3.91 \(\int (d x)^{3/2} \text{PolyLog}(3,a x^q) \, dx\)

Optimal. Leaf size=125 \[ -\frac{16 a d q^3 \sqrt{d x} x^{q+2} \text{Hypergeometric2F1}\left (1,\frac{q+\frac{5}{2}}{q},\frac{1}{2} \left (\frac{5}{q}+4\right ),a x^q\right )}{125 (2 q+5)}-\frac{4 q (d x)^{5/2} \text{PolyLog}\left (2,a x^q\right )}{25 d}+\frac{2 (d x)^{5/2} \text{PolyLog}\left (3,a x^q\right )}{5 d}-\frac{8 q^2 (d x)^{5/2} \log \left (1-a x^q\right )}{125 d} \]

[Out]

(-16*a*d*q^3*x^(2 + q)*Sqrt[d*x]*Hypergeometric2F1[1, (5/2 + q)/q, (4 + 5/q)/2, a*x^q])/(125*(5 + 2*q)) - (8*q
^2*(d*x)^(5/2)*Log[1 - a*x^q])/(125*d) - (4*q*(d*x)^(5/2)*PolyLog[2, a*x^q])/(25*d) + (2*(d*x)^(5/2)*PolyLog[3
, a*x^q])/(5*d)

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Rubi [A]  time = 0.0748302, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {6591, 2455, 20, 364} \[ -\frac{4 q (d x)^{5/2} \text{PolyLog}\left (2,a x^q\right )}{25 d}+\frac{2 (d x)^{5/2} \text{PolyLog}\left (3,a x^q\right )}{5 d}-\frac{16 a d q^3 \sqrt{d x} x^{q+2} \, _2F_1\left (1,\frac{q+\frac{5}{2}}{q};\frac{1}{2} \left (4+\frac{5}{q}\right );a x^q\right )}{125 (2 q+5)}-\frac{8 q^2 (d x)^{5/2} \log \left (1-a x^q\right )}{125 d} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(3/2)*PolyLog[3, a*x^q],x]

[Out]

(-16*a*d*q^3*x^(2 + q)*Sqrt[d*x]*Hypergeometric2F1[1, (5/2 + q)/q, (4 + 5/q)/2, a*x^q])/(125*(5 + 2*q)) - (8*q
^2*(d*x)^(5/2)*Log[1 - a*x^q])/(125*d) - (4*q*(d*x)^(5/2)*PolyLog[2, a*x^q])/(25*d) + (2*(d*x)^(5/2)*PolyLog[3
, a*x^q])/(5*d)

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 2455

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

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int (d x)^{3/2} \text{Li}_3\left (a x^q\right ) \, dx &=\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^q\right )}{5 d}-\frac{1}{5} (2 q) \int (d x)^{3/2} \text{Li}_2\left (a x^q\right ) \, dx\\ &=-\frac{4 q (d x)^{5/2} \text{Li}_2\left (a x^q\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^q\right )}{5 d}-\frac{1}{25} \left (4 q^2\right ) \int (d x)^{3/2} \log \left (1-a x^q\right ) \, dx\\ &=-\frac{8 q^2 (d x)^{5/2} \log \left (1-a x^q\right )}{125 d}-\frac{4 q (d x)^{5/2} \text{Li}_2\left (a x^q\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^q\right )}{5 d}-\frac{\left (8 a q^3\right ) \int \frac{x^{-1+q} (d x)^{5/2}}{1-a x^q} \, dx}{125 d}\\ &=-\frac{8 q^2 (d x)^{5/2} \log \left (1-a x^q\right )}{125 d}-\frac{4 q (d x)^{5/2} \text{Li}_2\left (a x^q\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^q\right )}{5 d}-\frac{\left (8 a d q^3 \sqrt{d x}\right ) \int \frac{x^{\frac{3}{2}+q}}{1-a x^q} \, dx}{125 \sqrt{x}}\\ &=-\frac{16 a d q^3 x^{2+q} \sqrt{d x} \, _2F_1\left (1,\frac{\frac{5}{2}+q}{q};\frac{1}{2} \left (4+\frac{5}{q}\right );a x^q\right )}{125 (5+2 q)}-\frac{8 q^2 (d x)^{5/2} \log \left (1-a x^q\right )}{125 d}-\frac{4 q (d x)^{5/2} \text{Li}_2\left (a x^q\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^q\right )}{5 d}\\ \end{align*}

Mathematica [C]  time = 0.0297026, size = 50, normalized size = 0.4 \[ -\frac{x (d x)^{3/2} G_{5,5}^{1,5}\left (-a x^q|\begin{array}{c} 1,1,1,1,1-\frac{5}{2 q} \\ 1,0,0,0,-\frac{5}{2 q} \\\end{array}\right )}{q} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(d*x)^(3/2)*PolyLog[3, a*x^q],x]

[Out]

-((x*(d*x)^(3/2)*MeijerG[{{1, 1, 1, 1, 1 - 5/(2*q)}, {}}, {{1}, {0, 0, 0, -5/(2*q)}}, -(a*x^q)])/q)

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Maple [C]  time = 0.379, size = 145, normalized size = 1.2 \begin{align*} -{\frac{1}{q} \left ( dx \right ) ^{{\frac{3}{2}}} \left ( -a \right ) ^{-{\frac{5}{2\,q}}} \left ({\frac{8\,{q}^{3}\ln \left ( 1-a{x}^{q} \right ) }{125}{x}^{{\frac{5}{2}}} \left ( -a \right ) ^{{\frac{5}{2\,q}}}}+{\frac{4\,{q}^{2}{\it polylog} \left ( 2,a{x}^{q} \right ) }{25}{x}^{{\frac{5}{2}}} \left ( -a \right ) ^{{\frac{5}{2\,q}}}}-2\,{\frac{q{x}^{5/2} \left ( 1+2/5\,q \right ){\it polylog} \left ( 3,a{x}^{q} \right ) }{5+2\,q} \left ( -a \right ) ^{5/2\,{q}^{-1}}}+{\frac{8\,{q}^{3}a}{125}{x}^{{\frac{5}{2}}+q} \left ( -a \right ) ^{{\frac{5}{2\,q}}}{\it LerchPhi} \left ( a{x}^{q},1,{\frac{5+2\,q}{2\,q}} \right ) } \right ){x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(3/2)*polylog(3,a*x^q),x)

[Out]

-(d*x)^(3/2)/x^(3/2)*(-a)^(-5/2/q)/q*(8/125*q^3*x^(5/2)*(-a)^(5/2/q)*ln(1-a*x^q)+4/25*q^2*x^(5/2)*(-a)^(5/2/q)
*polylog(2,a*x^q)-2*q/(5+2*q)*x^(5/2)*(-a)^(5/2/q)*(1+2/5*q)*polylog(3,a*x^q)+8/125*q^3*x^(5/2+q)*a*(-a)^(5/2/
q)*LerchPhi(a*x^q,1,1/2*(5+2*q)/q))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -16 \, d^{\frac{3}{2}} q^{4} \int \frac{x^{\frac{3}{2}}}{125 \,{\left (a^{2}{\left (2 \, q - 5\right )} x^{2 \, q} - 2 \, a{\left (2 \, q - 5\right )} x^{q} + 2 \, q - 5\right )}}\,{d x} - \frac{2 \,{\left (50 \,{\left ({\left (2 \, q^{2} - 5 \, q\right )} a d^{\frac{3}{2}} x x^{q} -{\left (2 \, q^{2} - 5 \, q\right )} d^{\frac{3}{2}} x\right )} x^{\frac{3}{2}}{\rm Li}_2\left (a x^{q}\right ) + 20 \,{\left ({\left (2 \, q^{3} - 5 \, q^{2}\right )} a d^{\frac{3}{2}} x x^{q} -{\left (2 \, q^{3} - 5 \, q^{2}\right )} d^{\frac{3}{2}} x\right )} x^{\frac{3}{2}} \log \left (-a x^{q} + 1\right ) - 125 \,{\left (a d^{\frac{3}{2}}{\left (2 \, q - 5\right )} x x^{q} - d^{\frac{3}{2}}{\left (2 \, q - 5\right )} x\right )} x^{\frac{3}{2}}{\rm Li}_{3}(a x^{q}) + 8 \,{\left (2 \, d^{\frac{3}{2}} q^{4} x -{\left (2 \, q^{4} - 5 \, q^{3}\right )} a d^{\frac{3}{2}} x x^{q}\right )} x^{\frac{3}{2}}\right )}}{625 \,{\left (a{\left (2 \, q - 5\right )} x^{q} - 2 \, q + 5\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(3/2)*polylog(3,a*x^q),x, algorithm="maxima")

[Out]

-16*d^(3/2)*q^4*integrate(1/125*x^(3/2)/(a^2*(2*q - 5)*x^(2*q) - 2*a*(2*q - 5)*x^q + 2*q - 5), x) - 2/625*(50*
((2*q^2 - 5*q)*a*d^(3/2)*x*x^q - (2*q^2 - 5*q)*d^(3/2)*x)*x^(3/2)*dilog(a*x^q) + 20*((2*q^3 - 5*q^2)*a*d^(3/2)
*x*x^q - (2*q^3 - 5*q^2)*d^(3/2)*x)*x^(3/2)*log(-a*x^q + 1) - 125*(a*d^(3/2)*(2*q - 5)*x*x^q - d^(3/2)*(2*q -
5)*x)*x^(3/2)*polylog(3, a*x^q) + 8*(2*d^(3/2)*q^4*x - (2*q^4 - 5*q^3)*a*d^(3/2)*x*x^q)*x^(3/2))/(a*(2*q - 5)*
x^q - 2*q + 5)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(3/2)*polylog(3,a*x^q),x, algorithm="fricas")

[Out]

integral(sqrt(d*x)*d*x*polylog(3, a*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)**(3/2)*polylog(3,a*x**q),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{\frac{3}{2}}{\rm Li}_{3}(a x^{q})\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(3/2)*polylog(3,a*x^q),x, algorithm="giac")

[Out]

integrate((d*x)^(3/2)*polylog(3, a*x^q), x)