\(\int \frac {(c+a^2 c x^2)^{5/2}}{\arctan (a x)^{5/2}} \, dx\) [1086]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^{5/2}} \, dx=\text {Int}\left (\frac {\left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^{5/2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^{5/2}} \, dx=\int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^{5/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^{5/2}} \, dx=\int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^{5/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.33 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate((a^2*c*x^2+c)^(5/2)/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 = 255.05 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^{5/2}} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{\arctan \left (a x\right )^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^{5/2}} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^{5/2}}{{\mathrm {atan}\left (a\,x\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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