\(\int \frac {x (c+a^2 c x^2)^3}{\arctan (a x)^{3/2}} \, dx\) [981]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^{3/2}} \, dx=\text {Int}\left (\frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^{3/2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^{3/2}} \, dx=\int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^{3/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.98 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^{3/2}} \, dx=\int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.69 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 4.91 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.09 \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^{3/2}} \, dx=c^{3} \left (\int \frac {x}{\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx + \int \frac {3 a^{2} x^{3}}{\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx + \int \frac {3 a^{4} x^{5}}{\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx + \int \frac {a^{6} x^{7}}{\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

c**3*(Integral(x/atan(a*x)**(3/2), x) + Integral(3*a**2*x**3/atan(a*x)**(3/2), x) + Integral(3*a**4*x**5/atan(
a*x)**(3/2), x) + Integral(a**6*x**7/atan(a*x)**(3/2), x))

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^{3/2}} \, dx=\int \frac {x\,{\left (c\,a^2\,x^2+c\right )}^3}{{\mathrm {atan}\left (a\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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