\(\int x^3 (c+a^2 c x^2)^{3/2} \arctan (a x) \, dx\) [208]

Optimal result

Integrand size = 22, antiderivative size = 217 \[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\frac {3 c x \sqrt {c+a^2 c x^2}}{112 a^3}-\frac {23 c x^3 \sqrt {c+a^2 c x^2}}{840 a}-\frac {1}{42} a c x^5 \sqrt {c+a^2 c x^2}-\frac {2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{35 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{35 a^2}+\frac {8}{35} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {17 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{560 a^4} \]

[Out]

17/560*c^(3/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))/a^4+3/112*c*x*(a^2*c*x^2+c)^(1/2)/a^3-23/840*c*x^3*(a^
2*c*x^2+c)^(1/2)/a-1/42*a*c*x^5*(a^2*c*x^2+c)^(1/2)-2/35*c*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^4+1/35*c*x^2*arct
an(a*x)*(a^2*c*x^2+c)^(1/2)/a^2+8/35*c*x^4*arctan(a*x)*(a^2*c*x^2+c)^(1/2)+1/7*a^2*c*x^6*arctan(a*x)*(a^2*c*x^
2+c)^(1/2)

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5070, 5066, 5072, 327, 223, 212, 5050} \[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\frac {c x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{35 a^2}+\frac {1}{7} a^2 c x^6 \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {8}{35} c x^4 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{42} a c x^5 \sqrt {a^2 c x^2+c}-\frac {23 c x^3 \sqrt {a^2 c x^2+c}}{840 a}-\frac {2 c \arctan (a x) \sqrt {a^2 c x^2+c}}{35 a^4}+\frac {17 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{560 a^4}+\frac {3 c x \sqrt {a^2 c x^2+c}}{112 a^3} \]

[In]

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

[Out]

(3*c*x*Sqrt[c + a^2*c*x^2])/(112*a^3) - (23*c*x^3*Sqrt[c + a^2*c*x^2])/(840*a) - (a*c*x^5*Sqrt[c + a^2*c*x^2])
/42 - (2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(35*a^4) + (c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(35*a^2) + (8*c
*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/35 + (a^2*c*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/7 + (17*c^(3/2)*ArcTanh
[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(560*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 \sqrt {c+a^2 c x^2} \arctan (a x) \, dx+\left (a^2 c\right ) \int x^5 \sqrt {c+a^2 c x^2} \arctan (a x) \, dx \\ & = \frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{5} c^2 \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{5} \left (a c^2\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{7} \left (a^2 c^2\right ) \int \frac {x^5 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{7} \left (a^3 c^2\right ) \int \frac {x^6}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {c x^3 \sqrt {c+a^2 c x^2}}{20 a}-\frac {1}{42} a c x^5 \sqrt {c+a^2 c x^2}+\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{15 a^2}+\frac {8}{35} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{35} \left (4 c^2\right ) \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (2 c^2\right ) \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}-\frac {c^2 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx}{15 a}+\frac {\left (3 c^2\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx}{20 a}-\frac {1}{35} \left (a c^2\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{42} \left (5 a c^2\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \frac {c x \sqrt {c+a^2 c x^2}}{24 a^3}-\frac {23 c x^3 \sqrt {c+a^2 c x^2}}{840 a}-\frac {1}{42} a c x^5 \sqrt {c+a^2 c x^2}-\frac {2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{15 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{35 a^2}+\frac {8}{35} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {c^2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{30 a^3}-\frac {\left (3 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{40 a^3}+\frac {\left (2 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^3}+\frac {\left (8 c^2\right ) \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{105 a^2}+\frac {\left (3 c^2\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx}{140 a}+\frac {\left (4 c^2\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx}{105 a}-\frac {\left (5 c^2\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx}{56 a} \\ & = \frac {3 c x \sqrt {c+a^2 c x^2}}{112 a^3}-\frac {23 c x^3 \sqrt {c+a^2 c x^2}}{840 a}-\frac {1}{42} a c x^5 \sqrt {c+a^2 c x^2}-\frac {2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{35 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{35 a^2}+\frac {8}{35} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {\left (3 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{280 a^3}-\frac {\left (2 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{105 a^3}+\frac {c^2 \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 (5 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{112 a^3}-\frac {\left (3 c^2\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}-\frac {\left (8 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{105 a^3}+\frac {\left (2 c^2\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 {3 c x \sqrt {c+a^2 c x^2}}{112 a^3}-\frac {23 c x^3 \sqrt {c+a^2 c x^2}}{840 a}-\frac {1}{42} a c x^5 \sqrt {c+a^2 c x^2}-\frac {2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{35 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{35 a^2}+\frac {8}{35} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {11 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{120 a^4}-\frac {\left (3 c^2\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^2\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^2\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^2\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 {3 c x \sqrt {c+a^2 c x^2}}{112 a^3}-\frac {23 c x^3 \sqrt {c+a^2 c x^2}}{840 a}-\frac {1}{42} a c x^5 \sqrt {c+a^2 c x^2}-\frac {2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{35 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{35 a^2}+\frac {8}{35} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {17 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{560 a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.55 \[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\frac {a c x \sqrt {c+a^2 c x^2} \left (45-46 a^2 x^2-40 a^4 x^4\right )+48 c \left (1+a^2 x^2\right )^2 \left (-2+5 a^2 x^2\right ) \sqrt {c+a^2 c x^2} \arctan (a x)+51 c^{3/2} \log \left (a c x+\sqrt {c} \sqrt {c+a^2 c x^2}\right )}{1680 a^4} \]

[In]

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

[Out]

(a*c*x*Sqrt[c + a^2*c*x^2]*(45 - 46*a^2*x^2 - 40*a^4*x^4) + 48*c*(1 + a^2*x^2)^2*(-2 + 5*a^2*x^2)*Sqrt[c + a^2
*c*x^2]*ArcTan[a*x] + 51*c^(3/2)*Log[a*c*x + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/(1680*a^4)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (240 a^{6} \arctan \left (a x \right ) x^{6}-40 a^{5} x^{5}+384 \arctan \left (a x \right ) a^{4} x^{4}-46 a^{3} x^{3}+48 a^{2} \arctan \left (a x \right ) x^{2}+45 a x -96 \arctan \left (a x \right )\right )}{1680 a^{4}}+\frac {17 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right )}{560 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {17 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right )}{560 a^{4} \sqrt {a^{2} x^{2}+1}}\)	\(199\)

[In]

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

[Out]

1/1680*c/a^4*(c*(a*x-I)*(I+a*x))^(1/2)*(240*a^6*arctan(a*x)*x^6-40*a^5*x^5+384*arctan(a*x)*a^4*x^4-46*a^3*x^3+
48*a^2*arctan(a*x)*x^2+45*a*x-96*arctan(a*x))+17/560*c/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)-17/560*c/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 = 0.26 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.54 \[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\frac {51 \, c^{\frac {3}{2}} \log \left (-2 \, a^{2} c x^{2} - 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right ) - 2 \, {\left (40 \, a^{5} c x^{5} + 46 \, a^{3} c x^{3} - 45 \, a c x - 48 \, {\left (5 \, a^{6} c x^{6} + 8 \, a^{4} c x^{4} + a^{2} c x^{2} - 2 \, c\right )} \arctan \left (a x\right )\right )} \sqrt {a^{2} c x^{2} + c}}{3360 \, a^{4}} \]

[In]

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

[Out]

1/3360*(51*c^(3/2)*log(-2*a^2*c*x^2 - 2*sqrt(a^2*c*x^2 + c)*a*sqrt(c)*x - c) - 2*(40*a^5*c*x^5 + 46*a^3*c*x^3
- 45*a*c*x - 48*(5*a^6*c*x^6 + 8*a^4*c*x^4 + a^2*c*x^2 - 2*c)*arctan(a*x))*sqrt(a^2*c*x^2 + c))/a^4

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.99 \[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=-\frac {1}{1680} \, {\left ({\left (5 \, {\left (\frac {8 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}{a^{2}} - \frac {6 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{4}} + \frac {3 \, \sqrt {a^{2} x^{2} + 1} x}{a^{4}} + \frac {3 \, \operatorname {arsinh}\left (a x\right )}{a^{5}}\right )} c + \frac {18 \, c {\left (\frac {2 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{2}} - \frac {\sqrt {a^{2} x^{2} + 1} x}{a^{2}} - \frac {\operatorname {arsinh}\left (a x\right )}{a^{3}}\right )}}{a^{2}} - \frac {48 \, {\left (\sqrt {a^{2} x^{2} + 1} x + \frac {\operatorname {arsinh}\left (a x\right )}{a}\right )} c}{a^{4}}\right )} a - 48 \, {\left (5 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c x^{4} + \frac {3 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c x^{2}}{a^{2}} - \frac {2 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{a^{4}}\right )} \arctan \left (a x\right )\right )} \sqrt {c} \]

[In]

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

[Out]

-1/1680*((5*(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*arcsi
nh(a*x)/a^5)*c + 18*c*(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 - 48*(sqr
t(a^2*x^2 + 1)*x + arcsinh(a*x)/a)*c/a^4)*a - 48*(5*(a^2*x^2 + 1)^(3/2)*c*x^4 + 3*(a^2*x^2 + 1)^(3/2)*c*x^2/a^
2 - 2*(a^2*x^2 + 1)^(3/2)*c/a^4)*arctan(a*x))*sqrt(c)

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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