\(\int x^m (c+a^2 c x^2)^2 \arctan (a x)^3 \, dx\) [458]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\text {Int}\left (x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^3,x\right ) \]

[Out]

Unintegrable(x^m*(a^2*c*x^2+c)^2*arctan(a*x)^3,x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.52 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 3.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int x^{m} \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{3}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{m} \arctan \left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*x^m*arctan(a*x)^3, x)

Sympy [N/A]

Not integrable

Time = 28.47 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.41 \[ \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=c^{2} \left (\int x^{m} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 2 a^{2} x^{2} x^{m} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{4} x^{4} x^{m} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

c**2*(Integral(x**m*atan(a*x)**3, x) + Integral(2*a**2*x**2*x**m*atan(a*x)**3, x) + Integral(a**4*x**4*x**m*at
an(a*x)**3, x))

Maxima [N/A]

Not integrable

Time = 13.16 (sec) , antiderivative size = 952, normalized size of antiderivative = 43.27 \[ \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{m} \arctan \left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

1/32*(4*((a^4*c^2*m^2 + 4*a^4*c^2*m + 3*a^4*c^2)*x^5 + 2*(a^2*c^2*m^2 + 6*a^2*c^2*m + 5*a^2*c^2)*x^3 + (c^2*m^
2 + 8*c^2*m + 15*c^2)*x)*x^m*arctan(a*x)^3 - 3*((a^4*c^2*m^2 + 4*a^4*c^2*m + 3*a^4*c^2)*x^5 + 2*(a^2*c^2*m^2 +
6*a^2*c^2*m + 5*a^2*c^2)*x^3 + (c^2*m^2 + 8*c^2*m + 15*c^2)*x)*x^m*arctan(a*x)*log(a^2*x^2 + 1)^2 + 32*(m^3 +
9*m^2 + 23*m + 15)*integrate(1/32*(28*((a^6*c^2*m^3 + 9*a^6*c^2*m^2 + 23*a^6*c^2*m + 15*a^6*c^2)*x^6 + c^2*m^
3 + 3*(a^4*c^2*m^3 + 9*a^4*c^2*m^2 + 23*a^4*c^2*m + 15*a^4*c^2)*x^4 + 9*c^2*m^2 + 23*c^2*m + 3*(a^2*c^2*m^3 +
9*a^2*c^2*m^2 + 23*a^2*c^2*m + 15*a^2*c^2)*x^2 + 15*c^2)*x^m*arctan(a*x)^3 - 12*((a^5*c^2*m^2 + 4*a^5*c^2*m +
3*a^5*c^2)*x^5 + 2*(a^3*c^2*m^2 + 6*a^3*c^2*m + 5*a^3*c^2)*x^3 + (a*c^2*m^2 + 8*a*c^2*m + 15*a*c^2)*x)*x^m*arc
tan(a*x)^2 + 12*((a^6*c^2*m^2 + 4*a^6*c^2*m + 3*a^6*c^2)*x^6 + 2*(a^4*c^2*m^2 + 6*a^4*c^2*m + 5*a^4*c^2)*x^4 +
(a^2*c^2*m^2 + 8*a^2*c^2*m + 15*a^2*c^2)*x^2)*x^m*arctan(a*x)*log(a^2*x^2 + 1) + 3*(((a^6*c^2*m^3 + 9*a^6*c^2
*m^2 + 23*a^6*c^2*m + 15*a^6*c^2)*x^6 + c^2*m^3 + 3*(a^4*c^2*m^3 + 9*a^4*c^2*m^2 + 23*a^4*c^2*m + 15*a^4*c^2)*
x^4 + 9*c^2*m^2 + 23*c^2*m + 3*(a^2*c^2*m^3 + 9*a^2*c^2*m^2 + 23*a^2*c^2*m + 15*a^2*c^2)*x^2 + 15*c^2)*x^m*arc
tan(a*x) + ((a^5*c^2*m^2 + 4*a^5*c^2*m + 3*a^5*c^2)*x^5 + 2*(a^3*c^2*m^2 + 6*a^3*c^2*m + 5*a^3*c^2)*x^3 + (a*c
^2*m^2 + 8*a*c^2*m + 15*a*c^2)*x)*x^m)*log(a^2*x^2 + 1)^2)/(m^3 + (a^2*m^3 + 9*a^2*m^2 + 23*a^2*m + 15*a^2)*x^
2 + 9*m^2 + 23*m + 15), x))/(m^3 + 9*m^2 + 23*m + 15)

Giac [N/A]

Not integrable

Time = 149.21 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{m} \arctan \left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int x^m\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]

[In]

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

[Out]

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