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

Optimal. Leaf size=130 \[ \frac{15 \text{Unintegrable}\left (\frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{a^2 c x^2+c}},x\right )}{8 a^2}-\frac{\text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{5/2}}{\sqrt{a^2 c x^2+c}},x\right )}{2 a^2}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{2 a^2 c}-\frac{5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{4 a^3 c} \]

[Out]

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

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Rubi [A]  time = 0.253795, 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 = {} \[ \int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 2.46559, size = 0, normalized size = 0. \[ \int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 3.563, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \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(x^2*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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