3.428 \(\int x^3 (c+a^2 c x^2)^{5/2} \tan ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=798 \[ \frac{1}{9} a^4 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^8-\frac{1}{24} a^3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^7+\frac{19}{63} a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^6+\frac{1}{84} a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^6-\frac{103 a c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^5}{1008}-\frac{1}{504} a c^2 \sqrt{a^2 c x^2+c} x^5+\frac{5}{21} c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^4+\frac{67 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^4}{2520}-\frac{205 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^3}{4032 a}-\frac{c^2 \sqrt{a^2 c x^2+c} x^3}{240 a}+\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^2}{63 a^2}-\frac{47 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^2}{30240 a^2}+\frac{47 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x}{896 a^3}+\frac{85 c^2 \sqrt{a^2 c x^2+c} x}{12096 a^3}-\frac{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{63 a^4}-\frac{115 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{1344 a^4 \sqrt{a^2 c x^2+c}}-\frac{6157 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{60480 a^4}+\frac{1433 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{15120 a^4}+\frac{115 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{1344 a^4 \sqrt{a^2 c x^2+c}}-\frac{115 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{1344 a^4 \sqrt{a^2 c x^2+c}}-\frac{115 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{1344 a^4 \sqrt{a^2 c x^2+c}}+\frac{115 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{1344 a^4 \sqrt{a^2 c x^2+c}} \]

[Out]

(85*c^2*x*Sqrt[c + a^2*c*x^2])/(12096*a^3) - (c^2*x^3*Sqrt[c + a^2*c*x^2])/(240*a) - (a*c^2*x^5*Sqrt[c + a^2*c
*x^2])/504 - (6157*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(60480*a^4) - (47*c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a
*x])/(30240*a^2) + (67*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/2520 + (a^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan
[a*x])/84 + (47*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(896*a^3) - (205*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a
*x]^2)/(4032*a) - (103*a*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/1008 - (a^3*c^2*x^7*Sqrt[c + a^2*c*x^2]*Ar
cTan[a*x]^2)/24 - (((115*I)/1344)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^4*Sqrt[c +
a^2*c*x^2]) - (2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(63*a^4) + (c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3
)/(63*a^2) + (5*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/21 + (19*a^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x
]^3)/63 + (a^4*c^2*x^8*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/9 + (1433*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2
*c*x^2]])/(15120*a^4) + (((115*I)/1344)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/
(a^4*Sqrt[c + a^2*c*x^2]) - (((115*I)/1344)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])
/(a^4*Sqrt[c + a^2*c*x^2]) - (115*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(1344*a^4*Sqrt[c +
a^2*c*x^2]) + (115*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(1344*a^4*Sqrt[c + a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 19.6636, antiderivative size = 798, normalized size of antiderivative = 1., number of steps used = 547, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4950, 4952, 4930, 217, 206, 4890, 4888, 4181, 2531, 2282, 6589, 321} \[ \frac{1}{9} a^4 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^8-\frac{1}{24} a^3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^7+\frac{19}{63} a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^6+\frac{1}{84} a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^6-\frac{103 a c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^5}{1008}-\frac{1}{504} a c^2 \sqrt{a^2 c x^2+c} x^5+\frac{5}{21} c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^4+\frac{67 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^4}{2520}-\frac{205 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^3}{4032 a}-\frac{c^2 \sqrt{a^2 c x^2+c} x^3}{240 a}+\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^2}{63 a^2}-\frac{47 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^2}{30240 a^2}+\frac{47 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x}{896 a^3}+\frac{85 c^2 \sqrt{a^2 c x^2+c} x}{12096 a^3}-\frac{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{63 a^4}-\frac{115 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{1344 a^4 \sqrt{a^2 c x^2+c}}-\frac{6157 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{60480 a^4}+\frac{1433 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{15120 a^4}+\frac{115 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{1344 a^4 \sqrt{a^2 c x^2+c}}-\frac{115 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{1344 a^4 \sqrt{a^2 c x^2+c}}-\frac{115 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{1344 a^4 \sqrt{a^2 c x^2+c}}+\frac{115 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{1344 a^4 \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3,x]

[Out]

(85*c^2*x*Sqrt[c + a^2*c*x^2])/(12096*a^3) - (c^2*x^3*Sqrt[c + a^2*c*x^2])/(240*a) - (a*c^2*x^5*Sqrt[c + a^2*c
*x^2])/504 - (6157*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(60480*a^4) - (47*c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a
*x])/(30240*a^2) + (67*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/2520 + (a^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan
[a*x])/84 + (47*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(896*a^3) - (205*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a
*x]^2)/(4032*a) - (103*a*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/1008 - (a^3*c^2*x^7*Sqrt[c + a^2*c*x^2]*Ar
cTan[a*x]^2)/24 - (((115*I)/1344)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^4*Sqrt[c +
a^2*c*x^2]) - (2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(63*a^4) + (c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3
)/(63*a^2) + (5*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/21 + (19*a^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x
]^3)/63 + (a^4*c^2*x^8*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/9 + (1433*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2
*c*x^2]])/(15120*a^4) + (((115*I)/1344)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/
(a^4*Sqrt[c + a^2*c*x^2]) - (((115*I)/1344)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])
/(a^4*Sqrt[c + a^2*c*x^2]) - (115*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(1344*a^4*Sqrt[c +
a^2*c*x^2]) + (115*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(1344*a^4*Sqrt[c + a^2*c*x^2])

Rule 4950

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Dist[
d, Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Dist[(c^2*d)/f^2, Int[(f*x)^(m + 2)*(d + e*
x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] &&
IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))

Rule 4952

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x])^p)/(c^2*d*m), x] + (-Dist[(b*f*p)/(c*m), Int[((f*x)^(m -
1)*(a + b*ArcTan[c*x])^(p - 1))/Sqrt[d + e*x^2], x], x] - Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a +
b*ArcTan[c*x])^p)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && Gt
Q[m, 1]

Rule 4930

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[((d + e*x^2)^
(q + 1)*(a + b*ArcTan[c*x])^p)/(2*e*(q + 1)), x] - Dist[(b*p)/(2*c*(q + 1)), Int[(d + e*x^2)^q*(a + b*ArcTan[c
*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 4890

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/Sq
rt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*
d] && IGtQ[p, 0] &&  !GtQ[d, 0]

Rule 4888

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[1/(c*Sqrt[d]), Subst
[Int[(a + b*x)^p*Sec[x], x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] &
& GtQ[d, 0]

Rule 4181

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

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]

Rubi steps

\begin{align*} \int x^3 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3 \, dx &=c \int x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx+\left (a^2 c\right ) \int x^5 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx\\ \end{align*}

Mathematica [A]  time = 6.47453, size = 850, normalized size = 1.07 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3,x]

[Out]

(c^2*Sqrt[c + a^2*c*x^2]*(774144*((-11*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 10*ArcTanh[(a*x)/Sqrt[1 +
a^2*x^2]] + (11*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (11*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTa
n[a*x])] - 11*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 11*PolyLog[3, I*E^(I*ArcTan[a*x])]) + 256*((-16407*I)*ArcTa
n[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 12788*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + (16407*I)*ArcTan[a*x]*PolyLog[2,
(-I)*E^(I*ArcTan[a*x])] - (16407*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 16407*PolyLog[3, (-I)*E^(I*
ArcTan[a*x])] + 16407*PolyLog[3, I*E^(I*ArcTan[a*x])]) - 16128*(1 + a^2*x^2)^(5/2)*((48*a*x)/(1 + a^2*x^2)^2 +
32*ArcTan[a*x]^3*(-1 + 5*Cos[2*ArcTan[a*x]]) + 6*ArcTan[a*x]*(25 + 36*Cos[2*ArcTan[a*x]] + 11*Cos[4*ArcTan[a*
x]]) + ArcTan[a*x]^2*(6*Sin[2*ArcTan[a*x]] - 33*Sin[4*ArcTan[a*x]])) + 576*(64*((309*I)*ArcTan[E^(I*ArcTan[a*x
])]*ArcTan[a*x]^2 - 259*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - (309*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x
])] + (309*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 309*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 309*PolyL
og[3, I*E^(I*ArcTan[a*x])]) + (1 + a^2*x^2)^(7/2)*(64*ArcTan[a*x]^3*(57 - 28*Cos[2*ArcTan[a*x]] + 35*Cos[4*Arc
Tan[a*x]]) + (8*ArcTan[a*x]*(647 + 764*Cos[2*ArcTan[a*x]] + 309*Cos[4*ArcTan[a*x]]))/(1 + a^2*x^2) + 4*(101*Si
n[2*ArcTan[a*x]] + 88*Sin[4*ArcTan[a*x]] + 25*Sin[6*ArcTan[a*x]]) - 3*ArcTan[a*x]^2*(211*Sin[2*ArcTan[a*x]] -
60*Sin[4*ArcTan[a*x]] + 103*Sin[6*ArcTan[a*x]]))) - (1 + a^2*x^2)^(9/2)*(1536*ArcTan[a*x]^3*(-178 + 711*Cos[2*
ArcTan[a*x]] - 126*Cos[4*ArcTan[a*x]] + 105*Cos[6*ArcTan[a*x]]) + (8*ArcTan[a*x]*(87630 + 153529*Cos[2*ArcTan[
a*x]] + 59266*Cos[4*ArcTan[a*x]] + 16407*Cos[6*ArcTan[a*x]]))/(1 + a^2*x^2) + 74932*Sin[2*ArcTan[a*x]] + 77908
*Sin[4*ArcTan[a*x]] + 36612*Sin[6*ArcTan[a*x]] + 3*ArcTan[a*x]^2*(13074*Sin[2*ArcTan[a*x]] - 26742*Sin[4*ArcTa
n[a*x]] + 6362*Sin[6*ArcTan[a*x]] - 5469*Sin[8*ArcTan[a*x]]) + 7238*Sin[8*ArcTan[a*x]])))/(15482880*a^4*Sqrt[1
+ a^2*x^2])

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Maple [A]  time = 4.576, size = 525, normalized size = 0.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3,x)

[Out]

1/120960*c^2/a^4*(c*(a*x-I)*(a*x+I))^(1/2)*(13440*arctan(a*x)^3*x^8*a^8-5040*arctan(a*x)^2*x^7*a^7+36480*arcta
n(a*x)^3*x^6*a^6+1440*arctan(a*x)*x^6*a^6-12360*arctan(a*x)^2*x^5*a^5+28800*arctan(a*x)^3*x^4*a^4-240*a^5*x^5+
3216*arctan(a*x)*x^4*a^4-6150*arctan(a*x)^2*x^3*a^3+1920*arctan(a*x)^3*x^2*a^2-504*a^3*x^3-188*arctan(a*x)*a^2
*x^2+6345*arctan(a*x)^2*x*a-3840*arctan(a*x)^3+850*a*x-12314*arctan(a*x))+115/8064*c^2*(c*(a*x-I)*(a*x+I))^(1/
2)*(I*arctan(a*x)^3-3*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*polylog(2,-I*(1+I*a*x)
/(a^2*x^2+1)^(1/2))-6*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^4/(a^2*x^2+1)^(1/2)-115/8064*c^2*(c*(a*x-I)
*(a*x+I))^(1/2)*(I*arctan(a*x)^3+6*I*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*arctan(a*x)^2*ln(1
-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^4/(a^2*x^2+1)^(1/2)-1433/7560*I*
c^2/a^4*(c*(a*x-I)*(a*x+I))^(1/2)*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*x^2+1)^(1/2)

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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(x^3*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{4} c^{2} x^{7} + 2 \, a^{2} c^{2} x^{5} + c^{2} x^{3}\right )} \sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3,x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^7 + 2*a^2*c^2*x^5 + c^2*x^3)*sqrt(a^2*c*x^2 + c)*arctan(a*x)^3, x)

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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(x**3*(a**2*c*x**2+c)**(5/2)*atan(a*x)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError