3.1.78 \(\int (d x)^{5/2} \text {PolyLog}(3,a x^2) \, dx\) [78]

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

[Out]

128/1029*d*(d*x)^(3/2)/a+128/2401*(d*x)^(7/2)/d+64/343*d^(5/2)*arctan(a^(1/4)*(d*x)^(1/2)/d^(1/2))/a^(7/4)-64/
343*d^(5/2)*arctanh(a^(1/4)*(d*x)^(1/2)/d^(1/2))/a^(7/4)-32/343*(d*x)^(7/2)*ln(-a*x^2+1)/d-8/49*(d*x)^(7/2)*po
lylog(2,a*x^2)/d+2/7*(d*x)^(7/2)*polylog(3,a*x^2)/d

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Rubi [A]
time = 0.09, antiderivative size = 161, normalized size of antiderivative = 1.00, 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 = {6726, 2505, 16, 327, 335, 304, 211, 214} \begin {gather*} \frac {64 d^{5/2} \text {ArcTan}\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 {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 {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} \end {gather*}

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 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 211

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

Rule 214

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

Rule 304

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 327

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 335

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 2505

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 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\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) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.07, size = 89, normalized size = 0.55 \begin {gather*} -\frac {11 d (d x)^{3/2} \Gamma \left (\frac {11}{4}\right ) \left (-448-192 a x^2+448 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};a x^2\right )+336 a x^2 \log \left (1-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 )\right )}{14406 a \Gamma \left (\frac {15}{4}\right )} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.16, size = 155, normalized size = 0.96

method	result	size
meijerg	\(-\frac {\left (d x \right )^{\frac {5}{2}} \left (\frac {4 x^{\frac {3}{2}} \left (-a \right )^{\frac {11}{4}} \left (2112 a \,x^{2}+4928\right )}{79233 a^{2}}+\frac {64 x^{\frac {3}{2}} \left (-a \right )^{\frac {11}{4}} \left (\ln \left (1-\left (a \,x^{2}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (a \,x^{2}\right )^{\frac {1}{4}}\right )+2 \arctan \left (\left (a \,x^{2}\right )^{\frac {1}{4}}\right )\right )}{343 a^{2} \left (a \,x^{2}\right )^{\frac {3}{4}}}-\frac {64 x^{\frac {7}{2}} \left (-a \right )^{\frac {11}{4}} \ln \left (-a \,x^{2}+1\right )}{343 a}-\frac {16 x^{\frac {7}{2}} \left (-a \right )^{\frac {11}{4}} \polylog \left (2, a \,x^{2}\right )}{49 a}+\frac {4 x^{\frac {7}{2}} \left (-a \right )^{\frac {11}{4}} \polylog \left (3, a \,x^{2}\right )}{7 a}\right )}{2 x^{\frac {5}{2}} \left (-a \right )^{\frac {7}{4}}}\)	\(155\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)/a*polylog(2,a*x^2)+4/7*x^(7/2)*(-a)^(11/4)/a*polylog(3,a*x^2))

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Maxima [A]
time = 0.47, size = 178, normalized size = 1.11 \begin {gather*} \frac {2 \, {\left (\frac {336 \, d^{4} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {d x} \sqrt {a}}{\sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d} \sqrt {a}} + \frac {\log \left (\frac {\sqrt {d x} \sqrt {a} - \sqrt {\sqrt {a} d}}{\sqrt {d x} \sqrt {a} + \sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d} \sqrt {a}}\right )}}{a} - \frac {588 \, \left (d x\right )^{\frac {7}{2}} a {\rm Li}_2\left (a x^{2}\right ) + 336 \, \left (d x\right )^{\frac {7}{2}} a \log \left (-a d^{2} x^{2} + d^{2}\right ) - 1029 \, \left (d x\right )^{\frac {7}{2}} a {\rm Li}_{3}(a x^{2}) - 96 \, \left (d x\right )^{\frac {7}{2}} {\left (7 \, a \log \left (d\right ) + 2 \, a\right )} - 448 \, \left (d x\right )^{\frac {3}{2}} d^{2}}{a}\right )}}{7203 \, d} \end {gather*}

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]

2/7203*(336*d^4*(2*arctan(sqrt(d*x)*sqrt(a)/sqrt(sqrt(a)*d))/(sqrt(sqrt(a)*d)*sqrt(a)) + log((sqrt(d*x)*sqrt(a
) - sqrt(sqrt(a)*d))/(sqrt(d*x)*sqrt(a) + sqrt(sqrt(a)*d)))/(sqrt(sqrt(a)*d)*sqrt(a)))/a - (588*(d*x)^(7/2)*a*
dilog(a*x^2) + 336*(d*x)^(7/2)*a*log(-a*d^2*x^2 + d^2) - 1029*(d*x)^(7/2)*a*polylog(3, a*x^2) - 96*(d*x)^(7/2)
*(7*a*log(d) + 2*a) - 448*(d*x)^(3/2)*d^2)/a)/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (116) = 232\).
time = 0.40, size = 237, normalized size = 1.47 \begin {gather*} \frac {2 \, {\left (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 ) - 4 \, {\left (147 \, a d^{2} x^{3} {\rm Li}_2\left (a x^{2}\right ) + 84 \, a d^{2} x^{3} \log \left (-a x^{2} + 1\right ) - 48 \, a d^{2} x^{3} - 112 \, d^{2} x\right )} \sqrt {d x}\right )}}{7203 \, a} \end {gather*}

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*(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 + sqrt(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(3
2768*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^1
0/a^7)^(3/4)*a^5) - 4*(147*a*d^2*x^3*dilog(a*x^2) + 84*a*d^2*x^3*log(-a*x^2 + 1) - 48*a*d^2*x^3 - 112*d^2*x)*s
qrt(d*x))/a

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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^{2}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3, a*x^2)*(d*x)^(5/2),x)

[Out]

int(polylog(3, a*x^2)*(d*x)^(5/2), x)

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