\(\int \frac {x^m (c+a^2 c x^2)}{\sqrt {\arctan (a x)}} \, dx\) [914]

Optimal result

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

[Out]

Unintegrable(x^m*(a^2*c*x^2+c)/arctan(a*x)^(1/2),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 = {} \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\sqrt {\arctan (a x)}} \, dx=\int \frac {x^m \left (c+a^2 c x^2\right )}{\sqrt {\arctan (a x)}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\sqrt {\arctan (a x)}} \, dx=\int \frac {x^m \left (c+a^2 c x^2\right )}{\sqrt {\arctan (a x)}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\sqrt {\arctan (a x)}} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} x^{m}}{\sqrt {\arctan \left (a x\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 13.75 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\sqrt {\arctan (a x)}} \, dx=c \left (\int \frac {x^{m}}{\sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx + \int \frac {a^{2} x^{2} x^{m}}{\sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx\right ) \]

[In]

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

[Out]

c*(Integral(x**m/sqrt(atan(a*x)), x) + Integral(a**2*x**2*x**m/sqrt(atan(a*x)), x))

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 64.26 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\sqrt {\arctan (a x)}} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} x^{m}}{\sqrt {\arctan \left (a x\right )}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\sqrt {\arctan (a x)}} \, dx=\int \frac {x^m\,\left (c\,a^2\,x^2+c\right )}{\sqrt {\mathrm {atan}\left (a\,x\right )}} \,d x \]

[In]

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

[Out]

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