3.729 \(\int x^m (c+a^2 c x^2)^{5/2} \sqrt {\tan ^{-1}(a x)} \, dx\)

Optimal. Leaf size=29 \[ \text {Int}\left (x^m \left (a^2 c x^2+c\right )^{5/2} \sqrt {\tan ^{-1}(a x)},x\right ) \]

[Out]

Unintegrable(x^m*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(1/2),x)

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Rubi [A]  time = 0.11, 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 = {} \[ \int x^m \left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.48, size = 0, normalized size = 0.00 \[ \int x^m \left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {a^{2} c x^{2} + c} x^{m} \sqrt {\arctan \left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(1/2),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)*x^m*sqrt(arctan(a*x)), x)

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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)^(5/2)*arctan(a*x)^(1/2),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

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maple [A]  time = 3.42, size = 0, normalized size = 0.00 \[ \int x^{m} \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \sqrt {\arctan \left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(1/2),x)

[Out]

int(x^m*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(1/2),x)

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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(x^m*(a^2*c*x^2+c)^(5/2)*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.

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int x^m\,\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*atan(a*x)^(1/2)*(c + a^2*c*x^2)^(5/2),x)

[Out]

int(x^m*atan(a*x)^(1/2)*(c + a^2*c*x^2)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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