3.92 \(\int \sqrt {d x} \text {Li}_3(a x^q) \, dx\)

Optimal. Leaf size=124 \[ -\frac {16 a q^3 \sqrt {d x} x^{q+1} \, _2F_1\left (1,\frac {q+\frac {3}{2}}{q};\frac {1}{2} \left (4+\frac {3}{q}\right );a x^q\right )}{27 (2 q+3)}-\frac {4 q (d x)^{3/2} \text {Li}_2\left (a x^q\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^q\right )}{3 d}-\frac {8 q^2 (d x)^{3/2} \log \left (1-a x^q\right )}{27 d} \]

[Out]

-8/27*q^2*(d*x)^(3/2)*ln(1-a*x^q)/d-4/9*q*(d*x)^(3/2)*polylog(2,a*x^q)/d+2/3*(d*x)^(3/2)*polylog(3,a*x^q)/d-16
/27*a*q^3*x^(1+q)*hypergeom([1, (3/2+q)/q],[2+3/2/q],a*x^q)*(d*x)^(1/2)/(3+2*q)

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Rubi [A]  time = 0.07, antiderivative size = 124, normalized size of antiderivative = 1.00, 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} \[ -\frac {4 q (d x)^{3/2} \text {PolyLog}\left (2,a x^q\right )}{9 d}+\frac {2 (d x)^{3/2} \text {PolyLog}\left (3,a x^q\right )}{3 d}-\frac {16 a q^3 \sqrt {d x} x^{q+1} \, _2F_1\left (1,\frac {q+\frac {3}{2}}{q};\frac {1}{2} \left (4+\frac {3}{q}\right );a x^q\right )}{27 (2 q+3)}-\frac {8 q^2 (d x)^{3/2} \log \left (1-a x^q\right )}{27 d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*x]*PolyLog[3, a*x^q],x]

[Out]

(-16*a*q^3*x^(1 + q)*Sqrt[d*x]*Hypergeometric2F1[1, (3/2 + q)/q, (4 + 3/q)/2, a*x^q])/(27*(3 + 2*q)) - (8*q^2*
(d*x)^(3/2)*Log[1 - a*x^q])/(27*d) - (4*q*(d*x)^(3/2)*PolyLog[2, a*x^q])/(9*d) + (2*(d*x)^(3/2)*PolyLog[3, a*x
^q])/(3*d)

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])

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 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]

Rubi steps

\begin {align*} \int \sqrt {d x} \text {Li}_3\left (a x^q\right ) \, dx &=\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^q\right )}{3 d}-\frac {1}{3} (2 q) \int \sqrt {d x} \text {Li}_2\left (a x^q\right ) \, dx\\ &=-\frac {4 q (d x)^{3/2} \text {Li}_2\left (a x^q\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^q\right )}{3 d}-\frac {1}{9} \left (4 q^2\right ) \int \sqrt {d x} \log \left (1-a x^q\right ) \, dx\\ &=-\frac {8 q^2 (d x)^{3/2} \log \left (1-a x^q\right )}{27 d}-\frac {4 q (d x)^{3/2} \text {Li}_2\left (a x^q\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^q\right )}{3 d}-\frac {\left (8 a q^3\right ) \int \frac {x^{-1+q} (d x)^{3/2}}{1-a x^q} \, dx}{27 d}\\ &=-\frac {8 q^2 (d x)^{3/2} \log \left (1-a x^q\right )}{27 d}-\frac {4 q (d x)^{3/2} \text {Li}_2\left (a x^q\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^q\right )}{3 d}-\frac {\left (8 a q^3 \sqrt {d x}\right ) \int \frac {x^{\frac {1}{2}+q}}{1-a x^q} \, dx}{27 \sqrt {x}}\\ &=-\frac {16 a q^3 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 )}{27 (3+2 q)}-\frac {8 q^2 (d x)^{3/2} \log \left (1-a x^q\right )}{27 d}-\frac {4 q (d x)^{3/2} \text {Li}_2\left (a x^q\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^q\right )}{3 d}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 50, normalized size = 0.40 \[ -\frac {x \sqrt {d x} G_{5,5}^{1,5}\left (-a x^q|\begin {array}{c} 1,1,1,1,1-\frac {3}{2 q} \\ 1,0,0,0,-\frac {3}{2 q} \\\end {array}\right )}{q} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[d*x]*PolyLog[3, a*x^q],x]

[Out]

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

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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {d x} {\rm polylog}\left (3, a x^{q}\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(1/2)*polylog(3,a*x^q),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {d x} {\rm Li}_{3}(a x^{q})\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(1/2)*polylog(3,a*x^q),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.28, size = 145, normalized size = 1.17 \[ -\frac {\sqrt {d x}\, \left (-a \right )^{-\frac {3}{2 q}} \left (\frac {8 q^{3} x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{2 q}} \ln \left (1-a \,x^{q}\right )}{27}+\frac {4 q^{2} x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{2 q}} \polylog \left (2, a \,x^{q}\right )}{9}-\frac {2 q \,x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{2 q}} \left (1+\frac {2 q}{3}\right ) \polylog \left (3, a \,x^{q}\right )}{3+2 q}+\frac {8 q^{3} x^{\frac {3}{2}+q} a \left (-a \right )^{\frac {3}{2 q}} \Phi \left (a \,x^{q}, 1, \frac {3+2 q}{2 q}\right )}{27}\right )}{\sqrt {x}\, q} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(1/2)*polylog(3,a*x^q),x)

[Out]

-(d*x)^(1/2)/x^(1/2)*(-a)^(-3/2/q)/q*(8/27*q^3*x^(3/2)*(-a)^(3/2/q)*ln(1-a*x^q)+4/9*q^2*x^(3/2)*(-a)^(3/2/q)*p
olylog(2,a*x^q)-2*q/(3+2*q)*x^(3/2)*(-a)^(3/2/q)*(1+2/3*q)*polylog(3,a*x^q)+8/27*q^3*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 = 0.00, size = 0, normalized size = 0.00 \[ -16 \, \sqrt {d} q^{4} \int \frac {\sqrt {x}}{27 \, {\left (a^{2} {\left (2 \, q - 3\right )} x^{2 \, q} - 2 \, a {\left (2 \, q - 3\right )} x^{q} + 2 \, q - 3\right )}}\,{d x} - \frac {2 \, {\left (18 \, {\left ({\left (2 \, q^{2} - 3 \, q\right )} a \sqrt {d} x x^{q} - {\left (2 \, q^{2} - 3 \, q\right )} \sqrt {d} x\right )} \sqrt {x} {\rm Li}_2\left (a x^{q}\right ) + 12 \, {\left ({\left (2 \, q^{3} - 3 \, q^{2}\right )} a \sqrt {d} x x^{q} - {\left (2 \, q^{3} - 3 \, q^{2}\right )} \sqrt {d} x\right )} \sqrt {x} \log \left (-a x^{q} + 1\right ) - 27 \, {\left (a \sqrt {d} {\left (2 \, q - 3\right )} x x^{q} - \sqrt {d} {\left (2 \, q - 3\right )} x\right )} \sqrt {x} {\rm Li}_{3}(a x^{q}) + 8 \, {\left (2 \, \sqrt {d} q^{4} x - {\left (2 \, q^{4} - 3 \, q^{3}\right )} a \sqrt {d} x x^{q}\right )} \sqrt {x}\right )}}{81 \, {\left (a {\left (2 \, q - 3\right )} x^{q} - 2 \, q + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(1/2)*polylog(3,a*x^q),x, algorithm="maxima")

[Out]

-16*sqrt(d)*q^4*integrate(1/27*sqrt(x)/(a^2*(2*q - 3)*x^(2*q) - 2*a*(2*q - 3)*x^q + 2*q - 3), x) - 2/81*(18*((
2*q^2 - 3*q)*a*sqrt(d)*x*x^q - (2*q^2 - 3*q)*sqrt(d)*x)*sqrt(x)*dilog(a*x^q) + 12*((2*q^3 - 3*q^2)*a*sqrt(d)*x
*x^q - (2*q^3 - 3*q^2)*sqrt(d)*x)*sqrt(x)*log(-a*x^q + 1) - 27*(a*sqrt(d)*(2*q - 3)*x*x^q - sqrt(d)*(2*q - 3)*
x)*sqrt(x)*polylog(3, a*x^q) + 8*(2*sqrt(d)*q^4*x - (2*q^4 - 3*q^3)*a*sqrt(d)*x*x^q)*sqrt(x))/(a*(2*q - 3)*x^q
- 2*q + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {d\,x}\,\mathrm {polylog}\left (3,a\,x^q\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(1/2)*polylog(3, a*x^q),x)

[Out]

int((d*x)^(1/2)*polylog(3, a*x^q), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {d x} \operatorname {Li}_{3}\left (a x^{q}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(1/2)*polylog(3,a*x**q),x)

[Out]

Integral(sqrt(d*x)*polylog(3, a*x**q), x)

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