\(\int \frac {\arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx\) [897]

Optimal result

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

[Out]

Unintegrable(arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),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 = {} \[ \int \frac {\arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {\arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {\arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 5.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83

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

[In]

int(arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x)

[Out]

int(arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x)

Fricas [F(-2)]

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

[In]

integrate(arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 120.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]

[In]

integrate(atan(a*x)**(5/2)/(a**2*c*x**2+c)**(1/2),x)

[Out]

Integral(atan(a*x)**(5/2)/sqrt(c*(a**2*x**2 + 1)), x)

Maxima [F(-2)]

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

[In]

integrate(arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 213.37 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int \frac {\arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\arctan \left (a x\right )^{\frac {5}{2}}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]

[In]

integrate(arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{5/2}}{\sqrt {c\,a^2\,x^2+c}} \,d x \]

[In]

int(atan(a*x)^(5/2)/(c + a^2*c*x^2)^(1/2),x)

[Out]

int(atan(a*x)^(5/2)/(c + a^2*c*x^2)^(1/2), x)