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

Optimal. Leaf size=66 \[ \frac {\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{3 a^2 c}-\frac {\text {Int}\left (\sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)},x\right )}{2 a} \]

[Out]

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

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 6.63, size = 0, normalized size = 0.00 \[ \int x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2), 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(x*arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2),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(x*arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(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 = 2.96, size = 0, normalized size = 0.00 \[ \int x \arctan \left (a x \right )^{\frac {3}{2}} \sqrt {a^{2} c \,x^{2}+c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(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*arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(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 = -0.02 \[ \int x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}\,\sqrt {c\,a^2\,x^2+c} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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