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

Optimal. Leaf size=161 \[ -\frac{8 (d x)^{5/2} \text{PolyLog}\left (2,a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{PolyLog}\left (3,a x^2\right )}{5 d}-\frac{64 d^{3/2} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 a^{5/4}}-\frac{64 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 a^{5/4}}-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}+\frac{128 d \sqrt{d x}}{125 a}+\frac{128 (d x)^{5/2}}{625 d} \]

[Out]

(128*d*Sqrt[d*x])/(125*a) + (128*(d*x)^(5/2))/(625*d) - (64*d^(3/2)*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(125*
a^(5/4)) - (64*d^(3/2)*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(125*a^(5/4)) - (32*(d*x)^(5/2)*Log[1 - a*x^2])/(
125*d) - (8*(d*x)^(5/2)*PolyLog[2, a*x^2])/(25*d) + (2*(d*x)^(5/2)*PolyLog[3, a*x^2])/(5*d)

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Rubi [A]  time = 0.115868, 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, 212, 208, 205} \[ -\frac{8 (d x)^{5/2} \text{PolyLog}\left (2,a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{PolyLog}\left (3,a x^2\right )}{5 d}-\frac{64 d^{3/2} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 a^{5/4}}-\frac{64 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 a^{5/4}}-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}+\frac{128 d \sqrt{d x}}{125 a}+\frac{128 (d x)^{5/2}}{625 d} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(3/2)*PolyLog[3, a*x^2],x]

[Out]

(128*d*Sqrt[d*x])/(125*a) + (128*(d*x)^(5/2))/(625*d) - (64*d^(3/2)*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(125*
a^(5/4)) - (64*d^(3/2)*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(125*a^(5/4)) - (32*(d*x)^(5/2)*Log[1 - a*x^2])/(
125*d) - (8*(d*x)^(5/2)*PolyLog[2, a*x^2])/(25*d) + (2*(d*x)^(5/2)*PolyLog[3, a*x^2])/(5*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 212

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

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]

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]

Rubi steps

\begin{align*} \int (d x)^{3/2} \text{Li}_3\left (a x^2\right ) \, dx &=\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}-\frac{4}{5} \int (d x)^{3/2} \text{Li}_2\left (a x^2\right ) \, dx\\ &=-\frac{8 (d x)^{5/2} \text{Li}_2\left (a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}-\frac{16}{25} \int (d x)^{3/2} \log \left (1-a x^2\right ) \, dx\\ &=-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}-\frac{8 (d x)^{5/2} \text{Li}_2\left (a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}-\frac{(64 a) \int \frac{x (d x)^{5/2}}{1-a x^2} \, dx}{125 d}\\ &=-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}-\frac{8 (d x)^{5/2} \text{Li}_2\left (a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}-\frac{(64 a) \int \frac{(d x)^{7/2}}{1-a x^2} \, dx}{125 d^2}\\ &=\frac{128 (d x)^{5/2}}{625 d}-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}-\frac{8 (d x)^{5/2} \text{Li}_2\left (a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}-\frac{64}{125} \int \frac{(d x)^{3/2}}{1-a x^2} \, dx\\ &=\frac{128 d \sqrt{d x}}{125 a}+\frac{128 (d x)^{5/2}}{625 d}-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}-\frac{8 (d x)^{5/2} \text{Li}_2\left (a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}-\frac{\left (64 d^2\right ) \int \frac{1}{\sqrt{d x} \left (1-a x^2\right )} \, dx}{125 a}\\ &=\frac{128 d \sqrt{d x}}{125 a}+\frac{128 (d x)^{5/2}}{625 d}-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}-\frac{8 (d x)^{5/2} \text{Li}_2\left (a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}-\frac{(128 d) \operatorname{Subst}\left (\int \frac{1}{1-\frac{a x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{125 a}\\ &=\frac{128 d \sqrt{d x}}{125 a}+\frac{128 (d x)^{5/2}}{625 d}-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}-\frac{8 (d x)^{5/2} \text{Li}_2\left (a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}-\frac{\left (64 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{d-\sqrt{a} x^2} \, dx,x,\sqrt{d x}\right )}{125 a}-\frac{\left (64 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+\sqrt{a} x^2} \, dx,x,\sqrt{d x}\right )}{125 a}\\ &=\frac{128 d \sqrt{d x}}{125 a}+\frac{128 (d x)^{5/2}}{625 d}-\frac{64 d^{3/2} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 a^{5/4}}-\frac{64 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 a^{5/4}}-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}-\frac{8 (d x)^{5/2} \text{Li}_2\left (a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}\\ \end{align*}

Mathematica [C]  time = 0.0996985, size = 89, normalized size = 0.55 \[ -\frac{9 d \text{Gamma}\left (\frac{9}{4}\right ) \sqrt{d x} \left (320 \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},a x^2\right )+100 a x^2 \text{PolyLog}\left (2,a x^2\right )-125 a x^2 \text{PolyLog}\left (3,a x^2\right )-64 a x^2+80 a x^2 \log \left (1-a x^2\right )-320\right )}{1250 a \text{Gamma}\left (\frac{13}{4}\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-9*d*Sqrt[d*x]*Gamma[9/4]*(-320 - 64*a*x^2 + 320*Hypergeometric2F1[1/4, 1, 5/4, a*x^2] + 80*a*x^2*Log[1 - a*x
^2] + 100*a*x^2*PolyLog[2, a*x^2] - 125*a*x^2*PolyLog[3, a*x^2]))/(1250*a*Gamma[13/4])

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Maple [A]  time = 0.208, size = 155, normalized size = 1. \begin{align*} -{\frac{1}{2} \left ( dx \right ) ^{{\frac{3}{2}}} \left ({\frac{2304\,a{x}^{2}+11520}{5625\,{a}^{2}}\sqrt{x} \left ( -a \right ) ^{{\frac{9}{4}}}}+{\frac{64}{125\,{a}^{2}}\sqrt{x} \left ( -a \right ) ^{{\frac{9}{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 ){\frac{1}{\sqrt [4]{a{x}^{2}}}}}-{\frac{64\,\ln \left ( -a{x}^{2}+1 \right ) }{125\,a}{x}^{{\frac{5}{2}}} \left ( -a \right ) ^{{\frac{9}{4}}}}-{\frac{16\,{\it polylog} \left ( 2,a{x}^{2} \right ) }{25\,a}{x}^{{\frac{5}{2}}} \left ( -a \right ) ^{{\frac{9}{4}}}}+{\frac{4\,{\it polylog} \left ( 3,a{x}^{2} \right ) }{5\,a}{x}^{{\frac{5}{2}}} \left ( -a \right ) ^{{\frac{9}{4}}}} \right ){x}^{-{\frac{3}{2}}} \left ( -a \right ) ^{-{\frac{5}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(d*x)^(3/2)/x^(3/2)/(-a)^(5/4)*(4/5625*x^(1/2)*(-a)^(9/4)*(576*a*x^2+2880)/a^2+64/125*x^(1/2)*(-a)^(9/4)/
a^2/(a*x^2)^(1/4)*(ln(1-(a*x^2)^(1/4))-ln(1+(a*x^2)^(1/4))-2*arctan((a*x^2)^(1/4)))-64/125*x^(5/2)*(-a)^(9/4)/
a*ln(-a*x^2+1)-16/25*x^(5/2)*(-a)^(9/4)*polylog(2,a*x^2)/a+4/5*x^(5/2)*(-a)^(9/4)/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)^(3/2)*polylog(3,a*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 2.91474, size = 606, normalized size = 3.76 \begin{align*} -\frac{2 \,{\left (100 \, \sqrt{d x} a d x^{2}{\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 ) - 125 \, \sqrt{d x} a d x^{2}{\rm polylog}\left (3, a x^{2}\right ) - 320 \, a \left (\frac{d^{6}}{a^{5}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{d x} a^{4} d \left (\frac{d^{6}}{a^{5}}\right )^{\frac{3}{4}} - \sqrt{d^{3} x + a^{2} \sqrt{\frac{d^{6}}{a^{5}}}} a^{4} \left (\frac{d^{6}}{a^{5}}\right )^{\frac{3}{4}}}{d^{6}}\right ) + 80 \, a \left (\frac{d^{6}}{a^{5}}\right )^{\frac{1}{4}} \log \left (32 \, \sqrt{d x} d + 32 \, a \left (\frac{d^{6}}{a^{5}}\right )^{\frac{1}{4}}\right ) - 80 \, a \left (\frac{d^{6}}{a^{5}}\right )^{\frac{1}{4}} \log \left (32 \, \sqrt{d x} d - 32 \, a \left (\frac{d^{6}}{a^{5}}\right )^{\frac{1}{4}}\right ) + 16 \,{\left (5 \, a d x^{2} \log \left (-a x^{2} + 1\right ) - 4 \, a d x^{2} - 20 \, d\right )} \sqrt{d x}\right )}}{625 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/625*(100*sqrt(d*x)*a*d*x^2*\%iint(a, x, -log(-a*x^2 + 1)/a, -2*log(-a*x^2 + 1)/x) - 125*sqrt(d*x)*a*d*x^2*po
lylog(3, a*x^2) - 320*a*(d^6/a^5)^(1/4)*arctan(-(sqrt(d*x)*a^4*d*(d^6/a^5)^(3/4) - sqrt(d^3*x + a^2*sqrt(d^6/a
^5))*a^4*(d^6/a^5)^(3/4))/d^6) + 80*a*(d^6/a^5)^(1/4)*log(32*sqrt(d*x)*d + 32*a*(d^6/a^5)^(1/4)) - 80*a*(d^6/a
^5)^(1/4)*log(32*sqrt(d*x)*d - 32*a*(d^6/a^5)^(1/4)) + 16*(5*a*d*x^2*log(-a*x^2 + 1) - 4*a*d*x^2 - 20*d)*sqrt(
d*x))/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{\frac{3}{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)^(3/2)*polylog(3,a*x^2),x, algorithm="giac")

[Out]

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