\(\int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}} \, dx\) [1091]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}} \, dx=-\frac {2 \sqrt {c+a^2 c x^2}}{3 a c x \arctan (a x)^{3/2}}-\frac {2 \text {Int}\left (\frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}},x\right )}{3 a} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.14 (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 = {} \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}} \, dx \]

[In]

Int[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)),x]

[Out]

(-2*Sqrt[c + a^2*c*x^2])/(3*a*c*x*ArcTan[a*x]^(3/2)) - (2*Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3
/2)), x])/(3*a)

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

Mathematica [N/A]

Not integrable

Time = 4.72 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}} \, dx \]

[In]

Integrate[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)),x]

[Out]

Integrate[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x]

Maple [N/A] (verified)

Not integrable

Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

\[\int \frac {1}{x \arctan \left (a x \right )^{\frac {5}{2}} \sqrt {a^{2} c \,x^{2}+c}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/x/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 = 126.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(1/x/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 = 299.05 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}} \, dx=\int { \frac {1}{\sqrt {a^{2} c x^{2} + c} x \arctan \left (a x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/x/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 = 0.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,\sqrt {c\,a^2\,x^2+c}} \,d x \]

[In]

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

[Out]

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