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

Optimal. Leaf size=158 \[ -\frac{5 c \text{Unintegrable}\left (\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right )}{16 a}-\frac{5 c \text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{a^2 c x^2+c}},x\right )}{12 a}+\frac{\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{5/2}}{3 a^2 c}-\frac{5 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{12 a}+\frac{5 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{8 a^2} \]

[Out]

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

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Rubi [A]  time = 0.178721, 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 = {} \[ \int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 4.82122, size = 0, normalized size = 0. \[ \int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.995, size = 0, normalized size = 0. \begin{align*} \int x \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{2}}}\sqrt{{a}^{2}c{x}^{2}+c}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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