\(\int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{x} \, dx\) [882]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{x} \, dx=\text {Int}\left (\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (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 = {} \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{x} \, dx=\int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{x} \, dx=\int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 3.89 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 168.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{x} \, dx=\int \frac {\sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate(arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2)/x,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 = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{x} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{5/2}\,\sqrt {c\,a^2\,x^2+c}}{x} \,d x \]

[In]

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

[Out]

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