3.3.17 \(\int x^2 (c+a^2 c x^2)^{5/2} \text {ArcTan}(a x) \, dx\) [217]

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

[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 = 1.43, 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 = {5070, 5066, 5072, 267, 5010, 5006, 272, 45} \begin {gather*} \frac {5 c^2 x \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}{128 a^2}+\frac {17}{48} a^2 c^2 x^5 \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}+\frac {59}{192} c^2 x^3 \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}+\frac {1}{8} a^4 c^2 x^7 \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {ArcTan}\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 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 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} \end {gather*}

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 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 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 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 5066

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 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]

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(907\) vs. \(2(418)=836\).
time = 10.82, size = 907, normalized size = 2.17 \begin {gather*} \frac {c^2 \sqrt {c+a^2 c x^2} \left (-\frac {19067}{32} \left (1+a^2 x^2\right )^{7/2}-\frac {3829}{32} \left (1+a^2 x^2\right )^4 \cos (3 \text {ArcTan}(a x))-3150 i \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )+3150 i \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )-420 \left (1+a^2 x^2\right )^2 \left (-\frac {2}{\sqrt {1+a^2 x^2}}-6 \cos (3 \text {ArcTan}(a x))+3 \text {ArcTan}(a x) \left (-\frac {14 a x}{\sqrt {1+a^2 x^2}}+3 \log \left (1-i e^{i \text {ArcTan}(a x)}\right )+4 \cos (2 \text {ArcTan}(a x)) \left (\log \left (1-i e^{i \text {ArcTan}(a x)}\right )-\log \left (1+i e^{i \text {ArcTan}(a x)}\right )\right )+\cos (4 \text {ArcTan}(a x)) \left (\log \left (1-i e^{i \text {ArcTan}(a x)}\right )-\log \left (1+i e^{i \text {ArcTan}(a x)}\right )\right )-3 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )+2 \sin (3 \text {ArcTan}(a x))\right )\right )+7 \left (1+a^2 x^2\right )^3 \left (\frac {12}{\sqrt {1+a^2 x^2}}+110 \cos (3 \text {ArcTan}(a x))-90 \cos (5 \text {ArcTan}(a x))+15 \text {ArcTan}(a x) \left (\frac {156 a x}{\sqrt {1+a^2 x^2}}+30 \log \left (1-i e^{i \text {ArcTan}(a x)}\right )+3 \cos (6 \text {ArcTan}(a x)) \log \left (1-i e^{i \text {ArcTan}(a x)}\right )+45 \cos (2 \text {ArcTan}(a x)) \left (\log \left (1-i e^{i \text {ArcTan}(a x)}\right )-\log \left (1+i e^{i \text {ArcTan}(a x)}\right )\right )+18 \cos (4 \text {ArcTan}(a x)) \left (\log \left (1-i e^{i \text {ArcTan}(a x)}\right )-\log \left (1+i e^{i \text {ArcTan}(a x)}\right )\right )-30 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )-3 \cos (6 \text {ArcTan}(a x)) \log \left (1+i e^{i \text {ArcTan}(a x)}\right )-94 \sin (3 \text {ArcTan}(a x))+6 \sin (5 \text {ArcTan}(a x))\right )\right )-\frac {35}{64} \left (1+a^2 x^2\right )^4 \left (314 \cos (5 \text {ArcTan}(a x))-90 \cos (7 \text {ArcTan}(a x))+3 \text {ArcTan}(a x) \left (-\frac {3530 a x}{\sqrt {1+a^2 x^2}}+525 \log \left (1-i e^{i \text {ArcTan}(a x)}\right )+120 \cos (6 \text {ArcTan}(a x)) \log \left (1-i e^{i \text {ArcTan}(a x)}\right )+15 \cos (8 \text {ArcTan}(a x)) \log \left (1-i e^{i \text {ArcTan}(a x)}\right )+840 \cos (2 \text {ArcTan}(a x)) \left (\log \left (1-i e^{i \text {ArcTan}(a x)}\right )-\log \left (1+i e^{i \text {ArcTan}(a x)}\right )\right )+420 \cos (4 \text {ArcTan}(a x)) \left (\log \left (1-i e^{i \text {ArcTan}(a x)}\right )-\log \left (1+i e^{i \text {ArcTan}(a x)}\right )\right )-525 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )-120 \cos (6 \text {ArcTan}(a x)) \log \left (1+i e^{i \text {ArcTan}(a x)}\right )-15 \cos (8 \text {ArcTan}(a x)) \log \left (1+i e^{i \text {ArcTan}(a x)}\right )+1790 \sin (3 \text {ArcTan}(a x))-794 \sin (5 \text {ArcTan}(a x))+30 \sin (7 \text {ArcTan}(a x))\right )\right )\right )}{80640 a^3 \sqrt {1+a^2 x^2}} \end {gather*}

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 + a^2*c*x^2]*((-19067*(1 + a^2*x^2)^(7/2))/32 - (3829*(1 + a^2*x^2)^4*Cos[3*ArcTan[a*x]])/32 - (31
50*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (3150*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] - 420*(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*ArcTa
n[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*ArcTan[a*x])] + 2*Sin[3*
ArcTan[a*x]])) + 7*(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*C
os[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]])) - (35
*(1 + a^2*x^2)^4*(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*ArcT
an[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*ArcTan[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])

________________________________________________________________________________________

Maple [A]
time = 0.55, size = 245, normalized size = 0.59

method	result	size
default	\(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (5040 \arctan \left (a x \right ) a^{7} x^{7}-720 a^{6} x^{6}+14280 \arctan \left (a x \right ) a^{5} x^{5}-1992 a^{4} x^{4}+12390 \arctan \left (a x \right ) a^{3} x^{3}-1474 a^{2} x^{2}+1575 \arctan \left (a x \right ) a x +1373\right )}{40320 a^{3}}+\frac {5 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) c^{2}}{128 \sqrt {a^{2} x^{2}+1}\, a^{3}}\)	\(245\)

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________