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

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 2.03111, antiderivative size = 418, normalized size of antiderivative = 1., 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 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 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 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 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^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 = 15.4626, 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])

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Maple [A]  time = 0.503, size = 245, normalized size = 0.6 \begin{align*}{\frac{{c}^{2} \left ( 5040\,\arctan \left ( ax \right ){x}^{7}{a}^{7}-720\,{x}^{6}{a}^{6}+14280\,\arctan \left ( ax \right ){x}^{5}{a}^{5}-1992\,{a}^{4}{x}^{4}+12390\,\arctan \left ( ax \right ){x}^{3}{a}^{3}-1474\,{a}^{2}{x}^{2}+1575\,\arctan \left ( ax \right ) xa+1373 \right ) }{40320\,{a}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{5\,{c}^{2}}{128\,{a}^{3}}\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^2*(a^2*c*x^2+c)^(5/2)*arctan(a*x),x)

[Out]

1/40320*c^2/a^3*(c*(a*x-I)*(a*x+I))^(1/2)*(5040*arctan(a*x)*x^7*a^7-720*x^6*a^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)*(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^3/(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^2*(a^2*c*x^2+c)^(5/2)*arctan(a*x),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^{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 ) \end{align*}

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)

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

[Out]

Exception raised: TypeError