\(\int \frac {x (c+a^2 c x^2)^{3/2}}{\arctan (a x)^3} \, dx\) [647]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (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 = {} \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^3} \, dx=\int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 6.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^3} \, dx=\int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 50.58 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 6.78 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^3} \, dx=\int \frac {x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 97.47 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^3} \, dx=\int \frac {x\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{{\mathrm {atan}\left (a\,x\right )}^3} \,d x \]

[In]

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

[Out]

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