3.1.95 \(\int \frac {\text {PolyLog}(3,a x^q)}{(d x)^{5/2}} \, dx\) [95]

Optimal. Leaf size=129 \[ -\frac {16 a q^3 x^{-1+q} \, _2F_1\left (1,\frac {1}{2} \left (2-\frac {3}{q}\right );\frac {1}{2} \left (4-\frac {3}{q}\right );a x^q\right )}{27 d^2 (3-2 q) \sqrt {d x}}+\frac {8 q^2 \log \left (1-a x^q\right )}{27 d (d x)^{3/2}}-\frac {4 q \text {PolyLog}\left (2,a x^q\right )}{9 d (d x)^{3/2}}-\frac {2 \text {PolyLog}\left (3,a x^q\right )}{3 d (d x)^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 129, 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 = {6726, 2505, 20, 371} \begin {gather*} -\frac {16 a q^3 x^{q-1} \, _2F_1\left (1,\frac {1}{2} \left (2-\frac {3}{q}\right );\frac {1}{2} \left (4-\frac {3}{q}\right );a x^q\right )}{27 d^2 (3-2 q) \sqrt {d x}}-\frac {4 q \text {Li}_2\left (a x^q\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3\left (a x^q\right )}{3 d (d x)^{3/2}}+\frac {8 q^2 \log \left (1-a x^q\right )}{27 d (d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[PolyLog[3, a*x^q]/(d*x)^(5/2),x]

[Out]

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

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2505

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 6726

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 \frac {\text {Li}_3\left (a x^q\right )}{(d x)^{5/2}} \, dx &=-\frac {2 \text {Li}_3\left (a x^q\right )}{3 d (d x)^{3/2}}+\frac {1}{3} (2 q) \int \frac {\text {Li}_2\left (a x^q\right )}{(d x)^{5/2}} \, dx\\ &=-\frac {4 q \text {Li}_2\left (a x^q\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3\left (a x^q\right )}{3 d (d x)^{3/2}}-\frac {1}{9} \left (4 q^2\right ) \int \frac {\log \left (1-a x^q\right )}{(d x)^{5/2}} \, dx\\ &=\frac {8 q^2 \log \left (1-a x^q\right )}{27 d (d x)^{3/2}}-\frac {4 q \text {Li}_2\left (a x^q\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3\left (a x^q\right )}{3 d (d x)^{3/2}}+\frac {\left (8 a q^3\right ) \int \frac {x^{-1+q}}{(d x)^{3/2} \left (1-a x^q\right )} \, dx}{27 d}\\ &=\frac {8 q^2 \log \left (1-a x^q\right )}{27 d (d x)^{3/2}}-\frac {4 q \text {Li}_2\left (a x^q\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3\left (a x^q\right )}{3 d (d x)^{3/2}}+\frac {\left (8 a q^3 \sqrt {x}\right ) \int \frac {x^{-\frac {5}{2}+q}}{1-a x^q} \, dx}{27 d^2 \sqrt {d x}}\\ &=-\frac {16 a q^3 x^{-1+q} \, _2F_1\left (1,\frac {1}{2} \left (2-\frac {3}{q}\right );\frac {1}{2} \left (4-\frac {3}{q}\right );a x^q\right )}{27 d^2 (3-2 q) \sqrt {d x}}+\frac {8 q^2 \log \left (1-a x^q\right )}{27 d (d x)^{3/2}}-\frac {4 q \text {Li}_2\left (a x^q\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3\left (a x^q\right )}{3 d (d x)^{3/2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
time = 0.02, size = 50, normalized size = 0.39 \begin {gather*} -\frac {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 (d x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[3, a*x^q]/(d*x)^(5/2),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 5.
time = 0.24, size = 145, normalized size = 1.12

method	result	size
meijerg	\(-\frac {x^{\frac {5}{2}} \left (-a \right )^{\frac {3}{2 q}} \left (-\frac {8 q^{3} \left (-a \right )^{-\frac {3}{2 q}} \ln \left (1-a \,x^{q}\right )}{27 x^{\frac {3}{2}}}+\frac {4 q^{2} \left (-a \right )^{-\frac {3}{2 q}} \polylog \left (2, a \,x^{q}\right )}{9 x^{\frac {3}{2}}}-\frac {2 q \left (-a \right )^{-\frac {3}{2 q}} \left (1-\frac {2 q}{3}\right ) \polylog \left (3, a \,x^{q}\right )}{\left (-3+2 q \right ) x^{\frac {3}{2}}}-\frac {8 q^{3} x^{q -\frac {3}{2}} a \left (-a \right )^{-\frac {3}{2 q}} \Phi \left (a \,x^{q}, 1, \frac {-3+2 q}{2 q}\right )}{27}\right )}{\left (d x \right )^{\frac {5}{2}} q}\)	\(145\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3,a*x^q)/(d*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/(d*x)^(5/2)*x^(5/2)*(-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)*polylog(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^(q-3/2)*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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x^q)/(d*x)^(5/2),x, algorithm="maxima")

[Out]

16*q^4*integrate(1/27/((a^2*d^(5/2)*(2*q + 3)*x^(2*q) - 2*a*d^(5/2)*(2*q + 3)*x^q + d^(5/2)*(2*q + 3))*x^(5/2)
), x) - 2/81*(18*((2*q^2 + 3*q)*a*x*x^q - (2*q^2 + 3*q)*x)*dilog(a*x^q)/x^(5/2) - 12*((2*q^3 + 3*q^2)*a*x*x^q
- (2*q^3 + 3*q^2)*x)*log(-a*x^q + 1)/x^(5/2) + 27*(a*(2*q + 3)*x*x^q - (2*q + 3)*x)*polylog(3, a*x^q)/x^(5/2)
+ 8*(2*q^4*x - (2*q^4 + 3*q^3)*a*x*x^q)/x^(5/2))/(a*d^(5/2)*(2*q + 3)*x^q - d^(5/2)*(2*q + 3))

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x^q)/(d*x)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {Li}_{3}\left (a x^{q}\right )}{\left (d x\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x**q)/(d*x)**(5/2),x)

[Out]

Integral(polylog(3, a*x**q)/(d*x)**(5/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x^q)/(d*x)^(5/2),x, algorithm="giac")

[Out]

integrate(polylog(3, a*x^q)/(d*x)^(5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {polylog}\left (3,a\,x^q\right )}{{\left (d\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3, a*x^q)/(d*x)^(5/2),x)

[Out]

int(polylog(3, a*x^q)/(d*x)^(5/2), x)

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