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

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {x^m \sqrt {a^2 c x^2+c}}{\tan ^{-1}(a x)},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.10, 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 = {} \[ \int \frac {x^m \sqrt {c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {x^m \sqrt {c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} x^{m}}{\arctan \left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 1.06, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \sqrt {a^{2} c \,x^{2}+c}}{\arctan \left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________