3.574 \(\int \frac {(c+a^2 c x^2)^{5/2}}{\tan ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=24 \[ \text {Int}\left (\frac {\left (a^2 c x^2+c\right )^{5/2}}{\tan ^{-1}(a x)^2},x\right ) \]

[Out]

Unintegrable((a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x)

________________________________________________________________________________________

Rubi [A]  time = 0.04, 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 = {} \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^2,x]

[Out]

Defer[Int][(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^2, x]

Rubi steps

\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\tan ^{-1}(a x)^2} \, dx &=\int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\tan ^{-1}(a x)^2} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.88, size = 0, normalized size = 0.00 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^2,x]

[Out]

Integrate[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^2, x]

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {a^{2} c x^{2} + c}}{\arctan \left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*sqrt(a^2*c*x^2 + c)/arctan(a*x)^2, x)

________________________________________________________________________________________

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)^(5/2)/arctan(a*x)^2,x, algorithm="giac")

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 1.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{\arctan \left (a x \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x)

[Out]

int((a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{\arctan \left (a x\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x, algorithm="maxima")

[Out]

integrate((a^2*c*x^2 + c)^(5/2)/arctan(a*x)^2, x)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (c\,a^2\,x^2+c\right )}^{5/2}}{{\mathrm {atan}\left (a\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + a^2*c*x^2)^(5/2)/atan(a*x)^2,x)

[Out]

int((c + a^2*c*x^2)^(5/2)/atan(a*x)^2, x)

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*c*x**2+c)**(5/2)/atan(a*x)**2,x)

[Out]

Integral((c*(a**2*x**2 + 1))**(5/2)/atan(a*x)**2, x)

________________________________________________________________________________________