3.10.49 \(\int \frac {x (c+a^2 c x^2)^{3/2}}{\sqrt {\text {ArcTan}(a x)}} \, dx\) [949]

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {x \left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\text {ArcTan}(a x)}},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.06, 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 = {} \begin {gather*} \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\text {ArcTan}(a x)}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\tan ^{-1}(a x)}} \, dx &=\int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\tan ^{-1}(a x)}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 2.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\text {ArcTan}(a x)}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 1.38, size = 0, normalized size = 0.00 \[\int \frac {x \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\sqrt {\arctan \left (a x \right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a^2*c*x^2+c)^(3/2)/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.

________________________________________________________________________________________

Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}{\sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{\sqrt {\mathrm {atan}\left (a\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________