Integrand size = 13, antiderivative size = 153 \[ \int (d x)^{5/2} \operatorname {PolyLog}(3,a x) \, dx=\frac {16 d^2 \sqrt {d x}}{343 a^3}+\frac {16 d (d x)^{3/2}}{1029 a^2}+\frac {16 (d x)^{5/2}}{1715 a}+\frac {16 (d x)^{7/2}}{2401 d}-\frac {16 d^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{343 a^{7/2}}-\frac {8 (d x)^{7/2} \log (1-a x)}{343 d}-\frac {4 (d x)^{7/2} \operatorname {PolyLog}(2,a x)}{49 d}+\frac {2 (d x)^{7/2} \operatorname {PolyLog}(3,a x)}{7 d} \]
16/1029*d*(d*x)^(3/2)/a^2+16/1715*(d*x)^(5/2)/a+16/2401*(d*x)^(7/2)/d-16/3 43*d^(5/2)*arctanh(a^(1/2)*(d*x)^(1/2)/d^(1/2))/a^(7/2)-8/343*(d*x)^(7/2)* ln(-a*x+1)/d-4/49*(d*x)^(7/2)*polylog(2,a*x)/d+2/7*(d*x)^(7/2)*polylog(3,a *x)/d+16/343*d^2*(d*x)^(1/2)/a^3
Time = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.64 \[ \int (d x)^{5/2} \operatorname {PolyLog}(3,a x) \, dx=\frac {2 (d x)^{5/2} \left (\frac {8 \left (105+35 a x+21 a^2 x^2+15 a^3 x^3\right )}{a^3}-\frac {840 \text {arctanh}\left (\sqrt {a} \sqrt {x}\right )}{a^{7/2} \sqrt {x}}-420 x^3 \log (1-a x)-1470 x^3 \operatorname {PolyLog}(2,a x)+5145 x^3 \operatorname {PolyLog}(3,a x)\right )}{36015 x^2} \]
(2*(d*x)^(5/2)*((8*(105 + 35*a*x + 21*a^2*x^2 + 15*a^3*x^3))/a^3 - (840*Ar cTanh[Sqrt[a]*Sqrt[x]])/(a^(7/2)*Sqrt[x]) - 420*x^3*Log[1 - a*x] - 1470*x^ 3*PolyLog[2, a*x] + 5145*x^3*PolyLog[3, a*x]))/(36015*x^2)
Time = 0.38 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {7145, 7145, 25, 2842, 60, 60, 60, 60, 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.
\(\displaystyle \int (d x)^{5/2} \operatorname {PolyLog}(3,a x) \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {2 (d x)^{7/2} \operatorname {PolyLog}(3,a x)}{7 d}-\frac {2}{7} \int (d x)^{5/2} \operatorname {PolyLog}(2,a x)dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {2 (d x)^{7/2} \operatorname {PolyLog}(3,a x)}{7 d}-\frac {2}{7} \left (\frac {2 (d x)^{7/2} \operatorname {PolyLog}(2,a x)}{7 d}-\frac {2}{7} \int -(d x)^{5/2} \log (1-a x)dx\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 (d x)^{7/2} \operatorname {PolyLog}(3,a x)}{7 d}-\frac {2}{7} \left (\frac {2}{7} \int (d x)^{5/2} \log (1-a x)dx+\frac {2 (d x)^{7/2} \operatorname {PolyLog}(2,a x)}{7 d}\right )\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle \frac {2 (d x)^{7/2} \operatorname {PolyLog}(3,a x)}{7 d}-\frac {2}{7} \left (\frac {2}{7} \left (\frac {2 a \int \frac {(d x)^{7/2}}{1-a x}dx}{7 d}+\frac {2 (d x)^{7/2} \log (1-a x)}{7 d}\right )+\frac {2 (d x)^{7/2} \operatorname {PolyLog}(2,a x)}{7 d}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {2 (d x)^{7/2} \operatorname {PolyLog}(3,a x)}{7 d}-\frac {2}{7} \left (\frac {2}{7} \left (\frac {2 a \left (\frac {d \int \frac {(d x)^{5/2}}{1-a x}dx}{a}-\frac {2 (d x)^{7/2}}{7 a}\right )}{7 d}+\frac {2 (d x)^{7/2} \log (1-a x)}{7 d}\right )+\frac {2 (d x)^{7/2} \operatorname {PolyLog}(2,a x)}{7 d}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {2 (d x)^{7/2} \operatorname {PolyLog}(3,a x)}{7 d}-\frac {2}{7} \left (\frac {2}{7} \left (\frac {2 a \left (\frac {d \left (\frac {d \int \frac {(d x)^{3/2}}{1-a x}dx}{a}-\frac {2 (d x)^{5/2}}{5 a}\right )}{a}-\frac {2 (d x)^{7/2}}{7 a}\right )}{7 d}+\frac {2 (d x)^{7/2} \log (1-a x)}{7 d}\right )+\frac {2 (d x)^{7/2} \operatorname {PolyLog}(2,a x)}{7 d}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {2 (d x)^{7/2} \operatorname {PolyLog}(3,a x)}{7 d}-\frac {2}{7} \left (\frac {2}{7} \left (\frac {2 a \left (\frac {d \left (\frac {d \left (\frac {d \int \frac {\sqrt {d x}}{1-a x}dx}{a}-\frac {2 (d x)^{3/2}}{3 a}\right )}{a}-\frac {2 (d x)^{5/2}}{5 a}\right )}{a}-\frac {2 (d x)^{7/2}}{7 a}\right )}{7 d}+\frac {2 (d x)^{7/2} \log (1-a x)}{7 d}\right )+\frac {2 (d x)^{7/2} \operatorname {PolyLog}(2,a x)}{7 d}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {2 (d x)^{7/2} \operatorname {PolyLog}(3,a x)}{7 d}-\frac {2}{7} \left (\frac {2}{7} \left (\frac {2 a \left (\frac {d \left (\frac {d \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {d x} (1-a x)}dx}{a}-\frac {2 \sqrt {d x}}{a}\right )}{a}-\frac {2 (d x)^{3/2}}{3 a}\right )}{a}-\frac {2 (d x)^{5/2}}{5 a}\right )}{a}-\frac {2 (d x)^{7/2}}{7 a}\right )}{7 d}+\frac {2 (d x)^{7/2} \log (1-a x)}{7 d}\right )+\frac {2 (d x)^{7/2} \operatorname {PolyLog}(2,a x)}{7 d}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {2 (d x)^{7/2} \operatorname {PolyLog}(3,a x)}{7 d}-\frac {2}{7} \left (\frac {2}{7} \left (\frac {2 a \left (\frac {d \left (\frac {d \left (\frac {d \left (\frac {2 \int \frac {1}{1-a x}d\sqrt {d x}}{a}-\frac {2 \sqrt {d x}}{a}\right )}{a}-\frac {2 (d x)^{3/2}}{3 a}\right )}{a}-\frac {2 (d x)^{5/2}}{5 a}\right )}{a}-\frac {2 (d x)^{7/2}}{7 a}\right )}{7 d}+\frac {2 (d x)^{7/2} \log (1-a x)}{7 d}\right )+\frac {2 (d x)^{7/2} \operatorname {PolyLog}(2,a x)}{7 d}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 (d x)^{7/2} \operatorname {PolyLog}(3,a x)}{7 d}-\frac {2}{7} \left (\frac {2}{7} \left (\frac {2 a \left (\frac {d \left (\frac {d \left (\frac {d \left (\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{a^{3/2}}-\frac {2 \sqrt {d x}}{a}\right )}{a}-\frac {2 (d x)^{3/2}}{3 a}\right )}{a}-\frac {2 (d x)^{5/2}}{5 a}\right )}{a}-\frac {2 (d x)^{7/2}}{7 a}\right )}{7 d}+\frac {2 (d x)^{7/2} \log (1-a x)}{7 d}\right )+\frac {2 (d x)^{7/2} \operatorname {PolyLog}(2,a x)}{7 d}\right )\) |
(-2*((2*((2*a*((-2*(d*x)^(7/2))/(7*a) + (d*((-2*(d*x)^(5/2))/(5*a) + (d*(( -2*(d*x)^(3/2))/(3*a) + (d*((-2*Sqrt[d*x])/a + (2*Sqrt[d]*ArcTanh[(Sqrt[a] *Sqrt[d*x])/Sqrt[d]])/a^(3/2)))/a))/a))/a))/(7*d) + (2*(d*x)^(7/2)*Log[1 - a*x])/(7*d)))/7 + (2*(d*x)^(7/2)*PolyLog[2, a*x])/(7*d)))/7 + (2*(d*x)^(7 /2)*PolyLog[3, a*x])/(7*d)
3.1.65.3.1 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/(b*(m + n + 1))), x] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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 = 0.24 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.97
method | result | size |
meijerg | \(\frac {\left (d x \right )^{\frac {5}{2}} \left (\frac {2 \sqrt {x}\, \left (-a \right )^{\frac {9}{2}} \left (360 a^{3} x^{3}+504 a^{2} x^{2}+840 a x +2520\right )}{108045 a^{4}}+\frac {8 \sqrt {x}\, \left (-a \right )^{\frac {9}{2}} \left (\ln \left (1-\sqrt {a x}\right )-\ln \left (1+\sqrt {a x}\right )\right )}{343 a^{4} \sqrt {a x}}-\frac {8 x^{\frac {7}{2}} \left (-a \right )^{\frac {9}{2}} \ln \left (-a x +1\right )}{343 a}-\frac {4 x^{\frac {7}{2}} \left (-a \right )^{\frac {9}{2}} \operatorname {polylog}\left (2, a x \right )}{49 a}+\frac {2 x^{\frac {7}{2}} \left (-a \right )^{\frac {9}{2}} \operatorname {polylog}\left (3, a x \right )}{7 a}\right )}{x^{\frac {5}{2}} \left (-a \right )^{\frac {5}{2}} a}\) | \(149\) |
(d*x)^(5/2)/x^(5/2)/(-a)^(5/2)/a*(2/108045*x^(1/2)*(-a)^(9/2)*(360*a^3*x^3 +504*a^2*x^2+840*a*x+2520)/a^4+8/343*x^(1/2)*(-a)^(9/2)/a^4/(a*x)^(1/2)*(l n(1-(a*x)^(1/2))-ln(1+(a*x)^(1/2)))-8/343*x^(7/2)*(-a)^(9/2)/a*ln(-a*x+1)- 4/49*x^(7/2)*(-a)^(9/2)/a*polylog(2,a*x)+2/7*x^(7/2)*(-a)^(9/2)/a*polylog( 3,a*x))
Time = 0.29 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.82 \[ \int (d x)^{5/2} \operatorname {PolyLog}(3,a x) \, dx=\left [\frac {2 \, {\left (5145 \, \sqrt {d x} a^{3} d^{2} x^{3} {\rm polylog}\left (3, a x\right ) + 420 \, d^{2} \sqrt {\frac {d}{a}} \log \left (\frac {a d x - 2 \, \sqrt {d x} a \sqrt {\frac {d}{a}} + d}{a x - 1}\right ) - 2 \, {\left (735 \, a^{3} d^{2} x^{3} {\rm Li}_2\left (a x\right ) + 210 \, a^{3} d^{2} x^{3} \log \left (-a x + 1\right ) - 60 \, a^{3} d^{2} x^{3} - 84 \, a^{2} d^{2} x^{2} - 140 \, a d^{2} x - 420 \, d^{2}\right )} \sqrt {d x}\right )}}{36015 \, a^{3}}, \frac {2 \, {\left (5145 \, \sqrt {d x} a^{3} d^{2} x^{3} {\rm polylog}\left (3, a x\right ) + 840 \, d^{2} \sqrt {-\frac {d}{a}} \arctan \left (\frac {\sqrt {d x} a \sqrt {-\frac {d}{a}}}{d}\right ) - 2 \, {\left (735 \, a^{3} d^{2} x^{3} {\rm Li}_2\left (a x\right ) + 210 \, a^{3} d^{2} x^{3} \log \left (-a x + 1\right ) - 60 \, a^{3} d^{2} x^{3} - 84 \, a^{2} d^{2} x^{2} - 140 \, a d^{2} x - 420 \, d^{2}\right )} \sqrt {d x}\right )}}{36015 \, a^{3}}\right ] \]
[2/36015*(5145*sqrt(d*x)*a^3*d^2*x^3*polylog(3, a*x) + 420*d^2*sqrt(d/a)*l og((a*d*x - 2*sqrt(d*x)*a*sqrt(d/a) + d)/(a*x - 1)) - 2*(735*a^3*d^2*x^3*d ilog(a*x) + 210*a^3*d^2*x^3*log(-a*x + 1) - 60*a^3*d^2*x^3 - 84*a^2*d^2*x^ 2 - 140*a*d^2*x - 420*d^2)*sqrt(d*x))/a^3, 2/36015*(5145*sqrt(d*x)*a^3*d^2 *x^3*polylog(3, a*x) + 840*d^2*sqrt(-d/a)*arctan(sqrt(d*x)*a*sqrt(-d/a)/d) - 2*(735*a^3*d^2*x^3*dilog(a*x) + 210*a^3*d^2*x^3*log(-a*x + 1) - 60*a^3* d^2*x^3 - 84*a^2*d^2*x^2 - 140*a*d^2*x - 420*d^2)*sqrt(d*x))/a^3]
\[ \int (d x)^{5/2} \operatorname {PolyLog}(3,a x) \, dx=\int \left (d x\right )^{\frac {5}{2}} \operatorname {Li}_{3}\left (a x\right )\, dx \]
Time = 0.30 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.02 \[ \int (d x)^{5/2} \operatorname {PolyLog}(3,a x) \, dx=\frac {2 \, {\left (\frac {420 \, d^{4} \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d} a^{3}} - \frac {1470 \, \left (d x\right )^{\frac {7}{2}} a^{3} {\rm Li}_2\left (a x\right ) + 420 \, \left (d x\right )^{\frac {7}{2}} a^{3} \log \left (-a d x + d\right ) - 5145 \, \left (d x\right )^{\frac {7}{2}} a^{3} {\rm Li}_{3}(a x) - 168 \, \left (d x\right )^{\frac {5}{2}} a^{2} d - 60 \, {\left (7 \, a^{3} \log \left (d\right ) + 2 \, a^{3}\right )} \left (d x\right )^{\frac {7}{2}} - 280 \, \left (d x\right )^{\frac {3}{2}} a d^{2} - 840 \, \sqrt {d x} d^{3}}{a^{3}}\right )}}{36015 \, d} \]
2/36015*(420*d^4*log((sqrt(d*x)*a - sqrt(a*d))/(sqrt(d*x)*a + sqrt(a*d)))/ (sqrt(a*d)*a^3) - (1470*(d*x)^(7/2)*a^3*dilog(a*x) + 420*(d*x)^(7/2)*a^3*l og(-a*d*x + d) - 5145*(d*x)^(7/2)*a^3*polylog(3, a*x) - 168*(d*x)^(5/2)*a^ 2*d - 60*(7*a^3*log(d) + 2*a^3)*(d*x)^(7/2) - 280*(d*x)^(3/2)*a*d^2 - 840* sqrt(d*x)*d^3)/a^3)/d
\[ \int (d x)^{5/2} \operatorname {PolyLog}(3,a x) \, dx=\int { \left (d x\right )^{\frac {5}{2}} {\rm Li}_{3}(a x) \,d x } \]
Timed out. \[ \int (d x)^{5/2} \operatorname {PolyLog}(3,a x) \, dx=\int {\left (d\,x\right )}^{5/2}\,\mathrm {polylog}\left (3,a\,x\right ) \,d x \]