\(\int \frac {x (c+a^2 c x^2)^{3/2}}{\arctan (a x)^{3/2}} \, dx\) [1012]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^{3/2}} \, dx=\text {Int}\left (\frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^{3/2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (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 = {} \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^{3/2}} \, dx=\int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^{3/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 6.83 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^{3/2}} \, dx=\int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 2.79 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 62.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^{3/2}} \, dx=\int \frac {x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}{\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F(-1)]

Timed out. \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)^{3/2}} \, dx=\int \frac {x\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{{\mathrm {atan}\left (a\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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