3.216 \(\int x^3 (c+a^2 c x^2)^{5/2} \tan ^{-1}(a x) \, dx\)

Optimal. Leaf size=289 \[ -\frac{1}{72} a^3 c^2 x^7 \sqrt{a^2 c x^2+c}-\frac{103 a c^2 x^5 \sqrt{a^2 c x^2+c}}{3024}-\frac{205 c^2 x^3 \sqrt{a^2 c x^2+c}}{12096 a}+\frac{47 c^2 x \sqrt{a^2 c x^2+c}}{2688 a^3}+\frac{1}{9} a^4 c^2 x^8 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{19}{63} a^2 c^2 x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{5}{21} c^2 x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{c^2 x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{63 a^2}-\frac{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{63 a^4}+\frac{115 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{8064 a^4} \]

[Out]

(47*c^2*x*Sqrt[c + a^2*c*x^2])/(2688*a^3) - (205*c^2*x^3*Sqrt[c + a^2*c*x^2])/(12096*a) - (103*a*c^2*x^5*Sqrt[
c + a^2*c*x^2])/3024 - (a^3*c^2*x^7*Sqrt[c + a^2*c*x^2])/72 - (2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(63*a^4)
+ (c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(63*a^2) + (5*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/21 + (19*a
^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/63 + (a^4*c^2*x^8*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/9 + (115*c^(5/2
)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(8064*a^4)

________________________________________________________________________________________

Rubi [A]  time = 1.97681, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 76, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {4950, 4946, 4952, 321, 217, 206, 4930} \[ -\frac{1}{72} a^3 c^2 x^7 \sqrt{a^2 c x^2+c}-\frac{103 a c^2 x^5 \sqrt{a^2 c x^2+c}}{3024}-\frac{205 c^2 x^3 \sqrt{a^2 c x^2+c}}{12096 a}+\frac{47 c^2 x \sqrt{a^2 c x^2+c}}{2688 a^3}+\frac{1}{9} a^4 c^2 x^8 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{19}{63} a^2 c^2 x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{5}{21} c^2 x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{c^2 x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{63 a^2}-\frac{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{63 a^4}+\frac{115 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{8064 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(47*c^2*x*Sqrt[c + a^2*c*x^2])/(2688*a^3) - (205*c^2*x^3*Sqrt[c + a^2*c*x^2])/(12096*a) - (103*a*c^2*x^5*Sqrt[
c + a^2*c*x^2])/3024 - (a^3*c^2*x^7*Sqrt[c + a^2*c*x^2])/72 - (2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(63*a^4)
+ (c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(63*a^2) + (5*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/21 + (19*a
^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/63 + (a^4*c^2*x^8*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/9 + (115*c^(5/2
)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(8064*a^4)

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 321

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 217

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 206

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

Rule 4930

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]

Rubi steps

\begin{align*} \int x^3 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x) \, dx &=c \int x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x) \, dx+\left (a^2 c\right ) \int x^5 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x) \, dx\\ &=c^2 \int x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx+2 \left (\left (a^2 c^2\right ) \int x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx\right )+\left (a^4 c^2\right ) \int x^7 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx\\ &=\frac{1}{5} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{9} a^4 c^2 x^8 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{5} c^3 \int \frac{x^3 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{5} \left (a c^3\right ) \int \frac{x^4}{\sqrt{c+a^2 c x^2}} \, dx+2 \left (\frac{1}{7} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{7} \left (a^2 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{7} \left (a^3 c^3\right ) \int \frac{x^6}{\sqrt{c+a^2 c x^2}} \, dx\right )+\frac{1}{9} \left (a^4 c^3\right ) \int \frac{x^7 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{9} \left (a^5 c^3\right ) \int \frac{x^8}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{c^2 x^3 \sqrt{c+a^2 c x^2}}{20 a}-\frac{1}{72} a^3 c^2 x^7 \sqrt{c+a^2 c x^2}+\frac{c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{15 a^2}+\frac{1}{5} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{63} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{9} a^4 c^2 x^8 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{\left (2 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^2}-\frac{c^3 \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{15 a}+\frac{\left (3 c^3\right ) \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{20 a}+2 \left (-\frac{1}{42} a c^2 x^5 \sqrt{c+a^2 c x^2}+\frac{1}{35} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{7} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{35} \left (4 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{35} \left (a c^3\right ) \int \frac{x^4}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{42} \left (5 a c^3\right ) \int \frac{x^4}{\sqrt{c+a^2 c x^2}} \, dx\right )-\frac{1}{21} \left (2 a^2 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{63} \left (a^3 c^3\right ) \int \frac{x^6}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{72} \left (7 a^3 c^3\right ) \int \frac{x^6}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{c^2 x \sqrt{c+a^2 c x^2}}{24 a^3}-\frac{c^2 x^3 \sqrt{c+a^2 c x^2}}{20 a}+\frac{41 a c^2 x^5 \sqrt{c+a^2 c x^2}}{3024}-\frac{1}{72} a^3 c^2 x^7 \sqrt{c+a^2 c x^2}-\frac{2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{15 a^4}+\frac{c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{15 a^2}+\frac{19}{105} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{63} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{9} a^4 c^2 x^8 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{105} \left (8 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{c^3 \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{30 a^3}-\frac{\left (3 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{40 a^3}+\frac{\left (2 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^3}+2 \left (\frac{19 c^2 x^3 \sqrt{c+a^2 c x^2}}{840 a}-\frac{1}{42} a c^2 x^5 \sqrt{c+a^2 c x^2}-\frac{4 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{105 a^2}+\frac{1}{35} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{7} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{\left (8 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{105 a^2}+\frac{\left (3 c^3\right ) \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{140 a}+\frac{\left (4 c^3\right ) \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{105 a}-\frac{\left (5 c^3\right ) \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{56 a}\right )+\frac{1}{378} \left (5 a c^3\right ) \int \frac{x^4}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{105} \left (2 a c^3\right ) \int \frac{x^4}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{432} \left (35 a c^3\right ) \int \frac{x^4}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{c^2 x \sqrt{c+a^2 c x^2}}{24 a^3}-\frac{3761 c^2 x^3 \sqrt{c+a^2 c x^2}}{60480 a}+\frac{41 a c^2 x^5 \sqrt{c+a^2 c x^2}}{3024}-\frac{1}{72} a^3 c^2 x^7 \sqrt{c+a^2 c x^2}-\frac{2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{15 a^4}+\frac{29 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{315 a^2}+\frac{19}{105} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{63} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{9} a^4 c^2 x^8 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{c^3 \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{30 a^3}-\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{40 a^3}+2 \left (-\frac{5 c^2 x \sqrt{c+a^2 c x^2}}{336 a^3}+\frac{19 c^2 x^3 \sqrt{c+a^2 c x^2}}{840 a}-\frac{1}{42} a c^2 x^5 \sqrt{c+a^2 c x^2}+\frac{8 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{105 a^4}-\frac{4 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{105 a^2}+\frac{1}{35} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{7} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{\left (3 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{280 a^3}-\frac{\left (2 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{105 a^3}+\frac{\left (5 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{112 a^3}-\frac{\left (8 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{105 a^3}\right )+\frac{\left (2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{15 a^3}-\frac{\left (16 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{315 a^2}-\frac{\left (5 c^3\right ) \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{504 a}-\frac{c^3 \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{70 a}-\frac{\left (8 c^3\right ) \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{315 a}+\frac{\left (35 c^3\right ) \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{576 a}\\ &=\frac{127 c^2 x \sqrt{c+a^2 c x^2}}{2688 a^3}-\frac{3761 c^2 x^3 \sqrt{c+a^2 c x^2}}{60480 a}+\frac{41 a c^2 x^5 \sqrt{c+a^2 c x^2}}{3024}-\frac{1}{72} a^3 c^2 x^7 \sqrt{c+a^2 c x^2}-\frac{58 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{315 a^4}+\frac{29 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{315 a^2}+\frac{19}{105} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{63} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{9} a^4 c^2 x^8 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{11 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{120 a^4}+\frac{\left (5 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{1008 a^3}+\frac{c^3 \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{140 a^3}+\frac{\left (4 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{315 a^3}-\frac{\left (35 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{1152 a^3}+\frac{\left (16 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{315 a^3}+2 \left (-\frac{5 c^2 x \sqrt{c+a^2 c x^2}}{336 a^3}+\frac{19 c^2 x^3 \sqrt{c+a^2 c x^2}}{840 a}-\frac{1}{42} a c^2 x^5 \sqrt{c+a^2 c x^2}+\frac{8 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{105 a^4}-\frac{4 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{105 a^2}+\frac{1}{35} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{7} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{280 a^3}-\frac{\left (2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{105 a^3}+\frac{\left (5 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{112 a^3}-\frac{\left (8 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{105 a^3}\right )\\ &=\frac{127 c^2 x \sqrt{c+a^2 c x^2}}{2688 a^3}-\frac{3761 c^2 x^3 \sqrt{c+a^2 c x^2}}{60480 a}+\frac{41 a c^2 x^5 \sqrt{c+a^2 c x^2}}{3024}-\frac{1}{72} a^3 c^2 x^7 \sqrt{c+a^2 c x^2}-\frac{58 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{315 a^4}+\frac{29 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{315 a^2}+\frac{19}{105} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{63} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{9} a^4 c^2 x^8 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{11 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{120 a^4}+2 \left (-\frac{5 c^2 x \sqrt{c+a^2 c x^2}}{336 a^3}+\frac{19 c^2 x^3 \sqrt{c+a^2 c x^2}}{840 a}-\frac{1}{42} a c^2 x^5 \sqrt{c+a^2 c x^2}+\frac{8 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{105 a^4}-\frac{4 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{105 a^2}+\frac{1}{35} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{7} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{103 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{1680 a^4}\right )+\frac{\left (5 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{1008 a^3}+\frac{c^3 \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{140 a^3}+\frac{\left (4 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{315 a^3}-\frac{\left (35 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{1152 a^3}+\frac{\left (16 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{315 a^3}\\ &=\frac{127 c^2 x \sqrt{c+a^2 c x^2}}{2688 a^3}-\frac{3761 c^2 x^3 \sqrt{c+a^2 c x^2}}{60480 a}+\frac{41 a c^2 x^5 \sqrt{c+a^2 c x^2}}{3024}-\frac{1}{72} a^3 c^2 x^7 \sqrt{c+a^2 c x^2}-\frac{58 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{315 a^4}+\frac{29 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{315 a^2}+\frac{19}{105} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{63} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{9} a^4 c^2 x^8 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{5519 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{40320 a^4}+2 \left (-\frac{5 c^2 x \sqrt{c+a^2 c x^2}}{336 a^3}+\frac{19 c^2 x^3 \sqrt{c+a^2 c x^2}}{840 a}-\frac{1}{42} a c^2 x^5 \sqrt{c+a^2 c x^2}+\frac{8 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{105 a^4}-\frac{4 c^2 x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{105 a^2}+\frac{1}{35} c^2 x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{7} a^2 c^2 x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{103 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{1680 a^4}\right )\\ \end{align*}

Mathematica [A]  time = 0.238495, size = 129, normalized size = 0.45 \[ \frac{c^2 \left (-a x \left (336 a^6 x^6+824 a^4 x^4+410 a^2 x^2-423\right ) \sqrt{a^2 c x^2+c}+345 \sqrt{c} \log \left (\sqrt{c} \sqrt{a^2 c x^2+c}+a c x\right )+384 \left (7 a^2 x^2-2\right ) \left (a^2 x^2+1\right )^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)\right )}{24192 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(-(a*x*Sqrt[c + a^2*c*x^2]*(-423 + 410*a^2*x^2 + 824*a^4*x^4 + 336*a^6*x^6)) + 384*(1 + a^2*x^2)^3*(-2 +
7*a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + 345*Sqrt[c]*Log[a*c*x + Sqrt[c]*Sqrt[c + a^2*c*x^2]]))/(24192*a^4
)

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Maple [C]  time = 0.934, size = 225, normalized size = 0.8 \begin{align*}{\frac{{c}^{2} \left ( 2688\,\arctan \left ( ax \right ){x}^{8}{a}^{8}-336\,{x}^{7}{a}^{7}+7296\,\arctan \left ( ax \right ){x}^{6}{a}^{6}-824\,{a}^{5}{x}^{5}+5760\,\arctan \left ( ax \right ){x}^{4}{a}^{4}-410\,{a}^{3}{x}^{3}+384\,\arctan \left ( ax \right ){a}^{2}{x}^{2}+423\,ax-768\,\arctan \left ( ax \right ) \right ) }{24192\,{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{115\,{c}^{2}}{8064\,{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }\ln \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-i \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{115\,{c}^{2}}{8064\,{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }\ln \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+i \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24192*c^2/a^4*(c*(a*x-I)*(a*x+I))^(1/2)*(2688*arctan(a*x)*x^8*a^8-336*x^7*a^7+7296*arctan(a*x)*x^6*a^6-824*a
^5*x^5+5760*arctan(a*x)*x^4*a^4-410*a^3*x^3+384*arctan(a*x)*a^2*x^2+423*a*x-768*arctan(a*x))-115/8064*c^2/a^4*
(c*(a*x-I)*(a*x+I))^(1/2)*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-I)/(a^2*x^2+1)^(1/2)+115/8064*c^2/a^4*(c*(a*x-I)*(a*x
+I))^(1/2)*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+I)/(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^3*(a^2*c*x^2+c)^(5/2)*arctan(a*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.2985, size = 360, normalized size = 1.25 \begin{align*} \frac{345 \, c^{\frac{5}{2}} \log \left (-2 \, a^{2} c x^{2} - 2 \, \sqrt{a^{2} c x^{2} + c} a \sqrt{c} x - c\right ) - 2 \,{\left (336 \, a^{7} c^{2} x^{7} + 824 \, a^{5} c^{2} x^{5} + 410 \, a^{3} c^{2} x^{3} - 423 \, a c^{2} x - 384 \,{\left (7 \, a^{8} c^{2} x^{8} + 19 \, a^{6} c^{2} x^{6} + 15 \, a^{4} c^{2} x^{4} + a^{2} c^{2} x^{2} - 2 \, c^{2}\right )} \arctan \left (a x\right )\right )} \sqrt{a^{2} c x^{2} + c}}{48384 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48384*(345*c^(5/2)*log(-2*a^2*c*x^2 - 2*sqrt(a^2*c*x^2 + c)*a*sqrt(c)*x - c) - 2*(336*a^7*c^2*x^7 + 824*a^5*
c^2*x^5 + 410*a^3*c^2*x^3 - 423*a*c^2*x - 384*(7*a^8*c^2*x^8 + 19*a^6*c^2*x^6 + 15*a^4*c^2*x^4 + a^2*c^2*x^2 -
2*c^2)*arctan(a*x))*sqrt(a^2*c*x^2 + c))/a^4

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

[Out]

Timed out

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Giac [A]  time = 1.22306, size = 366, normalized size = 1.27 \begin{align*} \frac{{\left (\frac{21 \,{\left (3 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} - 5 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} c\right )}}{a^{2}} + \frac{6 \,{\left (15 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{7}{2}} - 42 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} c + 35 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} c^{2}\right )}}{a^{2} c} + \frac{35 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{9}{2}} - 135 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{7}{2}} c + 189 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} c^{2} - 105 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} c^{3}}{a^{2} c^{2}}\right )} \arctan \left (a x\right )}{315 \, a^{2}} - \frac{\sqrt{a^{2} c x^{2} + c}{\left (2 \,{\left (205 \, a^{2} c^{2} + 4 \,{\left (42 \, a^{6} c^{2} x^{2} + 103 \, a^{4} c^{2}\right )} x^{2}\right )} x^{2} - 423 \, c^{2}\right )} x + \frac{345 \, c^{\frac{5}{2}} \log \left ({\left | -\sqrt{a^{2} c} x + \sqrt{a^{2} c x^{2} + c} \right |}\right )}{{\left | a \right |}}}{24192 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*(21*(3*(a^2*c*x^2 + c)^(5/2) - 5*(a^2*c*x^2 + c)^(3/2)*c)/a^2 + 6*(15*(a^2*c*x^2 + c)^(7/2) - 42*(a^2*c*
x^2 + c)^(5/2)*c + 35*(a^2*c*x^2 + c)^(3/2)*c^2)/(a^2*c) + (35*(a^2*c*x^2 + c)^(9/2) - 135*(a^2*c*x^2 + c)^(7/
2)*c + 189*(a^2*c*x^2 + c)^(5/2)*c^2 - 105*(a^2*c*x^2 + c)^(3/2)*c^3)/(a^2*c^2))*arctan(a*x)/a^2 - 1/24192*(sq
rt(a^2*c*x^2 + c)*(2*(205*a^2*c^2 + 4*(42*a^6*c^2*x^2 + 103*a^4*c^2)*x^2)*x^2 - 423*c^2)*x + 345*c^(5/2)*log(a
bs(-sqrt(a^2*c)*x + sqrt(a^2*c*x^2 + c)))/abs(a))/a^3