Integrand size = 13, antiderivative size = 108 \[ \int \frac {\operatorname {PolyLog}(3,a x)}{(d x)^{5/2}} \, dx=-\frac {16 a}{27 d^2 \sqrt {d x}}+\frac {16 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{27 d^{5/2}}+\frac {8 \log (1-a x)}{27 d (d x)^{3/2}}-\frac {4 \operatorname {PolyLog}(2,a x)}{9 d (d x)^{3/2}}-\frac {2 \operatorname {PolyLog}(3,a x)}{3 d (d x)^{3/2}} \]
16/27*a^(3/2)*arctanh(a^(1/2)*(d*x)^(1/2)/d^(1/2))/d^(5/2)+8/27*ln(-a*x+1) /d/(d*x)^(3/2)-4/9*polylog(2,a*x)/d/(d*x)^(3/2)-2/3*polylog(3,a*x)/d/(d*x) ^(3/2)-16/27*a/d^2/(d*x)^(1/2)
Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.59 \[ \int \frac {\operatorname {PolyLog}(3,a x)}{(d x)^{5/2}} \, dx=-\frac {2 x \left (8 a x-8 a^{3/2} x^{3/2} \text {arctanh}\left (\sqrt {a} \sqrt {x}\right )-4 \log (1-a x)+6 \operatorname {PolyLog}(2,a x)+9 \operatorname {PolyLog}(3,a x)\right )}{27 (d x)^{5/2}} \]
(-2*x*(8*a*x - 8*a^(3/2)*x^(3/2)*ArcTanh[Sqrt[a]*Sqrt[x]] - 4*Log[1 - a*x] + 6*PolyLog[2, a*x] + 9*PolyLog[3, a*x]))/(27*(d*x)^(5/2))
Time = 0.35 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {7145, 7145, 25, 2842, 61, 73, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\operatorname {PolyLog}(3,a x)}{(d x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {2}{3} \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{5/2}}dx-\frac {2 \operatorname {PolyLog}(3,a x)}{3 d (d x)^{3/2}}\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {2}{3} \left (\frac {2}{3} \int -\frac {\log (1-a x)}{(d x)^{5/2}}dx-\frac {2 \operatorname {PolyLog}(2,a x)}{3 d (d x)^{3/2}}\right )-\frac {2 \operatorname {PolyLog}(3,a x)}{3 d (d x)^{3/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2}{3} \left (-\frac {2}{3} \int \frac {\log (1-a x)}{(d x)^{5/2}}dx-\frac {2 \operatorname {PolyLog}(2,a x)}{3 d (d x)^{3/2}}\right )-\frac {2 \operatorname {PolyLog}(3,a x)}{3 d (d x)^{3/2}}\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle \frac {2}{3} \left (-\frac {2}{3} \left (-\frac {2 a \int \frac {1}{(d x)^{3/2} (1-a x)}dx}{3 d}-\frac {2 \log (1-a x)}{3 d (d x)^{3/2}}\right )-\frac {2 \operatorname {PolyLog}(2,a x)}{3 d (d x)^{3/2}}\right )-\frac {2 \operatorname {PolyLog}(3,a x)}{3 d (d x)^{3/2}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {2}{3} \left (-\frac {2}{3} \left (-\frac {2 a \left (\frac {a \int \frac {1}{\sqrt {d x} (1-a x)}dx}{d}-\frac {2}{d \sqrt {d x}}\right )}{3 d}-\frac {2 \log (1-a x)}{3 d (d x)^{3/2}}\right )-\frac {2 \operatorname {PolyLog}(2,a x)}{3 d (d x)^{3/2}}\right )-\frac {2 \operatorname {PolyLog}(3,a x)}{3 d (d x)^{3/2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {2}{3} \left (-\frac {2}{3} \left (-\frac {2 a \left (\frac {2 a \int \frac {1}{1-a x}d\sqrt {d x}}{d^2}-\frac {2}{d \sqrt {d x}}\right )}{3 d}-\frac {2 \log (1-a x)}{3 d (d x)^{3/2}}\right )-\frac {2 \operatorname {PolyLog}(2,a x)}{3 d (d x)^{3/2}}\right )-\frac {2 \operatorname {PolyLog}(3,a x)}{3 d (d x)^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2}{3} \left (-\frac {2}{3} \left (-\frac {2 a \left (\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2}{d \sqrt {d x}}\right )}{3 d}-\frac {2 \log (1-a x)}{3 d (d x)^{3/2}}\right )-\frac {2 \operatorname {PolyLog}(2,a x)}{3 d (d x)^{3/2}}\right )-\frac {2 \operatorname {PolyLog}(3,a x)}{3 d (d x)^{3/2}}\) |
(2*((-2*((-2*a*(-2/(d*Sqrt[d*x]) + (2*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[d*x])/ Sqrt[d]])/d^(3/2)))/(3*d) - (2*Log[1 - a*x])/(3*d*(d*x)^(3/2))))/3 - (2*Po lyLog[2, a*x])/(3*d*(d*x)^(3/2))))/3 - (2*PolyLog[3, a*x])/(3*d*(d*x)^(3/2 ))
3.1.70.3.1 Defintions of rubi rules used
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] - Simp[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] && 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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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] - Simp[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]
Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbo l] :> Simp[(d*x)^(m + 1)*(PolyLog[n, a*(b*x^p)^q]/(d*(m + 1))), x] - Simp[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]
Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.13
method | result | size |
meijerg | \(\frac {x^{\frac {5}{2}} \left (-a \right )^{\frac {5}{2}} \left (-\frac {16}{27 \sqrt {x}\, \sqrt {-a}}-\frac {8 \sqrt {x}\, a \left (\ln \left (1-\sqrt {a x}\right )-\ln \left (1+\sqrt {a x}\right )\right )}{27 \sqrt {-a}\, \sqrt {a x}}+\frac {8 \ln \left (-a x +1\right )}{27 x^{\frac {3}{2}} \sqrt {-a}\, a}-\frac {4 \operatorname {polylog}\left (2, a x \right )}{9 x^{\frac {3}{2}} \sqrt {-a}\, a}-\frac {2 \operatorname {polylog}\left (3, a x \right )}{3 x^{\frac {3}{2}} \sqrt {-a}\, a}\right )}{\left (d x \right )^{\frac {5}{2}} a}\) | \(122\) |
1/(d*x)^(5/2)*x^(5/2)*(-a)^(5/2)/a*(-16/27/x^(1/2)/(-a)^(1/2)-8/27*x^(1/2) /(-a)^(1/2)*a/(a*x)^(1/2)*(ln(1-(a*x)^(1/2))-ln(1+(a*x)^(1/2)))+8/27/x^(3/ 2)/(-a)^(1/2)/a*ln(-a*x+1)-4/9/x^(3/2)/(-a)^(1/2)/a*polylog(2,a*x)-2/3/x^( 3/2)/(-a)^(1/2)/a*polylog(3,a*x))
Time = 0.27 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.62 \[ \int \frac {\operatorname {PolyLog}(3,a x)}{(d x)^{5/2}} \, dx=\left [\frac {2 \, {\left (4 \, a d x^{2} \sqrt {\frac {a}{d}} \log \left (\frac {a x + 2 \, \sqrt {d x} \sqrt {\frac {a}{d}} + 1}{a x - 1}\right ) - 2 \, {\left (4 \, a x + 3 \, {\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )} \sqrt {d x} - 9 \, \sqrt {d x} {\rm polylog}\left (3, a x\right )\right )}}{27 \, d^{3} x^{2}}, -\frac {2 \, {\left (8 \, a d x^{2} \sqrt {-\frac {a}{d}} \arctan \left (\frac {\sqrt {d x} \sqrt {-\frac {a}{d}}}{a x}\right ) + 2 \, {\left (4 \, a x + 3 \, {\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )} \sqrt {d x} + 9 \, \sqrt {d x} {\rm polylog}\left (3, a x\right )\right )}}{27 \, d^{3} x^{2}}\right ] \]
[2/27*(4*a*d*x^2*sqrt(a/d)*log((a*x + 2*sqrt(d*x)*sqrt(a/d) + 1)/(a*x - 1) ) - 2*(4*a*x + 3*dilog(a*x) - 2*log(-a*x + 1))*sqrt(d*x) - 9*sqrt(d*x)*pol ylog(3, a*x))/(d^3*x^2), -2/27*(8*a*d*x^2*sqrt(-a/d)*arctan(sqrt(d*x)*sqrt (-a/d)/(a*x)) + 2*(4*a*x + 3*dilog(a*x) - 2*log(-a*x + 1))*sqrt(d*x) + 9*s qrt(d*x)*polylog(3, a*x))/(d^3*x^2)]
\[ \int \frac {\operatorname {PolyLog}(3,a x)}{(d x)^{5/2}} \, dx=\int \frac {\operatorname {Li}_{3}\left (a x\right )}{\left (d x\right )^{\frac {5}{2}}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.90 \[ \int \frac {\operatorname {PolyLog}(3,a x)}{(d x)^{5/2}} \, dx=-\frac {2 \, {\left (\frac {4 \, a^{2} \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d} d} + \frac {8 \, a d x + 6 \, d {\rm Li}_2\left (a x\right ) - 4 \, d \log \left (-a d x + d\right ) + 4 \, d \log \left (d\right ) + 9 \, d {\rm Li}_{3}(a x)}{\left (d x\right )^{\frac {3}{2}} d}\right )}}{27 \, d} \]
-2/27*(4*a^2*log((sqrt(d*x)*a - sqrt(a*d))/(sqrt(d*x)*a + sqrt(a*d)))/(sqr t(a*d)*d) + (8*a*d*x + 6*d*dilog(a*x) - 4*d*log(-a*d*x + d) + 4*d*log(d) + 9*d*polylog(3, a*x))/((d*x)^(3/2)*d))/d
\[ \int \frac {\operatorname {PolyLog}(3,a x)}{(d x)^{5/2}} \, dx=\int { \frac {{\rm Li}_{3}(a x)}{\left (d x\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\operatorname {PolyLog}(3,a x)}{(d x)^{5/2}} \, dx=\int \frac {\mathrm {polylog}\left (3,a\,x\right )}{{\left (d\,x\right )}^{5/2}} \,d x \]