3.2.50 \(\int \frac {\text {PolyLog}(3,e (\frac {a+b x}{c+d x})^n)}{(a+b x) (c+d x)} \, dx\) [150]

Optimal. Leaf size=33 \[ \frac {\text {PolyLog}\left (4,e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d) n} \]

[Out]

polylog(4,e*((b*x+a)/(d*x+c))^n)/(-a*d+b*c)/n

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Rubi [A]
time = 0.04, antiderivative size = 33, 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 = {6745} \begin {gather*} \frac {\text {Li}_4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[PolyLog[3, e*((a + b*x)/(c + d*x))^n]/((a + b*x)*(c + d*x)),x]

[Out]

PolyLog[4, e*((a + b*x)/(c + d*x))^n]/((b*c - a*d)*n)

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin {align*} \int \frac {\text {Li}_3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx &=\frac {\text {Li}_4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d) n}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 32, normalized size = 0.97 \begin {gather*} \frac {\text {PolyLog}\left (4,e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b c n-a d n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[3, e*((a + b*x)/(c + d*x))^n]/((a + b*x)*(c + d*x)),x]

[Out]

PolyLog[4, e*((a + b*x)/(c + d*x))^n]/(b*c*n - a*d*n)

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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \frac {\polylog \left (3, e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{\left (b x +a \right ) \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3,e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x)

[Out]

int(polylog(3,e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x)

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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]

integrate(polylog(3,e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

-1/6*(3*(n*log(b*x + a)^2 - 2*n*log(b*x + a)*log(d*x + c) + n*log(d*x + c)^2)*dilog(e^(n*log(b*x + a) - n*log(
d*x + c) + 1)) + (n^2*log(b*x + a)^3 - 3*n^2*log(b*x + a)^2*log(d*x + c) + 3*n^2*log(b*x + a)*log(d*x + c)^2 -
n^2*log(d*x + c)^3)*log((d*x + c)^n - e^(n*log(b*x + a) + 1)) - (n^2*log(b*x + a)^3 - 3*n^2*log(b*x + a)^2*lo
g(d*x + c) + 3*n^2*log(b*x + a)*log(d*x + c)^2 - n^2*log(d*x + c)^3)*log((d*x + c)^n) - 6*(log(b*x + a) - log(
d*x + c))*polylog(3, e^(n*log(b*x + a) - n*log(d*x + c) + 1)))/(b*c - a*d) + integrate(1/6*(n^3*e*log(b*x + a)
^3 - 3*n^3*e*log(b*x + a)^2*log(d*x + c) + 3*n^3*e*log(b*x + a)*log(d*x + c)^2 - n^3*e*log(d*x + c)^3)*(b*x +
a)^n/((b*d*x^2*e + a*c*e + (b*c + a*d)*x*e)*(b*x + a)^n - (b*d*x^2 + a*c + (b*c + a*d)*x)*(d*x + c)^n), x)

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Fricas [A]
time = 0.42, size = 34, normalized size = 1.03 \begin {gather*} \frac {{\rm polylog}\left (4, \left (\frac {b x + a}{d x + c}\right )^{n} e\right )}{{\left (b c - a d\right )} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

polylog(4, ((b*x + a)/(d*x + c))^n*e)/((b*c - a*d)*n)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {Li}_{3}\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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,e*((b*x+a)/(d*x+c))**n)/(b*x+a)/(d*x+c),x)

[Out]

Integral(polylog(3, e*(a/(c + d*x) + b*x/(c + d*x))**n)/((a + b*x)*(c + d*x)), x)

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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]

integrate(polylog(3,e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

integrate(polylog(3, e*((b*x + a)/(d*x + c))^n)/((b*x + a)*(d*x + c)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {polylog}\left (3,e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3, e*((a + b*x)/(c + d*x))^n)/((a + b*x)*(c + d*x)),x)

[Out]

int(polylog(3, e*((a + b*x)/(c + d*x))^n)/((a + b*x)*(c + d*x)), x)

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