3.63 \(\int \frac{\text{PolyLog}(2,a x)}{(d x)^{5/2}} \, dx\)

Optimal. Leaf size=89 \[ -\frac{2 \text{PolyLog}(2,a x)}{3 d (d x)^{3/2}}+\frac{8 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{9 d^{5/2}}-\frac{8 a}{9 d^2 \sqrt{d x}}+\frac{4 \log (1-a x)}{9 d (d x)^{3/2}} \]

[Out]

(-8*a)/(9*d^2*Sqrt[d*x]) + (8*a^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/(9*d^(5/2)) + (4*Log[1 - a*x])/(9*
d*(d*x)^(3/2)) - (2*PolyLog[2, a*x])/(3*d*(d*x)^(3/2))

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Rubi [A]  time = 0.0504307, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {6591, 2395, 51, 63, 206} \[ -\frac{2 \text{PolyLog}(2,a x)}{3 d (d x)^{3/2}}+\frac{8 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{9 d^{5/2}}-\frac{8 a}{9 d^2 \sqrt{d x}}+\frac{4 \log (1-a x)}{9 d (d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x]/(d*x)^(5/2),x]

[Out]

(-8*a)/(9*d^2*Sqrt[d*x]) + (8*a^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/(9*d^(5/2)) + (4*Log[1 - a*x])/(9*
d*(d*x)^(3/2)) - (2*PolyLog[2, a*x])/(3*d*(d*x)^(3/2))

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 \frac{\text{Li}_2(a x)}{(d x)^{5/2}} \, dx &=-\frac{2 \text{Li}_2(a x)}{3 d (d x)^{3/2}}-\frac{2}{3} \int \frac{\log (1-a x)}{(d x)^{5/2}} \, dx\\ &=\frac{4 \log (1-a x)}{9 d (d x)^{3/2}}-\frac{2 \text{Li}_2(a x)}{3 d (d x)^{3/2}}+\frac{(4 a) \int \frac{1}{(d x)^{3/2} (1-a x)} \, dx}{9 d}\\ &=-\frac{8 a}{9 d^2 \sqrt{d x}}+\frac{4 \log (1-a x)}{9 d (d x)^{3/2}}-\frac{2 \text{Li}_2(a x)}{3 d (d x)^{3/2}}+\frac{\left (4 a^2\right ) \int \frac{1}{\sqrt{d x} (1-a x)} \, dx}{9 d^2}\\ &=-\frac{8 a}{9 d^2 \sqrt{d x}}+\frac{4 \log (1-a x)}{9 d (d x)^{3/2}}-\frac{2 \text{Li}_2(a x)}{3 d (d x)^{3/2}}+\frac{\left (8 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{a x^2}{d}} \, dx,x,\sqrt{d x}\right )}{9 d^3}\\ &=-\frac{8 a}{9 d^2 \sqrt{d x}}+\frac{8 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{9 d^{5/2}}+\frac{4 \log (1-a x)}{9 d (d x)^{3/2}}-\frac{2 \text{Li}_2(a x)}{3 d (d x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0741523, size = 57, normalized size = 0.64 \[ -\frac{2 x \left (3 \text{PolyLog}(2,a x)-4 a^{3/2} x^{3/2} \tanh ^{-1}\left (\sqrt{a} \sqrt{x}\right )+4 a x-2 \log (1-a x)\right )}{9 (d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x*(4*a*x - 4*a^(3/2)*x^(3/2)*ArcTanh[Sqrt[a]*Sqrt[x]] - 2*Log[1 - a*x] + 3*PolyLog[2, a*x]))/(9*(d*x)^(5/2
))

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Maple [A]  time = 0.055, size = 76, normalized size = 0.9 \begin{align*} -{\frac{2\,{\it polylog} \left ( 2,ax \right ) }{3\,d} \left ( dx \right ) ^{-{\frac{3}{2}}}}+{\frac{4}{9\,d}\ln \left ({\frac{-adx+d}{d}} \right ) \left ( dx \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,a}{9\,{d}^{2}}{\frac{1}{\sqrt{dx}}}}+{\frac{8\,{a}^{2}}{9\,{d}^{2}}{\it Artanh} \left ({a\sqrt{dx}{\frac{1}{\sqrt{ad}}}} \right ){\frac{1}{\sqrt{ad}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,a*x)/(d*x)^(5/2),x)

[Out]

-2/3*polylog(2,a*x)/d/(d*x)^(3/2)+4/9/d/(d*x)^(3/2)*ln((-a*d*x+d)/d)-8/9*a/d^2/(d*x)^(1/2)+8/9/d^2*a^2/(a*d)^(
1/2)*arctanh(a*(d*x)^(1/2)/(a*d)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.66517, size = 365, normalized size = 4.1 \begin{align*} \left [\frac{2 \,{\left (2 \, a d x^{2} \sqrt{\frac{a}{d}} \log \left (\frac{a x + 2 \, \sqrt{d x} \sqrt{\frac{a}{d}} + 1}{a x - 1}\right ) -{\left (4 \, a x + 3 \,{\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )} \sqrt{d x}\right )}}{9 \, d^{3} x^{2}}, -\frac{2 \,{\left (4 \, a d x^{2} \sqrt{-\frac{a}{d}} \arctan \left (\frac{\sqrt{d x} \sqrt{-\frac{a}{d}}}{a x}\right ) +{\left (4 \, a x + 3 \,{\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )} \sqrt{d x}\right )}}{9 \, d^{3} x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2/9*(2*a*d*x^2*sqrt(a/d)*log((a*x + 2*sqrt(d*x)*sqrt(a/d) + 1)/(a*x - 1)) - (4*a*x + 3*dilog(a*x) - 2*log(-a*
x + 1))*sqrt(d*x))/(d^3*x^2), -2/9*(4*a*d*x^2*sqrt(-a/d)*arctan(sqrt(d*x)*sqrt(-a/d)/(a*x)) + (4*a*x + 3*dilog
(a*x) - 2*log(-a*x + 1))*sqrt(d*x))/(d^3*x^2)]

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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(polylog(2,a*x)/(d*x)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_2\left (a x\right )}{\left (d x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(dilog(a*x)/(d*x)^(5/2), x)