3.78 \(\int (d x)^{5/2} \text{PolyLog}(3,a x^2) \, dx\)

Optimal. Leaf size=161 \[ -\frac{8 (d x)^{7/2} \text{PolyLog}\left (2,a x^2\right )}{49 d}+\frac{2 (d x)^{7/2} \text{PolyLog}\left (3,a x^2\right )}{7 d}+\frac{64 d^{5/2} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{343 a^{7/4}}-\frac{64 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{343 a^{7/4}}-\frac{32 (d x)^{7/2} \log \left (1-a x^2\right )}{343 d}+\frac{128 d (d x)^{3/2}}{1029 a}+\frac{128 (d x)^{7/2}}{2401 d} \]

[Out]

(128*d*(d*x)^(3/2))/(1029*a) + (128*(d*x)^(7/2))/(2401*d) + (64*d^(5/2)*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(
343*a^(7/4)) - (64*d^(5/2)*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(343*a^(7/4)) - (32*(d*x)^(7/2)*Log[1 - a*x^2
])/(343*d) - (8*(d*x)^(7/2)*PolyLog[2, a*x^2])/(49*d) + (2*(d*x)^(7/2)*PolyLog[3, a*x^2])/(7*d)

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Rubi [A]  time = 0.126749, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {6591, 2455, 16, 321, 329, 298, 205, 208} \[ -\frac{8 (d x)^{7/2} \text{PolyLog}\left (2,a x^2\right )}{49 d}+\frac{2 (d x)^{7/2} \text{PolyLog}\left (3,a x^2\right )}{7 d}+\frac{64 d^{5/2} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{343 a^{7/4}}-\frac{64 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{343 a^{7/4}}-\frac{32 (d x)^{7/2} \log \left (1-a x^2\right )}{343 d}+\frac{128 d (d x)^{3/2}}{1029 a}+\frac{128 (d x)^{7/2}}{2401 d} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(5/2)*PolyLog[3, a*x^2],x]

[Out]

(128*d*(d*x)^(3/2))/(1029*a) + (128*(d*x)^(7/2))/(2401*d) + (64*d^(5/2)*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(
343*a^(7/4)) - (64*d^(5/2)*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(343*a^(7/4)) - (32*(d*x)^(7/2)*Log[1 - a*x^2
])/(343*d) - (8*(d*x)^(7/2)*PolyLog[2, a*x^2])/(49*d) + (2*(d*x)^(7/2)*PolyLog[3, a*x^2])/(7*d)

Rule 6591

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]

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int (d x)^{5/2} \text{Li}_3\left (a x^2\right ) \, dx &=\frac{2 (d x)^{7/2} \text{Li}_3\left (a x^2\right )}{7 d}-\frac{4}{7} \int (d x)^{5/2} \text{Li}_2\left (a x^2\right ) \, dx\\ &=-\frac{8 (d x)^{7/2} \text{Li}_2\left (a x^2\right )}{49 d}+\frac{2 (d x)^{7/2} \text{Li}_3\left (a x^2\right )}{7 d}-\frac{16}{49} \int (d x)^{5/2} \log \left (1-a x^2\right ) \, dx\\ &=-\frac{32 (d x)^{7/2} \log \left (1-a x^2\right )}{343 d}-\frac{8 (d x)^{7/2} \text{Li}_2\left (a x^2\right )}{49 d}+\frac{2 (d x)^{7/2} \text{Li}_3\left (a x^2\right )}{7 d}-\frac{(64 a) \int \frac{x (d x)^{7/2}}{1-a x^2} \, dx}{343 d}\\ &=-\frac{32 (d x)^{7/2} \log \left (1-a x^2\right )}{343 d}-\frac{8 (d x)^{7/2} \text{Li}_2\left (a x^2\right )}{49 d}+\frac{2 (d x)^{7/2} \text{Li}_3\left (a x^2\right )}{7 d}-\frac{(64 a) \int \frac{(d x)^{9/2}}{1-a x^2} \, dx}{343 d^2}\\ &=\frac{128 (d x)^{7/2}}{2401 d}-\frac{32 (d x)^{7/2} \log \left (1-a x^2\right )}{343 d}-\frac{8 (d x)^{7/2} \text{Li}_2\left (a x^2\right )}{49 d}+\frac{2 (d x)^{7/2} \text{Li}_3\left (a x^2\right )}{7 d}-\frac{64}{343} \int \frac{(d x)^{5/2}}{1-a x^2} \, dx\\ &=\frac{128 d (d x)^{3/2}}{1029 a}+\frac{128 (d x)^{7/2}}{2401 d}-\frac{32 (d x)^{7/2} \log \left (1-a x^2\right )}{343 d}-\frac{8 (d x)^{7/2} \text{Li}_2\left (a x^2\right )}{49 d}+\frac{2 (d x)^{7/2} \text{Li}_3\left (a x^2\right )}{7 d}-\frac{\left (64 d^2\right ) \int \frac{\sqrt{d x}}{1-a x^2} \, dx}{343 a}\\ &=\frac{128 d (d x)^{3/2}}{1029 a}+\frac{128 (d x)^{7/2}}{2401 d}-\frac{32 (d x)^{7/2} \log \left (1-a x^2\right )}{343 d}-\frac{8 (d x)^{7/2} \text{Li}_2\left (a x^2\right )}{49 d}+\frac{2 (d x)^{7/2} \text{Li}_3\left (a x^2\right )}{7 d}-\frac{(128 d) \operatorname{Subst}\left (\int \frac{x^2}{1-\frac{a x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{343 a}\\ &=\frac{128 d (d x)^{3/2}}{1029 a}+\frac{128 (d x)^{7/2}}{2401 d}-\frac{32 (d x)^{7/2} \log \left (1-a x^2\right )}{343 d}-\frac{8 (d x)^{7/2} \text{Li}_2\left (a x^2\right )}{49 d}+\frac{2 (d x)^{7/2} \text{Li}_3\left (a x^2\right )}{7 d}-\frac{\left (64 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{d-\sqrt{a} x^2} \, dx,x,\sqrt{d x}\right )}{343 a^{3/2}}+\frac{\left (64 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+\sqrt{a} x^2} \, dx,x,\sqrt{d x}\right )}{343 a^{3/2}}\\ &=\frac{128 d (d x)^{3/2}}{1029 a}+\frac{128 (d x)^{7/2}}{2401 d}+\frac{64 d^{5/2} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{343 a^{7/4}}-\frac{64 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{343 a^{7/4}}-\frac{32 (d x)^{7/2} \log \left (1-a x^2\right )}{343 d}-\frac{8 (d x)^{7/2} \text{Li}_2\left (a x^2\right )}{49 d}+\frac{2 (d x)^{7/2} \text{Li}_3\left (a x^2\right )}{7 d}\\ \end{align*}

Mathematica [C]  time = 0.103023, size = 89, normalized size = 0.55 \[ -\frac{11 d \text{Gamma}\left (\frac{11}{4}\right ) (d x)^{3/2} \left (448 \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},a x^2\right )+588 a x^2 \text{PolyLog}\left (2,a x^2\right )-1029 a x^2 \text{PolyLog}\left (3,a x^2\right )-192 a x^2+336 a x^2 \log \left (1-a x^2\right )-448\right )}{14406 a \text{Gamma}\left (\frac{15}{4}\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-11*d*(d*x)^(3/2)*Gamma[11/4]*(-448 - 192*a*x^2 + 448*Hypergeometric2F1[3/4, 1, 7/4, a*x^2] + 336*a*x^2*Log[1
- a*x^2] + 588*a*x^2*PolyLog[2, a*x^2] - 1029*a*x^2*PolyLog[3, a*x^2]))/(14406*a*Gamma[15/4])

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Maple [A]  time = 0.193, size = 155, normalized size = 1. \begin{align*} -{\frac{1}{2} \left ( dx \right ) ^{{\frac{5}{2}}} \left ({\frac{8448\,a{x}^{2}+19712}{79233\,{a}^{2}}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{11}{4}}}}+{\frac{64}{343\,{a}^{2}}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{11}{4}}} \left ( \ln \left ( 1-\sqrt [4]{a{x}^{2}} \right ) -\ln \left ( 1+\sqrt [4]{a{x}^{2}} \right ) +2\,\arctan \left ( \sqrt [4]{a{x}^{2}} \right ) \right ) \left ( a{x}^{2} \right ) ^{-{\frac{3}{4}}}}-{\frac{64\,\ln \left ( -a{x}^{2}+1 \right ) }{343\,a}{x}^{{\frac{7}{2}}} \left ( -a \right ) ^{{\frac{11}{4}}}}-{\frac{16\,{\it polylog} \left ( 2,a{x}^{2} \right ) }{49\,a}{x}^{{\frac{7}{2}}} \left ( -a \right ) ^{{\frac{11}{4}}}}+{\frac{4\,{\it polylog} \left ( 3,a{x}^{2} \right ) }{7\,a}{x}^{{\frac{7}{2}}} \left ( -a \right ) ^{{\frac{11}{4}}}} \right ){x}^{-{\frac{5}{2}}} \left ( -a \right ) ^{-{\frac{7}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(d*x)^(5/2)/x^(5/2)/(-a)^(7/4)*(4/79233*x^(3/2)*(-a)^(11/4)*(2112*a*x^2+4928)/a^2+64/343*x^(3/2)*(-a)^(11
/4)/a^2/(a*x^2)^(3/4)*(ln(1-(a*x^2)^(1/4))-ln(1+(a*x^2)^(1/4))+2*arctan((a*x^2)^(1/4)))-64/343*x^(7/2)*(-a)^(1
1/4)/a*ln(-a*x^2+1)-16/49*x^(7/2)*(-a)^(11/4)*polylog(2,a*x^2)/a+4/7*x^(7/2)*(-a)^(11/4)/a*polylog(3,a*x^2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 3.10612, size = 682, normalized size = 4.24 \begin{align*} -\frac{2 \,{\left (588 \, \sqrt{d x} a d^{2} x^{3}{\rm \%iint}\left (a, x, -\frac{\log \left (-a x^{2} + 1\right )}{a}, -\frac{2 \, \log \left (-a x^{2} + 1\right )}{x}\right ) - 1029 \, \sqrt{d x} a d^{2} x^{3}{\rm polylog}\left (3, a x^{2}\right ) + 1344 \, \left (\frac{d^{10}}{a^{7}}\right )^{\frac{1}{4}} a \arctan \left (-\frac{\left (\frac{d^{10}}{a^{7}}\right )^{\frac{1}{4}} \sqrt{d x} a^{2} d^{7} - \sqrt{d^{15} x + \sqrt{\frac{d^{10}}{a^{7}}} a^{3} d^{10}} \left (\frac{d^{10}}{a^{7}}\right )^{\frac{1}{4}} a^{2}}{d^{10}}\right ) + 336 \, \left (\frac{d^{10}}{a^{7}}\right )^{\frac{1}{4}} a \log \left (32768 \, \sqrt{d x} d^{7} + 32768 \, \left (\frac{d^{10}}{a^{7}}\right )^{\frac{3}{4}} a^{5}\right ) - 336 \, \left (\frac{d^{10}}{a^{7}}\right )^{\frac{1}{4}} a \log \left (32768 \, \sqrt{d x} d^{7} - 32768 \, \left (\frac{d^{10}}{a^{7}}\right )^{\frac{3}{4}} a^{5}\right ) + 16 \,{\left (21 \, a d^{2} x^{3} \log \left (-a x^{2} + 1\right ) - 12 \, a d^{2} x^{3} - 28 \, d^{2} x\right )} \sqrt{d x}\right )}}{7203 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/7203*(588*sqrt(d*x)*a*d^2*x^3*\%iint(a, x, -log(-a*x^2 + 1)/a, -2*log(-a*x^2 + 1)/x) - 1029*sqrt(d*x)*a*d^2*
x^3*polylog(3, a*x^2) + 1344*(d^10/a^7)^(1/4)*a*arctan(-((d^10/a^7)^(1/4)*sqrt(d*x)*a^2*d^7 - sqrt(d^15*x + sq
rt(d^10/a^7)*a^3*d^10)*(d^10/a^7)^(1/4)*a^2)/d^10) + 336*(d^10/a^7)^(1/4)*a*log(32768*sqrt(d*x)*d^7 + 32768*(d
^10/a^7)^(3/4)*a^5) - 336*(d^10/a^7)^(1/4)*a*log(32768*sqrt(d*x)*d^7 - 32768*(d^10/a^7)^(3/4)*a^5) + 16*(21*a*
d^2*x^3*log(-a*x^2 + 1) - 12*a*d^2*x^3 - 28*d^2*x)*sqrt(d*x))/a

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{\frac{5}{2}}{\rm Li}_{3}(a x^{2})\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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