3.67 \(\int \sqrt{d x} \text{PolyLog}(3,a x) \, dx\)

Optimal. Leaf size=121 \[ -\frac{4 (d x)^{3/2} \text{PolyLog}(2,a x)}{9 d}+\frac{2 (d x)^{3/2} \text{PolyLog}(3,a x)}{3 d}-\frac{16 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{27 a^{3/2}}+\frac{16 \sqrt{d x}}{27 a}-\frac{8 (d x)^{3/2} \log (1-a x)}{27 d}+\frac{16 (d x)^{3/2}}{81 d} \]

[Out]

(16*Sqrt[d*x])/(27*a) + (16*(d*x)^(3/2))/(81*d) - (16*Sqrt[d]*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/(27*a^(3/2
)) - (8*(d*x)^(3/2)*Log[1 - a*x])/(27*d) - (4*(d*x)^(3/2)*PolyLog[2, a*x])/(9*d) + (2*(d*x)^(3/2)*PolyLog[3, a
*x])/(3*d)

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Rubi [A]  time = 0.066747, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {6591, 2395, 50, 63, 206} \[ -\frac{4 (d x)^{3/2} \text{PolyLog}(2,a x)}{9 d}+\frac{2 (d x)^{3/2} \text{PolyLog}(3,a x)}{3 d}-\frac{16 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{27 a^{3/2}}+\frac{16 \sqrt{d x}}{27 a}-\frac{8 (d x)^{3/2} \log (1-a x)}{27 d}+\frac{16 (d x)^{3/2}}{81 d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*x]*PolyLog[3, a*x],x]

[Out]

(16*Sqrt[d*x])/(27*a) + (16*(d*x)^(3/2))/(81*d) - (16*Sqrt[d]*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/(27*a^(3/2
)) - (8*(d*x)^(3/2)*Log[1 - a*x])/(27*d) - (4*(d*x)^(3/2)*PolyLog[2, a*x])/(9*d) + (2*(d*x)^(3/2)*PolyLog[3, a
*x])/(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 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \sqrt{d x} \text{Li}_3(a x) \, dx &=\frac{2 (d x)^{3/2} \text{Li}_3(a x)}{3 d}-\frac{2}{3} \int \sqrt{d x} \text{Li}_2(a x) \, dx\\ &=-\frac{4 (d x)^{3/2} \text{Li}_2(a x)}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_3(a x)}{3 d}-\frac{4}{9} \int \sqrt{d x} \log (1-a x) \, dx\\ &=-\frac{8 (d x)^{3/2} \log (1-a x)}{27 d}-\frac{4 (d x)^{3/2} \text{Li}_2(a x)}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_3(a x)}{3 d}-\frac{(8 a) \int \frac{(d x)^{3/2}}{1-a x} \, dx}{27 d}\\ &=\frac{16 (d x)^{3/2}}{81 d}-\frac{8 (d x)^{3/2} \log (1-a x)}{27 d}-\frac{4 (d x)^{3/2} \text{Li}_2(a x)}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_3(a x)}{3 d}-\frac{8}{27} \int \frac{\sqrt{d x}}{1-a x} \, dx\\ &=\frac{16 \sqrt{d x}}{27 a}+\frac{16 (d x)^{3/2}}{81 d}-\frac{8 (d x)^{3/2} \log (1-a x)}{27 d}-\frac{4 (d x)^{3/2} \text{Li}_2(a x)}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_3(a x)}{3 d}-\frac{(8 d) \int \frac{1}{\sqrt{d x} (1-a x)} \, dx}{27 a}\\ &=\frac{16 \sqrt{d x}}{27 a}+\frac{16 (d x)^{3/2}}{81 d}-\frac{8 (d x)^{3/2} \log (1-a x)}{27 d}-\frac{4 (d x)^{3/2} \text{Li}_2(a x)}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_3(a x)}{3 d}-\frac{16 \operatorname{Subst}\left (\int \frac{1}{1-\frac{a x^2}{d}} \, dx,x,\sqrt{d x}\right )}{27 a}\\ &=\frac{16 \sqrt{d x}}{27 a}+\frac{16 (d x)^{3/2}}{81 d}-\frac{16 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{27 a^{3/2}}-\frac{8 (d x)^{3/2} \log (1-a x)}{27 d}-\frac{4 (d x)^{3/2} \text{Li}_2(a x)}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_3(a x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.188443, size = 73, normalized size = 0.6 \[ \frac{2}{81} \sqrt{d x} \left (-18 x \text{PolyLog}(2,a x)+27 x \text{PolyLog}(3,a x)+4 \left (-\frac{6 \tanh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{3/2} \sqrt{x}}-3 x \log (1-a x)+\frac{6}{a}+2 x\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d*x]*(4*(6/a + 2*x - (6*ArcTanh[Sqrt[a]*Sqrt[x]])/(a^(3/2)*Sqrt[x]) - 3*x*Log[1 - a*x]) - 18*x*PolyLog
[2, a*x] + 27*x*PolyLog[3, a*x]))/81

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Maple [A]  time = 0.058, size = 133, normalized size = 1.1 \begin{align*}{\frac{1}{a}\sqrt{dx} \left ({\frac{80\,ax+240}{405\,{a}^{2}}\sqrt{x} \left ( -a \right ) ^{{\frac{5}{2}}}}+{\frac{8}{27\,{a}^{2}}\sqrt{x} \left ( -a \right ) ^{{\frac{5}{2}}} \left ( \ln \left ( 1-\sqrt{ax} \right ) -\ln \left ( 1+\sqrt{ax} \right ) \right ){\frac{1}{\sqrt{ax}}}}-{\frac{8\,\ln \left ( -ax+1 \right ) }{27\,a}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{5}{2}}}}-{\frac{4\,{\it polylog} \left ( 2,ax \right ) }{9\,a}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{\it polylog} \left ( 3,ax \right ) }{3\,a}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{5}{2}}}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x)^(1/2)/x^(1/2)/(-a)^(1/2)/a*(2/405*x^(1/2)*(-a)^(5/2)*(40*a*x+120)/a^2+8/27*x^(1/2)*(-a)^(5/2)/a^2/(a*x)^
(1/2)*(ln(1-(a*x)^(1/2))-ln(1+(a*x)^(1/2)))-8/27*x^(3/2)*(-a)^(5/2)/a*ln(-a*x+1)-4/9*x^(3/2)*(-a)^(5/2)*polylo
g(2,a*x)/a+2/3*x^(3/2)*(-a)^(5/2)/a*polylog(3,a*x))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 2.8478, size = 593, normalized size = 4.9 \begin{align*} \left [-\frac{2 \,{\left (18 \, \sqrt{d x} a x{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) - 27 \, \sqrt{d x} a x{\rm polylog}\left (3, a x\right ) + 4 \,{\left (3 \, a x \log \left (-a x + 1\right ) - 2 \, a x - 6\right )} \sqrt{d x} - 12 \, \sqrt{\frac{d}{a}} \log \left (\frac{a d x - 2 \, \sqrt{d x} a \sqrt{\frac{d}{a}} + d}{a x - 1}\right )\right )}}{81 \, a}, -\frac{2 \,{\left (18 \, \sqrt{d x} a x{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) - 27 \, \sqrt{d x} a x{\rm polylog}\left (3, a x\right ) + 4 \,{\left (3 \, a x \log \left (-a x + 1\right ) - 2 \, a x - 6\right )} \sqrt{d x} - 24 \, \sqrt{-\frac{d}{a}} \arctan \left (\frac{\sqrt{d x} a \sqrt{-\frac{d}{a}}}{d}\right )\right )}}{81 \, a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-2/81*(18*sqrt(d*x)*a*x*\%iint(a, x, -log(-a*x + 1)/a, -log(-a*x + 1)/x) - 27*sqrt(d*x)*a*x*polylog(3, a*x) +
4*(3*a*x*log(-a*x + 1) - 2*a*x - 6)*sqrt(d*x) - 12*sqrt(d/a)*log((a*d*x - 2*sqrt(d*x)*a*sqrt(d/a) + d)/(a*x -
1)))/a, -2/81*(18*sqrt(d*x)*a*x*\%iint(a, x, -log(-a*x + 1)/a, -log(-a*x + 1)/x) - 27*sqrt(d*x)*a*x*polylog(3,
a*x) + 4*(3*a*x*log(-a*x + 1) - 2*a*x - 6)*sqrt(d*x) - 24*sqrt(-d/a)*arctan(sqrt(d*x)*a*sqrt(-d/a)/d))/a]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d x} \operatorname{Li}_{3}\left (a x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d x}{\rm Li}_{3}(a x)\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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