\(\int x^3 (c+a^2 c x^2)^{5/2} \arctan (a x) \, dx\) [216]

Optimal result

Integrand size = 22, antiderivative size = 289 \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x) \, dx=\frac {47 c^2 x \sqrt {c+a^2 c x^2}}{2688 a^3}-\frac {205 c^2 x^3 \sqrt {c+a^2 c x^2}}{12096 a}-\frac {103 a c^2 x^5 \sqrt {c+a^2 c x^2}}{3024}-\frac {1}{72} a^3 c^2 x^7 \sqrt {c+a^2 c x^2}-\frac {2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{63 a^4}+\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{63 a^2}+\frac {5}{21} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {19}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {115 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{8064 a^4} \]

[Out]

115/8064*c^(5/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))/a^4+47/2688*c^2*x*(a^2*c*x^2+c)^(1/2)/a^3-205/12096*
c^2*x^3*(a^2*c*x^2+c)^(1/2)/a-103/3024*a*c^2*x^5*(a^2*c*x^2+c)^(1/2)-1/72*a^3*c^2*x^7*(a^2*c*x^2+c)^(1/2)-2/63
*c^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^4+1/63*c^2*x^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^2+5/21*c^2*x^4*arctan(
a*x)*(a^2*c*x^2+c)^(1/2)+19/63*a^2*c^2*x^6*arctan(a*x)*(a^2*c*x^2+c)^(1/2)+1/9*a^4*c^2*x^8*arctan(a*x)*(a^2*c*
x^2+c)^(1/2)

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 76, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5070, 5066, 5072, 327, 223, 212, 5050} \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x) \, dx=\frac {c^2 x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{63 a^2}+\frac {19}{63} a^2 c^2 x^6 \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {5}{21} c^2 x^4 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {103 a c^2 x^5 \sqrt {a^2 c x^2+c}}{3024}-\frac {205 c^2 x^3 \sqrt {a^2 c x^2+c}}{12096 a}-\frac {2 c^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{63 a^4}+\frac {1}{9} a^4 c^2 x^8 \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {115 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{8064 a^4}+\frac {47 c^2 x \sqrt {a^2 c x^2+c}}{2688 a^3}-\frac {1}{72} a^3 c^2 x^7 \sqrt {a^2 c x^2+c} \]

[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 212

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

Rule 223

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 327

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 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 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*} \text {integral}& = c \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx+\left (a^2 c\right ) \int x^5 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx \\ & = c^2 \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x) \, dx+2 \left (\left (a^2 c^2\right ) \int x^5 \sqrt {c+a^2 c x^2} \arctan (a x) \, dx\right )+\left (a^4 c^2\right ) \int x^7 \sqrt {c+a^2 c x^2} \arctan (a x) \, dx \\ & = \frac {1}{5} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{5} c^3 \int \frac {x^3 \arctan (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} \arctan (a x)+\frac {1}{7} \left (a^2 c^3\right ) \int \frac {x^5 \arctan (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 \arctan (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} \arctan (a x)}{15 a^2}+\frac {1}{5} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {\left (2 c^3\right ) \int \frac {x \arctan (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} \arctan (a x)+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{35} \left (4 c^3\right ) \int \frac {x^3 \arctan (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 \arctan (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} \arctan (a x)}{15 a^4}+\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{15 a^2}+\frac {19}{105} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{105} \left (8 c^3\right ) \int \frac {x^3 \arctan (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} \arctan (a x)}{105 a^2}+\frac {1}{35} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {\left (8 c^3\right ) \int \frac {x \arctan (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} \arctan (a x)}{15 a^4}+\frac {29 c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{315 a^2}+\frac {19}{105} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {c^3 \text {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 ) \text {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} \arctan (a x)}{105 a^4}-\frac {4 c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{105 a^2}+\frac {1}{35} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (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 ) \text {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 \arctan (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} \arctan (a x)}{315 a^4}+\frac {29 c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{315 a^2}+\frac {19}{105} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {11 c^{5/2} \text {arctanh}\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} \arctan (a x)}{105 a^4}-\frac {4 c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{105 a^2}+\frac {1}{35} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {\left (3 c^3\right ) \text {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 ) \text {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 ) \text {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 ) \text {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} \arctan (a x)}{315 a^4}+\frac {29 c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{315 a^2}+\frac {19}{105} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {11 c^{5/2} \text {arctanh}\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} \arctan (a x)}{105 a^4}-\frac {4 c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{105 a^2}+\frac {1}{35} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {103 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{1680 a^4}\right )+\frac {\left (5 c^3\right ) \text {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 \text {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 ) \text {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 ) \text {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 ) \text {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} \arctan (a x)}{315 a^4}+\frac {29 c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{315 a^2}+\frac {19}{105} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {5519 c^{5/2} \text {arctanh}\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} \arctan (a x)}{105 a^4}-\frac {4 c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{105 a^2}+\frac {1}{35} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {103 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{1680 a^4}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.45 \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x) \, dx=\frac {c^2 \left (-a x \sqrt {c+a^2 c x^2} \left (-423+410 a^2 x^2+824 a^4 x^4+336 a^6 x^6\right )+384 \left (1+a^2 x^2\right )^3 \left (-2+7 a^2 x^2\right ) \sqrt {c+a^2 c x^2} \arctan (a x)+345 \sqrt {c} \log \left (a c x+\sqrt {c} \sqrt {c+a^2 c x^2}\right )\right )}{24192 a^4} \]

[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
)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 5.16 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.78

method	result	size
default	\(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (2688 \arctan \left (a x \right ) a^{8} x^{8}-336 a^{7} x^{7}+7296 a^{6} \arctan \left (a x \right ) x^{6}-824 a^{5} x^{5}+5760 \arctan \left (a x \right ) a^{4} x^{4}-410 a^{3} x^{3}+384 a^{2} \arctan \left (a x \right ) x^{2}+423 a x -768 \arctan \left (a x \right )\right )}{24192 a^{4}}+\frac {115 c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right )}{8064 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {115 c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right )}{8064 a^{4} \sqrt {a^{2} x^{2}+1}}\)	\(225\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.53 \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x) \, dx=\frac {345 \, c^{\frac {5}{2}} \log \left (-2 \, a^{2} c x^{2} - 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right ) - 2 \, {\left (336 \, a^{7} c^{2} x^{7} + 824 \, a^{5} c^{2} x^{5} + 410 \, a^{3} c^{2} x^{3} - 423 \, a c^{2} x - 384 \, {\left (7 \, a^{8} c^{2} x^{8} + 19 \, a^{6} c^{2} x^{6} + 15 \, a^{4} c^{2} x^{4} + a^{2} c^{2} x^{2} - 2 \, c^{2}\right )} \arctan \left (a x\right )\right )} \sqrt {a^{2} c x^{2} + c}}{48384 \, a^{4}} \]

[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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.17 \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x) \, dx=-\frac {1}{24192} \, {\left ({\left (7 \, {\left (\frac {48 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{5}}{a^{2}} - \frac {40 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}{a^{4}} + \frac {30 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{6}} - \frac {15 \, \sqrt {a^{2} x^{2} + 1} x}{a^{6}} - \frac {15 \, \operatorname {arsinh}\left (a x\right )}{a^{7}}\right )} a^{2} c^{2} + 96 \, {\left (\frac {8 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}{a^{2}} - \frac {6 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{4}} + \frac {3 \, \sqrt {a^{2} x^{2} + 1} x}{a^{4}} + \frac {3 \, \operatorname {arsinh}\left (a x\right )}{a^{5}}\right )} c^{2} + \frac {144 \, c^{2} {\left (\frac {2 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{2}} - \frac {\sqrt {a^{2} x^{2} + 1} x}{a^{2}} - \frac {\operatorname {arsinh}\left (a x\right )}{a^{3}}\right )}}{a^{2}} - \frac {384 \, {\left (\sqrt {a^{2} x^{2} + 1} x + \frac {\operatorname {arsinh}\left (a x\right )}{a}\right )} c^{2}}{a^{4}}\right )} a - 384 \, {\left (7 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2} c^{2} x^{6} + 12 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2} x^{4} + \frac {3 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2} x^{2}}{a^{2}} - \frac {2 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}{a^{4}}\right )} \arctan \left (a x\right )\right )} \sqrt {c} \]

[In]

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

[Out]

-1/24192*((7*(48*(a^2*x^2 + 1)^(3/2)*x^5/a^2 - 40*(a^2*x^2 + 1)^(3/2)*x^3/a^4 + 30*(a^2*x^2 + 1)^(3/2)*x/a^6 -
15*sqrt(a^2*x^2 + 1)*x/a^6 - 15*arcsinh(a*x)/a^7)*a^2*c^2 + 96*(8*(a^2*x^2 + 1)^(3/2)*x^3/a^2 - 6*(a^2*x^2 +
1)^(3/2)*x/a^4 + 3*sqrt(a^2*x^2 + 1)*x/a^4 + 3*arcsinh(a*x)/a^5)*c^2 + 144*c^2*(2*(a^2*x^2 + 1)^(3/2)*x/a^2 -
sqrt(a^2*x^2 + 1)*x/a^2 - arcsinh(a*x)/a^3)/a^2 - 384*(sqrt(a^2*x^2 + 1)*x + arcsinh(a*x)/a)*c^2/a^4)*a - 384*
(7*(a^2*x^2 + 1)^(3/2)*a^2*c^2*x^6 + 12*(a^2*x^2 + 1)^(3/2)*c^2*x^4 + 3*(a^2*x^2 + 1)^(3/2)*c^2*x^2/a^2 - 2*(a
^2*x^2 + 1)^(3/2)*c^2/a^4)*arctan(a*x))*sqrt(c)

Giac [F(-2)]

Exception generated. \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x) \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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