Optimal. Leaf size=35 \[ \frac {\text {Li}_{n+1}\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6610} \[ \frac {\text {PolyLog}\left (n+1,e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6610
Rubi steps
\begin {align*} \int \frac {\text {Li}_n\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx &=\frac {\text {Li}_{1+n}\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d) n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 34, normalized size = 0.97 \[ \frac {\text {Li}_{n+1}\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b c n-a d n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm polylog}\left (n, e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{b d x^{2} + a c + {\left (b c + a d\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_{n}(e \left (\frac {b x + a}{d x + c}\right )^{n})}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\polylog \left (n , e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{\left (b x +a \right ) \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_{n}(e \left (\frac {b x + a}{d x + c}\right )^{n})}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {polylog}\left (n,e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {Li}_{n}\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n}\right )}{\left (a + b x\right ) \left (c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________