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

Optimal. Leaf size=136 \[ \frac{3}{8} a^2 \text{Unintegrable}\left (\frac{1}{x \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right )-\frac{1}{2} a^2 \text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{a^2 c x^2+c}},x\right )-\frac{3 a \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{4 c x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{2 c x^2} \]

[Out]

(-3*a*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(4*c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(2*c*x^2) + (3*
a^2*Unintegrable[1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/8 - (a^2*Unintegrable[ArcTan[a*x]^(3/2)/(x*S
qrt[c + a^2*c*x^2]), x])/2

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Rubi [A]  time = 0.422847, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\tan ^{-1}(a x)^{3/2}}{x^3 \sqrt{c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-3*a*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(4*c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(2*c*x^2) + (3*
a^2*Defer[Int][1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/8 - (a^2*Defer[Int][ArcTan[a*x]^(3/2)/(x*Sqrt[
c + a^2*c*x^2]), x])/2

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)^{3/2}}{x^3 \sqrt{c+a^2 c x^2}} \, dx &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{2 c x^2}+\frac{1}{4} (3 a) \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \sqrt{c+a^2 c x^2}} \, dx-\frac{1}{2} a^2 \int \frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{3 a \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}{4 c x}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{2 c x^2}+\frac{1}{8} \left (3 a^2\right ) \int \frac{1}{x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx-\frac{1}{2} a^2 \int \frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{c+a^2 c x^2}} \, dx\\ \end{align*}

Mathematica [A]  time = 3.95466, size = 0, normalized size = 0. \[ \int \frac{\tan ^{-1}(a x)^{3/2}}{x^3 \sqrt{c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.401, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )^{\frac{3}{2}}}{\sqrt{a^{2} c x^{2} + c} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arctan(a*x)^(3/2)/(sqrt(a^2*c*x^2 + c)*x^3), x)