\(\int x (c+a^2 c x^2)^3 \sqrt {\arctan (a x)} \, dx\) [695]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)} \, dx=\frac {c^3 \left (1+a^2 x^2\right )^4 \sqrt {\arctan (a x)}}{8 a^2}-\frac {\text {Int}\left (\frac {\left (c+a^2 c x^2\right )^3}{\sqrt {\arctan (a x)}},x\right )}{16 a} \]

[Out]

1/8*c^3*(a^2*x^2+1)^4*arctan(a*x)^(1/2)/a^2-1/16*Unintegrable((a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x)/a

Rubi [N/A]

Not integrable

Time = 0.04 (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 = {} \[ \int x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)} \, dx=\int x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)} \, dx \]

[In]

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

[Out]

(c^3*(1 + a^2*x^2)^4*Sqrt[ArcTan[a*x]])/(8*a^2) - Defer[Int][(c + a^2*c*x^2)^3/Sqrt[ArcTan[a*x]], x]/(16*a)

Rubi steps \begin{align*} \text {integral}& = \frac {c^3 \left (1+a^2 x^2\right )^4 \sqrt {\arctan (a x)}}{8 a^2}-\frac {\int \frac {\left (c+a^2 c x^2\right )^3}{\sqrt {\arctan (a x)}} \, dx}{16 a} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)} \, dx=\int x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 4.62 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x*(a^2*c*x^2+c)^3*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 = 6.43 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.09 \[ \int x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)} \, dx=c^{3} \left (\int x \sqrt {\operatorname {atan}{\left (a x \right )}}\, dx + \int 3 a^{2} x^{3} \sqrt {\operatorname {atan}{\left (a x \right )}}\, dx + \int 3 a^{4} x^{5} \sqrt {\operatorname {atan}{\left (a x \right )}}\, dx + \int a^{6} x^{7} \sqrt {\operatorname {atan}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(x*(a^2*c*x^2+c)^3*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 = 56.69 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x \sqrt {\arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)} \, dx=\int x\,\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \]

[In]

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

[Out]

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