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

Optimal. Leaf size=79 \[ \frac{5}{8} c \text{Unintegrable}\left (\sqrt{\tan ^{-1}(a x)},x\right )+\frac{2}{3} c \text{Unintegrable}\left (\tan ^{-1}(a x)^{5/2},x\right )+\frac{1}{3} c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{5/2}-\frac{5 c \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}{12 a} \]

[Out]

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

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Rubi [A]  time = 0.0249849, 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 = {} \[ \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.56711, size = 0, normalized size = 0. \[ \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.323, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}c{x}^{2}+c \right ) \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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