\(\int x^2 (c+a^2 c x^2)^{5/2} \arctan (a x)^3 \, dx\) [429]

Optimal result

Integrand size = 24, antiderivative size = 1019 \[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\frac {13 c^2 \sqrt {c+a^2 c x^2}}{6720 a^3}-\frac {3 c \left (c+a^2 c x^2\right )^{3/2}}{560 a^3}-\frac {\left (c+a^2 c x^2\right )^{5/2}}{280 a^3}+\frac {43 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)}{1344 a^2}+\frac {29}{560} c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{56} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1373 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{13440 a^3}-\frac {737 c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{6720 a}-\frac {83}{560} a c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3}{56} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {5 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^3}{128 a^2}+\frac {59}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {17}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {397 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{840 a^3 \sqrt {c+a^2 c x^2}}-\frac {15 i c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{128 a^3 \sqrt {c+a^2 c x^2}}+\frac {15 i c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{128 a^3 \sqrt {c+a^2 c x^2}}-\frac {397 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {c+a^2 c x^2}}+\frac {397 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {c+a^2 c x^2}}+\frac {15 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {15 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {15 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {15 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}} \]

[Out]

-3/560*c*(a^2*c*x^2+c)^(3/2)/a^3-1/280*(a^2*c*x^2+c)^(5/2)/a^3+397/840*I*c^3*arctan(a*x)*arctan((1+I*a*x)^(1/2
)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)-15/128*I*c^3*arctan(a*x)^2*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)+15/128*I*c^3*arctan(a*x)^2*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)-397/1680*I*c^3*polylog(2,-I*(1+I*a*x)^(1/2)/(1-I
*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)+397/1680*I*c^3*polylog(2,I*(1+I*a*x)^(1/2)/(1-I*a*x)^(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((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^3*(a
^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)+15/64*c^3*arctan(a*x)*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)-15/64*c^3*arctan(a*x)*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)-15/64*I*c^3*polylog(4,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)+15/64*I*c^3*polylog(4,-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
)+13/6720*c^2*(a^2*c*x^2+c)^(1/2)/a^3+43/1344*c^2*x*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^2+29/560*c^2*x^3*arctan(
a*x)*(a^2*c*x^2+c)^(1/2)+1/56*a^2*c^2*x^5*arctan(a*x)*(a^2*c*x^2+c)^(1/2)+1373/13440*c^2*arctan(a*x)^2*(a^2*c*
x^2+c)^(1/2)/a^3-737/6720*c^2*x^2*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a-83/560*a*c^2*x^4*arctan(a*x)^2*(a^2*c*x^
2+c)^(1/2)-3/56*a^3*c^2*x^6*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)+5/128*c^2*x*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/a^
2+59/192*c^2*x^3*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)+17/48*a^2*c^2*x^5*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)+1/8*a^4
*c^2*x^7*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 10.84 (sec) , antiderivative size = 1019, normalized size of antiderivative = 1.00, number of steps used = 293, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5070, 5072, 5050, 5010, 5006, 5008, 4266, 2611, 6744, 2320, 6724, 267, 272, 45} \[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\frac {1}{8} a^4 c^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7-\frac {3}{56} a^3 c^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6+\frac {17}{48} a^2 c^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5+\frac {1}{56} a^2 c^2 \sqrt {a^2 c x^2+c} \arctan (a x) x^5-\frac {83}{560} a c^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4+\frac {59}{192} c^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3 x^3+\frac {29}{560} c^2 \sqrt {a^2 c x^2+c} \arctan (a x) x^3-\frac {737 c^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2 x^2}{6720 a}+\frac {5 c^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3 x}{128 a^2}+\frac {43 c^2 \sqrt {a^2 c x^2+c} \arctan (a x) x}{1344 a^2}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{64 a^3 \sqrt {a^2 c x^2+c}}-\frac {\left (a^2 c x^2+c\right )^{5/2}}{280 a^3}+\frac {1373 c^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{13440 a^3}-\frac {3 c \left (a^2 c x^2+c\right )^{3/2}}{560 a^3}+\frac {397 i c^3 \sqrt {a^2 x^2+1} \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{840 a^3 \sqrt {a^2 c x^2+c}}-\frac {15 i c^3 \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{128 a^3 \sqrt {a^2 c x^2+c}}+\frac {15 i c^3 \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{128 a^3 \sqrt {a^2 c x^2+c}}-\frac {397 i c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {a^2 c x^2+c}}+\frac {397 i c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {a^2 c x^2+c}}+\frac {15 c^3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}-\frac {15 c^3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {15 i c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}-\frac {15 i c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {13 c^2 \sqrt {a^2 c x^2+c}}{6720 a^3} \]

[In]

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

[Out]

(13*c^2*Sqrt[c + a^2*c*x^2])/(6720*a^3) - (3*c*(c + a^2*c*x^2)^(3/2))/(560*a^3) - (c + a^2*c*x^2)^(5/2)/(280*a
^3) + (43*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(1344*a^2) + (29*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/560
+ (a^2*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/56 + (1373*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(13440*a^3)
- (737*c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(6720*a) - (83*a*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)
/560 - (3*a^3*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/56 + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(128
*a^2) + (59*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/192 + (17*a^2*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3
)/48 + (a^4*c^2*x^7*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/8 + (((5*I)/64)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTa
n[a*x])]*ArcTan[a*x]^3)/(a^3*Sqrt[c + a^2*c*x^2]) + (((397*I)/840)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sq
rt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - (((15*I)/128)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*
PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (((15*I)/128)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x
]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (((397*I)/1680)*c^3*Sqrt[1 + a^2*x^2]*PolyLog
[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + (((397*I)/1680)*c^3*Sqrt[1 + a^2*x^2]
*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + (15*c^3*Sqrt[1 + a^2*x^2]*ArcTan
[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2]) - (15*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]
*PolyLog[3, I*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2]) + (((15*I)/64)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4,
(-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (((15*I)/64)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*Arc
Tan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2])

Rule 45

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])

Rule 267

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 272

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 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 5006

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 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]

Rule 6744

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

Rubi steps \begin{align*} \text {integral}& = c \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx+\left (a^2 c\right ) \int x^4 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx \\ & = c^2 \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx+2 \left (\left (a^2 c^2\right ) \int x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx\right )+\left (a^4 c^2\right ) \int x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx \\ & = c^3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac {x^4 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {x^6 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^3\right ) \int \frac {x^4 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {x^6 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^6 c^3\right ) \int \frac {x^8 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \frac {c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 a^2}+\frac {1}{4} c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {1}{4} \left (3 c^3\right ) \int \frac {x^2 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {c^3 \int \frac {\arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {\left (3 c^3\right ) \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a}-\frac {1}{4} \left (3 a c^3\right ) \int \frac {x^3 \arctan (a x)^2}{\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)^3}{\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)^3+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {1}{4} \left (3 c^3\right ) \int \frac {x^2 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{4} \left (3 a c^3\right ) \int \frac {x^3 \arctan (a x)^2}{\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)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a^3 c^3\right ) \int \frac {x^5 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right )-\frac {1}{2} \left (a^3 c^3\right ) \int \frac {x^5 \arctan (a x)^2}{\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)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{8} \left (3 a^5 c^3\right ) \int \frac {x^7 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3}-\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{4 a}-\frac {1}{10} a c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3}{56} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^3}{8 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{2} c^3 \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (5 c^3\right ) \int \frac {x^2 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (3 c^3\right ) \int \frac {\arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {\left (3 c^3\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}+\frac {c^3 \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a}+\frac {\left (9 c^3\right ) \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{8 a}+\frac {1}{5} \left (2 a c^3\right ) \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (5 a c^3\right ) \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{5} \left (a^2 c^3\right ) \int \frac {x^4 \arctan (a x)}{\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)^2}{4 a}-\frac {1}{10} a c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^3}{8 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{2} c^3 \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (5 c^3\right ) \int \frac {x^2 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (3 c^3\right ) \int \frac {\arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {c^3 \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a}+\frac {\left (9 c^3\right ) \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{8 a}+\frac {1}{5} \left (2 a c^3\right ) \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (5 a c^3\right ) \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{5} \left (a^2 c^3\right ) \int \frac {x^4 \arctan (a x)}{\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)^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{28} \left (9 a^3 c^3\right ) \int \frac {x^5 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{16} \left (7 a^3 c^3\right ) \int \frac {x^5 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{28} \left (3 a^4 c^3\right ) \int \frac {x^6 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^3}{\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 [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(6517\) vs. \(2(1019)=2038\).

Time = 24.69 (sec) , antiderivative size = 6517, normalized size of antiderivative = 6.40 \[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [A] (verified)

Time = 12.63 (sec) , antiderivative size = 566, normalized size of antiderivative = 0.56

method	result	size
default	\(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (1680 \arctan \left (a x \right )^{3} a^{7} x^{7}-720 a^{6} x^{6} \arctan \left (a x \right )^{2}+4760 \arctan \left (a x \right )^{3} a^{5} x^{5}+240 \arctan \left (a x \right ) a^{5} x^{5}-1992 a^{4} \arctan \left (a x \right )^{2} x^{4}+4130 \arctan \left (a x \right )^{3} a^{3} x^{3}-48 a^{4} x^{4}+696 \arctan \left (a x \right ) x^{3} a^{3}-1474 x^{2} \arctan \left (a x \right )^{2} a^{2}+525 \arctan \left (a x \right )^{3} a x -168 a^{2} x^{2}+430 x \arctan \left (a x \right ) a +1373 \arctan \left (a x \right )^{2}-94\right )}{13440 a^{3}}+\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (525 \arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-525 \arctan \left (a x \right )^{3} \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} \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+1575 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3176 \arctan \left (a x \right ) \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 (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3176 \arctan \left (a x \right ) \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 (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3150 i \operatorname {polylog}\left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3150 i \operatorname {polylog}\left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3176 i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3176 i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{13440 a^{3} \sqrt {a^{2} x^{2}+1}}\)	\(566\)

[In]

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

[Out]

1/13440*c^2/a^3*(c*(a*x-I)*(I+a*x))^(1/2)*(1680*arctan(a*x)^3*a^7*x^7-720*a^6*x^6*arctan(a*x)^2+4760*arctan(a*
x)^3*a^5*x^5+240*arctan(a*x)*a^5*x^5-1992*a^4*arctan(a*x)^2*x^4+4130*arctan(a*x)^3*a^3*x^3-48*a^4*x^4+696*arct
an(a*x)*x^3*a^3-1474*x^2*arctan(a*x)^2*a^2+525*arctan(a*x)^3*a*x-168*a^2*x^2+430*x*arctan(a*x)*a+1373*arctan(a
*x)^2-94)+1/13440*c^2*(c*(a*x-I)*(I+a*x))^(1/2)*(525*arctan(a*x)^3*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-525*arc
tan(a*x)^3*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-1575*I*arctan(a*x)^2*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+
1575*I*arctan(a*x)^2*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3176*arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1
/2))+3150*arctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3176*arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^
(1/2))-3150*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3150*I*polylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/
2))-3150*I*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3176*I*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3176*I*dilog
(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^3/(a^2*x^2+1)^(1/2)

Fricas [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{3} \,d x } \]

[In]

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

Sympy [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\int x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}\, dx \]

[In]

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

[Out]

Integral(x**2*(c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**3, x)

Maxima [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\int x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]

[In]

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

[Out]

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