3.221 \(\int (d x)^m (a+b \log (c x^n)) \text{PolyLog}(2,e x^q) \, dx\)

Optimal. Leaf size=177 \[ \frac{(d x)^{m+1} \text{PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}+\frac{q \text{Unintegrable}\left ((d x)^m \log \left (1-e x^q\right ) \left (a+b \log \left (c x^n\right )\right ),x\right )}{m+1}-\frac{b n (d x)^{m+1} \text{PolyLog}\left (2,e x^q\right )}{d (m+1)^2}-\frac{b e n q^2 x^{q+1} (d x)^m \, _2F_1\left (1,\frac{m+q+1}{q};\frac{m+2 q+1}{q};e x^q\right )}{(m+1)^3 (m+q+1)}-\frac{b n q (d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)^3} \]

[Out]

-((b*e*n*q^2*x^(1 + q)*(d*x)^m*Hypergeometric2F1[1, (1 + m + q)/q, (1 + m + 2*q)/q, e*x^q])/((1 + m)^3*(1 + m
+ q))) - (b*n*q*(d*x)^(1 + m)*Log[1 - e*x^q])/(d*(1 + m)^3) - (b*n*(d*x)^(1 + m)*PolyLog[2, e*x^q])/(d*(1 + m)
^2) + ((d*x)^(1 + m)*(a + b*Log[c*x^n])*PolyLog[2, e*x^q])/(d*(1 + m)) + (q*Unintegrable[(d*x)^m*(a + b*Log[c*
x^n])*Log[1 - e*x^q], x])/(1 + m)

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Rubi [A]  time = 0.103605, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,e x^q\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(d*x)^m*(a + b*Log[c*x^n])*PolyLog[2, e*x^q],x]

[Out]

-((b*e*n*q^2*x^(1 + q)*(d*x)^m*Hypergeometric2F1[1, (1 + m + q)/q, (1 + m + 2*q)/q, e*x^q])/((1 + m)^3*(1 + m
+ q))) - (b*n*q*(d*x)^(1 + m)*Log[1 - e*x^q])/(d*(1 + m)^3) - (b*n*(d*x)^(1 + m)*PolyLog[2, e*x^q])/(d*(1 + m)
^2) + ((d*x)^(1 + m)*(a + b*Log[c*x^n])*PolyLog[2, e*x^q])/(d*(1 + m)) + (q*Defer[Int][(d*x)^m*(a + b*Log[c*x^
n])*Log[1 - e*x^q], x])/(1 + m)

Rubi steps

\begin{align*} \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (e x^q\right ) \, dx &=-\frac{b n (d x)^{1+m} \text{Li}_2\left (e x^q\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (e x^q\right )}{d (1+m)}+\frac{q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{1+m}-\frac{(b n q) \int (d x)^m \log \left (1-e x^q\right ) \, dx}{(1+m)^2}\\ &=-\frac{b n q (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^3}-\frac{b n (d x)^{1+m} \text{Li}_2\left (e x^q\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (e x^q\right )}{d (1+m)}+\frac{q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{1+m}-\frac{\left (b e n q^2\right ) \int \frac{x^{-1+q} (d x)^{1+m}}{1-e x^q} \, dx}{d (1+m)^3}\\ &=-\frac{b n q (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^3}-\frac{b n (d x)^{1+m} \text{Li}_2\left (e x^q\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (e x^q\right )}{d (1+m)}+\frac{q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{1+m}-\frac{\left (b e n q^2 x^{-m} (d x)^m\right ) \int \frac{x^{m+q}}{1-e x^q} \, dx}{(1+m)^3}\\ &=-\frac{b e n q^2 x^{1+q} (d x)^m \, _2F_1\left (1,\frac{1+m+q}{q};\frac{1+m+2 q}{q};e x^q\right )}{(1+m)^3 (1+m+q)}-\frac{b n q (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^3}-\frac{b n (d x)^{1+m} \text{Li}_2\left (e x^q\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (e x^q\right )}{d (1+m)}+\frac{q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{1+m}\\ \end{align*}

Mathematica [A]  time = 0.107549, size = 0, normalized size = 0. \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,e x^q\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(d*x)^m*(a + b*Log[c*x^n])*PolyLog[2, e*x^q],x]

[Out]

Integrate[(d*x)^m*(a + b*Log[c*x^n])*PolyLog[2, e*x^q], x]

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Maple [A]  time = 0.267, size = 867, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a+b*ln(c*x^n))*polylog(2,e*x^q),x)

[Out]

-(d*x)^m*x^(-m)*(-e)^(-1/q*m-1/q)*a/q*(-q^2*x^(1+m)*(-e)^(1/q*m+1/q)/(1+m)^2*ln(1-e*x^q)-q*x^(1+m)*(-e)^(1/q*m
+1/q)/(1+m)*polylog(2,e*x^q)-q^2*x^(1+m+q)*e*(-e)^(1/q*m+1/q)/(1+m)^2*LerchPhi(e*x^q,1,(1+m+q)/q))-(d*x)^m*x^(
-m)*(-e)^(-1/q*m-1/q)*b*ln(c)/q*(-q^2*x^(1+m)*(-e)^(1/q*m+1/q)/(1+m)^2*ln(1-e*x^q)-q*x^(1+m)*(-e)^(1/q*m+1/q)/
(1+m)*polylog(2,e*x^q)-q^2*x^(1+m+q)*e*(-e)^(1/q*m+1/q)/(1+m)^2*LerchPhi(e*x^q,1,(1+m+q)/q))+(ln(-e)/q^2*(-e)^
(-1/q*m-1/q)*(d*x)^m*x^(-m)*b*n*(-q^2*x^m*(-e)^(1/q*m+1/q)/(1+m)^2*ln(1-e*x^q)-q*x^m*(-e)^(1/q*m+1/q)/(1+m)*po
lylog(2,e*x^q)-q^2*x^(q+m)*e*(-e)^(1/q*m+1/q)/(1+m)^2*LerchPhi(e*x^q,1,(1+m+q)/q))-(-e)^(-1/q*m-1/q)*(d*x)^m*x
^(-m)*b*n/q*(-q^2*ln(x)*x^m*(-e)^(1/q*m+1/q)/(1+m)^2*ln(1-e*x^q)-q*ln(-e)*x^m*(-e)^(1/q*m+1/q)/(1+m)^2*ln(1-e*
x^q)+2*q^2*x^m*(-e)^(1/q*m+1/q)/(1+m)^3*ln(1-e*x^q)-q*ln(x)*x^m*(-e)^(1/q*m+1/q)/(1+m)*polylog(2,e*x^q)-ln(-e)
*x^m*(-e)^(1/q*m+1/q)/(1+m)*polylog(2,e*x^q)+q*x^m*(-e)^(1/q*m+1/q)/(1+m)^2*polylog(2,e*x^q)-q^2*x^(q+m)*e*ln(
x)*(-e)^(1/q*m+1/q)/(1+m)^2*LerchPhi(e*x^q,1,(1+m+q)/q)-q*x^(q+m)*e*ln(-e)*(-e)^(1/q*m+1/q)/(1+m)^2*LerchPhi(e
*x^q,1,(1+m+q)/q)+2*q^2*x^(q+m)*e*(-e)^(1/q*m+1/q)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q)+q*x^(q+m)*e*(-e)^(1/q*m
+1/q)/(1+m)^2*LerchPhi(e*x^q,2,(1+m+q)/q)))*x

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left ({\left (b d^{m} m^{2} + 2 \, b d^{m} m + b d^{m}\right )} x x^{m} \log \left (x^{n}\right ) +{\left ({\left (b \log \left (c\right ) + a\right )} d^{m} m^{2} + 2 \,{\left (b \log \left (c\right ) + a\right )} d^{m} m +{\left (b \log \left (c\right ) + a\right )} d^{m} -{\left (b d^{m} m + b d^{m}\right )} n\right )} x x^{m}\right )}{\rm Li}_2\left (e x^{q}\right ) +{\left ({\left (b d^{m} m + b d^{m}\right )} q x x^{m} \log \left (x^{n}\right ) +{\left ({\left (b \log \left (c\right ) + a\right )} d^{m} m - 2 \, b d^{m} n +{\left (b \log \left (c\right ) + a\right )} d^{m}\right )} q x x^{m}\right )} \log \left (-e x^{q} + 1\right )}{m^{3} + 3 \, m^{2} + 3 \, m + 1} - \int -\frac{{\left (b d^{m} e m + b d^{m} e\right )} q^{2} e^{\left (m \log \left (x\right ) + q \log \left (x\right )\right )} \log \left (x^{n}\right ) +{\left ({\left (b \log \left (c\right ) + a\right )} d^{m} e m - 2 \, b d^{m} e n +{\left (b \log \left (c\right ) + a\right )} d^{m} e\right )} q^{2} e^{\left (m \log \left (x\right ) + q \log \left (x\right )\right )}}{m^{3} + 3 \, m^{2} -{\left (e m^{3} + 3 \, e m^{2} + 3 \, e m + e\right )} x^{q} + 3 \, m + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*log(c*x^n))*polylog(2,e*x^q),x, algorithm="maxima")

[Out]

(((b*d^m*m^2 + 2*b*d^m*m + b*d^m)*x*x^m*log(x^n) + ((b*log(c) + a)*d^m*m^2 + 2*(b*log(c) + a)*d^m*m + (b*log(c
) + a)*d^m - (b*d^m*m + b*d^m)*n)*x*x^m)*dilog(e*x^q) + ((b*d^m*m + b*d^m)*q*x*x^m*log(x^n) + ((b*log(c) + a)*
d^m*m - 2*b*d^m*n + (b*log(c) + a)*d^m)*q*x*x^m)*log(-e*x^q + 1))/(m^3 + 3*m^2 + 3*m + 1) - integrate(-((b*d^m
*e*m + b*d^m*e)*q^2*e^(m*log(x) + q*log(x))*log(x^n) + ((b*log(c) + a)*d^m*e*m - 2*b*d^m*e*n + (b*log(c) + a)*
d^m*e)*q^2*e^(m*log(x) + q*log(x)))/(m^3 + 3*m^2 - (e*m^3 + 3*e*m^2 + 3*e*m + e)*x^q + 3*m + 1), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d x\right )^{m} b{\rm Li}_2\left (e x^{q}\right ) \log \left (c x^{n}\right ) + \left (d x\right )^{m} a{\rm Li}_2\left (e x^{q}\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*log(c*x^n))*polylog(2,e*x^q),x, algorithm="fricas")

[Out]

integral((d*x)^m*b*dilog(e*x^q)*log(c*x^n) + (d*x)^m*a*dilog(e*x^q), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(a+b*ln(c*x**n))*polylog(2,e*x**q),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )} \left (d x\right )^{m}{\rm Li}_2\left (e x^{q}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*log(c*x^n))*polylog(2,e*x^q),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*(d*x)^m*dilog(e*x^q), x)