Optimal. Leaf size=69 \[ \frac {a x}{e}-\frac {b n x}{e}+\frac {b x \log \left (c x^n\right )}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^2}-\frac {b d n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {45, 2393, 2332,
2354, 2438} \begin {gather*} -\frac {b d n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^2}-\frac {d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {a x}{e}+\frac {b x \log \left (c x^n\right )}{e}-\frac {b n x}{e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 2332
Rule 2354
Rule 2393
Rule 2438
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx &=\int \left (\frac {a+b \log \left (c x^n\right )}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}\right ) \, dx\\ &=\frac {\int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e}-\frac {d \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e}\\ &=\frac {a x}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^2}+\frac {b \int \log \left (c x^n\right ) \, dx}{e}+\frac {(b d n) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^2}\\ &=\frac {a x}{e}-\frac {b n x}{e}+\frac {b x \log \left (c x^n\right )}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^2}-\frac {b d n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 66, normalized size = 0.96 \begin {gather*} \frac {a e x-b e n x-a d \log \left (1+\frac {e x}{d}\right )+b \log \left (c x^n\right ) \left (e x-d \log \left (1+\frac {e x}{d}\right )\right )-b d n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.13, size = 343, normalized size = 4.97
method | result | size |
risch | \(\frac {b \ln \left (x^{n}\right ) x}{e}-\frac {b \ln \left (x^{n}\right ) d \ln \left (e x +d \right )}{e^{2}}-\frac {b n x}{e}-\frac {b n d}{e^{2}}+\frac {b n d \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{2}}+\frac {b n d \dilog \left (-\frac {e x}{d}\right )}{e^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x}{2 e}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d \ln \left (e x +d \right )}{2 e^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) d \ln \left (e x +d \right )}{2 e^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d \ln \left (e x +d \right )}{2 e^{2}}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x}{2 e}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d \ln \left (e x +d \right )}{2 e^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x}{2 e}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x}{2 e}+\frac {b \ln \left (c \right ) x}{e}-\frac {b \ln \left (c \right ) d \ln \left (e x +d \right )}{e^{2}}+\frac {a x}{e}-\frac {a d \ln \left (e x +d \right )}{e^{2}}\) | \(343\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 8.69, size = 163, normalized size = 2.36 \begin {gather*} - \frac {a d \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{e} + \frac {a x}{e} + \frac {b d n \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \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 {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{e} - \frac {b d \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} - \frac {b n x}{e} + \frac {b x \log {\left (c x^{n} \right )}}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________