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

Optimal. Leaf size=115 \[ -\frac{5 c \text{Unintegrable}\left (\frac{1}{\sqrt{\tan ^{-1}(a x)}},x\right )}{64 a}-\frac{5 c \text{Unintegrable}\left (\tan ^{-1}(a x)^{3/2},x\right )}{12 a}+\frac{c \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{5/2}}{4 a^2}-\frac{5 c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}{24 a}+\frac{5 c \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}}{32 a^2} \]

[Out]

(5*c*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(32*a^2) - (5*c*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(24*a) + (c*(1 + a^2*
x^2)^2*ArcTan[a*x]^(5/2))/(4*a^2) - (5*c*Unintegrable[1/Sqrt[ArcTan[a*x]], x])/(64*a) - (5*c*Unintegrable[ArcT
an[a*x]^(3/2), x])/(12*a)

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Rubi [A]  time = 0.0551784, 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 = {} \[ \int x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(5*c*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(32*a^2) - (5*c*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(24*a) + (c*(1 + a^2*
x^2)^2*ArcTan[a*x]^(5/2))/(4*a^2) - (5*c*Defer[Int][1/Sqrt[ArcTan[a*x]], x])/(64*a) - (5*c*Defer[Int][ArcTan[a
*x]^(3/2), x])/(12*a)

Rubi steps

\begin{align*} \int x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2} \, dx &=\frac{c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{5/2}}{4 a^2}-\frac{5 \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2} \, dx}{8 a}\\ &=\frac{5 c \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}{32 a^2}-\frac{5 c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}{24 a}+\frac{c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{5/2}}{4 a^2}-\frac{(5 c) \int \frac{1}{\sqrt{\tan ^{-1}(a x)}} \, dx}{64 a}-\frac{(5 c) \int \tan ^{-1}(a x)^{3/2} \, dx}{12 a}\\ \end{align*}

Mathematica [A]  time = 1.92466, size = 0, normalized size = 0. \[ \int x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.377, size = 0, normalized size = 0. \begin{align*} \int x \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(x*(a^2*c*x^2+c)*arctan(a*x)^(5/2),x)

[Out]

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

[Out]

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