3.88 \(\int \frac{\text{PolyLog}(2,a x^q)}{\sqrt{d x}} \, dx\)

Optimal. Leaf size=93 \[ \frac{8 a q^2 \sqrt{d x} x^q \text{Hypergeometric2F1}\left (1,\frac{q+\frac{1}{2}}{q},\frac{1}{2} \left (\frac{1}{q}+4\right ),a x^q\right )}{d (2 q+1)}+\frac{2 \sqrt{d x} \text{PolyLog}\left (2,a x^q\right )}{d}+\frac{4 q \sqrt{d x} \log \left (1-a x^q\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.0540382, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {6591, 2455, 20, 364} \[ \frac{2 \sqrt{d x} \text{PolyLog}\left (2,a x^q\right )}{d}+\frac{8 a q^2 \sqrt{d x} x^q \, _2F_1\left (1,\frac{q+\frac{1}{2}}{q};\frac{1}{2} \left (4+\frac{1}{q}\right );a x^q\right )}{d (2 q+1)}+\frac{4 q \sqrt{d x} \log \left (1-a x^q\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x^q]/Sqrt[d*x],x]

[Out]

(8*a*q^2*x^q*Sqrt[d*x]*Hypergeometric2F1[1, (1/2 + q)/q, (4 + q^(-1))/2, a*x^q])/(d*(1 + 2*q)) + (4*q*Sqrt[d*x
]*Log[1 - a*x^q])/d + (2*Sqrt[d*x]*PolyLog[2, a*x^q])/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 \frac{\text{Li}_2\left (a x^q\right )}{\sqrt{d x}} \, dx &=\frac{2 \sqrt{d x} \text{Li}_2\left (a x^q\right )}{d}+(2 q) \int \frac{\log \left (1-a x^q\right )}{\sqrt{d x}} \, dx\\ &=\frac{4 q \sqrt{d x} \log \left (1-a x^q\right )}{d}+\frac{2 \sqrt{d x} \text{Li}_2\left (a x^q\right )}{d}+\frac{\left (4 a q^2\right ) \int \frac{x^{-1+q} \sqrt{d x}}{1-a x^q} \, dx}{d}\\ &=\frac{4 q \sqrt{d x} \log \left (1-a x^q\right )}{d}+\frac{2 \sqrt{d x} \text{Li}_2\left (a x^q\right )}{d}+\frac{\left (4 a q^2 \sqrt{d x}\right ) \int \frac{x^{-\frac{1}{2}+q}}{1-a x^q} \, dx}{d \sqrt{x}}\\ &=\frac{8 a q^2 x^q \sqrt{d x} \, _2F_1\left (1,\frac{\frac{1}{2}+q}{q};\frac{1}{2} \left (4+\frac{1}{q}\right );a x^q\right )}{d (1+2 q)}+\frac{4 q \sqrt{d x} \log \left (1-a x^q\right )}{d}+\frac{2 \sqrt{d x} \text{Li}_2\left (a x^q\right )}{d}\\ \end{align*}

Mathematica [C]  time = 0.0209061, size = 48, normalized size = 0.52 \[ -\frac{x G_{4,4}^{1,4}\left (-a x^q|\begin{array}{c} 1,1,1,1-\frac{1}{2 q} \\ 1,0,0,-\frac{1}{2 q} \\\end{array}\right )}{q \sqrt{d x}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[PolyLog[2, a*x^q]/Sqrt[d*x],x]

[Out]

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

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Maple [C]  time = 0.232, size = 109, normalized size = 1.2 \begin{align*} -{\frac{1}{q}\sqrt{x} \left ( -a \right ) ^{-{\frac{1}{2\,q}}} \left ( -4\,{q}^{2}\sqrt{x} \left ( -a \right ) ^{1/2\,{q}^{-1}}\ln \left ( 1-a{x}^{q} \right ) -2\,q\sqrt{x} \left ( -a \right ) ^{1/2\,{q}^{-1}}{\it polylog} \left ( 2,a{x}^{q} \right ) -4\,{q}^{2}{x}^{1/2+q}a \left ( -a \right ) ^{1/2\,{q}^{-1}}{\it LerchPhi} \left ( a{x}^{q},1,1/2\,{\frac{1+2\,q}{q}} \right ) \right ){\frac{1}{\sqrt{dx}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,a*x^q)/(d*x)^(1/2),x)

[Out]

-1/(d*x)^(1/2)*x^(1/2)*(-a)^(-1/2/q)/q*(-4*q^2*x^(1/2)*(-a)^(1/2/q)*ln(1-a*x^q)-2*q*x^(1/2)*(-a)^(1/2/q)*polyl
og(2,a*x^q)-4*q^2*x^(1/2+q)*a*(-a)^(1/2/q)*LerchPhi(a*x^q,1,1/2*(1+2*q)/q))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x^q)/(d*x)^(1/2),x, algorithm="maxima")

[Out]

8*q^3*integrate(1/(((2*a^2*sqrt(d)*q - a^2*sqrt(d))*x^(2*q) - 2*(2*a*sqrt(d)*q - a*sqrt(d))*x^q + 2*sqrt(d)*q
- sqrt(d))*sqrt(x)), x) - 2*(((2*a*sqrt(d)*q - a*sqrt(d))*x*x^q - (2*sqrt(d)*q - sqrt(d))*x)*dilog(a*x^q)/sqrt
(x) + 2*((2*a*sqrt(d)*q^2 - a*sqrt(d)*q)*x*x^q - (2*sqrt(d)*q^2 - sqrt(d)*q)*x)*log(-a*x^q + 1)/sqrt(x) + 4*(2
*sqrt(d)*q^3*x - (2*a*sqrt(d)*q^3 - a*sqrt(d)*q^2)*x*x^q)/sqrt(x))/(2*d*q - (2*a*d*q - a*d)*x^q - d)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x}{\rm Li}_2\left (a x^{q}\right )}{d x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x^q)/(d*x)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*x)*dilog(a*x^q)/(d*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x**q)/(d*x)**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_2\left (a x^{q}\right )}{\sqrt{d x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x^q)/(d*x)^(1/2),x, algorithm="giac")

[Out]

integrate(dilog(a*x^q)/sqrt(d*x), x)