3.765 \(\int (c+a^2 c x^2)^2 \tan ^{-1}(a x)^{3/2} \, dx\)

Optimal. Leaf size=169 \[ \frac{3}{80} c \text{Unintegrable}\left (\frac{a^2 c x^2+c}{\sqrt{\tan ^{-1}(a x)}},x\right )+\frac{1}{10} c^2 \text{Unintegrable}\left (\frac{1}{\sqrt{\tan ^{-1}(a x)}},x\right )+\frac{8}{15} c^2 \text{Unintegrable}\left (\tan ^{-1}(a x)^{3/2},x\right )+\frac{1}{5} c^2 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}+\frac{4}{15} c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}-\frac{3 c^2 \left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)}}{40 a}-\frac{c^2 \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}}{5 a} \]

[Out]

-(c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(5*a) - (3*c^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(40*a) + (4*c^2*x*(1
+ a^2*x^2)*ArcTan[a*x]^(3/2))/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/5 + (c^2*Unintegrable[1/Sqrt[ArcT
an[a*x]], x])/10 + (3*c*Unintegrable[(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x])/80 + (8*c^2*Unintegrable[ArcTan[a*
x]^(3/2), x])/15

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Rubi [A]  time = 0.0663589, 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 = {} \[ \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2),x]

[Out]

-(c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(5*a) - (3*c^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(40*a) + (4*c^2*x*(1
+ a^2*x^2)*ArcTan[a*x]^(3/2))/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/5 + (c^2*Defer[Int][1/Sqrt[ArcTan
[a*x]], x])/10 + (3*c*Defer[Int][(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x])/80 + (8*c^2*Defer[Int][ArcTan[a*x]^(3/
2), x])/15

Rubi steps

\begin{align*} \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2} \, dx &=-\frac{3 c^2 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}{40 a}+\frac{1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}+\frac{1}{80} (3 c) \int \frac{c+a^2 c x^2}{\sqrt{\tan ^{-1}(a x)}} \, dx+\frac{1}{5} (4 c) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2} \, dx\\ &=-\frac{c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}{5 a}-\frac{3 c^2 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}{40 a}+\frac{4}{15} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}+\frac{1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}+\frac{1}{80} (3 c) \int \frac{c+a^2 c x^2}{\sqrt{\tan ^{-1}(a x)}} \, dx+\frac{1}{10} c^2 \int \frac{1}{\sqrt{\tan ^{-1}(a x)}} \, dx+\frac{1}{15} \left (8 c^2\right ) \int \tan ^{-1}(a x)^{3/2} \, dx\\ \end{align*}

Mathematica [A]  time = 2.12461, size = 0, normalized size = 0. \[ \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2),x]

[Out]

Integrate[(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2), x]

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Maple [A]  time = 0.431, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}c{x}^{2}+c \right ) ^{2} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*c*x^2+c)^2*arctan(a*x)^(3/2),x)

[Out]

int((a^2*c*x^2+c)^2*arctan(a*x)^(3/2),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)^2*arctan(a*x)^(3/2),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)^2*arctan(a*x)^(3/2),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)**2*atan(a*x)**(3/2),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 )}^{2} \arctan \left (a x\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^2*arctan(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate((a^2*c*x^2 + c)^2*arctan(a*x)^(3/2), x)