3.801 \(\int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{x} \, dx\)

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{x},x\right ) \]

[Out]

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

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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 = {} \[ \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 3.36, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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

[Out]

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

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

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maple [A]  time = 1.72, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (a x \right )^{\frac {3}{2}} \sqrt {a^{2} c \,x^{2}+c}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/x,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(arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/x,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 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^{3/2}\,\sqrt {c\,a^2\,x^2+c}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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