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

Optimal. Leaf size=100 \[ \frac{8 a q^2 \sqrt{d x} x^{q+1} \text{Hypergeometric2F1}\left (1,\frac{q+\frac{3}{2}}{q},\frac{1}{2} \left (\frac{3}{q}+4\right ),a x^q\right )}{9 (2 q+3)}+\frac{2 (d x)^{3/2} \text{PolyLog}\left (2,a x^q\right )}{3 d}+\frac{4 q (d x)^{3/2} \log \left (1-a x^q\right )}{9 d} \]

[Out]

(8*a*q^2*x^(1 + q)*Sqrt[d*x]*Hypergeometric2F1[1, (3/2 + q)/q, (4 + 3/q)/2, a*x^q])/(9*(3 + 2*q)) + (4*q*(d*x)
^(3/2)*Log[1 - a*x^q])/(9*d) + (2*(d*x)^(3/2)*PolyLog[2, a*x^q])/(3*d)

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Rubi [A]  time = 0.0546737, antiderivative size = 100, 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} \[ \frac{2 (d x)^{3/2} \text{PolyLog}\left (2,a x^q\right )}{3 d}+\frac{8 a q^2 \sqrt{d x} x^{q+1} \, _2F_1\left (1,\frac{q+\frac{3}{2}}{q};\frac{1}{2} \left (4+\frac{3}{q}\right );a x^q\right )}{9 (2 q+3)}+\frac{4 q (d x)^{3/2} \log \left (1-a x^q\right )}{9 d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*x]*PolyLog[2, a*x^q],x]

[Out]

(8*a*q^2*x^(1 + q)*Sqrt[d*x]*Hypergeometric2F1[1, (3/2 + q)/q, (4 + 3/q)/2, a*x^q])/(9*(3 + 2*q)) + (4*q*(d*x)
^(3/2)*Log[1 - a*x^q])/(9*d) + (2*(d*x)^(3/2)*PolyLog[2, a*x^q])/(3*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 \sqrt{d x} \text{Li}_2\left (a x^q\right ) \, dx &=\frac{2 (d x)^{3/2} \text{Li}_2\left (a x^q\right )}{3 d}+\frac{1}{3} (2 q) \int \sqrt{d x} \log \left (1-a x^q\right ) \, dx\\ &=\frac{4 q (d x)^{3/2} \log \left (1-a x^q\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_2\left (a x^q\right )}{3 d}+\frac{\left (4 a q^2\right ) \int \frac{x^{-1+q} (d x)^{3/2}}{1-a x^q} \, dx}{9 d}\\ &=\frac{4 q (d x)^{3/2} \log \left (1-a x^q\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_2\left (a x^q\right )}{3 d}+\frac{\left (4 a q^2 \sqrt{d x}\right ) \int \frac{x^{\frac{1}{2}+q}}{1-a x^q} \, dx}{9 \sqrt{x}}\\ &=\frac{8 a q^2 x^{1+q} \sqrt{d x} \, _2F_1\left (1,\frac{\frac{3}{2}+q}{q};\frac{1}{2} \left (4+\frac{3}{q}\right );a x^q\right )}{9 (3+2 q)}+\frac{4 q (d x)^{3/2} \log \left (1-a x^q\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_2\left (a x^q\right )}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.106325, size = 82, normalized size = 0.82 \[ \frac{2 x \sqrt{d x} \left (4 a q^2 x^q \text{Hypergeometric2F1}\left (1,\frac{q+\frac{3}{2}}{q},\frac{3}{2 q}+2,a x^q\right )+(2 q+3) \left (3 \text{PolyLog}\left (2,a x^q\right )+2 q \log \left (1-a x^q\right )\right )\right )}{9 (2 q+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[d*x]*(4*a*q^2*x^q*Hypergeometric2F1[1, (3/2 + q)/q, 2 + 3/(2*q), a*x^q] + (3 + 2*q)*(2*q*Log[1 - a*x
^q] + 3*PolyLog[2, a*x^q])))/(9*(3 + 2*q))

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Maple [C]  time = 0.229, size = 121, normalized size = 1.2 \begin{align*} -{\frac{1}{q}\sqrt{dx} \left ( -a \right ) ^{-{\frac{3}{2\,q}}} \left ( -{\frac{4\,{q}^{2}\ln \left ( 1-a{x}^{q} \right ) }{9}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{3}{2\,q}}}}-2\,{\frac{q{x}^{3/2} \left ( 1+2/3\,q \right ){\it polylog} \left ( 2,a{x}^{q} \right ) }{3+2\,q} \left ( -a \right ) ^{3/2\,{q}^{-1}}}-{\frac{4\,{q}^{2}a}{9}{x}^{{\frac{3}{2}}+q} \left ( -a \right ) ^{{\frac{3}{2\,q}}}{\it LerchPhi} \left ( a{x}^{q},1,{\frac{3+2\,q}{2\,q}} \right ) } \right ){\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*x)^(1/2)/x^(1/2)*(-a)^(-3/2/q)/q*(-4/9*q^2*x^(3/2)*(-a)^(3/2/q)*ln(1-a*x^q)-2*q/(3+2*q)*x^(3/2)*(-a)^(3/2/
q)*(1+2/3*q)*polylog(2,a*x^q)-4/9*q^2*x^(3/2+q)*a*(-a)^(3/2/q)*LerchPhi(a*x^q,1,1/2*(3+2*q)/q))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 8 \, \sqrt{d} q^{3} \int \frac{\sqrt{x}}{9 \,{\left ({\left (2 \, a^{2} q - 3 \, a^{2}\right )} x^{2 \, q} - 2 \,{\left (2 \, a q - 3 \, a\right )} x^{q} + 2 \, q - 3\right )}}\,{d x} + \frac{2 \,{\left (9 \,{\left ({\left (2 \, a \sqrt{d} q - 3 \, a \sqrt{d}\right )} x x^{q} -{\left (2 \, \sqrt{d} q - 3 \, \sqrt{d}\right )} x\right )} \sqrt{x}{\rm Li}_2\left (a x^{q}\right ) + 6 \,{\left ({\left (2 \, a \sqrt{d} q^{2} - 3 \, a \sqrt{d} q\right )} x x^{q} -{\left (2 \, \sqrt{d} q^{2} - 3 \, \sqrt{d} q\right )} x\right )} \sqrt{x} \log \left (-a x^{q} + 1\right ) + 4 \,{\left (2 \, \sqrt{d} q^{3} x -{\left (2 \, a \sqrt{d} q^{3} - 3 \, a \sqrt{d} q^{2}\right )} x x^{q}\right )} \sqrt{x}\right )}}{27 \,{\left ({\left (2 \, a q - 3 \, a\right )} x^{q} - 2 \, q + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8*sqrt(d)*q^3*integrate(1/9*sqrt(x)/((2*a^2*q - 3*a^2)*x^(2*q) - 2*(2*a*q - 3*a)*x^q + 2*q - 3), x) + 2/27*(9*
((2*a*sqrt(d)*q - 3*a*sqrt(d))*x*x^q - (2*sqrt(d)*q - 3*sqrt(d))*x)*sqrt(x)*dilog(a*x^q) + 6*((2*a*sqrt(d)*q^2
- 3*a*sqrt(d)*q)*x*x^q - (2*sqrt(d)*q^2 - 3*sqrt(d)*q)*x)*sqrt(x)*log(-a*x^q + 1) + 4*(2*sqrt(d)*q^3*x - (2*a
*sqrt(d)*q^3 - 3*a*sqrt(d)*q^2)*x*x^q)*sqrt(x))/((2*a*q - 3*a)*x^q - 2*q + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(d*x)*dilog(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)**(1/2)*polylog(2,a*x**q),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*x)*dilog(a*x^q), x)