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

Optimal. Leaf size=117 \[ \frac{15}{8} c \text{Unintegrable}\left (\frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{a^2 c x^2+c}},x\right )+\frac{1}{2} c \text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{5/2}}{\sqrt{a^2 c x^2+c}},x\right )+\frac{1}{2} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}-\frac{5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{4 a} \]

[Out]

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

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Rubi [A]  time = 0.103224, 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 = {} \[ \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.293931, size = 0, normalized size = 0. \[ \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.869, size = 0, normalized size = 0. \begin{align*} \int \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(arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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)*arctan(a*x)^(5/2), x)