3.900 \(\int \frac{\tan ^{-1}(a x)^{5/2}}{x^3 \sqrt{c+a^2 c x^2}} \, dx\)

Optimal. Leaf size=136 \[ \frac{15}{8} a^2 \text{Unintegrable}\left (\frac{\sqrt{\tan ^{-1}(a x)}}{x \sqrt{a^2 c x^2+c}},x\right )-\frac{1}{2} a^2 \text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{5/2}}{x \sqrt{a^2 c x^2+c}},x\right )-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{2 c x^2}-\frac{5 a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{4 c x} \]

[Out]

(-5*a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(4*c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(2*c*x^2) + (15
*a^2*Unintegrable[Sqrt[ArcTan[a*x]]/(x*Sqrt[c + a^2*c*x^2]), x])/8 - (a^2*Unintegrable[ArcTan[a*x]^(5/2)/(x*Sq
rt[c + a^2*c*x^2]), x])/2

________________________________________________________________________________________

Rubi [A]  time = 0.423259, 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 = {} \[ \int \frac{\tan ^{-1}(a x)^{5/2}}{x^3 \sqrt{c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[ArcTan[a*x]^(5/2)/(x^3*Sqrt[c + a^2*c*x^2]),x]

[Out]

(-5*a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(4*c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(2*c*x^2) + (15
*a^2*Defer[Int][Sqrt[ArcTan[a*x]]/(x*Sqrt[c + a^2*c*x^2]), x])/8 - (a^2*Defer[Int][ArcTan[a*x]^(5/2)/(x*Sqrt[c
+ a^2*c*x^2]), x])/2

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)^{5/2}}{x^3 \sqrt{c+a^2 c x^2}} \, dx &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{2 c x^2}+\frac{1}{4} (5 a) \int \frac{\tan ^{-1}(a x)^{3/2}}{x^2 \sqrt{c+a^2 c x^2}} \, dx-\frac{1}{2} a^2 \int \frac{\tan ^{-1}(a x)^{5/2}}{x \sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{5 a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{4 c x}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{2 c x^2}-\frac{1}{2} a^2 \int \frac{\tan ^{-1}(a x)^{5/2}}{x \sqrt{c+a^2 c x^2}} \, dx+\frac{1}{8} \left (15 a^2\right ) \int \frac{\sqrt{\tan ^{-1}(a x)}}{x \sqrt{c+a^2 c x^2}} \, dx\\ \end{align*}

Mathematica [A]  time = 4.51711, size = 0, normalized size = 0. \[ \int \frac{\tan ^{-1}(a x)^{5/2}}{x^3 \sqrt{c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[ArcTan[a*x]^(5/2)/(x^3*Sqrt[c + a^2*c*x^2]),x]

[Out]

Integrate[ArcTan[a*x]^(5/2)/(x^3*Sqrt[c + a^2*c*x^2]), x]

________________________________________________________________________________________

Maple [A]  time = 1.404, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \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^3/(a^2*c*x^2+c)^(1/2),x)

[Out]

int(arctan(a*x)^(5/2)/x^3/(a^2*c*x^2+c)^(1/2),x)

________________________________________________________________________________________

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^(5/2)/x^3/(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^3), x)