Optimal. Leaf size=110 \[ -\frac {e^2 \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac {a+b \log \left (c x^n\right )}{2 d x^2}+\frac {b e^2 n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^3}+\frac {b e n}{d^2 x}-\frac {b n}{4 d x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 135, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {44, 2351, 2304, 2301, 2317, 2391} \[ -\frac {b e^2 n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^3}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^3 n}-\frac {e^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac {a+b \log \left (c x^n\right )}{2 d x^2}+\frac {b e n}{d^2 x}-\frac {b n}{4 d x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 2301
Rule 2304
Rule 2317
Rule 2351
Rule 2391
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)} \, dx &=\int \left (\frac {a+b \log \left (c x^n\right )}{d x^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^2 x^2}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d^2}+\frac {e^2 \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{d^3}-\frac {e^3 \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^3}\\ &=-\frac {b n}{4 d x^2}+\frac {b e n}{d^2 x}-\frac {a+b \log \left (c x^n\right )}{2 d x^2}+\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^3 n}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^3}+\frac {\left (b e^2 n\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^3}\\ &=-\frac {b n}{4 d x^2}+\frac {b e n}{d^2 x}-\frac {a+b \log \left (c x^n\right )}{2 d x^2}+\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^3 n}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^3}-\frac {b e^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{d^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.21, size = 124, normalized size = 1.13 \[ -\frac {\frac {2 d^2 \left (a+b \log \left (c x^n\right )\right )}{x^2}+4 e^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 d e \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {2 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}+\frac {b d^2 n}{x^2}+4 b e^2 n \text {Li}_2\left (-\frac {e x}{d}\right )-\frac {4 b d e n}{x}}{4 d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{e x^{4} + d x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.19, size = 689, normalized size = 6.26 \[ \frac {b \,e^{2} n \dilog \left (-\frac {e x}{d}\right )}{d^{3}}+\frac {b e \ln \left (x^{n}\right )}{d^{2} x}-\frac {b \,e^{2} \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{d^{3}}+\frac {b \,e^{2} \ln \relax (x ) \ln \left (x^{n}\right )}{d^{3}}+\frac {b e \ln \relax (c )}{d^{2} x}-\frac {b \,e^{2} \ln \relax (c ) \ln \left (e x +d \right )}{d^{3}}+\frac {b \,e^{2} \ln \relax (c ) \ln \relax (x )}{d^{3}}-\frac {b \,e^{2} n \ln \relax (x )^{2}}{2 d^{3}}+\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{4 d \,x^{2}}-\frac {i \pi b \,e^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x +d \right )}{2 d^{3}}+\frac {i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 d^{2} x}+\frac {i \pi b e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 d^{2} x}+\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{4 d \,x^{2}}-\frac {i \pi b \,e^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x +d \right )}{2 d^{3}}+\frac {i \pi b \,e^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )}{2 d^{3}}+\frac {i \pi b \,e^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )}{2 d^{3}}-\frac {b \ln \left (x^{n}\right )}{2 d \,x^{2}}+\frac {b \,e^{2} n \ln \left (-\frac {e x}{d}\right ) \ln \left (e x +d \right )}{d^{3}}+\frac {a \,e^{2} \ln \relax (x )}{d^{3}}+\frac {a e}{d^{2} x}-\frac {a \,e^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {b \ln \relax (c )}{2 d \,x^{2}}-\frac {a}{2 d \,x^{2}}-\frac {i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 d \,x^{2}}-\frac {i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 d^{2} x}-\frac {i \pi b \,e^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \relax (x )}{2 d^{3}}+\frac {i \pi b \,e^{2} \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 d^{3}}-\frac {b n}{4 d \,x^{2}}-\frac {i \pi b \,e^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (x )}{2 d^{3}}+\frac {i \pi b \,e^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (e x +d \right )}{2 d^{3}}-\frac {i \pi b e \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 d^{2} x}-\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 d \,x^{2}}+\frac {b e n}{d^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, a {\left (\frac {2 \, e^{2} \log \left (e x + d\right )}{d^{3}} - \frac {2 \, e^{2} \log \relax (x)}{d^{3}} - \frac {2 \, e x - d}{d^{2} x^{2}}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{e x^{4} + d x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 77.20, size = 246, normalized size = 2.24 \[ - \frac {a}{2 d x^{2}} + \frac {a e}{d^{2} x} - \frac {a e^{3} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{d^{3}} + \frac {a e^{2} \log {\relax (x )}}{d^{3}} - \frac {b n}{4 d x^{2}} - \frac {b \log {\left (c x^{n} \right )}}{2 d x^{2}} + \frac {b e n}{d^{2} x} + \frac {b e \log {\left (c x^{n} \right )}}{d^{2} x} + \frac {b e^{3} n \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} \log {\relax (d )} \log {\relax (x )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (d )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (d )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (d )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{d^{3}} - \frac {b e^{3} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{3}} - \frac {b e^{2} n \log {\relax (x )}^{2}}{2 d^{3}} + \frac {b e^{2} \log {\relax (x )} \log {\left (c x^{n} \right )}}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________