3.692 \(\int (c+a^2 c x^2)^2 \sqrt {\tan ^{-1}(a x)} \, dx\)

Optimal. Leaf size=24 \[ \text {Int}\left (\left (a^2 c x^2+c\right )^2 \sqrt {\tan ^{-1}(a x)},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)} \, dx &=\int \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.85, size = 0, normalized size = 0.00 \[ \int \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 2.69, size = 0, normalized size = 0.00 \[ \int \left (a^{2} c \,x^{2}+c \right )^{2} \sqrt {\arctan \left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{2} \left (\int 2 a^{2} x^{2} \sqrt {\operatorname {atan}{\left (a x \right )}}\, dx + \int a^{4} x^{4} \sqrt {\operatorname {atan}{\left (a x \right )}}\, dx + \int \sqrt {\operatorname {atan}{\left (a x \right )}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________