\(\int \frac {x^m (c+a^2 c x^2)^{3/2}}{\sqrt {\arctan (a x)}} \, dx\) [948]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\arctan (a x)}} \, dx=\text {Int}\left (\frac {x^m \left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\arctan (a x)}},x\right ) \]

[Out]

Unintegrable(x^m*(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 26, 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 = {} \[ \int \frac {x^m \left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\arctan (a x)}} \, dx=\int \frac {x^m \left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\arctan (a x)}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\arctan (a x)}} \, dx=\int \frac {x^m \left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\arctan (a x)}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 6.79 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

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

[In]

int(x^m*(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2),x)

[Out]

int(x^m*(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2),x)

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\arctan (a x)}} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{m}}{\sqrt {\arctan \left (a x\right )}} \,d x } \]

[In]

integrate(x^m*(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(x^m*(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

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

[In]

integrate(x^m*(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\arctan (a x)}} \, dx=\int \frac {x^m\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{\sqrt {\mathrm {atan}\left (a\,x\right )}} \,d x \]

[In]

int((x^m*(c + a^2*c*x^2)^(3/2))/atan(a*x)^(1/2),x)

[Out]

int((x^m*(c + a^2*c*x^2)^(3/2))/atan(a*x)^(1/2), x)