\(\int \frac {x (c+a^2 c x^2)^{5/2}}{\arctan (a x)^2} \, dx\) [573]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {x \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^2} \, dx=\text {Int}\left (\frac {x \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (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 = {} \[ \int \frac {x \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^2} \, dx=\int \frac {x \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^2} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^2} \, dx=\int \frac {x \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 62.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

\[\int \frac {x \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{\arctan \left (a x \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18 \[ \int \frac {x \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}{\arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

integral((a^4*c^2*x^5 + 2*a^2*c^2*x^3 + c^2*x)*sqrt(a^2*c*x^2 + c)/arctan(a*x)^2, x)

Sympy [N/A]

Not integrable

Time = 29.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^2} \, dx=\int \frac {x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}{\arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 89.07 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}{\arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^2} \, dx=\int \frac {x\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{{\mathrm {atan}\left (a\,x\right )}^2} \,d x \]

[In]

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

[Out]

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