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

Optimal. Leaf size=578 \[ \frac{115 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{4032 a^4 \sqrt{a^2 c x^2+c}}-\frac{115 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{4032 a^4 \sqrt{a^2 c x^2+c}}-\frac{115 c^2 \sqrt{a^2 c x^2+c}}{4032 a^4}+\frac{1}{9} a^4 c^2 x^8 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{1}{36} a^3 c^2 x^7 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{19}{63} a^2 c^2 x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{103 a c^2 x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{1512}+\frac{5}{21} c^2 x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{205 c^2 x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{6048 a}+\frac{c^2 x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{63 a^2}+\frac{47 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{1344 a^3}-\frac{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{63 a^4}-\frac{115 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2016 a^4 \sqrt{a^2 c x^2+c}}+\frac{\left (a^2 c x^2+c\right )^{7/2}}{252 a^4 c}-\frac{23 \left (a^2 c x^2+c\right )^{5/2}}{7560 a^4}-\frac{115 c \left (a^2 c x^2+c\right )^{3/2}}{18144 a^4} \]

[Out]

(-115*c^2*Sqrt[c + a^2*c*x^2])/(4032*a^4) - (115*c*(c + a^2*c*x^2)^(3/2))/(18144*a^4) - (23*(c + a^2*c*x^2)^(5
/2))/(7560*a^4) + (c + a^2*c*x^2)^(7/2)/(252*a^4*c) + (47*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(1344*a^3) -
(205*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(6048*a) - (103*a*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/1512
- (a^3*c^2*x^7*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/36 - (2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(63*a^4) + (c^2
*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(63*a^2) + (5*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/21 + (19*a^2*
c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/63 + (a^4*c^2*x^8*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/9 - (((115*I)/
2016)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2]) + (
((115*I)/4032)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x
^2]) - (((115*I)/4032)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^
2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 10.7013, antiderivative size = 578, normalized size of antiderivative = 1., number of steps used = 203, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4950, 4952, 261, 4890, 4886, 4930, 266, 43} \[ \frac{115 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{4032 a^4 \sqrt{a^2 c x^2+c}}-\frac{115 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{4032 a^4 \sqrt{a^2 c x^2+c}}-\frac{115 c^2 \sqrt{a^2 c x^2+c}}{4032 a^4}+\frac{1}{9} a^4 c^2 x^8 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{1}{36} a^3 c^2 x^7 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{19}{63} a^2 c^2 x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{103 a c^2 x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{1512}+\frac{5}{21} c^2 x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{205 c^2 x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{6048 a}+\frac{c^2 x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{63 a^2}+\frac{47 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{1344 a^3}-\frac{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{63 a^4}-\frac{115 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2016 a^4 \sqrt{a^2 c x^2+c}}+\frac{\left (a^2 c x^2+c\right )^{7/2}}{252 a^4 c}-\frac{23 \left (a^2 c x^2+c\right )^{5/2}}{7560 a^4}-\frac{115 c \left (a^2 c x^2+c\right )^{3/2}}{18144 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-115*c^2*Sqrt[c + a^2*c*x^2])/(4032*a^4) - (115*c*(c + a^2*c*x^2)^(3/2))/(18144*a^4) - (23*(c + a^2*c*x^2)^(5
/2))/(7560*a^4) + (c + a^2*c*x^2)^(7/2)/(252*a^4*c) + (47*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(1344*a^3) -
(205*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(6048*a) - (103*a*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/1512
- (a^3*c^2*x^7*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/36 - (2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(63*a^4) + (c^2
*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(63*a^2) + (5*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/21 + (19*a^2*
c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/63 + (a^4*c^2*x^8*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/9 - (((115*I)/
2016)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2]) + (
((115*I)/4032)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x
^2]) - (((115*I)/4032)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(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 261

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

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 4886

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(-2*I*(a + b*ArcTan[c*x])*
ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]])/(c*Sqrt[d]), x] + (Simp[(I*b*PolyLog[2, -((I*Sqrt[1 + I*c*x])/Sqrt[1
- I*c*x])])/(c*Sqrt[d]), x] - Simp[(I*b*PolyLog[2, (I*Sqrt[1 + I*c*x])/Sqrt[1 - I*c*x]])/(c*Sqrt[d]), x]) /; F
reeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^3 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2 \, dx &=c \int x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx+\left (a^2 c\right ) \int x^5 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx\\ &=c^2 \int x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx+2 \left (\left (a^2 c^2\right ) \int x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\right )+\left (a^4 c^2\right ) \int x^7 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=c^3 \int \frac{x^3 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac{x^7 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac{x^7 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx\right )+\left (a^6 c^3\right ) \int \frac{x^9 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^2}+\frac{1}{5} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{7} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{9} a^4 c^2 x^8 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{1}{5} \left (4 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx-\frac{\left (2 c^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{3 a^2}-\frac{\left (2 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{3 a}-\frac{1}{5} \left (2 a c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{7} \left (6 a^2 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+2 \left (\frac{1}{5} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{7} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{1}{5} \left (4 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{5} \left (2 a c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{7} \left (6 a^2 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{7} \left (2 a^3 c^3\right ) \int \frac{x^6 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx\right )-\frac{1}{7} \left (2 a^3 c^3\right ) \int \frac{x^6 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{9} \left (8 a^4 c^3\right ) \int \frac{x^7 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{9} \left (2 a^5 c^3\right ) \int \frac{x^8 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 a^3}-\frac{c^2 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}-\frac{1}{21} a c^2 x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{36} a^3 c^2 x^7 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^4}+\frac{c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac{1}{35} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{63} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{9} a^4 c^2 x^8 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{10} c^3 \int \frac{x^3}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{35} \left (24 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\frac{c^3 \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{3 a^3}+\frac{\left (4 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{3 a^3}+\frac{c^3 \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{3 a^2}+\frac{\left (8 c^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^2}+\frac{\left (3 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{10 a}+\frac{\left (8 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{15 a}+\frac{1}{21} \left (5 a c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{35} \left (12 a c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{21} \left (a^2 c^3\right ) \int \frac{x^5}{\sqrt{c+a^2 c x^2}} \, dx+2 \left (-\frac{c^2 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}-\frac{1}{21} a c^2 x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{4 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac{1}{35} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{7} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{10} c^3 \int \frac{x^3}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{35} \left (24 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\frac{\left (8 c^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^2}+\frac{\left (3 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{10 a}+\frac{\left (8 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{15 a}+\frac{1}{21} \left (5 a c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{35} \left (12 a c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{21} \left (a^2 c^3\right ) \int \frac{x^5}{\sqrt{c+a^2 c x^2}} \, dx\right )+\frac{1}{21} \left (16 a^2 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{36} \left (7 a^3 c^3\right ) \int \frac{x^6 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{63} \left (16 a^3 c^3\right ) \int \frac{x^6 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{36} \left (a^4 c^3\right ) \int \frac{x^7}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{c^2 \sqrt{c+a^2 c x^2}}{3 a^4}+\frac{c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}+\frac{19 c^2 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}+\frac{41 a c^2 x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{1512}-\frac{1}{36} a^3 c^2 x^7 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^4}+\frac{31 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}+\frac{19}{105} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{63} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{9} a^4 c^2 x^8 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{20} c^3 \operatorname{Subst}\left (\int \frac{x}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )-\frac{1}{84} \left (5 c^3\right ) \int \frac{x^3}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{35} \left (3 c^3\right ) \int \frac{x^3}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{105} \left (64 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx-\frac{\left (3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{20 a^3}-\frac{\left (4 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^3}-\frac{\left (16 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^3}-\frac{\left (3 c^3\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{20 a^2}-\frac{\left (4 c^3\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^2}-\frac{\left (16 c^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^2}-\frac{\left (5 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{28 a}-\frac{\left (9 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{35 a}-\frac{\left (16 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{35 a}-\frac{1}{216} \left (35 a c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{189} \left (40 a c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{105} \left (32 a c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{42} \left (a^2 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )+2 \left (\frac{5 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}+\frac{19 c^2 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac{1}{21} a c^2 x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{8 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^4}-\frac{4 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}+\frac{1}{35} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{7} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{20} c^3 \operatorname{Subst}\left (\int \frac{x}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )-\frac{1}{84} \left (5 c^3\right ) \int \frac{x^3}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{35} \left (3 c^3\right ) \int \frac{x^3}{\sqrt{c+a^2 c x^2}} \, dx-\frac{\left (3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{20 a^3}-\frac{\left (4 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^3}-\frac{\left (16 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^3}-\frac{\left (3 c^3\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{20 a^2}-\frac{\left (4 c^3\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^2}-\frac{\left (16 c^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^2}-\frac{\left (5 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{28 a}-\frac{\left (9 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{35 a}-\frac{\left (16 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{35 a}+\frac{1}{42} \left (a^2 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )\right )-\frac{1}{216} \left (7 a^2 c^3\right ) \int \frac{x^5}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{189} \left (8 a^2 c^3\right ) \int \frac{x^5}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{72} \left (a^4 c^3\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )+\frac{\left (c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{3 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (4 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{3 a^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{c^2 \sqrt{c+a^2 c x^2}}{12 a^4}-\frac{61 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{168 a^3}-\frac{3761 c^2 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{30240 a}+\frac{41 a c^2 x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{1512}-\frac{1}{36} a^3 c^2 x^7 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{62 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^4}+\frac{29 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{315 a^2}+\frac{19}{105} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{63} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{9} a^4 c^2 x^8 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{10 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 a^4 \sqrt{c+a^2 c x^2}}+\frac{5 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 a^4 \sqrt{c+a^2 c x^2}}-\frac{5 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 a^4 \sqrt{c+a^2 c x^2}}-\frac{1}{168} \left (5 c^3\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )+\frac{1}{864} \left (35 c^3\right ) \int \frac{x^3}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{70} \left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )+\frac{1}{20} c^3 \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \sqrt{c+a^2 c x}}+\frac{\sqrt{c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )+\frac{1}{189} \left (10 c^3\right ) \int \frac{x^3}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{105} \left (8 c^3\right ) \int \frac{x^3}{\sqrt{c+a^2 c x^2}} \, dx+\frac{\left (5 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{56 a^3}+\frac{\left (9 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{70 a^3}+\frac{\left (8 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^3}+\frac{\left (32 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^3}+\frac{\left (5 c^3\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{56 a^2}+\frac{\left (9 c^3\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{70 a^2}+\frac{\left (8 c^3\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^2}+\frac{\left (128 c^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{315 a^2}+\frac{\left (35 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{288 a}+\frac{\left (10 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{63 a}+\frac{\left (8 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{35 a}+\frac{\left (128 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{315 a}-\frac{1}{432} \left (7 a^2 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )-\frac{1}{189} \left (4 a^2 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )+\frac{1}{42} \left (a^2 c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^4 \sqrt{c+a^2 c x}}-\frac{2 \sqrt{c+a^2 c x}}{a^4 c}+\frac{\left (c+a^2 c x\right )^{3/2}}{a^4 c^2}\right ) \, dx,x,x^2\right )+\frac{1}{72} \left (a^4 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^6 \sqrt{c+a^2 c x}}+\frac{3 \sqrt{c+a^2 c x}}{a^6 c}-\frac{3 \left (c+a^2 c x\right )^{3/2}}{a^6 c^2}+\frac{\left (c+a^2 c x\right )^{5/2}}{a^6 c^3}\right ) \, dx,x,x^2\right )-\frac{\left (3 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{20 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (4 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{15 a^3 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{5 c^2 \sqrt{c+a^2 c x^2}}{12 a^4}-\frac{5 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{168 a^3}+\frac{19 c^2 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac{1}{21} a c^2 x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{8 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^4}-\frac{4 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}+\frac{1}{35} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{7} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{1}{168} \left (5 c^3\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )-\frac{1}{70} \left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )+\frac{1}{20} c^3 \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \sqrt{c+a^2 c x}}+\frac{\sqrt{c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )+\frac{\left (5 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{56 a^3}+\frac{\left (9 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{70 a^3}+\frac{\left (8 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^3}+\frac{\left (32 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^3}+\frac{\left (5 c^3\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{56 a^2}+\frac{\left (9 c^3\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{70 a^2}+\frac{\left (8 c^3\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^2}+\frac{1}{42} \left (a^2 c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^4 \sqrt{c+a^2 c x}}-\frac{2 \sqrt{c+a^2 c x}}{a^4 c}+\frac{\left (c+a^2 c x\right )^{3/2}}{a^4 c^2}\right ) \, dx,x,x^2\right )-\frac{\left (3 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{20 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (4 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{15 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (16 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{15 a^3 \sqrt{c+a^2 c x^2}}\right )-\frac{\left (16 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{15 a^3 \sqrt{c+a^2 c x^2}}\\ &=\frac{713 c^2 \sqrt{c+a^2 c x^2}}{2520 a^4}+\frac{37 c \left (c+a^2 c x^2\right )^{3/2}}{1260 a^4}-\frac{\left (c+a^2 c x^2\right )^{5/2}}{140 a^4}+\frac{\left (c+a^2 c x^2\right )^{7/2}}{252 a^4 c}+\frac{127 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{1344 a^3}-\frac{3761 c^2 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{30240 a}+\frac{41 a c^2 x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{1512}-\frac{1}{36} a^3 c^2 x^7 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{58 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{315 a^4}+\frac{29 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{315 a^2}+\frac{19}{105} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{63} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{9} a^4 c^2 x^8 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{11 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{30 a^4 \sqrt{c+a^2 c x^2}}+\frac{11 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 a^4 \sqrt{c+a^2 c x^2}}-\frac{11 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (35 c^3\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )}{1728}+\frac{1}{189} \left (5 c^3\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )-\frac{1}{168} \left (5 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \sqrt{c+a^2 c x}}+\frac{\sqrt{c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )+\frac{1}{105} \left (4 c^3\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )-\frac{1}{70} \left (3 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \sqrt{c+a^2 c x}}+\frac{\sqrt{c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )-\frac{\left (35 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{576 a^3}-\frac{\left (5 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{63 a^3}-\frac{\left (4 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^3}-\frac{\left (64 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{315 a^3}-\frac{\left (256 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{315 a^3}-\frac{\left (35 c^3\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{576 a^2}-\frac{\left (5 c^3\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{63 a^2}-\frac{\left (4 c^3\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^2}-\frac{\left (64 c^3\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{315 a^2}-\frac{1}{432} \left (7 a^2 c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^4 \sqrt{c+a^2 c x}}-\frac{2 \sqrt{c+a^2 c x}}{a^4 c}+\frac{\left (c+a^2 c x\right )^{3/2}}{a^4 c^2}\right ) \, dx,x,x^2\right )-\frac{1}{189} \left (4 a^2 c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^4 \sqrt{c+a^2 c x}}-\frac{2 \sqrt{c+a^2 c x}}{a^4 c}+\frac{\left (c+a^2 c x\right )^{3/2}}{a^4 c^2}\right ) \, dx,x,x^2\right )+\frac{\left (5 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{56 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (9 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{70 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (8 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{35 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (32 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{35 a^3 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{19 c^2 \sqrt{c+a^2 c x^2}}{840 a^4}+\frac{c \left (c+a^2 c x^2\right )^{3/2}}{630 a^4}+\frac{\left (c+a^2 c x^2\right )^{5/2}}{105 a^4}-\frac{5 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{168 a^3}+\frac{19 c^2 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac{1}{21} a c^2 x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{8 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^4}-\frac{4 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}+\frac{1}{35} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{7} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{89 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{30 a^4 \sqrt{c+a^2 c x^2}}-\frac{89 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 a^4 \sqrt{c+a^2 c x^2}}+\frac{89 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 a^4 \sqrt{c+a^2 c x^2}}-\frac{1}{168} \left (5 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \sqrt{c+a^2 c x}}+\frac{\sqrt{c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )-\frac{1}{70} \left (3 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \sqrt{c+a^2 c x}}+\frac{\sqrt{c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )+\frac{\left (5 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{56 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (9 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{70 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (8 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{35 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (32 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{35 a^3 \sqrt{c+a^2 c x^2}}\right )\\ &=-\frac{6299 c^2 \sqrt{c+a^2 c x^2}}{60480 a^4}+\frac{349 c \left (c+a^2 c x^2\right )^{3/2}}{11340 a^4}-\frac{167 \left (c+a^2 c x^2\right )^{5/2}}{7560 a^4}+\frac{\left (c+a^2 c x^2\right )^{7/2}}{252 a^4 c}+\frac{127 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{1344 a^3}-\frac{3761 c^2 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{30240 a}+\frac{41 a c^2 x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{1512}-\frac{1}{36} a^3 c^2 x^7 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{58 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{315 a^4}+\frac{29 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{315 a^2}+\frac{19}{105} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{63} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{9} a^4 c^2 x^8 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{1297 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{420 a^4 \sqrt{c+a^2 c x^2}}+\frac{1297 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{840 a^4 \sqrt{c+a^2 c x^2}}-\frac{1297 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{840 a^4 \sqrt{c+a^2 c x^2}}+2 \left (\frac{103 c^2 \sqrt{c+a^2 c x^2}}{840 a^4}-\frac{59 c \left (c+a^2 c x^2\right )^{3/2}}{1260 a^4}+\frac{\left (c+a^2 c x^2\right )^{5/2}}{105 a^4}-\frac{5 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{168 a^3}+\frac{19 c^2 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac{1}{21} a c^2 x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{8 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^4}-\frac{4 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}+\frac{1}{35} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{7} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{103 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{420 a^4 \sqrt{c+a^2 c x^2}}-\frac{103 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{840 a^4 \sqrt{c+a^2 c x^2}}+\frac{103 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{840 a^4 \sqrt{c+a^2 c x^2}}\right )+\frac{\left (35 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \sqrt{c+a^2 c x}}+\frac{\sqrt{c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )}{1728}+\frac{1}{189} \left (5 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \sqrt{c+a^2 c x}}+\frac{\sqrt{c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )+\frac{1}{105} \left (4 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \sqrt{c+a^2 c x}}+\frac{\sqrt{c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )-\frac{\left (35 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{576 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (5 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{63 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (4 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{35 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (64 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{315 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (256 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{315 a^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{5519 c^2 \sqrt{c+a^2 c x^2}}{20160 a^4}+\frac{7921 c \left (c+a^2 c x^2\right )^{3/2}}{90720 a^4}-\frac{167 \left (c+a^2 c x^2\right )^{5/2}}{7560 a^4}+\frac{\left (c+a^2 c x^2\right )^{7/2}}{252 a^4 c}+\frac{127 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{1344 a^3}-\frac{3761 c^2 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{30240 a}+\frac{41 a c^2 x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{1512}-\frac{1}{36} a^3 c^2 x^7 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{58 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{315 a^4}+\frac{29 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{315 a^2}+\frac{19}{105} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{63} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{9} a^4 c^2 x^8 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{5519 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{10080 a^4 \sqrt{c+a^2 c x^2}}+\frac{5519 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{20160 a^4 \sqrt{c+a^2 c x^2}}-\frac{5519 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{20160 a^4 \sqrt{c+a^2 c x^2}}+2 \left (\frac{103 c^2 \sqrt{c+a^2 c x^2}}{840 a^4}-\frac{59 c \left (c+a^2 c x^2\right )^{3/2}}{1260 a^4}+\frac{\left (c+a^2 c x^2\right )^{5/2}}{105 a^4}-\frac{5 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{168 a^3}+\frac{19 c^2 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac{1}{21} a c^2 x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{8 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^4}-\frac{4 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}+\frac{1}{35} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{7} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{103 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{420 a^4 \sqrt{c+a^2 c x^2}}-\frac{103 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{840 a^4 \sqrt{c+a^2 c x^2}}+\frac{103 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{840 a^4 \sqrt{c+a^2 c x^2}}\right )\\ \end{align*}

Mathematica [B]  time = 8.25716, size = 1320, normalized size = 2.28 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((c + a^2*c*x^2)^(5/2)*(-48384*(50 - 32*ArcTan[a*x]^2 + 72*Cos[2*ArcTan[a*x]] + 160*ArcTan[a*x]^2*Cos[2*ArcTan
[a*x]] + 22*Cos[4*ArcTan[a*x]] - (110*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 55*ArcTan[
a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 11*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan
[a*x])] + (110*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]
*Log[1 + I*E^(I*ArcTan[a*x])] + 11*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((176*I)*Poly
Log[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + ((176*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^
(5/2) + 4*ArcTan[a*x]*Sin[2*ArcTan[a*x]] - 22*ArcTan[a*x]*Sin[4*ArcTan[a*x]]) + 576*(1 + a^2*x^2)*(4116 + 1094
4*ArcTan[a*x]^2 + 6262*Cos[2*ArcTan[a*x]] - 5376*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 2764*Cos[4*ArcTan[a*x]] +
6720*ArcTan[a*x]^2*Cos[4*ArcTan[a*x]] + 618*Cos[6*ArcTan[a*x]] - (10815*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x]
)])/Sqrt[1 + a^2*x^2] - 6489*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 2163*ArcTan[a*x]*Co
s[5*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 309*ArcTan[a*x]*Cos[7*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])
] + (10815*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 6489*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*L
og[1 + I*E^(I*ArcTan[a*x])] + 2163*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 309*ArcTan[a*
x]*Cos[7*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((19776*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x
^2)^(7/2) + ((19776*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(7/2) - 1266*ArcTan[a*x]*Sin[2*ArcTan[a*
x]] + 360*ArcTan[a*x]*Sin[4*ArcTan[a*x]] - 618*ArcTan[a*x]*Sin[6*ArcTan[a*x]]) - (1 + a^2*x^2)^2*(657578 - 820
224*ArcTan[a*x]^2 + 1083168*Cos[2*ArcTan[a*x]] + 3276288*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 576936*Cos[4*ArcTa
n[a*x]] - 580608*ArcTan[a*x]^2*Cos[4*ArcTan[a*x]] + 184160*Cos[6*ArcTan[a*x]] + 483840*ArcTan[a*x]^2*Cos[6*Arc
Tan[a*x]] + 32814*Cos[8*ArcTan[a*x]] - (2067282*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] -
1378188*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 590652*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Lo
g[1 - I*E^(I*ArcTan[a*x])] - 147663*ArcTan[a*x]*Cos[7*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 16407*ArcTan
[a*x]*Cos[9*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (2067282*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqr
t[1 + a^2*x^2] + 1378188*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 590652*ArcTan[a*x]*Cos[
5*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 147663*ArcTan[a*x]*Cos[7*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x]
)] + 16407*ArcTan[a*x]*Cos[9*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((4200192*I)*PolyLog[2, (-I)*E^(I*Arc
Tan[a*x])])/(1 + a^2*x^2)^(9/2) + ((4200192*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(9/2) + 78444*Ar
cTan[a*x]*Sin[2*ArcTan[a*x]] - 160452*ArcTan[a*x]*Sin[4*ArcTan[a*x]] + 38172*ArcTan[a*x]*Sin[6*ArcTan[a*x]] -
32814*ArcTan[a*x]*Sin[8*ArcTan[a*x]])))/(46448640*a^4)

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Maple [A]  time = 0.957, size = 309, normalized size = 0.5 \begin{align*}{\frac{{c}^{2} \left ( 20160\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{8}{a}^{8}-5040\,\arctan \left ( ax \right ){x}^{7}{a}^{7}+54720\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{6}{a}^{6}+720\,{x}^{6}{a}^{6}-12360\,\arctan \left ( ax \right ){x}^{5}{a}^{5}+43200\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{4}{a}^{4}+1608\,{a}^{4}{x}^{4}-6150\,\arctan \left ( ax \right ){x}^{3}{a}^{3}+2880\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}-94\,{a}^{2}{x}^{2}+6345\,\arctan \left ( ax \right ) xa-5760\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-6157 \right ) }{181440\,{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{115\,{c}^{2}}{4032\,{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( \arctan \left ( ax \right ) \ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -\arctan \left ( ax \right ) \ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +i{\it dilog} \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -i{\it dilog} \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \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)^2,x)

[Out]

1/181440*c^2/a^4*(c*(a*x-I)*(a*x+I))^(1/2)*(20160*arctan(a*x)^2*x^8*a^8-5040*arctan(a*x)*x^7*a^7+54720*arctan(
a*x)^2*x^6*a^6+720*x^6*a^6-12360*arctan(a*x)*x^5*a^5+43200*arctan(a*x)^2*x^4*a^4+1608*a^4*x^4-6150*arctan(a*x)
*x^3*a^3+2880*arctan(a*x)^2*x^2*a^2-94*a^2*x^2+6345*arctan(a*x)*x*a-5760*arctan(a*x)^2-6157)+115/4032*c^2*(c*(
a*x-I)*(a*x+I))^(1/2)*(arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1
)^(1/2))+I*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-I*dilog(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^4/(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)^2,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 )^{2}, 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)^2,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)^2, 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)**2,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)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError