Optimal. Leaf size=104 \[ -\frac {f \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {a x}{g}+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac {b f n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}-\frac {b n x}{g} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {43, 2416, 2389, 2295, 2394, 2393, 2391} \[ -\frac {b f n \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^2}-\frac {f \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {a x}{g}+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac {b n x}{g} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx &=\int \left (\frac {a+b \log \left (c (d+e x)^n\right )}{g}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (f+g x)}\right ) \, dx\\ &=\frac {\int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g}-\frac {f \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{g}\\ &=\frac {a x}{g}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}+\frac {b \int \log \left (c (d+e x)^n\right ) \, dx}{g}+\frac {(b e f n) \int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g^2}\\ &=\frac {a x}{g}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}+\frac {b \operatorname {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e g}+\frac {(b f n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g^2}\\ &=\frac {a x}{g}-\frac {b n x}{g}+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}-\frac {b f n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 95, normalized size = 0.91 \[ \frac {-f \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+a g x+\frac {b g (d+e x) \log \left (c (d+e x)^n\right )}{e}-b f n \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )-b g n x}{g^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x \log \left ({\left (e x + d\right )}^{n} c\right ) + a x}{g x + f}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x}{g x + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.31, size = 463, normalized size = 4.45 \[ \frac {i \pi b f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{2 g^{2}}-\frac {i \pi b f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (g x +f \right )}{2 g^{2}}-\frac {i \pi b f \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (g x +f \right )}{2 g^{2}}+\frac {i \pi b f \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} \ln \left (g x +f \right )}{2 g^{2}}-\frac {i \pi b x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )}{2 g}+\frac {i \pi b x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2 g}+\frac {i \pi b x \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2 g}-\frac {i \pi b x \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2 g}+\frac {b f n \ln \left (\frac {d g -e f +\left (g x +f \right ) e}{d g -e f}\right ) \ln \left (g x +f \right )}{g^{2}}+\frac {b d n \ln \left (d g -e f +\left (g x +f \right ) e \right )}{e g}+\frac {b f n \dilog \left (\frac {d g -e f +\left (g x +f \right ) e}{d g -e f}\right )}{g^{2}}-\frac {b f \ln \relax (c ) \ln \left (g x +f \right )}{g^{2}}-\frac {b f \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{g^{2}}-\frac {b n x}{g}+\frac {b x \ln \relax (c )}{g}+\frac {b x \ln \left (\left (e x +d \right )^{n}\right )}{g}-\frac {a f \ln \left (g x +f \right )}{g^{2}}+\frac {a x}{g}-\frac {b f n}{g^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ a {\left (\frac {x}{g} - \frac {f \log \left (g x + f\right )}{g^{2}}\right )} + b \int \frac {x \log \left ({\left (e x + d\right )}^{n}\right ) + x \log \relax (c)}{g x + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{f+g\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )}{f + g x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________