3.71 \(\int \frac{\text{PolyLog}(3,a x)}{(d x)^{7/2}} \, dx\)

Optimal. Leaf size=125 \[ -\frac{4 \text{PolyLog}(2,a x)}{25 d (d x)^{5/2}}-\frac{2 \text{PolyLog}(3,a x)}{5 d (d x)^{5/2}}-\frac{16 a^2}{125 d^3 \sqrt{d x}}+\frac{16 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 d^{7/2}}-\frac{16 a}{375 d^2 (d x)^{3/2}}+\frac{8 \log (1-a x)}{125 d (d x)^{5/2}} \]

[Out]

(-16*a)/(375*d^2*(d*x)^(3/2)) - (16*a^2)/(125*d^3*Sqrt[d*x]) + (16*a^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]
])/(125*d^(7/2)) + (8*Log[1 - a*x])/(125*d*(d*x)^(5/2)) - (4*PolyLog[2, a*x])/(25*d*(d*x)^(5/2)) - (2*PolyLog[
3, a*x])/(5*d*(d*x)^(5/2))

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Rubi [A]  time = 0.0763217, antiderivative size = 125, 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, 51, 63, 206} \[ -\frac{4 \text{PolyLog}(2,a x)}{25 d (d x)^{5/2}}-\frac{2 \text{PolyLog}(3,a x)}{5 d (d x)^{5/2}}-\frac{16 a^2}{125 d^3 \sqrt{d x}}+\frac{16 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 d^{7/2}}-\frac{16 a}{375 d^2 (d x)^{3/2}}+\frac{8 \log (1-a x)}{125 d (d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[3, a*x]/(d*x)^(7/2),x]

[Out]

(-16*a)/(375*d^2*(d*x)^(3/2)) - (16*a^2)/(125*d^3*Sqrt[d*x]) + (16*a^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]
])/(125*d^(7/2)) + (8*Log[1 - a*x])/(125*d*(d*x)^(5/2)) - (4*PolyLog[2, a*x])/(25*d*(d*x)^(5/2)) - (2*PolyLog[
3, a*x])/(5*d*(d*x)^(5/2))

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}_3(a x)}{(d x)^{7/2}} \, dx &=-\frac{2 \text{Li}_3(a x)}{5 d (d x)^{5/2}}+\frac{2}{5} \int \frac{\text{Li}_2(a x)}{(d x)^{7/2}} \, dx\\ &=-\frac{4 \text{Li}_2(a x)}{25 d (d x)^{5/2}}-\frac{2 \text{Li}_3(a x)}{5 d (d x)^{5/2}}-\frac{4}{25} \int \frac{\log (1-a x)}{(d x)^{7/2}} \, dx\\ &=\frac{8 \log (1-a x)}{125 d (d x)^{5/2}}-\frac{4 \text{Li}_2(a x)}{25 d (d x)^{5/2}}-\frac{2 \text{Li}_3(a x)}{5 d (d x)^{5/2}}+\frac{(8 a) \int \frac{1}{(d x)^{5/2} (1-a x)} \, dx}{125 d}\\ &=-\frac{16 a}{375 d^2 (d x)^{3/2}}+\frac{8 \log (1-a x)}{125 d (d x)^{5/2}}-\frac{4 \text{Li}_2(a x)}{25 d (d x)^{5/2}}-\frac{2 \text{Li}_3(a x)}{5 d (d x)^{5/2}}+\frac{\left (8 a^2\right ) \int \frac{1}{(d x)^{3/2} (1-a x)} \, dx}{125 d^2}\\ &=-\frac{16 a}{375 d^2 (d x)^{3/2}}-\frac{16 a^2}{125 d^3 \sqrt{d x}}+\frac{8 \log (1-a x)}{125 d (d x)^{5/2}}-\frac{4 \text{Li}_2(a x)}{25 d (d x)^{5/2}}-\frac{2 \text{Li}_3(a x)}{5 d (d x)^{5/2}}+\frac{\left (8 a^3\right ) \int \frac{1}{\sqrt{d x} (1-a x)} \, dx}{125 d^3}\\ &=-\frac{16 a}{375 d^2 (d x)^{3/2}}-\frac{16 a^2}{125 d^3 \sqrt{d x}}+\frac{8 \log (1-a x)}{125 d (d x)^{5/2}}-\frac{4 \text{Li}_2(a x)}{25 d (d x)^{5/2}}-\frac{2 \text{Li}_3(a x)}{5 d (d x)^{5/2}}+\frac{\left (16 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{a x^2}{d}} \, dx,x,\sqrt{d x}\right )}{125 d^4}\\ &=-\frac{16 a}{375 d^2 (d x)^{3/2}}-\frac{16 a^2}{125 d^3 \sqrt{d x}}+\frac{16 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 d^{7/2}}+\frac{8 \log (1-a x)}{125 d (d x)^{5/2}}-\frac{4 \text{Li}_2(a x)}{25 d (d x)^{5/2}}-\frac{2 \text{Li}_3(a x)}{5 d (d x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.13028, size = 72, normalized size = 0.58 \[ -\frac{2 x \left (30 \text{PolyLog}(2,a x)+75 \text{PolyLog}(3,a x)+24 a^2 x^2-24 a^{5/2} x^{5/2} \tanh ^{-1}\left (\sqrt{a} \sqrt{x}\right )+8 a x-12 \log (1-a x)\right )}{375 (d x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[3, a*x]/(d*x)^(7/2),x]

[Out]

(-2*x*(8*a*x + 24*a^2*x^2 - 24*a^(5/2)*x^(5/2)*ArcTanh[Sqrt[a]*Sqrt[x]] - 12*Log[1 - a*x] + 30*PolyLog[2, a*x]
+ 75*PolyLog[3, a*x]))/(375*(d*x)^(7/2))

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Maple [A]  time = 0.062, size = 135, normalized size = 1.1 \begin{align*}{\frac{1}{a}{x}^{{\frac{7}{2}}} \left ( -a \right ) ^{{\frac{7}{2}}} \left ( -{\frac{16}{375}{x}^{-{\frac{3}{2}}} \left ( -a \right ) ^{-{\frac{3}{2}}}}-{\frac{16\,a}{125}{\frac{1}{\sqrt{x}}} \left ( -a \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,{a}^{2}}{125}\sqrt{x} \left ( \ln \left ( 1-\sqrt{ax} \right ) -\ln \left ( 1+\sqrt{ax} \right ) \right ) \left ( -a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ax}}}}+{\frac{8\,\ln \left ( -ax+1 \right ) }{125\,a}{x}^{-{\frac{5}{2}}} \left ( -a \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,{\it polylog} \left ( 2,ax \right ) }{25\,a}{x}^{-{\frac{5}{2}}} \left ( -a \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,{\it polylog} \left ( 3,ax \right ) }{5\,a}{x}^{-{\frac{5}{2}}} \left ( -a \right ) ^{-{\frac{3}{2}}}} \right ) \left ( dx \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3,a*x)/(d*x)^(7/2),x)

[Out]

1/(d*x)^(7/2)*x^(7/2)*(-a)^(7/2)/a*(-16/375/x^(3/2)/(-a)^(3/2)-16/125/x^(1/2)/(-a)^(3/2)*a-8/125*x^(1/2)/(-a)^
(3/2)*a^2/(a*x)^(1/2)*(ln(1-(a*x)^(1/2))-ln(1+(a*x)^(1/2)))+8/125/x^(5/2)/(-a)^(3/2)/a*ln(-a*x+1)-4/25/x^(5/2)
/(-a)^(3/2)*polylog(2,a*x)/a-2/5/x^(5/2)/(-a)^(3/2)/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(polylog(3,a*x)/(d*x)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 2.85994, size = 629, normalized size = 5.03 \begin{align*} \left [\frac{2 \,{\left (12 \, a^{2} d x^{3} \sqrt{\frac{a}{d}} \log \left (\frac{a x + 2 \, \sqrt{d x} \sqrt{\frac{a}{d}} + 1}{a x - 1}\right ) - 4 \,{\left (6 \, a^{2} x^{2} + 2 \, a x - 3 \, \log \left (-a x + 1\right )\right )} \sqrt{d x} - 30 \, \sqrt{d x}{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) - 75 \, \sqrt{d x}{\rm polylog}\left (3, a x\right )\right )}}{375 \, d^{4} x^{3}}, -\frac{2 \,{\left (24 \, a^{2} d x^{3} \sqrt{-\frac{a}{d}} \arctan \left (\frac{\sqrt{d x} \sqrt{-\frac{a}{d}}}{a x}\right ) + 4 \,{\left (6 \, a^{2} x^{2} + 2 \, a x - 3 \, \log \left (-a x + 1\right )\right )} \sqrt{d x} + 30 \, \sqrt{d x}{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) + 75 \, \sqrt{d x}{\rm polylog}\left (3, a x\right )\right )}}{375 \, d^{4} x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x)/(d*x)^(7/2),x, algorithm="fricas")

[Out]

[2/375*(12*a^2*d*x^3*sqrt(a/d)*log((a*x + 2*sqrt(d*x)*sqrt(a/d) + 1)/(a*x - 1)) - 4*(6*a^2*x^2 + 2*a*x - 3*log
(-a*x + 1))*sqrt(d*x) - 30*sqrt(d*x)*\%iint(a, x, -log(-a*x + 1)/a, -log(-a*x + 1)/x) - 75*sqrt(d*x)*polylog(3,
a*x))/(d^4*x^3), -2/375*(24*a^2*d*x^3*sqrt(-a/d)*arctan(sqrt(d*x)*sqrt(-a/d)/(a*x)) + 4*(6*a^2*x^2 + 2*a*x -
3*log(-a*x + 1))*sqrt(d*x) + 30*sqrt(d*x)*\%iint(a, x, -log(-a*x + 1)/a, -log(-a*x + 1)/x) + 75*sqrt(d*x)*polyl
og(3, a*x))/(d^4*x^3)]

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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(3,a*x)/(d*x)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_{3}(a x)}{\left (d x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x)/(d*x)^(7/2),x, algorithm="giac")

[Out]

integrate(polylog(3, a*x)/(d*x)^(7/2), x)