\(\int \frac {x^m (c+a^2 c x^2)}{\arctan (a x)^{5/2}} \, dx\) [1042]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (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 = {} \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^{5/2}} \, dx=\int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^{5/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.98 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^{5/2}} \, dx=\int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^{5/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 4.52 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^{5/2}} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} x^{m}}{\arctan \left (a x\right )^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^{5/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 11.19 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^{5/2}} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} x^{m}}{\arctan \left (a x\right )^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.87 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^{5/2}} \, dx=\int \frac {x^m\,\left (c\,a^2\,x^2+c\right )}{{\mathrm {atan}\left (a\,x\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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