3 \, b^{3} c^{3} n^{3} \log \relax (f)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \log \left (e {\left (f^{{\left (b x + a\right )} c}\right )}^{n} + d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.35, size = 916, normalized size = 5.87 \[ \frac {x^{3} \ln \left (e \left (f^{\left (b x +a \right ) c}\right )^{n}+d \right )}{3}-\frac {x^{3} \ln \left (e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}+d \right )}{3}-\frac {x^{2} \ln \left (f^{\left (b x +a \right ) c}\right ) \ln \left (\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}+d}{d}\right )}{b c \ln \relax (f )}+\frac {x^{2} \ln \left (f^{\left (b x +a \right ) c}\right ) \ln \left (e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}+d \right )}{b c \ln \relax (f )}-\frac {x^{2} \dilog \left (\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}+d}{d}\right )}{b c n \ln \relax (f )}+\frac {2 x \ln \left (f^{\left (b x +a \right ) c}\right )^{2} \ln \left (\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}+d}{d}\right )}{b^{2} c^{2} \ln \relax (f )^{2}}-\frac {x \ln \left (f^{\left (b x +a \right ) c}\right )^{2} \ln \left (\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}}{d}+1\right )}{b^{2} c^{2} \ln \relax (f )^{2}}-\frac {x \ln \left (f^{\left (b x +a \right ) c}\right )^{2} \ln \left (e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}+d \right )}{b^{2} c^{2} \ln \relax (f )^{2}}+\frac {2 x \dilog \left (\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}+d}{d}\right ) \ln \left (f^{\left (b x +a \right ) c}\right )}{b^{2} c^{2} n \ln \relax (f )^{2}}-\frac {2 x \polylog \left (2, -\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}}{d}\right ) \ln \left (f^{\left (b x +a \right ) c}\right )}{b^{2} c^{2} n \ln \relax (f )^{2}}-\frac {\ln \left (f^{\left (b x +a \right ) c}\right )^{3} \ln \left (\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}+d}{d}\right )}{b^{3} c^{3} \ln \relax (f )^{3}}+\frac {2 \ln \left (f^{\left (b x +a \right ) c}\right )^{3} \ln \left (\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}}{d}+1\right )}{3 b^{3} c^{3} \ln \relax (f )^{3}}+\frac {\ln \left (f^{\left (b x +a \right ) c}\right )^{3} \ln \left (e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}+d \right )}{3 b^{3} c^{3} \ln \relax (f )^{3}}+\frac {2 x \polylog \left (3, -\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}}{d}\right )}{b^{2} c^{2} n^{2} \ln \relax (f )^{2}}-\frac {\dilog \left (\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}+d}{d}\right ) \ln \left (f^{\left (b x +a \right ) c}\right )^{2}}{b^{3} c^{3} n \ln \relax (f )^{3}}+\frac {\polylog \left (2, -\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}}{d}\right ) \ln \left (f^{\left (b x +a \right ) c}\right )^{2}}{b^{3} c^{3} n \ln \relax (f )^{3}}-\frac {2 \polylog \left (4, -\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}}{d}\right )}{b^{3} c^{3} n^{3} \ln \relax (f )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.87, size = 165, normalized size = 1.06 \[ \frac {1}{3} \, x^{3} \log \left (e f^{{\left (b x + a\right )} c n} + d\right ) - \frac {b^{3} c^{3} n^{3} x^{3} \log \left (\frac {e f^{b c n x} f^{a c n}}{d} + 1\right ) \log \relax (f)^{3} + 3 \, b^{2} c^{2} n^{2} x^{2} {\rm Li}_2\left (-\frac {e f^{b c n x} f^{a c n}}{d}\right ) \log \relax (f)^{2} - 6 \, b c n x \log \relax (f) {\rm Li}_{3}(-\frac {e f^{b c n x} f^{a c n}}{d}) + 6 \, {\rm Li}_{4}(-\frac {e f^{b c n x} f^{a c n}}{d})}{3 \, b^{3} c^{3} n^{3} \log \relax (f)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\ln \left (d+e\,{\left (f^{c\,\left (a+b\,x\right )}\right )}^n\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {b c e n e^{a c n \log {\relax (f )}} \log {\relax (f )} \int \frac {x^{3} e^{b c n x \log {\relax (f )}}}{d + e e^{a c n \log {\relax (f )}} e^{b c n x \log {\relax (f )}}}\, dx}{3} + \frac {x^{3} \log {\left (d + e \left (f^{c \left (a + b x\right )}\right )^{n} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________