3.1100 \(\int \frac{1}{x^3 (c+a^2 c x^2)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=199 \[ 16 a^2 \text{Unintegrable}\left (\frac{1}{x \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right )+\frac{92}{3} \text{Unintegrable}\left (\frac{1}{x^3 \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right )+\frac{16 \text{Unintegrable}\left (\frac{1}{x^5 \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right )}{a^2}+\frac{16}{3 c x^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}+\frac{4}{a^2 c x^4 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}} \]

[Out]

-2/(3*a*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) + 4/(a^2*c*x^4*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + 1
6/(3*c*x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (16*Unintegrable[1/(x^5*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[
a*x]]), x])/a^2 + (92*Unintegrable[1/(x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/3 + 16*a^2*Unintegrabl
e[1/(x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]

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Rubi [A]  time = 0.84894, 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 = {} \[ \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]

[Out]

-2/(3*a*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) + 4/(a^2*c*x^4*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + 1
6/(3*c*x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (16*Defer[Int][1/(x^5*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*
x]]), x])/a^2 + (92*Defer[Int][1/(x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/3 + 16*a^2*Defer[Int][1/(x
*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]

Rubi steps

\begin{align*} \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac{2}{3 a c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac{2 \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx}{a}-\frac{1}{3} (8 a) \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2}{3 a c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}+\frac{4}{a^2 c x^4 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+\frac{16}{3 c x^2 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+\frac{32}{3} \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+20 \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{16 \int \frac{1}{x^5 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a^2}+\left (16 a^2\right ) \int \frac{1}{x \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx\\ \end{align*}

Mathematica [A]  time = 15.0857, size = 0, normalized size = 0. \[ \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]

[Out]

Integrate[1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x]

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Maple [A]  time = 0.983, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}} \left ( \arctan \left ( ax \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x^{3} \arctan \left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x, algorithm="giac")

[Out]

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