3.1.66 \(\int (d x)^{3/2} \operatorname {PolyLog}(3,a x) \, dx\) [66]

3.1.66.1 Optimal result

Integrand size = 13, antiderivative size = 136 \[ \int (d x)^{3/2} \operatorname {PolyLog}(3,a x) \, dx=\frac {16 d \sqrt {d x}}{125 a^2}+\frac {16 (d x)^{3/2}}{375 a}+\frac {16 (d x)^{5/2}}{625 d}-\frac {16 d^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{125 a^{5/2}}-\frac {8 (d x)^{5/2} \log (1-a x)}{125 d}-\frac {4 (d x)^{5/2} \operatorname {PolyLog}(2,a x)}{25 d}+\frac {2 (d x)^{5/2} \operatorname {PolyLog}(3,a x)}{5 d} \]

output

16/375*(d*x)^(3/2)/a+16/625*(d*x)^(5/2)/d-16/125*d^(3/2)*arctanh(a^(1/2)*( 
d*x)^(1/2)/d^(1/2))/a^(5/2)-8/125*(d*x)^(5/2)*ln(-a*x+1)/d-4/25*(d*x)^(5/2 
)*polylog(2,a*x)/d+2/5*(d*x)^(5/2)*polylog(3,a*x)/d+16/125*d*(d*x)^(1/2)/a 
^2
 

3.1.66.2 Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.65 \[ \int (d x)^{3/2} \operatorname {PolyLog}(3,a x) \, dx=\frac {2 d \sqrt {d x} \left (4 \left (\frac {30}{a^2}+\frac {10 x}{a}+6 x^2-\frac {30 \text {arctanh}\left (\sqrt {a} \sqrt {x}\right )}{a^{5/2} \sqrt {x}}-15 x^2 \log (1-a x)\right )-150 x^2 \operatorname {PolyLog}(2,a x)+375 x^2 \operatorname {PolyLog}(3,a x)\right )}{1875} \]

input

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

output

(2*d*Sqrt[d*x]*(4*(30/a^2 + (10*x)/a + 6*x^2 - (30*ArcTanh[Sqrt[a]*Sqrt[x] 
])/(a^(5/2)*Sqrt[x]) - 15*x^2*Log[1 - a*x]) - 150*x^2*PolyLog[2, a*x] + 37 
5*x^2*PolyLog[3, a*x]))/1875
 

3.1.66.3 Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.19, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {7145, 7145, 25, 2842, 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.

\(\displaystyle \int (d x)^{3/2} \operatorname {PolyLog}(3,a x) \, dx\)

\(\Big \downarrow \) 7145

\(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}(3,a x)}{5 d}-\frac {2}{5} \int (d x)^{3/2} \operatorname {PolyLog}(2,a x)dx\)

\(\Big \downarrow \) 7145

\(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}(3,a x)}{5 d}-\frac {2}{5} \left (\frac {2 (d x)^{5/2} \operatorname {PolyLog}(2,a x)}{5 d}-\frac {2}{5} \int -(d x)^{3/2} \log (1-a x)dx\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}(3,a x)}{5 d}-\frac {2}{5} \left (\frac {2}{5} \int (d x)^{3/2} \log (1-a x)dx+\frac {2 (d x)^{5/2} \operatorname {PolyLog}(2,a x)}{5 d}\right )\)

\(\Big \downarrow \) 2842

\(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}(3,a x)}{5 d}-\frac {2}{5} \left (\frac {2}{5} \left (\frac {2 a \int \frac {(d x)^{5/2}}{1-a x}dx}{5 d}+\frac {2 (d x)^{5/2} \log (1-a x)}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}(2,a x)}{5 d}\right )\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}(3,a x)}{5 d}-\frac {2}{5} \left (\frac {2}{5} \left (\frac {2 a \left (\frac {d \int \frac {(d x)^{3/2}}{1-a x}dx}{a}-\frac {2 (d x)^{5/2}}{5 a}\right )}{5 d}+\frac {2 (d x)^{5/2} \log (1-a x)}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}(2,a x)}{5 d}\right )\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}(3,a x)}{5 d}-\frac {2}{5} \left (\frac {2}{5} \left (\frac {2 a \left (\frac {d \left (\frac {d \int \frac {\sqrt {d x}}{1-a x}dx}{a}-\frac {2 (d x)^{3/2}}{3 a}\right )}{a}-\frac {2 (d x)^{5/2}}{5 a}\right )}{5 d}+\frac {2 (d x)^{5/2} \log (1-a x)}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}(2,a x)}{5 d}\right )\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}(3,a x)}{5 d}-\frac {2}{5} \left (\frac {2}{5} \left (\frac {2 a \left (\frac {d \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {d x} (1-a x)}dx}{a}-\frac {2 \sqrt {d x}}{a}\right )}{a}-\frac {2 (d x)^{3/2}}{3 a}\right )}{a}-\frac {2 (d x)^{5/2}}{5 a}\right )}{5 d}+\frac {2 (d x)^{5/2} \log (1-a x)}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}(2,a x)}{5 d}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}(3,a x)}{5 d}-\frac {2}{5} \left (\frac {2}{5} \left (\frac {2 a \left (\frac {d \left (\frac {d \left (\frac {2 \int \frac {1}{1-a x}d\sqrt {d x}}{a}-\frac {2 \sqrt {d x}}{a}\right )}{a}-\frac {2 (d x)^{3/2}}{3 a}\right )}{a}-\frac {2 (d x)^{5/2}}{5 a}\right )}{5 d}+\frac {2 (d x)^{5/2} \log (1-a x)}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}(2,a x)}{5 d}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}(3,a x)}{5 d}-\frac {2}{5} \left (\frac {2}{5} \left (\frac {2 a \left (\frac {d \left (\frac {d \left (\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{a^{3/2}}-\frac {2 \sqrt {d x}}{a}\right )}{a}-\frac {2 (d x)^{3/2}}{3 a}\right )}{a}-\frac {2 (d x)^{5/2}}{5 a}\right )}{5 d}+\frac {2 (d x)^{5/2} \log (1-a x)}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}(2,a x)}{5 d}\right )\)

input

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

output

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

3.1.66.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 60

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]
 

rule 73

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]
 

rule 219

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])
 

rule 2842

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]
 

rule 7145

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]
 

3.1.66.4 Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.04

method	result	size
meijerg	\(\frac {\left (d x \right )^{\frac {3}{2}} \left (\frac {2 \sqrt {x}\, \left (-a \right )^{\frac {7}{2}} \left (168 a^{2} x^{2}+280 a x +840\right )}{13125 a^{3}}+\frac {8 \sqrt {x}\, \left (-a \right )^{\frac {7}{2}} \left (\ln \left (1-\sqrt {a x}\right )-\ln \left (1+\sqrt {a x}\right )\right )}{125 a^{3} \sqrt {a x}}-\frac {8 x^{\frac {5}{2}} \left (-a \right )^{\frac {7}{2}} \ln \left (-a x +1\right )}{125 a}-\frac {4 x^{\frac {5}{2}} \left (-a \right )^{\frac {7}{2}} \operatorname {polylog}\left (2, a x \right )}{25 a}+\frac {2 x^{\frac {5}{2}} \left (-a \right )^{\frac {7}{2}} \operatorname {polylog}\left (3, a x \right )}{5 a}\right )}{x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{2}} a}\)	\(141\)

input

int((d*x)^(3/2)*polylog(3,a*x),x,method=_RETURNVERBOSE)
 

output

(d*x)^(3/2)/x^(3/2)/(-a)^(3/2)/a*(2/13125*x^(1/2)*(-a)^(7/2)*(168*a^2*x^2+ 
280*a*x+840)/a^3+8/125*x^(1/2)*(-a)^(7/2)/a^3/(a*x)^(1/2)*(ln(1-(a*x)^(1/2 
))-ln(1+(a*x)^(1/2)))-8/125*x^(5/2)*(-a)^(7/2)/a*ln(-a*x+1)-4/25*x^(5/2)*( 
-a)^(7/2)/a*polylog(2,a*x)+2/5*x^(5/2)*(-a)^(7/2)/a*polylog(3,a*x))
 

3.1.66.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.68 \[ \int (d x)^{3/2} \operatorname {PolyLog}(3,a x) \, dx=\left [\frac {2 \, {\left (375 \, \sqrt {d x} a^{2} d x^{2} {\rm polylog}\left (3, a x\right ) + 60 \, d \sqrt {\frac {d}{a}} \log \left (\frac {a d x - 2 \, \sqrt {d x} a \sqrt {\frac {d}{a}} + d}{a x - 1}\right ) - 2 \, {\left (75 \, a^{2} d x^{2} {\rm Li}_2\left (a x\right ) + 30 \, a^{2} d x^{2} \log \left (-a x + 1\right ) - 12 \, a^{2} d x^{2} - 20 \, a d x - 60 \, d\right )} \sqrt {d x}\right )}}{1875 \, a^{2}}, \frac {2 \, {\left (375 \, \sqrt {d x} a^{2} d x^{2} {\rm polylog}\left (3, a x\right ) + 120 \, d \sqrt {-\frac {d}{a}} \arctan \left (\frac {\sqrt {d x} a \sqrt {-\frac {d}{a}}}{d}\right ) - 2 \, {\left (75 \, a^{2} d x^{2} {\rm Li}_2\left (a x\right ) + 30 \, a^{2} d x^{2} \log \left (-a x + 1\right ) - 12 \, a^{2} d x^{2} - 20 \, a d x - 60 \, d\right )} \sqrt {d x}\right )}}{1875 \, a^{2}}\right ] \]

input

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

output

[2/1875*(375*sqrt(d*x)*a^2*d*x^2*polylog(3, a*x) + 60*d*sqrt(d/a)*log((a*d 
*x - 2*sqrt(d*x)*a*sqrt(d/a) + d)/(a*x - 1)) - 2*(75*a^2*d*x^2*dilog(a*x) 
+ 30*a^2*d*x^2*log(-a*x + 1) - 12*a^2*d*x^2 - 20*a*d*x - 60*d)*sqrt(d*x))/ 
a^2, 2/1875*(375*sqrt(d*x)*a^2*d*x^2*polylog(3, a*x) + 120*d*sqrt(-d/a)*ar 
ctan(sqrt(d*x)*a*sqrt(-d/a)/d) - 2*(75*a^2*d*x^2*dilog(a*x) + 30*a^2*d*x^2 
*log(-a*x + 1) - 12*a^2*d*x^2 - 20*a*d*x - 60*d)*sqrt(d*x))/a^2]
 

3.1.66.6 Sympy [F]

\[ \int (d x)^{3/2} \operatorname {PolyLog}(3,a x) \, dx=\int \left (d x\right )^{\frac {3}{2}} \operatorname {Li}_{3}\left (a x\right )\, dx \]

input

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

output

Integral((d*x)**(3/2)*polylog(3, a*x), x)
 

3.1.66.7 Maxima [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.05 \[ \int (d x)^{3/2} \operatorname {PolyLog}(3,a x) \, dx=\frac {2 \, {\left (\frac {60 \, d^{3} \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d} a^{2}} - \frac {150 \, \left (d x\right )^{\frac {5}{2}} a^{2} {\rm Li}_2\left (a x\right ) + 60 \, \left (d x\right )^{\frac {5}{2}} a^{2} \log \left (-a d x + d\right ) - 375 \, \left (d x\right )^{\frac {5}{2}} a^{2} {\rm Li}_{3}(a x) - 12 \, {\left (5 \, a^{2} \log \left (d\right ) + 2 \, a^{2}\right )} \left (d x\right )^{\frac {5}{2}} - 40 \, \left (d x\right )^{\frac {3}{2}} a d - 120 \, \sqrt {d x} d^{2}}{a^{2}}\right )}}{1875 \, d} \]

input

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

output

2/1875*(60*d^3*log((sqrt(d*x)*a - sqrt(a*d))/(sqrt(d*x)*a + sqrt(a*d)))/(s 
qrt(a*d)*a^2) - (150*(d*x)^(5/2)*a^2*dilog(a*x) + 60*(d*x)^(5/2)*a^2*log(- 
a*d*x + d) - 375*(d*x)^(5/2)*a^2*polylog(3, a*x) - 12*(5*a^2*log(d) + 2*a^ 
2)*(d*x)^(5/2) - 40*(d*x)^(3/2)*a*d - 120*sqrt(d*x)*d^2)/a^2)/d
 

3.1.66.8 Giac [F]

\[ \int (d x)^{3/2} \operatorname {PolyLog}(3,a x) \, dx=\int { \left (d x\right )^{\frac {3}{2}} {\rm Li}_{3}(a x) \,d x } \]

input

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

output

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

3.1.66.9 Mupad [F(-1)]

Timed out. \[ \int (d x)^{3/2} \operatorname {PolyLog}(3,a x) \, dx=\int {\left (d\,x\right )}^{3/2}\,\mathrm {polylog}\left (3,a\,x\right ) \,d x \]

input

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

output

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