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

Optimal. Leaf size=146 \[ -\frac{8 (d x)^{3/2} \text{PolyLog}\left (2,a x^2\right )}{9 d}+\frac{2 (d x)^{3/2} \text{PolyLog}\left (3,a x^2\right )}{3 d}+\frac{64 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{27 a^{3/4}}-\frac{64 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{27 a^{3/4}}-\frac{32 (d x)^{3/2} \log \left (1-a x^2\right )}{27 d}+\frac{128 (d x)^{3/2}}{81 d} \]

[Out]

(128*(d*x)^(3/2))/(81*d) + (64*Sqrt[d]*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(27*a^(3/4)) - (64*Sqrt[d]*ArcTanh
[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(27*a^(3/4)) - (32*(d*x)^(3/2)*Log[1 - a*x^2])/(27*d) - (8*(d*x)^(3/2)*PolyLog[
2, a*x^2])/(9*d) + (2*(d*x)^(3/2)*PolyLog[3, a*x^2])/(3*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0974576, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{8 (d x)^{3/2} \text{PolyLog}\left (2,a x^2\right )}{9 d}+\frac{2 (d x)^{3/2} \text{PolyLog}\left (3,a x^2\right )}{3 d}+\frac{64 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{27 a^{3/4}}-\frac{64 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{27 a^{3/4}}-\frac{32 (d x)^{3/2} \log \left (1-a x^2\right )}{27 d}+\frac{128 (d x)^{3/2}}{81 d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*x]*PolyLog[3, a*x^2],x]

[Out]

(128*(d*x)^(3/2))/(81*d) + (64*Sqrt[d]*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(27*a^(3/4)) - (64*Sqrt[d]*ArcTanh
[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(27*a^(3/4)) - (32*(d*x)^(3/2)*Log[1 - a*x^2])/(27*d) - (8*(d*x)^(3/2)*PolyLog[
2, a*x^2])/(9*d) + (2*(d*x)^(3/2)*PolyLog[3, a*x^2])/(3*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 \sqrt{d x} \text{Li}_3\left (a x^2\right ) \, dx &=\frac{2 (d x)^{3/2} \text{Li}_3\left (a x^2\right )}{3 d}-\frac{4}{3} \int \sqrt{d x} \text{Li}_2\left (a x^2\right ) \, dx\\ &=-\frac{8 (d x)^{3/2} \text{Li}_2\left (a x^2\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_3\left (a x^2\right )}{3 d}-\frac{16}{9} \int \sqrt{d x} \log \left (1-a x^2\right ) \, dx\\ &=-\frac{32 (d x)^{3/2} \log \left (1-a x^2\right )}{27 d}-\frac{8 (d x)^{3/2} \text{Li}_2\left (a x^2\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_3\left (a x^2\right )}{3 d}-\frac{(64 a) \int \frac{x (d x)^{3/2}}{1-a x^2} \, dx}{27 d}\\ &=-\frac{32 (d x)^{3/2} \log \left (1-a x^2\right )}{27 d}-\frac{8 (d x)^{3/2} \text{Li}_2\left (a x^2\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_3\left (a x^2\right )}{3 d}-\frac{(64 a) \int \frac{(d x)^{5/2}}{1-a x^2} \, dx}{27 d^2}\\ &=\frac{128 (d x)^{3/2}}{81 d}-\frac{32 (d x)^{3/2} \log \left (1-a x^2\right )}{27 d}-\frac{8 (d x)^{3/2} \text{Li}_2\left (a x^2\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_3\left (a x^2\right )}{3 d}-\frac{64}{27} \int \frac{\sqrt{d x}}{1-a x^2} \, dx\\ &=\frac{128 (d x)^{3/2}}{81 d}-\frac{32 (d x)^{3/2} \log \left (1-a x^2\right )}{27 d}-\frac{8 (d x)^{3/2} \text{Li}_2\left (a x^2\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_3\left (a x^2\right )}{3 d}-\frac{128 \operatorname{Subst}\left (\int \frac{x^2}{1-\frac{a x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{27 d}\\ &=\frac{128 (d x)^{3/2}}{81 d}-\frac{32 (d x)^{3/2} \log \left (1-a x^2\right )}{27 d}-\frac{8 (d x)^{3/2} \text{Li}_2\left (a x^2\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_3\left (a x^2\right )}{3 d}-\frac{(64 d) \operatorname{Subst}\left (\int \frac{1}{d-\sqrt{a} x^2} \, dx,x,\sqrt{d x}\right )}{27 \sqrt{a}}+\frac{(64 d) \operatorname{Subst}\left (\int \frac{1}{d+\sqrt{a} x^2} \, dx,x,\sqrt{d x}\right )}{27 \sqrt{a}}\\ &=\frac{128 (d x)^{3/2}}{81 d}+\frac{64 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{27 a^{3/4}}-\frac{64 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{27 a^{3/4}}-\frac{32 (d x)^{3/2} \log \left (1-a x^2\right )}{27 d}-\frac{8 (d x)^{3/2} \text{Li}_2\left (a x^2\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_3\left (a x^2\right )}{3 d}\\ \end{align*}

Mathematica [C]  time = 0.0915019, size = 68, normalized size = 0.47 \[ -\frac{7 x \text{Gamma}\left (\frac{7}{4}\right ) \sqrt{d x} \left (64 \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},a x^2\right )+36 \text{PolyLog}\left (2,a x^2\right )-27 \text{PolyLog}\left (3,a x^2\right )+48 \log \left (1-a x^2\right )-64\right )}{162 \text{Gamma}\left (\frac{11}{4}\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[d*x]*PolyLog[3, a*x^2],x]

[Out]

(-7*x*Sqrt[d*x]*Gamma[7/4]*(-64 + 64*Hypergeometric2F1[3/4, 1, 7/4, a*x^2] + 48*Log[1 - a*x^2] + 36*PolyLog[2,
a*x^2] - 27*PolyLog[3, a*x^2]))/(162*Gamma[11/4])

________________________________________________________________________________________

Maple [A]  time = 0.224, size = 147, normalized size = 1. \begin{align*} -{\frac{1}{2}\sqrt{dx} \left ({\frac{256}{81\,a}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{7}{4}}}}+{\frac{64}{27\,a}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{7}{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 ) }{27\,a}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{7}{4}}}}-{\frac{16\,{\it polylog} \left ( 2,a{x}^{2} \right ) }{9\,a}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{7}{4}}}}+{\frac{4\,{\it polylog} \left ( 3,a{x}^{2} \right ) }{3\,a}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{7}{4}}}} \right ){\frac{1}{\sqrt{x}}} \left ( -a \right ) ^{-{\frac{3}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(1/2)*polylog(3,a*x^2),x)

[Out]

-1/2*(d*x)^(1/2)/x^(1/2)/(-a)^(3/4)*(256/81*x^(3/2)*(-a)^(7/4)/a+64/27*x^(3/2)*(-a)^(7/4)/a/(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)))-64/27*x^(3/2)*(-a)^(7/4)/a*ln(-a*x^2+1)-16/9*x^(
3/2)*(-a)^(7/4)*polylog(2,a*x^2)/a+4/3*x^(3/2)*(-a)^(7/4)/a*polylog(3,a*x^2))

________________________________________________________________________________________

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)^(1/2)*polylog(3,a*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [C]  time = 2.8515, size = 579, normalized size = 3.97 \begin{align*} -\frac{8}{9} \, \sqrt{d x} x{\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 ) + \frac{2}{3} \, \sqrt{d x} x{\rm polylog}\left (3, a x^{2}\right ) - \frac{32}{81} \, \sqrt{d x}{\left (3 \, x \log \left (-a x^{2} + 1\right ) - 4 \, x\right )} - \frac{128}{27} \, \left (\frac{d^{2}}{a^{3}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{d x} a d \left (\frac{d^{2}}{a^{3}}\right )^{\frac{1}{4}} - \sqrt{d^{3} x + a d^{2} \sqrt{\frac{d^{2}}{a^{3}}}} a \left (\frac{d^{2}}{a^{3}}\right )^{\frac{1}{4}}}{d^{2}}\right ) - \frac{32}{27} \, \left (\frac{d^{2}}{a^{3}}\right )^{\frac{1}{4}} \log \left (32768 \, a^{2} \left (\frac{d^{2}}{a^{3}}\right )^{\frac{3}{4}} + 32768 \, \sqrt{d x} d\right ) + \frac{32}{27} \, \left (\frac{d^{2}}{a^{3}}\right )^{\frac{1}{4}} \log \left (-32768 \, a^{2} \left (\frac{d^{2}}{a^{3}}\right )^{\frac{3}{4}} + 32768 \, \sqrt{d x} d\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(1/2)*polylog(3,a*x^2),x, algorithm="fricas")

[Out]

-8/9*sqrt(d*x)*x*\%iint(a, x, -log(-a*x^2 + 1)/a, -2*log(-a*x^2 + 1)/x) + 2/3*sqrt(d*x)*x*polylog(3, a*x^2) - 3
2/81*sqrt(d*x)*(3*x*log(-a*x^2 + 1) - 4*x) - 128/27*(d^2/a^3)^(1/4)*arctan(-(sqrt(d*x)*a*d*(d^2/a^3)^(1/4) - s
qrt(d^3*x + a*d^2*sqrt(d^2/a^3))*a*(d^2/a^3)^(1/4))/d^2) - 32/27*(d^2/a^3)^(1/4)*log(32768*a^2*(d^2/a^3)^(3/4)
+ 32768*sqrt(d*x)*d) + 32/27*(d^2/a^3)^(1/4)*log(-32768*a^2*(d^2/a^3)^(3/4) + 32768*sqrt(d*x)*d)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d x} \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)**(1/2)*polylog(3,a*x**2),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(1/2)*polylog(3,a*x^2),x, algorithm="giac")

[Out]

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