\(\int \frac {1}{x^3 (c+a^2 c x^2)^2 \arctan (a x)^{5/2}} \, dx\) [1065]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=-\frac {2}{3 a c^2 x^3 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {4}{a^2 c^2 x^4 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {20}{3 c^2 x^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {16 \text {Int}\left (\frac {1}{x^5 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}},x\right )}{a^2}+\frac {112}{3} \text {Int}\left (\frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}},x\right )+\frac {80}{3} a^2 \text {Int}\left (\frac {1}{x \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}},x\right ) \]

[Out]

-2/3/a/c^2/x^3/(a^2*x^2+1)/arctan(a*x)^(3/2)+4/a^2/c^2/x^4/(a^2*x^2+1)/arctan(a*x)^(1/2)+20/3/c^2/x^2/(a^2*x^2
+1)/arctan(a*x)^(1/2)+16*Unintegrable(1/x^5/(a^2*c*x^2+c)^2/arctan(a*x)^(1/2),x)/a^2+112/3*Unintegrable(1/x^3/
(a^2*c*x^2+c)^2/arctan(a*x)^(1/2),x)+80/3*a^2*Unintegrable(1/x/(a^2*c*x^2+c)^2/arctan(a*x)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx \]

[In]

Int[1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)),x]

[Out]

-2/(3*a*c^2*x^3*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)) + 4/(a^2*c^2*x^4*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + 20/(3*c^2
*x^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (16*Defer[Int][1/(x^5*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/a^2 +
(112*Defer[Int][1/(x^3*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/3 + (80*a^2*Defer[Int][1/(x*(c + a^2*c*x^2)^2
*Sqrt[ArcTan[a*x]]), x])/3

Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 a c^2 x^3 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {2 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx}{a}-\frac {1}{3} (10 a) \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx \\ & = -\frac {2}{3 a c^2 x^3 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {4}{a^2 c^2 x^4 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {20}{3 c^2 x^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {40}{3} \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx+24 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx+\frac {16 \int \frac {1}{x^5 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx}{a^2}+\frac {1}{3} \left (80 a^2\right ) \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 5.52 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx \]

[In]

Integrate[1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)),x]

[Out]

Integrate[1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x]

Maple [N/A] (verified)

Not integrable

Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

\[\int \frac {1}{x^{3} \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{\frac {5}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/x^3/(a^2*c*x^2+c)^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 [N/A]

Not integrable

Time = 22.68 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\frac {\int \frac {1}{a^{4} x^{7} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 2 a^{2} x^{5} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + x^{3} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{2}} \]

[In]

integrate(1/x**3/(a**2*c*x**2+c)**2/atan(a*x)**(5/2),x)

[Out]

Integral(1/(a**4*x**7*atan(a*x)**(5/2) + 2*a**2*x**5*atan(a*x)**(5/2) + x**3*atan(a*x)**(5/2)), x)/c**2

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(1/x^3/(a^2*c*x^2+c)^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 = 226.73 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{3} \arctan \left (a x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/x^3/(a^2*c*x^2+c)^2/arctan(a*x)^(5/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x^3\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

[In]

int(1/(x^3*atan(a*x)^(5/2)*(c + a^2*c*x^2)^2),x)

[Out]

int(1/(x^3*atan(a*x)^(5/2)*(c + a^2*c*x^2)^2), x)