Optimal. Leaf size=148 \[ -\frac {d^3 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac {a d^2 x}{e^3}+\frac {b d^2 x \log \left (c x^n\right )}{e^3}-\frac {b d^3 n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}-\frac {b d^2 n x}{e^3}+\frac {b d n x^2}{4 e^2}-\frac {b n x^3}{9 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {43, 2351, 2295, 2304, 2317, 2391} \[ -\frac {b d^3 n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^4}-\frac {d^3 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac {a d^2 x}{e^3}+\frac {b d^2 x \log \left (c x^n\right )}{e^3}-\frac {b d^2 n x}{e^3}+\frac {b d n x^2}{4 e^2}-\frac {b n x^3}{9 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2295
Rule 2304
Rule 2317
Rule 2351
Rule 2391
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx &=\int \left (\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {d x \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {d^2 \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^3}-\frac {d^3 \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^3}-\frac {d \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}+\frac {\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{e}\\ &=\frac {a d^2 x}{e^3}+\frac {b d n x^2}{4 e^2}-\frac {b n x^3}{9 e}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {\left (b d^2\right ) \int \log \left (c x^n\right ) \, dx}{e^3}+\frac {\left (b d^3 n\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^4}\\ &=\frac {a d^2 x}{e^3}-\frac {b d^2 n x}{e^3}+\frac {b d n x^2}{4 e^2}-\frac {b n x^3}{9 e}+\frac {b d^2 x \log \left (c x^n\right )}{e^3}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {b d^3 n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 142, normalized size = 0.96 \[ \frac {-36 a d^3 \log \left (\frac {e x}{d}+1\right )+36 a d^2 e x-18 a d e^2 x^2+12 a e^3 x^3+6 b \log \left (c x^n\right ) \left (e x \left (6 d^2-3 d e x+2 e^2 x^2\right )-6 d^3 \log \left (\frac {e x}{d}+1\right )\right )-36 b d^3 n \text {Li}_2\left (-\frac {e x}{d}\right )-36 b d^2 e n x+9 b d e^2 n x^2-4 b e^3 n x^3}{36 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{3} \log \left (c x^{n}\right ) + a x^{3}}{e x + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.23, size = 693, normalized size = 4.68 \[ \frac {b \,d^{2} x \ln \relax (c )}{e^{3}}-\frac {b \,d^{3} \ln \relax (c ) \ln \left (e x +d \right )}{e^{4}}-\frac {b d \,x^{2} \ln \relax (c )}{2 e^{2}}+\frac {a \,x^{3}}{3 e}+\frac {b \,d^{3} n \dilog \left (-\frac {e x}{d}\right )}{e^{4}}-\frac {i \pi b \,x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{6 e}+\frac {b \,d^{2} x \ln \left (x^{n}\right )}{e^{3}}-\frac {b \,d^{3} \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e^{4}}-\frac {b d \,x^{2} \ln \left (x^{n}\right )}{2 e^{2}}-\frac {i \pi b \,d^{2} x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 e^{3}}-\frac {i \pi b d \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 e^{2}}-\frac {i \pi b d \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 e^{2}}-\frac {i \pi b \,x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{6 e}-\frac {49 b \,d^{3} n}{36 e^{4}}-\frac {a d \,x^{2}}{2 e^{2}}-\frac {a \,d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {b \,x^{3} \ln \relax (c )}{3 e}+\frac {b \,d^{3} n \ln \left (-\frac {e x}{d}\right ) \ln \left (e x +d \right )}{e^{4}}+\frac {i \pi b \,d^{2} x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 e^{3}}-\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x +d \right )}{2 e^{4}}-\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x +d \right )}{2 e^{4}}+\frac {i \pi b \,d^{2} x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 e^{3}}+\frac {b \,x^{3} \ln \left (x^{n}\right )}{3 e}+\frac {i \pi b d \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{4 e^{2}}+\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (e x +d \right )}{2 e^{4}}-\frac {b n \,x^{3}}{9 e}+\frac {i \pi b d \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{4 e^{2}}+\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (e x +d \right )}{2 e^{4}}-\frac {i \pi b \,d^{2} x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 e^{3}}+\frac {i \pi b \,x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6 e}+\frac {i \pi b \,x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6 e}-\frac {b \,d^{2} n x}{e^{3}}+\frac {a \,d^{2} x}{e^{3}}+\frac {b d n \,x^{2}}{4 e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{6} \, a {\left (\frac {6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac {2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} + b \int \frac {x^{3} \log \relax (c) + x^{3} \log \left (x^{n}\right )}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 38.90, size = 248, normalized size = 1.68 \[ - \frac {a d^{3} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{3}} + \frac {a d^{2} x}{e^{3}} - \frac {a d x^{2}}{2 e^{2}} + \frac {a x^{3}}{3 e} + \frac {b d^{3} n \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} \log {\relax (d )} \log {\relax (x )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (d )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (d )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (d )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{e^{3}} - \frac {b d^{3} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{3}} - \frac {b d^{2} n x}{e^{3}} + \frac {b d^{2} x \log {\left (c x^{n} \right )}}{e^{3}} + \frac {b d n x^{2}}{4 e^{2}} - \frac {b d x^{2} \log {\left (c x^{n} \right )}}{2 e^{2}} - \frac {b n x^{3}}{9 e} + \frac {b x^{3} \log {\left (c x^{n} \right )}}{3 e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________