\(\int \frac {(c+a^2 c x^2)^{5/2}}{x \arctan (a x)} \, dx\) [504]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{x \arctan (a x)} \, dx=\text {Int}\left (\frac {\left (c+a^2 c x^2\right )^{5/2}}{x \arctan (a x)},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{x \arctan (a x)} \, dx=\int \frac {\left (c+a^2 c x^2\right )^{5/2}}{x \arctan (a x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.60 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{x \arctan (a x)} \, dx=\int \frac {\left (c+a^2 c x^2\right )^{5/2}}{x \arctan (a x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 98.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{x \arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{x \arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 15.95 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{x \arctan (a x)} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}{x \operatorname {atan}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{x \arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{x \arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 56.62 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{x \arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{x \arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{x \arctan (a x)} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x\,\mathrm {atan}\left (a\,x\right )} \,d x \]

[In]

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

[Out]

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