\(\int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx\) [899]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx=-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{c x}+\frac {5}{2} a \text {Int}\left (\frac {\arctan (a x)^{3/2}}{x \sqrt {c+a^2 c x^2}},x\right ) \]

[Out]

-arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2)/c/x+5/2*a*Unintegrable(arctan(a*x)^(3/2)/x/(a^2*c*x^2+c)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.13 (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 = {} \[ \int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx \]

[In]

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

[Out]

-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(c*x)) + (5*a*Defer[Int][ArcTan[a*x]^(3/2)/(x*Sqrt[c + a^2*c*x^2]),
x])/2

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{c x}+\frac {1}{2} (5 a) \int \frac {\arctan (a x)^{3/2}}{x \sqrt {c+a^2 c x^2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.56 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 5.74 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(arctan(a*x)^(5/2)/x^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 = 171.52 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{x^{2} \sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(arctan(a*x)^(5/2)/x^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 = 223.40 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\arctan \left (a x\right )^{\frac {5}{2}}}{\sqrt {a^{2} c x^{2} + c} x^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{5/2}}{x^2\,\sqrt {c\,a^2\,x^2+c}} \,d x \]

[In]

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

[Out]

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