\(\int x^2 (c+a^2 c x^2)^{5/2} \arctan (a x)^2 \, dx\) [324]

Optimal result

Integrand size = 24, antiderivative size = 638 \[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\frac {43 c^2 x \sqrt {c+a^2 c x^2}}{4032 a^2}+\frac {29 c^2 x^3 \sqrt {c+a^2 c x^2}}{1680}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}+\frac {1373 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{20160 a^3}-\frac {737 c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{10080 a}-\frac {83}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {5 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{128 a^2}+\frac {59}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {17}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {397 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{5040 a^3}-\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {5 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {5 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}} \]

[Out]

-397/5040*c^(5/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))/a^3+5/64*I*c^3*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*
arctan(a*x)^2*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)-5/64*I*c^3*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2
+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)+5/64*I*c^3*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^
(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)+5/64*c^3*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1
)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)-5/64*c^3*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c
*x^2+c)^(1/2)+43/4032*c^2*x*(a^2*c*x^2+c)^(1/2)/a^2+29/1680*c^2*x^3*(a^2*c*x^2+c)^(1/2)+1/168*a^2*c^2*x^5*(a^2
*c*x^2+c)^(1/2)+1373/20160*c^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^3-737/10080*c^2*x^2*arctan(a*x)*(a^2*c*x^2+c)
^(1/2)/a-83/840*a*c^2*x^4*arctan(a*x)*(a^2*c*x^2+c)^(1/2)-1/28*a^3*c^2*x^6*arctan(a*x)*(a^2*c*x^2+c)^(1/2)+5/1
28*c^2*x*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a^2+59/192*c^2*x^3*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)+17/48*a^2*c^2*
x^5*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)+1/8*a^4*c^2*x^7*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 5.89 (sec) , antiderivative size = 638, normalized size of antiderivative = 1.00, number of steps used = 238, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5070, 5072, 5050, 223, 212, 5010, 5008, 4266, 2611, 2320, 6724, 327} \[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=-\frac {737 c^2 x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{10080 a}+\frac {5 c^2 x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{128 a^2}+\frac {17}{48} a^2 c^2 x^5 \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {83}{840} a c^2 x^4 \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {59}{192} c^2 x^3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}+\frac {43 c^2 x \sqrt {a^2 c x^2+c}}{4032 a^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {a^2 c x^2+c}+\frac {29 c^2 x^3 \sqrt {a^2 c x^2+c}}{1680}+\frac {1}{8} a^4 c^2 x^7 \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {5 i c^3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {5 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}-\frac {5 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {1373 c^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{20160 a^3}-\frac {1}{28} a^3 c^2 x^6 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {397 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{5040 a^3} \]

[In]

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

[Out]

(43*c^2*x*Sqrt[c + a^2*c*x^2])/(4032*a^2) + (29*c^2*x^3*Sqrt[c + a^2*c*x^2])/1680 + (a^2*c^2*x^5*Sqrt[c + a^2*
c*x^2])/168 + (1373*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(20160*a^3) - (737*c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan
[a*x])/(10080*a) - (83*a*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/840 - (a^3*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTa
n[a*x])/28 + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(128*a^2) + (59*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*
x]^2)/192 + (17*a^2*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/48 + (a^4*c^2*x^7*Sqrt[c + a^2*c*x^2]*ArcTan[a*
x]^2)/8 + (((5*I)/64)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^3*Sqrt[c + a^2*c*x^2])
- (397*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(5040*a^3) - (((5*I)/64)*c^3*Sqrt[1 + a^2*x^2]*Arc
Tan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (((5*I)/64)*c^3*Sqrt[1 + a^2*x^2]*Arc
Tan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)
*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2]) - (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/
(64*a^3*Sqrt[c + a^2*c*x^2])

Rule 212

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

Rule 223

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 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 2320

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 2611

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 4266

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 5008

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 5010

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 5050

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 5070

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 5072

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 6724

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]

Rubi steps \begin{align*} \text {integral}& = c \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx+\left (a^2 c\right ) \int x^4 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx \\ & = c^2 \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx+2 \left (\left (a^2 c^2\right ) \int x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx\right )+\left (a^4 c^2\right ) \int x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx \\ & = c^3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac {x^4 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {x^6 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^3\right ) \int \frac {x^4 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {x^6 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^6 c^3\right ) \int \frac {x^8 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \frac {c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^2}+\frac {1}{4} c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{4} \left (3 c^3\right ) \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {c^3 \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {c^3 \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a}-\frac {1}{2} \left (a c^3\right ) \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{6} \left (5 a^2 c^3\right ) \int \frac {x^4 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\frac {1}{4} c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{4} \left (3 c^3\right ) \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a c^3\right ) \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{6} \left (5 a^2 c^3\right ) \int \frac {x^4 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (a^3 c^3\right ) \int \frac {x^5 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx\right )-\frac {1}{3} \left (a^3 c^3\right ) \int \frac {x^5 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{8} \left (7 a^4 c^3\right ) \int \frac {x^6 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{4} \left (a^5 c^3\right ) \int \frac {x^7 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{a^3}-\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{6} c^3 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (5 c^3\right ) \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (3 c^3\right ) \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}+\frac {c^3 \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}+\frac {\left (3 c^3\right ) \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a}+\frac {1}{15} \left (4 a c^3\right ) \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{12} \left (5 a c^3\right ) \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{15} \left (a^2 c^3\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (-\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {3 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{6} c^3 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (5 c^3\right ) \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (3 c^3\right ) \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {c^3 \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}+\frac {\left (3 c^3\right ) \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a}+\frac {1}{15} \left (4 a c^3\right ) \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{12} \left (5 a c^3\right ) \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{15} \left (a^2 c^3\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx\right )+\frac {1}{48} \left (35 a^2 c^3\right ) \int \frac {x^4 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{14} \left (3 a^3 c^3\right ) \int \frac {x^5 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{24} \left (7 a^3 c^3\right ) \int \frac {x^5 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{28} \left (a^4 c^3\right ) \int \frac {x^6}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2 \sqrt {c+a^2 c x^2}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.15 (sec) , antiderivative size = 759, normalized size of antiderivative = 1.19 \[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\frac {c^2 \sqrt {c+a^2 c x^2} \left (53760 a x \left (1+a^2 x^2\right )^{3/2}-25088 a x \left (1+a^2 x^2\right )^{5/2}+7006 a x \left (1+a^2 x^2\right )^{7/2}+53760 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)+5376 \left (1+a^2 x^2\right )^{5/2} \arctan (a x)-38134 \left (1+a^2 x^2\right )^{7/2} \arctan (a x)+564480 a x \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^2+524160 a x \left (1+a^2 x^2\right )^{5/2} \arctan (a x)^2+185325 a x \left (1+a^2 x^2\right )^{7/2} \arctan (a x)^2+201600 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-203264 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+161280 \left (1+a^2 x^2\right )^2 \arctan (a x) \cos (3 \arctan (a x))+49280 \left (1+a^2 x^2\right )^3 \arctan (a x) \cos (3 \arctan (a x))-7658 \left (1+a^2 x^2\right )^4 \arctan (a x) \cos (3 \arctan (a x))-40320 \left (1+a^2 x^2\right )^3 \arctan (a x) \cos (5 \arctan (a x))-10990 \left (1+a^2 x^2\right )^4 \arctan (a x) \cos (5 \arctan (a x))+3150 \left (1+a^2 x^2\right )^4 \arctan (a x) \cos (7 \arctan (a x))-201600 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+201600 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+201600 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-201600 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )+53760 \left (1+a^2 x^2\right )^2 \sin (3 \arctan (a x))-48384 \left (1+a^2 x^2\right )^3 \sin (3 \arctan (a x))+12246 \left (1+a^2 x^2\right )^4 \sin (3 \arctan (a x))-80640 \left (1+a^2 x^2\right )^2 \arctan (a x)^2 \sin (3 \arctan (a x))-315840 \left (1+a^2 x^2\right )^3 \arctan (a x)^2 \sin (3 \arctan (a x))-93975 \left (1+a^2 x^2\right )^4 \arctan (a x)^2 \sin (3 \arctan (a x))-23296 \left (1+a^2 x^2\right )^3 \sin (5 \arctan (a x))+7678 \left (1+a^2 x^2\right )^4 \sin (5 \arctan (a x))+20160 \left (1+a^2 x^2\right )^3 \arctan (a x)^2 \sin (5 \arctan (a x))+41685 \left (1+a^2 x^2\right )^4 \arctan (a x)^2 \sin (5 \arctan (a x))+2438 \left (1+a^2 x^2\right )^4 \sin (7 \arctan (a x))-1575 \left (1+a^2 x^2\right )^4 \arctan (a x)^2 \sin (7 \arctan (a x))\right )}{2580480 a^3 \sqrt {1+a^2 x^2}} \]

[In]

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

[Out]

(c^2*Sqrt[c + a^2*c*x^2]*(53760*a*x*(1 + a^2*x^2)^(3/2) - 25088*a*x*(1 + a^2*x^2)^(5/2) + 7006*a*x*(1 + a^2*x^
2)^(7/2) + 53760*(1 + a^2*x^2)^(3/2)*ArcTan[a*x] + 5376*(1 + a^2*x^2)^(5/2)*ArcTan[a*x] - 38134*(1 + a^2*x^2)^
(7/2)*ArcTan[a*x] + 564480*a*x*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2 + 524160*a*x*(1 + a^2*x^2)^(5/2)*ArcTan[a*x]^
2 + 185325*a*x*(1 + a^2*x^2)^(7/2)*ArcTan[a*x]^2 + (201600*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - 203264
*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + 161280*(1 + a^2*x^2)^2*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 49280*(1 + a^2*x^2
)^3*ArcTan[a*x]*Cos[3*ArcTan[a*x]] - 7658*(1 + a^2*x^2)^4*ArcTan[a*x]*Cos[3*ArcTan[a*x]] - 40320*(1 + a^2*x^2)
^3*ArcTan[a*x]*Cos[5*ArcTan[a*x]] - 10990*(1 + a^2*x^2)^4*ArcTan[a*x]*Cos[5*ArcTan[a*x]] + 3150*(1 + a^2*x^2)^
4*ArcTan[a*x]*Cos[7*ArcTan[a*x]] - (201600*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (201600*I)*ArcT
an[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 201600*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 201600*PolyLog[3, I*E^(I
*ArcTan[a*x])] + 53760*(1 + a^2*x^2)^2*Sin[3*ArcTan[a*x]] - 48384*(1 + a^2*x^2)^3*Sin[3*ArcTan[a*x]] + 12246*(
1 + a^2*x^2)^4*Sin[3*ArcTan[a*x]] - 80640*(1 + a^2*x^2)^2*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 315840*(1 + a^2*x
^2)^3*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 93975*(1 + a^2*x^2)^4*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 23296*(1 + a
^2*x^2)^3*Sin[5*ArcTan[a*x]] + 7678*(1 + a^2*x^2)^4*Sin[5*ArcTan[a*x]] + 20160*(1 + a^2*x^2)^3*ArcTan[a*x]^2*S
in[5*ArcTan[a*x]] + 41685*(1 + a^2*x^2)^4*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]] + 2438*(1 + a^2*x^2)^4*Sin[7*ArcTan
[a*x]] - 1575*(1 + a^2*x^2)^4*ArcTan[a*x]^2*Sin[7*ArcTan[a*x]]))/(2580480*a^3*Sqrt[1 + a^2*x^2])

Maple [A] (verified)

Time = 6.04 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.59

method	result	size
default	\(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (5040 \arctan \left (a x \right )^{2} a^{7} x^{7}-1440 a^{6} \arctan \left (a x \right ) x^{6}+14280 a^{5} \arctan \left (a x \right )^{2} x^{5}+240 a^{5} x^{5}-3984 \arctan \left (a x \right ) a^{4} x^{4}+12390 a^{3} \arctan \left (a x \right )^{2} x^{3}+696 a^{3} x^{3}-2948 a^{2} \arctan \left (a x \right ) x^{2}+1575 a \arctan \left (a x \right )^{2} x +430 a x +2746 \arctan \left (a x \right )\right )}{40320 a^{3}}-\frac {i c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (1575 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-1575 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3150 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3150 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3150 i \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3150 i \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6352 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{40320 a^{3} \sqrt {a^{2} x^{2}+1}}\)	\(376\)

[In]

int(x^2*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/40320*c^2/a^3*(c*(a*x-I)*(I+a*x))^(1/2)*(5040*arctan(a*x)^2*a^7*x^7-1440*a^6*arctan(a*x)*x^6+14280*a^5*arcta
n(a*x)^2*x^5+240*a^5*x^5-3984*arctan(a*x)*a^4*x^4+12390*a^3*arctan(a*x)^2*x^3+696*a^3*x^3-2948*a^2*arctan(a*x)
*x^2+1575*a*arctan(a*x)^2*x+430*a*x+2746*arctan(a*x))-1/40320*I*c^2*(c*(a*x-I)*(I+a*x))^(1/2)*(1575*I*arctan(a
*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-1575*I*arctan(a*x)^2*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3150*arctan
(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3150*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3150
*I*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3150*I*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6352*arctan((1+I*
a*x)/(a^2*x^2+1)^(1/2)))/a^3/(a^2*x^2+1)^(1/2)

Fricas [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{2} \,d x } \]

[In]

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

[Out]

integral((a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2)*sqrt(a^2*c*x^2 + c)*arctan(a*x)^2, x)

Sympy [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]

[In]

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

[Out]

Integral(x**2*(c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**2, x)

Maxima [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{2} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int x^2\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]

[In]

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

[Out]

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