3.1.65 \(\int (d x)^{5/2} \text {PolyLog}(3,a x) \, dx\) [65]

Optimal. Leaf size=153 \[ \frac {16 d^2 \sqrt {d x}}{343 a^3}+\frac {16 d (d x)^{3/2}}{1029 a^2}+\frac {16 (d x)^{5/2}}{1715 a}+\frac {16 (d x)^{7/2}}{2401 d}-\frac {16 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{343 a^{7/2}}-\frac {8 (d x)^{7/2} \log (1-a x)}{343 d}-\frac {4 (d x)^{7/2} \text {PolyLog}(2,a x)}{49 d}+\frac {2 (d x)^{7/2} \text {PolyLog}(3,a x)}{7 d} \]

[Out]

16/1029*d*(d*x)^(3/2)/a^2+16/1715*(d*x)^(5/2)/a+16/2401*(d*x)^(7/2)/d-16/343*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

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Rubi [A]
time = 0.07, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6726, 2442, 52, 65, 212} \begin {gather*} -\frac {16 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{343 a^{7/2}}+\frac {16 d^2 \sqrt {d x}}{343 a^3}+\frac {16 d (d x)^{3/2}}{1029 a^2}-\frac {4 (d x)^{7/2} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}+\frac {16 (d x)^{5/2}}{1715 a}-\frac {8 (d x)^{7/2} \log (1-a x)}{343 d}+\frac {16 (d x)^{7/2}}{2401 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^(5/2)*PolyLog[3, a*x],x]

[Out]

(16*d^2*Sqrt[d*x])/(343*a^3) + (16*d*(d*x)^(3/2))/(1029*a^2) + (16*(d*x)^(5/2))/(1715*a) + (16*(d*x)^(7/2))/(2
401*d) - (16*d^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/(343*a^(7/2)) - (8*(d*x)^(7/2)*Log[1 - a*x])/(343*d
) - (4*(d*x)^(7/2)*PolyLog[2, a*x])/(49*d) + (2*(d*x)^(7/2)*PolyLog[3, a*x])/(7*d)

Rule 52

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] + Dist[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

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 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2442

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 6726

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]

Rubi steps

\begin {align*} \int (d x)^{5/2} \text {Li}_3(a x) \, dx &=\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}-\frac {2}{7} \int (d x)^{5/2} \text {Li}_2(a x) \, dx\\ &=-\frac {4 (d x)^{7/2} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}-\frac {4}{49} \int (d x)^{5/2} \log (1-a x) \, dx\\ &=-\frac {8 (d x)^{7/2} \log (1-a x)}{343 d}-\frac {4 (d x)^{7/2} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}-\frac {(8 a) \int \frac {(d x)^{7/2}}{1-a x} \, dx}{343 d}\\ &=\frac {16 (d x)^{7/2}}{2401 d}-\frac {8 (d x)^{7/2} \log (1-a x)}{343 d}-\frac {4 (d x)^{7/2} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}-\frac {8}{343} \int \frac {(d x)^{5/2}}{1-a x} \, dx\\ &=\frac {16 (d x)^{5/2}}{1715 a}+\frac {16 (d x)^{7/2}}{2401 d}-\frac {8 (d x)^{7/2} \log (1-a x)}{343 d}-\frac {4 (d x)^{7/2} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}-\frac {(8 d) \int \frac {(d x)^{3/2}}{1-a x} \, dx}{343 a}\\ &=\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 {8 (d x)^{7/2} \log (1-a x)}{343 d}-\frac {4 (d x)^{7/2} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}-\frac {\left (8 d^2\right ) \int \frac {\sqrt {d x}}{1-a x} \, dx}{343 a^2}\\ &=\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 {8 (d x)^{7/2} \log (1-a x)}{343 d}-\frac {4 (d x)^{7/2} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}-\frac {\left (8 d^3\right ) \int \frac {1}{\sqrt {d x} (1-a x)} \, dx}{343 a^3}\\ &=\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 {8 (d x)^{7/2} \log (1-a x)}{343 d}-\frac {4 (d x)^{7/2} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}-\frac {\left (16 d^2\right ) \text {Subst}\left (\int \frac {1}{1-\frac {a x^2}{d}} \, dx,x,\sqrt {d x}\right )}{343 a^3}\\ &=\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} \tanh ^{-1}\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} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 98, normalized size = 0.64 \begin {gather*} \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 \tanh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{7/2} \sqrt {x}}-420 x^3 \log (1-a x)-1470 x^3 \text {PolyLog}(2,a x)+5145 x^3 \text {PolyLog}(3,a x)\right )}{36015 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(5/2)*PolyLog[3, a*x],x]

[Out]

(2*(d*x)^(5/2)*((8*(105 + 35*a*x + 21*a^2*x^2 + 15*a^3*x^3))/a^3 - (840*ArcTanh[Sqrt[a]*Sqrt[x]])/(a^(7/2)*Sqr
t[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)

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Maple [A]
time = 0.14, size = 149, normalized size = 0.97

method	result	size
meijerg	\(\frac {\left (d x \right )^{\frac {5}{2}} \left (\frac {2 \sqrt {x}\, \left (-a \right )^{\frac {9}{2}} \left (360 a^{3} x^{3}+504 a^{2} x^{2}+840 a x +2520\right )}{108045 a^{4}}+\frac {8 \sqrt {x}\, \left (-a \right )^{\frac {9}{2}} \left (\ln \left (1-\sqrt {a x}\right )-\ln \left (1+\sqrt {a x}\right )\right )}{343 a^{4} \sqrt {a x}}-\frac {8 x^{\frac {7}{2}} \left (-a \right )^{\frac {9}{2}} \ln \left (-a x +1\right )}{343 a}-\frac {4 x^{\frac {7}{2}} \left (-a \right )^{\frac {9}{2}} \polylog \left (2, a x \right )}{49 a}+\frac {2 x^{\frac {7}{2}} \left (-a \right )^{\frac {9}{2}} \polylog \left (3, a x \right )}{7 a}\right )}{x^{\frac {5}{2}} \left (-a \right )^{\frac {5}{2}} a}\)	\(149\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(5/2)*polylog(3,a*x),x,method=_RETURNVERBOSE)

[Out]

(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)*(ln(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))

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Maxima [A]
time = 0.48, size = 156, normalized size = 1.02 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*polylog(3,a*x),x, algorithm="maxima")

[Out]

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*log(-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

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Fricas [A]
time = 0.39, size = 279, normalized size = 1.82 \begin {gather*} \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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*polylog(3,a*x),x, algorithm="fricas")

[Out]

[2/36015*(5145*sqrt(d*x)*a^3*d^2*x^3*polylog(3, a*x) + 420*d^2*sqrt(d/a)*log((a*d*x - 2*sqrt(d*x)*a*sqrt(d/a)
+ d)/(a*x - 1)) - 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, 2/36015*(5145*sqrt(d*x)*a^3*d^2*x^3*polylog(3, a*x) + 840*d^2*sqr
t(-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]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d x\right )^{\frac {5}{2}} \operatorname {Li}_{3}\left (a x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(5/2)*polylog(3,a*x),x)

[Out]

Integral((d*x)**(5/2)*polylog(3, a*x), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*polylog(3,a*x),x, algorithm="giac")

[Out]

integrate((d*x)^(5/2)*polylog(3, a*x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (d\,x\right )}^{5/2}\,\mathrm {polylog}\left (3,a\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(5/2)*polylog(3, a*x),x)

[Out]

int((d*x)^(5/2)*polylog(3, a*x), x)

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