Integrand size = 15, antiderivative size = 100 \[ \int \sqrt {d x} \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\frac {8 a q^2 x^{1+q} \sqrt {d x} \operatorname {Hypergeometric2F1}\left (1,\frac {\frac {3}{2}+q}{q},\frac {1}{2} \left (4+\frac {3}{q}\right ),a x^q\right )}{9 (3+2 q)}+\frac {4 q (d x)^{3/2} \log \left (1-a x^q\right )}{9 d}+\frac {2 (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^q\right )}{3 d} \]
4/9*q*(d*x)^(3/2)*ln(1-a*x^q)/d+2/3*(d*x)^(3/2)*polylog(2,a*x^q)/d+8/9*a*q ^2*x^(1+q)*hypergeom([1, (3/2+q)/q],[2+3/2/q],a*x^q)*(d*x)^(1/2)/(3+2*q)
Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.82 \[ \int \sqrt {d x} \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\frac {2 x \sqrt {d x} \left (4 a q^2 x^q \operatorname {Hypergeometric2F1}\left (1,\frac {\frac {3}{2}+q}{q},2+\frac {3}{2 q},a x^q\right )+(3+2 q) \left (2 q \log \left (1-a x^q\right )+3 \operatorname {PolyLog}\left (2,a x^q\right )\right )\right )}{9 (3+2 q)} \]
(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))
Time = 0.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {7145, 25, 2905, 30, 888}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {d x} \operatorname {PolyLog}\left (2,a x^q\right ) \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {2 (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^q\right )}{3 d}-\frac {2}{3} q \int -\sqrt {d x} \log \left (1-a x^q\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2}{3} q \int \sqrt {d x} \log \left (1-a x^q\right )dx+\frac {2 (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^q\right )}{3 d}\) |
\(\Big \downarrow \) 2905 |
\(\displaystyle \frac {2}{3} q \left (\frac {2 a q \int \frac {x^{q-1} (d x)^{3/2}}{1-a x^q}dx}{3 d}+\frac {2 (d x)^{3/2} \log \left (1-a x^q\right )}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^q\right )}{3 d}\) |
\(\Big \downarrow \) 30 |
\(\displaystyle \frac {2}{3} q \left (\frac {2 a q \sqrt {d x} \int \frac {x^{q+\frac {1}{2}}}{1-a x^q}dx}{3 \sqrt {x}}+\frac {2 (d x)^{3/2} \log \left (1-a x^q\right )}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^q\right )}{3 d}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {2}{3} q \left (\frac {4 a q \sqrt {d x} x^{q+1} \operatorname {Hypergeometric2F1}\left (1,\frac {q+\frac {3}{2}}{q},\frac {1}{2} \left (4+\frac {3}{q}\right ),a x^q\right )}{3 (2 q+3)}+\frac {2 (d x)^{3/2} \log \left (1-a x^q\right )}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^q\right )}{3 d}\) |
(2*q*((4*a*q*x^(1 + q)*Sqrt[d*x]*Hypergeometric2F1[1, (3/2 + q)/q, (4 + 3/ q)/2, a*x^q])/(3*(3 + 2*q)) + (2*(d*x)^(3/2)*Log[1 - a*x^q])/(3*d)))/3 + ( 2*(d*x)^(3/2)*PolyLog[2, a*x^q])/(3*d)
3.1.87.3.1 Defintions of rubi rules used
Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & & !IntegerQ[p]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
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] - Simp[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]
Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbo l] :> Simp[(d*x)^(m + 1)*(PolyLog[n, a*(b*x^p)^q]/(d*(m + 1))), x] - Simp[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]
Result contains higher order function than in optimal. Order 9 vs. order 5.
Time = 0.50 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.21
method | result | size |
meijerg | \(-\frac {\sqrt {d x}\, \left (-a \right )^{-\frac {3}{2 q}} \left (-\frac {4 q^{2} x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{2 q}} \ln \left (1-a \,x^{q}\right )}{9}-\frac {2 q \,x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{2 q}} \left (1+\frac {2 q}{3}\right ) \operatorname {polylog}\left (2, a \,x^{q}\right )}{3+2 q}-\frac {4 q^{2} x^{\frac {3}{2}+q} a \left (-a \right )^{\frac {3}{2 q}} \operatorname {LerchPhi}\left (a \,x^{q}, 1, \frac {3+2 q}{2 q}\right )}{9}\right )}{\sqrt {x}\, q}\) | \(121\) |
-(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))
\[ \int \sqrt {d x} \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\int { \sqrt {d x} {\rm Li}_2\left (a x^{q}\right ) \,d x } \]
\[ \int \sqrt {d x} \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\int \sqrt {d x} \operatorname {Li}_{2}\left (a x^{q}\right )\, dx \]
\[ \int \sqrt {d x} \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\int { \sqrt {d x} {\rm Li}_2\left (a x^{q}\right ) \,d x } \]
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)*s qrt(x))/((2*a*q - 3*a)*x^q - 2*q + 3)
\[ \int \sqrt {d x} \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\int { \sqrt {d x} {\rm Li}_2\left (a x^{q}\right ) \,d x } \]
Timed out. \[ \int \sqrt {d x} \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\int \sqrt {d\,x}\,\mathrm {polylog}\left (2,a\,x^q\right ) \,d x \]