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

Optimal. Leaf size=65 \[ \frac {3}{2} a \text {Int}\left (\frac {\sqrt {\tan ^{-1}(a x)}}{x \sqrt {a^2 c x^2+c}},x\right )-\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{c x} \]

[Out]

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

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Rubi [A]  time = 0.21, 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 = {} \[ \int \frac {\tan ^{-1}(a x)^{3/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.35, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{-1}(a x)^{3/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[ArcTan[a*x]^(3/2)/(x^2*Sqrt[c + a^2*c*x^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(arctan(a*x)^(3/2)/x^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 [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(arctan(a*x)^(3/2)/x^2/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 1.96, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (a x \right )^{\frac {3}{2}}}{x^{2} \sqrt {a^{2} c \,x^{2}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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