\(\int \frac {x^3}{(c+a^2 c x^2)^2 \sqrt {\arctan (a x)}} \, dx\) [931]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx=\text {Int}\left (\frac {x^3}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (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 = {} \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx=\int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.13 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx=\int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 5.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x^3/(a^2*c*x^2+c)^2/arctan(a*x)^(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 = 1.80 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx=\frac {\int \frac {x^{3}}{a^{4} x^{4} \sqrt {\operatorname {atan}{\left (a x \right )}} + 2 a^{2} x^{2} \sqrt {\operatorname {atan}{\left (a x \right )}} + \sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx}{c^{2}} \]

[In]

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

[Out]

Integral(x**3/(a**4*x**4*sqrt(atan(a*x)) + 2*a**2*x**2*sqrt(atan(a*x)) + sqrt(atan(a*x))), x)/c**2

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 134.14 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx=\int { \frac {x^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} \sqrt {\arctan \left (a x\right )}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx=\int \frac {x^3}{\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

[In]

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

[Out]

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