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

Optimal. Leaf size=418 \[ \frac {5 c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{128 a^2}+\frac {17}{48} a^2 c^2 x^5 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {59}{192} c^2 x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {1}{8} a^4 c^2 x^7 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)-\frac {5 i c^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{128 a^3 \sqrt {a^2 c x^2+c}}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{128 a^3 \sqrt {a^2 c x^2+c}}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {5 c^2 \sqrt {a^2 c x^2+c}}{128 a^3}-\frac {\left (a^2 c x^2+c\right )^{7/2}}{56 a^3 c}+\frac {\left (a^2 c x^2+c\right )^{5/2}}{240 a^3}+\frac {5 c \left (a^2 c x^2+c\right )^{3/2}}{576 a^3} \]

[Out]

5/576*c*(a^2*c*x^2+c)^(3/2)/a^3+1/240*(a^2*c*x^2+c)^(5/2)/a^3-1/56*(a^2*c*x^2+c)^(7/2)/a^3/c+5/64*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)-5/128*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)+5/128*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)+5/128*c^2*(a^2*c*x^2+c)^(1/2)/a^3+5/128*c^2
*x*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^2+59/192*c^2*x^3*arctan(a*x)*(a^2*c*x^2+c)^(1/2)+17/48*a^2*c^2*x^5*arctan
(a*x)*(a^2*c*x^2+c)^(1/2)+1/8*a^4*c^2*x^7*arctan(a*x)*(a^2*c*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 2.03, antiderivative size = 418, normalized size of antiderivative = 1.00, number of steps used = 51, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4950, 4946, 4952, 261, 4890, 4886, 266, 43} \[ -\frac {5 i c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{128 a^3 \sqrt {a^2 c x^2+c}}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{128 a^3 \sqrt {a^2 c x^2+c}}+\frac {5 c^2 \sqrt {a^2 c x^2+c}}{128 a^3}+\frac {1}{8} a^4 c^2 x^7 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {17}{48} a^2 c^2 x^5 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {59}{192} c^2 x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {5 c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{128 a^2}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}-\frac {\left (a^2 c x^2+c\right )^{7/2}}{56 a^3 c}+\frac {\left (a^2 c x^2+c\right )^{5/2}}{240 a^3}+\frac {5 c \left (a^2 c x^2+c\right )^{3/2}}{576 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*c^2*Sqrt[c + a^2*c*x^2])/(128*a^3) + (5*c*(c + a^2*c*x^2)^(3/2))/(576*a^3) + (c + a^2*c*x^2)^(5/2)/(240*a^3
) - (c + a^2*c*x^2)^(7/2)/(56*a^3*c) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(128*a^2) + (59*c^2*x^3*Sqrt[
c + a^2*c*x^2]*ArcTan[a*x])/192 + (17*a^2*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/48 + (a^4*c^2*x^7*Sqrt[c +
a^2*c*x^2]*ArcTan[a*x])/8 + (((5*I)/64)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*
x]])/(a^3*Sqrt[c + a^2*c*x^2]) - (((5*I)/128)*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]) + (((5*I)/128)*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])

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])

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 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 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 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 4946

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

Rubi steps

\begin {align*} \int x^2 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x) \, dx &=c \int x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x) \, dx+\left (a^2 c\right ) \int x^4 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x) \, dx\\ &=c^2 \int x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x) \, dx+2 \left (\left (a^2 c^2\right ) \int x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x) \, dx\right )+\left (a^4 c^2\right ) \int x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x) \, dx\\ &=\frac {1}{4} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{4} c^3 \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{4} \left (a c^3\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{6} \left (a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{6} \left (a^3 c^3\right ) \int \frac {x^5}{\sqrt {c+a^2 c x^2}} \, dx\right )+\frac {1}{8} \left (a^4 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{8} \left (a^5 c^3\right ) \int \frac {x^7}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{8 a^2}+\frac {1}{4} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c^3 \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}-\frac {c^3 \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{8 a}-\frac {1}{8} \left (a c^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {1}{48} \left (5 a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{48} \left (a^3 c^3\right ) \int \frac {x^5}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{8} c^3 \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{24} \left (a c^3\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{12} \left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )\right )-\frac {1}{16} \left (a^5 c^3\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac {c^2 \sqrt {c+a^2 c x^2}}{8 a^3}+\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{8 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{64} \left (5 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{192} \left (5 a c^3\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{8} \left (a 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}{96} \left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )+2 \left (-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {c^3 \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{16 a^2}+\frac {c^3 \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{16 a}-\frac {1}{48} \left (a c^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {1}{12} \left (a^3 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 )\right )-\frac {1}{16} \left (a^5 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 (c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{8 a^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {c^2 \sqrt {c+a^2 c x^2}}{4 a^3}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2}}{24 a^3}+\frac {3 \left (c+a^2 c x^2\right )^{5/2}}{40 a^3}-\frac {\left (c+a^2 c x^2\right )^{7/2}}{56 a^3 c}+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {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 )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {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 )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {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 )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{128 a^2}-\frac {\left (5 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{128 a}+\frac {1}{384} \left (5 a c^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {1}{96} \left (a^3 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 )+2 \left (-\frac {5 c^2 \sqrt {c+a^2 c x^2}}{48 a^3}+\frac {c \left (c+a^2 c x^2\right )^{3/2}}{9 a^3}-\frac {\left (c+a^2 c x^2\right )^{5/2}}{30 a^3}-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{48} \left (a 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 (c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{16 a^2 \sqrt {c+a^2 c x^2}}\right )\\ &=\frac {73 c^2 \sqrt {c+a^2 c x^2}}{384 a^3}-\frac {7 c \left (c+a^2 c x^2\right )^{3/2}}{36 a^3}+\frac {17 \left (c+a^2 c x^2\right )^{5/2}}{240 a^3}-\frac {\left (c+a^2 c x^2\right )^{7/2}}{56 a^3 c}+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {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 )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {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 )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {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 )}{8 a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c^2 \sqrt {c+a^2 c x^2}}{16 a^3}+\frac {7 c \left (c+a^2 c x^2\right )^{3/2}}{72 a^3}-\frac {\left (c+a^2 c x^2\right )^{5/2}}{30 a^3}-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {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 )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {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 )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {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 )}{16 a^3 \sqrt {c+a^2 c x^2}}\right )+\frac {1}{384} \left (5 a 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}{128 a^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {21 c^2 \sqrt {c+a^2 c x^2}}{128 a^3}-\frac {107 c \left (c+a^2 c x^2\right )^{3/2}}{576 a^3}+\frac {17 \left (c+a^2 c x^2\right )^{5/2}}{240 a^3}-\frac {\left (c+a^2 c x^2\right )^{7/2}}{56 a^3 c}+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {21 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 )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {21 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 )}{128 a^3 \sqrt {c+a^2 c x^2}}+\frac {21 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 )}{128 a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c^2 \sqrt {c+a^2 c x^2}}{16 a^3}+\frac {7 c \left (c+a^2 c x^2\right )^{3/2}}{72 a^3}-\frac {\left (c+a^2 c x^2\right )^{5/2}}{30 a^3}-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {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 )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {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 )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {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 )}{16 a^3 \sqrt {c+a^2 c x^2}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 15.28, size = 1059, normalized size = 2.53 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*Sqrt[c*(1 + a^2*x^2)]*((-6*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (6*I)*PolyLog[2, I*E^(I*ArcTan[a*x])]
- ((1 + a^2*x^2)^2*(-2/Sqrt[1 + a^2*x^2] - 6*Cos[3*ArcTan[a*x]] + 3*ArcTan[a*x]*((-14*a*x)/Sqrt[1 + a^2*x^2] +
3*Log[1 - I*E^(I*ArcTan[a*x])] + 4*Cos[2*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a
*x])]) + Cos[4*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) - 3*Log[1 + I*E^(I*A
rcTan[a*x])] + 2*Sin[3*ArcTan[a*x]])))/4))/(48*a^3*Sqrt[1 + a^2*x^2]) + (c^2*Sqrt[c*(1 + a^2*x^2)]*((90*I)*Pol
yLog[2, (-I)*E^(I*ArcTan[a*x])] - (90*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] + ((1 + a^2*x^2)^3*(12/Sqrt[1 + a^2*x
^2] + 110*Cos[3*ArcTan[a*x]] - 90*Cos[5*ArcTan[a*x]] + 15*ArcTan[a*x]*((156*a*x)/Sqrt[1 + a^2*x^2] + 30*Log[1
- I*E^(I*ArcTan[a*x])] + 3*Cos[6*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + 45*Cos[2*ArcTan[a*x]]*(Log[1 - I*
E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + 18*Cos[4*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log
[1 + I*E^(I*ArcTan[a*x])]) - 30*Log[1 + I*E^(I*ArcTan[a*x])] - 3*Cos[6*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x]
)] - 94*Sin[3*ArcTan[a*x]] + 6*Sin[5*ArcTan[a*x]])))/16))/(720*a^3*Sqrt[1 + a^2*x^2]) + (c^2*Sqrt[c*(1 + a^2*x
^2)]*((-3150*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (3150*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] - ((1 + a^2*x^2)
^4*(38134/Sqrt[1 + a^2*x^2] + 7658*Cos[3*ArcTan[a*x]] + 35*(314*Cos[5*ArcTan[a*x]] - 90*Cos[7*ArcTan[a*x]] + 3
*ArcTan[a*x]*((-3530*a*x)/Sqrt[1 + a^2*x^2] + 525*Log[1 - I*E^(I*ArcTan[a*x])] + 120*Cos[6*ArcTan[a*x]]*Log[1
- I*E^(I*ArcTan[a*x])] + 15*Cos[8*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + 840*Cos[2*ArcTan[a*x]]*(Log[1 -
I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + 420*Cos[4*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] -
Log[1 + I*E^(I*ArcTan[a*x])]) - 525*Log[1 + I*E^(I*ArcTan[a*x])] - 120*Cos[6*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTa
n[a*x])] - 15*Cos[8*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 1790*Sin[3*ArcTan[a*x]] - 794*Sin[5*ArcTan[a*x
]] + 30*Sin[7*ArcTan[a*x]]))))/64))/(80640*a^3*Sqrt[1 + a^2*x^2])

________________________________________________________________________________________

fricas [F]  time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{6} + 2 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 1.35, size = 245, normalized size = 0.59 \[ \frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (5040 \arctan \left (a x \right ) x^{7} a^{7}-720 a^{6} x^{6}+14280 \arctan \left (a x \right ) x^{5} a^{5}-1992 a^{4} x^{4}+12390 \arctan \left (a x \right ) x^{3} a^{3}-1474 a^{2} x^{2}+1575 \arctan \left (a x \right ) x a +1373\right )}{40320 a^{3}}+\frac {5 c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{128 a^{3} \sqrt {a^{2} x^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40320*c^2/a^3*(c*(a*x-I)*(I+a*x))^(1/2)*(5040*arctan(a*x)*x^7*a^7-720*a^6*x^6+14280*arctan(a*x)*x^5*a^5-1992
*a^4*x^4+12390*arctan(a*x)*x^3*a^3-1474*a^2*x^2+1575*arctan(a*x)*x*a+1373)+5/128*c^2*(c*(a*x-I)*(I+a*x))^(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^3/(a^2*x^2+1)^(1/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}{\left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________