3.103 \(\int (d x)^m \text{PolyLog}(3,a x) \, dx\)

Optimal. Leaf size=102 \[ -\frac{a (d x)^{m+2} \text{Hypergeometric2F1}(1,m+2,m+3,a x)}{d^2 (m+1)^3 (m+2)}-\frac{(d x)^{m+1} \text{PolyLog}(2,a x)}{d (m+1)^2}+\frac{(d x)^{m+1} \text{PolyLog}(3,a x)}{d (m+1)}-\frac{\log (1-a x) (d x)^{m+1}}{d (m+1)^3} \]

[Out]

-((a*(d*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, a*x])/(d^2*(1 + m)^3*(2 + m))) - ((d*x)^(1 + m)*Log[1 -
a*x])/(d*(1 + m)^3) - ((d*x)^(1 + m)*PolyLog[2, a*x])/(d*(1 + m)^2) + ((d*x)^(1 + m)*PolyLog[3, a*x])/(d*(1 +
m))

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Rubi [A]  time = 0.0636762, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6591, 2395, 64} \[ -\frac{(d x)^{m+1} \text{PolyLog}(2,a x)}{d (m+1)^2}+\frac{(d x)^{m+1} \text{PolyLog}(3,a x)}{d (m+1)}-\frac{a (d x)^{m+2} \, _2F_1(1,m+2;m+3;a x)}{d^2 (m+1)^3 (m+2)}-\frac{\log (1-a x) (d x)^{m+1}}{d (m+1)^3} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*PolyLog[3, a*x],x]

[Out]

-((a*(d*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, a*x])/(d^2*(1 + m)^3*(2 + m))) - ((d*x)^(1 + m)*Log[1 -
a*x])/(d*(1 + m)^3) - ((d*x)^(1 + m)*PolyLog[2, a*x])/(d*(1 + m)^2) + ((d*x)^(1 + m)*PolyLog[3, a*x])/(d*(1 +
m))

Rule 6591

Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbol] :> Simp[((d*x)^(m + 1)*PolyLog[n
, a*(b*x^p)^q])/(d*(m + 1)), x] - Dist[(p*q)/(m + 1), Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ
[{a, b, d, m, p, q}, x] && NeQ[m, -1] && GtQ[n, 0]

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-(d/(b*c)), 0])))

Rubi steps

\begin{align*} \int (d x)^m \text{Li}_3(a x) \, dx &=\frac{(d x)^{1+m} \text{Li}_3(a x)}{d (1+m)}-\frac{\int (d x)^m \text{Li}_2(a x) \, dx}{1+m}\\ &=-\frac{(d x)^{1+m} \text{Li}_2(a x)}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_3(a x)}{d (1+m)}-\frac{\int (d x)^m \log (1-a x) \, dx}{(1+m)^2}\\ &=-\frac{(d x)^{1+m} \log (1-a x)}{d (1+m)^3}-\frac{(d x)^{1+m} \text{Li}_2(a x)}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_3(a x)}{d (1+m)}-\frac{a \int \frac{(d x)^{1+m}}{1-a x} \, dx}{d (1+m)^3}\\ &=-\frac{a (d x)^{2+m} \, _2F_1(1,2+m;3+m;a x)}{d^2 (1+m)^3 (2+m)}-\frac{(d x)^{1+m} \log (1-a x)}{d (1+m)^3}-\frac{(d x)^{1+m} \text{Li}_2(a x)}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_3(a x)}{d (1+m)}\\ \end{align*}

Mathematica [C]  time = 0.0598987, size = 88, normalized size = 0.86 \[ -\frac{x \text{Gamma}(m+2) (d x)^m \left (a (m+1) x \text{Gamma}(m+1) \, _2\tilde{F}_1(1,m+2;m+3;a x)+m^2 (-\text{PolyLog}(3,a x))-2 m \text{PolyLog}(3,a x)+(m+1) \text{PolyLog}(2,a x)-\text{PolyLog}(3,a x)+\log (1-a x)\right )}{(m+1)^4 \text{Gamma}(m+1)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(d*x)^m*PolyLog[3, a*x],x]

[Out]

-((x*(d*x)^m*Gamma[2 + m]*(a*(1 + m)*x*Gamma[1 + m]*HypergeometricPFQRegularized[{1, 2 + m}, {3 + m}, a*x] + L
og[1 - a*x] + (1 + m)*PolyLog[2, a*x] - PolyLog[3, a*x] - 2*m*PolyLog[3, a*x] - m^2*PolyLog[3, a*x]))/((1 + m)
^4*Gamma[1 + m]))

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Maple [C]  time = 0.388, size = 173, normalized size = 1.7 \begin{align*}{\frac{ \left ( dx \right ) ^{m}{x}^{-m} \left ( -a \right ) ^{-m}}{a} \left ({\frac{{x}^{m} \left ( -a \right ) ^{m} \left ( a{m}^{2}x+2\,amx+{m}^{2}+3\,m+2 \right ) }{ \left ( 1+m \right ) ^{4} \left ( 2+m \right ) m}}-{\frac{{x}^{1+m}a \left ( -a \right ) ^{m}\ln \left ( -ax+1 \right ) }{ \left ( 1+m \right ) ^{3}}}+{\frac{{x}^{1+m}a \left ( -a \right ) ^{m} \left ( -m-2 \right ){\it polylog} \left ( 2,ax \right ) }{ \left ( 1+m \right ) ^{2} \left ( 2+m \right ) }}+{\frac{{x}^{1+m}a \left ( -a \right ) ^{m}{\it polylog} \left ( 3,ax \right ) }{1+m}}+{\frac{{x}^{m} \left ( -a \right ) ^{m} \left ( -m-2 \right ){\it LerchPhi} \left ( ax,1,m \right ) }{ \left ( 1+m \right ) ^{3} \left ( 2+m \right ) }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*polylog(3,a*x),x)

[Out]

(d*x)^m*x^(-m)*(-a)^(-m)/a*(1/(2+m)*x^m*(-a)^m*(a*m^2*x+2*a*m*x+m^2+3*m+2)/(1+m)^4/m-x^(1+m)*a*(-a)^m/(1+m)^3*
ln(-a*x+1)+1/(2+m)*x^(1+m)*a*(-a)^m*(-m-2)/(1+m)^2*polylog(2,a*x)+x^(1+m)*a*(-a)^m/(1+m)*polylog(3,a*x)+1/(2+m
)*x^m*(-a)^m*(-m-2)/(1+m)^3*LerchPhi(a*x,1,m))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*polylog(3,a*x),x, algorithm="maxima")

[Out]

a*d^m*integrate(-x*x^m/(m^3 - (m^3 + 3*m^2 + 3*m + 1)*a*x + 3*m^2 + 3*m + 1), x) - (d^m*(m + 1)*x*x^m*dilog(a*
x) - (m^2 + 2*m + 1)*d^m*x*x^m*polylog(3, a*x) + d^m*x*x^m*log(-a*x + 1))/(m^3 + 3*m^2 + 3*m + 1)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d x\right )^{m}{\rm polylog}\left (3, a x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*polylog(3,a*x),x, algorithm="fricas")

[Out]

integral((d*x)^m*polylog(3, a*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{m} \operatorname{Li}_{3}\left (a x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*polylog(3,a*x),x)

[Out]

Integral((d*x)**m*polylog(3, a*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{m}{\rm Li}_{3}(a x)\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*polylog(3,a*x),x, algorithm="giac")

[Out]

integrate((d*x)^m*polylog(3, a*x), x)