3.901 \(\int \frac{\tan ^{-1}(a x)^{5/2}}{x^4 \sqrt{c+a^2 c x^2}} \, dx\)

Optimal. Leaf size=206 \[ \frac{5}{16} a^3 \text{Unintegrable}\left (\frac{1}{x \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right )-\frac{25}{12} a^3 \text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{a^2 c x^2+c}},x\right )+\frac{2 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{3 c x}-\frac{5 a^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{8 c x}-\frac{5 a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{12 c x^2}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{3 c x^3} \]

[Out]

(-5*a^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(8*c*x) - (5*a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(12*c*x^2
) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*c*x)
+ (5*a^3*Unintegrable[1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/16 - (25*a^3*Unintegrable[ArcTan[a*x]^(
3/2)/(x*Sqrt[c + a^2*c*x^2]), x])/12

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Rubi [A]  time = 0.743608, 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 = {} \[ \int \frac{\tan ^{-1}(a x)^{5/2}}{x^4 \sqrt{c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[ArcTan[a*x]^(5/2)/(x^4*Sqrt[c + a^2*c*x^2]),x]

[Out]

(-5*a^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(8*c*x) - (5*a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(12*c*x^2
) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*c*x)
+ (5*a^3*Defer[Int][1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/16 - (25*a^3*Defer[Int][ArcTan[a*x]^(3/2)
/(x*Sqrt[c + a^2*c*x^2]), x])/12

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)^{5/2}}{x^4 \sqrt{c+a^2 c x^2}} \, dx &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 c x^3}+\frac{1}{6} (5 a) \int \frac{\tan ^{-1}(a x)^{3/2}}{x^3 \sqrt{c+a^2 c x^2}} \, dx-\frac{1}{3} \left (2 a^2\right ) \int \frac{\tan ^{-1}(a x)^{5/2}}{x^2 \sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{5 a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{12 c x^2}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 c x^3}+\frac{2 a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 c x}+\frac{1}{8} \left (5 a^2\right ) \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \sqrt{c+a^2 c x^2}} \, dx-\frac{1}{12} \left (5 a^3\right ) \int \frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{c+a^2 c x^2}} \, dx-\frac{1}{3} \left (5 a^3\right ) \int \frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{5 a^2 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}{8 c x}-\frac{5 a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{12 c x^2}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 c x^3}+\frac{2 a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 c x}+\frac{1}{16} \left (5 a^3\right ) \int \frac{1}{x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx-\frac{1}{12} \left (5 a^3\right ) \int \frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{c+a^2 c x^2}} \, dx-\frac{1}{3} \left (5 a^3\right ) \int \frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{c+a^2 c x^2}} \, dx\\ \end{align*}

Mathematica [A]  time = 16.0537, size = 0, normalized size = 0. \[ \int \frac{\tan ^{-1}(a x)^{5/2}}{x^4 \sqrt{c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[ArcTan[a*x]^(5/2)/(x^4*Sqrt[c + a^2*c*x^2]),x]

[Out]

Integrate[ArcTan[a*x]^(5/2)/(x^4*Sqrt[c + a^2*c*x^2]), x]

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Maple [A]  time = 3.984, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{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)^(5/2)/x^4/(a^2*c*x^2+c)^(1/2),x)

[Out]

int(arctan(a*x)^(5/2)/x^4/(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)^(5/2)/x^4/(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)^(5/2)/x^4/(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)**(5/2)/x**4/(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{5}{2}}}{\sqrt{a^{2} c x^{2} + c} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^(5/2)/x^4/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate(arctan(a*x)^(5/2)/(sqrt(a^2*c*x^2 + c)*x^4), x)