Integrand size = 13, antiderivative size = 89 \[ \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{5/2}} \, dx=-\frac {8 a}{9 d^2 \sqrt {d x}}+\frac {8 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{9 d^{5/2}}+\frac {4 \log (1-a x)}{9 d (d x)^{3/2}}-\frac {2 \operatorname {PolyLog}(2,a x)}{3 d (d x)^{3/2}} \]
8/9*a^(3/2)*arctanh(a^(1/2)*(d*x)^(1/2)/d^(1/2))/d^(5/2)+4/9*ln(-a*x+1)/d/ (d*x)^(3/2)-2/3*polylog(2,a*x)/d/(d*x)^(3/2)-8/9*a/d^2/(d*x)^(1/2)
Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.64 \[ \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{5/2}} \, dx=-\frac {2 x \left (4 a x-4 a^{3/2} x^{3/2} \text {arctanh}\left (\sqrt {a} \sqrt {x}\right )-2 \log (1-a x)+3 \operatorname {PolyLog}(2,a x)\right )}{9 (d x)^{5/2}} \]
(-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))
Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {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.
\(\displaystyle \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {2}{3} \int -\frac {\log (1-a x)}{(d x)^{5/2}}dx-\frac {2 \operatorname {PolyLog}(2,a x)}{3 d (d x)^{3/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {2}{3} \int \frac {\log (1-a x)}{(d x)^{5/2}}dx-\frac {2 \operatorname {PolyLog}(2,a x)}{3 d (d x)^{3/2}}\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle -\frac {2}{3} \left (-\frac {2 a \int \frac {1}{(d x)^{3/2} (1-a x)}dx}{3 d}-\frac {2 \log (1-a x)}{3 d (d x)^{3/2}}\right )-\frac {2 \operatorname {PolyLog}(2,a x)}{3 d (d x)^{3/2}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {2}{3} \left (-\frac {2 a \left (\frac {a \int \frac {1}{\sqrt {d x} (1-a x)}dx}{d}-\frac {2}{d \sqrt {d x}}\right )}{3 d}-\frac {2 \log (1-a x)}{3 d (d x)^{3/2}}\right )-\frac {2 \operatorname {PolyLog}(2,a x)}{3 d (d x)^{3/2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {2}{3} \left (-\frac {2 a \left (\frac {2 a \int \frac {1}{1-a x}d\sqrt {d x}}{d^2}-\frac {2}{d \sqrt {d x}}\right )}{3 d}-\frac {2 \log (1-a x)}{3 d (d x)^{3/2}}\right )-\frac {2 \operatorname {PolyLog}(2,a x)}{3 d (d x)^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2}{3} \left (-\frac {2 a \left (\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2}{d \sqrt {d x}}\right )}{3 d}-\frac {2 \log (1-a x)}{3 d (d x)^{3/2}}\right )-\frac {2 \operatorname {PolyLog}(2,a x)}{3 d (d x)^{3/2}}\) |
(-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*PolyLo g[2, a*x])/(3*d*(d*x)^(3/2))
3.1.63.3.1 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 = 0.70 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {-\frac {2 \operatorname {polylog}\left (2, a x \right )}{3 \left (d x \right )^{\frac {3}{2}}}+\frac {4 \ln \left (\frac {-a d x +d}{d}\right )}{9 \left (d x \right )^{\frac {3}{2}}}+\frac {8 a \left (\frac {a \,\operatorname {arctanh}\left (\frac {a \sqrt {d x}}{\sqrt {a d}}\right )}{d \sqrt {a d}}-\frac {1}{d \sqrt {d x}}\right )}{9}}{d}\) | \(75\) |
default | \(\frac {-\frac {2 \operatorname {polylog}\left (2, a x \right )}{3 \left (d x \right )^{\frac {3}{2}}}+\frac {4 \ln \left (\frac {-a d x +d}{d}\right )}{9 \left (d x \right )^{\frac {3}{2}}}+\frac {8 a \left (\frac {a \,\operatorname {arctanh}\left (\frac {a \sqrt {d x}}{\sqrt {a d}}\right )}{d \sqrt {a d}}-\frac {1}{d \sqrt {d x}}\right )}{9}}{d}\) | \(75\) |
meijerg | \(\frac {x^{\frac {5}{2}} \left (-a \right )^{\frac {5}{2}} \left (-\frac {8}{9 \sqrt {x}\, \sqrt {-a}}-\frac {4 \sqrt {x}\, a \left (\ln \left (1-\sqrt {a x}\right )-\ln \left (1+\sqrt {a x}\right )\right )}{9 \sqrt {-a}\, \sqrt {a x}}+\frac {4 \ln \left (-a x +1\right )}{9 x^{\frac {3}{2}} \sqrt {-a}\, a}-\frac {2 \operatorname {polylog}\left (2, a x \right )}{3 x^{\frac {3}{2}} \sqrt {-a}\, a}\right )}{\left (d x \right )^{\frac {5}{2}} a}\) | \(104\) |
2/d*(-1/3*polylog(2,a*x)/(d*x)^(3/2)+2/9/(d*x)^(3/2)*ln((-a*d*x+d)/d)+4/9* a*(a/d/(a*d)^(1/2)*arctanh(a*(d*x)^(1/2)/(a*d)^(1/2))-1/d/(d*x)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.69 \[ \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{5/2}} \, dx=\left [\frac {2 \, {\left (2 \, a d x^{2} \sqrt {\frac {a}{d}} \log \left (\frac {a x + 2 \, \sqrt {d x} \sqrt {\frac {a}{d}} + 1}{a x - 1}\right ) - {\left (4 \, a x + 3 \, {\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )} \sqrt {d x}\right )}}{9 \, d^{3} x^{2}}, -\frac {2 \, {\left (4 \, a d x^{2} \sqrt {-\frac {a}{d}} \arctan \left (\frac {\sqrt {d x} \sqrt {-\frac {a}{d}}}{a x}\right ) + {\left (4 \, a x + 3 \, {\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )} \sqrt {d x}\right )}}{9 \, d^{3} x^{2}}\right ] \]
[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)]
\[ \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{5/2}} \, dx=\int \frac {\operatorname {Li}_{2}\left (a x\right )}{\left (d x\right )^{\frac {5}{2}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{5/2}} \, dx=-\frac {2 \, {\left (\frac {2 \, a^{2} \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d} d} + \frac {4 \, a d x + 3 \, d {\rm Li}_2\left (a x\right ) - 2 \, d \log \left (-a d x + d\right ) + 2 \, d \log \left (d\right )}{\left (d x\right )^{\frac {3}{2}} d}\right )}}{9 \, d} \]
-2/9*(2*a^2*log((sqrt(d*x)*a - sqrt(a*d))/(sqrt(d*x)*a + sqrt(a*d)))/(sqrt (a*d)*d) + (4*a*d*x + 3*d*dilog(a*x) - 2*d*log(-a*d*x + d) + 2*d*log(d))/( (d*x)^(3/2)*d))/d
\[ \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{5/2}} \, dx=\int { \frac {{\rm Li}_2\left (a x\right )}{\left (d x\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{5/2}} \, dx=\int \frac {\mathrm {polylog}\left (2,a\,x\right )}{{\left (d\,x\right )}^{5/2}} \,d x \]