3.222 \(\int (d x)^m (a+b \log (c x^n)) \text{PolyLog}(3,e x^q) \, dx\)

Optimal. Leaf size=244 \[ -\frac{q (d x)^{m+1} \text{PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)^2}+\frac{(d x)^{m+1} \text{PolyLog}\left (3,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac{q^2 \text{Unintegrable}\left ((d x)^m \log \left (1-e x^q\right ) \left (a+b \log \left (c x^n\right )\right ),x\right )}{(m+1)^2}+\frac{2 b n q (d x)^{m+1} \text{PolyLog}\left (2,e x^q\right )}{d (m+1)^3}-\frac{b n (d x)^{m+1} \text{PolyLog}\left (3,e x^q\right )}{d (m+1)^2}+\frac{2 b e n q^3 x^{q+1} (d x)^m \, _2F_1\left (1,\frac{m+q+1}{q};\frac{m+2 q+1}{q};e x^q\right )}{(m+1)^4 (m+q+1)}+\frac{2 b n q^2 (d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)^4} \]

[Out]

(2*b*e*n*q^3*x^(1 + q)*(d*x)^m*Hypergeometric2F1[1, (1 + m + q)/q, (1 + m + 2*q)/q, e*x^q])/((1 + m)^4*(1 + m
+ q)) + (2*b*n*q^2*(d*x)^(1 + m)*Log[1 - e*x^q])/(d*(1 + m)^4) + (2*b*n*q*(d*x)^(1 + m)*PolyLog[2, e*x^q])/(d*
(1 + m)^3) - (q*(d*x)^(1 + m)*(a + b*Log[c*x^n])*PolyLog[2, e*x^q])/(d*(1 + m)^2) - (b*n*(d*x)^(1 + m)*PolyLog
[3, e*x^q])/(d*(1 + m)^2) + ((d*x)^(1 + m)*(a + b*Log[c*x^n])*PolyLog[3, e*x^q])/(d*(1 + m)) - (q^2*Unintegrab
le[(d*x)^m*(a + b*Log[c*x^n])*Log[1 - e*x^q], x])/(1 + m)^2

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Rubi [A]  time = 0.216504, 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 = {} \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,e x^q\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(2*b*e*n*q^3*x^(1 + q)*(d*x)^m*Hypergeometric2F1[1, (1 + m + q)/q, (1 + m + 2*q)/q, e*x^q])/((1 + m)^4*(1 + m
+ q)) + (2*b*n*q^2*(d*x)^(1 + m)*Log[1 - e*x^q])/(d*(1 + m)^4) + (2*b*n*q*(d*x)^(1 + m)*PolyLog[2, e*x^q])/(d*
(1 + m)^3) - (q*(d*x)^(1 + m)*(a + b*Log[c*x^n])*PolyLog[2, e*x^q])/(d*(1 + m)^2) - (b*n*(d*x)^(1 + m)*PolyLog
[3, e*x^q])/(d*(1 + m)^2) + ((d*x)^(1 + m)*(a + b*Log[c*x^n])*PolyLog[3, e*x^q])/(d*(1 + m)) - (q^2*Defer[Int]
[(d*x)^m*(a + b*Log[c*x^n])*Log[1 - e*x^q], x])/(1 + m)^2

Rubi steps

\begin{align*} \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (e x^q\right ) \, dx &=-\frac{b n (d x)^{1+m} \text{Li}_3\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}_3\left (e x^q\right )}{d (1+m)}-\frac{q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (e x^q\right ) \, dx}{1+m}+\frac{(b n q) \int (d x)^m \text{Li}_2\left (e x^q\right ) \, dx}{(1+m)^2}\\ &=\frac{2 b n q (d x)^{1+m} \text{Li}_2\left (e x^q\right )}{d (1+m)^3}-\frac{q (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)^2}-\frac{b n (d x)^{1+m} \text{Li}_3\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}_3\left (e x^q\right )}{d (1+m)}-\frac{q^2 \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{(1+m)^2}+2 \frac{\left (b n q^2\right ) \int (d x)^m \log \left (1-e x^q\right ) \, dx}{(1+m)^3}\\ &=\frac{2 b n q (d x)^{1+m} \text{Li}_2\left (e x^q\right )}{d (1+m)^3}-\frac{q (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)^2}-\frac{b n (d x)^{1+m} \text{Li}_3\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}_3\left (e x^q\right )}{d (1+m)}-\frac{q^2 \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{(1+m)^2}+2 \left (\frac{b n q^2 (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^4}+\frac{\left (b e n q^3\right ) \int \frac{x^{-1+q} (d x)^{1+m}}{1-e x^q} \, dx}{d (1+m)^4}\right )\\ &=\frac{2 b n q (d x)^{1+m} \text{Li}_2\left (e x^q\right )}{d (1+m)^3}-\frac{q (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)^2}-\frac{b n (d x)^{1+m} \text{Li}_3\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}_3\left (e x^q\right )}{d (1+m)}-\frac{q^2 \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{(1+m)^2}+2 \left (\frac{b n q^2 (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^4}+\frac{\left (b e n q^3 x^{-m} (d x)^m\right ) \int \frac{x^{m+q}}{1-e x^q} \, dx}{(1+m)^4}\right )\\ &=2 \left (\frac{b e n q^3 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)^4 (1+m+q)}+\frac{b n q^2 (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^4}\right )+\frac{2 b n q (d x)^{1+m} \text{Li}_2\left (e x^q\right )}{d (1+m)^3}-\frac{q (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)^2}-\frac{b n (d x)^{1+m} \text{Li}_3\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}_3\left (e x^q\right )}{d (1+m)}-\frac{q^2 \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{(1+m)^2}\\ \end{align*}

Mathematica [A]  time = 0.0612201, size = 0, normalized size = 0. \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,e x^q\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.089, size = 1065, normalized size = 4.4 \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(3,e*x^q),x)

[Out]

-(d*x)^m*x^(-m)*(-e)^(-1/q*m-1/q)*a/q*(q^3*x^(1+m)*(-e)^(1/q*m+1/q)/(1+m)^3*ln(1-e*x^q)+q^2*x^(1+m)*(-e)^(1/q*
m+1/q)/(1+m)^2*polylog(2,e*x^q)-q*x^(1+m)*(-e)^(1/q*m+1/q)/(1+m)*polylog(3,e*x^q)+q^3*x^(1+m+q)*e*(-e)^(1/q*m+
1/q)/(1+m)^3*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^3*x^(1+m)*(-e)^(1/q*m+
1/q)/(1+m)^3*ln(1-e*x^q)+q^2*x^(1+m)*(-e)^(1/q*m+1/q)/(1+m)^2*polylog(2,e*x^q)-q*x^(1+m)*(-e)^(1/q*m+1/q)/(1+m
)*polylog(3,e*x^q)+q^3*x^(1+m+q)*e*(-e)^(1/q*m+1/q)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q))+(1/q^2*ln(-e)*(-e)^(-
1/q*m-1/q)*(d*x)^m*x^(-m)*b*n*(q^3*x^m*(-e)^(1/q*m+1/q)/(1+m)^3*ln(1-e*x^q)+q^2*x^m*(-e)^(1/q*m+1/q)/(1+m)^2*p
olylog(2,e*x^q)-q*x^m*(-e)^(1/q*m+1/q)/(1+m)*polylog(3,e*x^q)+q^3*x^(q+m)*e*(-e)^(1/q*m+1/q)/(1+m)^3*LerchPhi(
e*x^q,1,(1+m+q)/q))-(-e)^(-1/q*m-1/q)*(d*x)^m*x^(-m)*b*n/q*(q^3*ln(x)*x^m*(-e)^(1/q*m+1/q)/(1+m)^3*ln(1-e*x^q)
+q^2*ln(-e)*x^m*(-e)^(1/q*m+1/q)/(1+m)^3*ln(1-e*x^q)-3*q^3*x^m*(-e)^(1/q*m+1/q)/(1+m)^4*ln(1-e*x^q)+q^2*ln(x)*
x^m*(-e)^(1/q*m+1/q)/(1+m)^2*polylog(2,e*x^q)+q*ln(-e)*x^m*(-e)^(1/q*m+1/q)/(1+m)^2*polylog(2,e*x^q)-2*q^2*x^m
*(-e)^(1/q*m+1/q)/(1+m)^3*polylog(2,e*x^q)-q*ln(x)*x^m*(-e)^(1/q*m+1/q)/(1+m)*polylog(3,e*x^q)-ln(-e)*x^m*(-e)
^(1/q*m+1/q)/(1+m)*polylog(3,e*x^q)+q*x^m*(-e)^(1/q*m+1/q)/(1+m)^2*polylog(3,e*x^q)+q^3*x^(q+m)*e*ln(x)*(-e)^(
1/q*m+1/q)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q)+q^2*x^(q+m)*e*ln(-e)*(-e)^(1/q*m+1/q)/(1+m)^3*LerchPhi(e*x^q,1,
(1+m+q)/q)-3*q^3*x^(q+m)*e*(-e)^(1/q*m+1/q)/(1+m)^4*LerchPhi(e*x^q,1,(1+m+q)/q)-q^2*x^(q+m)*e*(-e)^(1/q*m+1/q)
/(1+m)^3*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 (m^{2} q + 2 \, m q + q\right )} b d^{m} x x^{m} \log \left (x^{n}\right ) +{\left ({\left (m^{2} q + 2 \, m q + q\right )} a d^{m} +{\left ({\left (m^{2} q + 2 \, m q + q\right )} d^{m} \log \left (c\right ) - 2 \,{\left (m n q + n q\right )} d^{m}\right )} b\right )} x x^{m}\right )}{\rm Li}_2\left (e x^{q}\right ) +{\left ({\left (m q^{2} + q^{2}\right )} b d^{m} x x^{m} \log \left (x^{n}\right ) +{\left ({\left (m q^{2} + q^{2}\right )} a d^{m} -{\left (3 \, d^{m} n q^{2} -{\left (m q^{2} + q^{2}\right )} d^{m} \log \left (c\right )\right )} b\right )} x x^{m}\right )} \log \left (-e x^{q} + 1\right ) -{\left ({\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} b d^{m} x x^{m} \log \left (x^{n}\right ) +{\left ({\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} a d^{m} +{\left ({\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} d^{m} \log \left (c\right ) -{\left (m^{2} n + 2 \, m n + n\right )} d^{m}\right )} b\right )} x x^{m}\right )}{\rm Li}_{3}(e x^{q})}{m^{4} + 4 \, m^{3} + 6 \, m^{2} + 4 \, m + 1} + \int -\frac{{\left (m q^{3} + q^{3}\right )} b d^{m} e e^{\left (m \log \left (x\right ) + q \log \left (x\right )\right )} \log \left (x^{n}\right ) +{\left ({\left (m q^{3} + q^{3}\right )} a d^{m} e -{\left (3 \, d^{m} e n q^{3} -{\left (m q^{3} + q^{3}\right )} d^{m} e \log \left (c\right )\right )} b\right )} e^{\left (m \log \left (x\right ) + q \log \left (x\right )\right )}}{m^{4} + 4 \, m^{3} -{\left (m^{4} + 4 \, m^{3} + 6 \, m^{2} + 4 \, m + 1\right )} e x^{q} + 6 \, m^{2} + 4 \, 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(3,e*x^q),x, algorithm="maxima")

[Out]

-(((m^2*q + 2*m*q + q)*b*d^m*x*x^m*log(x^n) + ((m^2*q + 2*m*q + q)*a*d^m + ((m^2*q + 2*m*q + q)*d^m*log(c) - 2
*(m*n*q + n*q)*d^m)*b)*x*x^m)*dilog(e*x^q) + ((m*q^2 + q^2)*b*d^m*x*x^m*log(x^n) + ((m*q^2 + q^2)*a*d^m - (3*d
^m*n*q^2 - (m*q^2 + q^2)*d^m*log(c))*b)*x*x^m)*log(-e*x^q + 1) - ((m^3 + 3*m^2 + 3*m + 1)*b*d^m*x*x^m*log(x^n)
+ ((m^3 + 3*m^2 + 3*m + 1)*a*d^m + ((m^3 + 3*m^2 + 3*m + 1)*d^m*log(c) - (m^2*n + 2*m*n + n)*d^m)*b)*x*x^m)*p
olylog(3, e*x^q))/(m^4 + 4*m^3 + 6*m^2 + 4*m + 1) + integrate(-((m*q^3 + q^3)*b*d^m*e*e^(m*log(x) + q*log(x))*
log(x^n) + ((m*q^3 + q^3)*a*d^m*e - (3*d^m*e*n*q^3 - (m*q^3 + q^3)*d^m*e*log(c))*b)*e^(m*log(x) + q*log(x)))/(
m^4 + 4*m^3 - (m^4 + 4*m^3 + 6*m^2 + 4*m + 1)*e*x^q + 6*m^2 + 4*m + 1), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (\left (d x\right )^{m} b \log \left (c x^{n}\right ) + \left (d x\right )^{m} a\right )}{\rm polylog}\left (3, 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(3,e*x^q),x, algorithm="fricas")

[Out]

integral(((d*x)^m*b*log(c*x^n) + (d*x)^m*a)*polylog(3, 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(3,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}_{3}(e x^{q})\,{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(3,e*x^q),x, algorithm="giac")

[Out]

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