3.852 \(\int (c+a^2 c x^2)^3 \tan ^{-1}(a x)^{5/2} \, dx\)

Optimal. Leaf size=255 \[ \frac{9}{56} c^2 \text{Unintegrable}\left (\left (a^2 c x^2+c\right ) \sqrt{\tan ^{-1}(a x)},x\right )+\frac{5}{56} c \text{Unintegrable}\left (\left (a^2 c x^2+c\right )^2 \sqrt{\tan ^{-1}(a x)},x\right )+\frac{3}{7} c^3 \text{Unintegrable}\left (\sqrt{\tan ^{-1}(a x)},x\right )+\frac{16}{35} c^3 \text{Unintegrable}\left (\tan ^{-1}(a x)^{5/2},x\right )+\frac{1}{7} c^3 x \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^{5/2}+\frac{6}{35} c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{5/2}+\frac{8}{35} c^3 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{5/2}-\frac{5 c^3 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^{3/2}}{84 a}-\frac{3 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}}{28 a}-\frac{2 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}{7 a} \]

[Out]

(-2*c^3*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(7*a) - (3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(28*a) - (5*c^3*(1
+ a^2*x^2)^3*ArcTan[a*x]^(3/2))/(84*a) + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^(5/2))/35 + (6*c^3*x*(1 + a^2*x^2)
^2*ArcTan[a*x]^(5/2))/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^(5/2))/7 + (3*c^3*Unintegrable[Sqrt[ArcTan[a*x]]
, x])/7 + (9*c^2*Unintegrable[(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x])/56 + (5*c*Unintegrable[(c + a^2*c*x^2)^2*
Sqrt[ArcTan[a*x]], x])/56 + (16*c^3*Unintegrable[ArcTan[a*x]^(5/2), x])/35

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Rubi [A]  time = 0.126401, 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 = {} \[ \int \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2),x]

[Out]

(-2*c^3*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(7*a) - (3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(28*a) - (5*c^3*(1
+ a^2*x^2)^3*ArcTan[a*x]^(3/2))/(84*a) + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^(5/2))/35 + (6*c^3*x*(1 + a^2*x^2)
^2*ArcTan[a*x]^(5/2))/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^(5/2))/7 + (3*c^3*Defer[Int][Sqrt[ArcTan[a*x]],
x])/7 + (9*c^2*Defer[Int][(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x])/56 + (5*c*Defer[Int][(c + a^2*c*x^2)^2*Sqrt[A
rcTan[a*x]], x])/56 + (16*c^3*Defer[Int][ArcTan[a*x]^(5/2), x])/35

Rubi steps

\begin{align*} \int \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2} \, dx &=-\frac{5 c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{3/2}}{84 a}+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{5/2}+\frac{1}{56} (5 c) \int \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)} \, dx+\frac{1}{7} (6 c) \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2} \, dx\\ &=-\frac{3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{28 a}-\frac{5 c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{3/2}}{84 a}+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{5/2}+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{5/2}+\frac{1}{56} (5 c) \int \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)} \, dx+\frac{1}{56} \left (9 c^2\right ) \int \left (c+a^2 c x^2\right ) \sqrt{\tan ^{-1}(a x)} \, dx+\frac{1}{35} \left (24 c^2\right ) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2} \, dx\\ &=-\frac{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}{7 a}-\frac{3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{28 a}-\frac{5 c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{3/2}}{84 a}+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{5/2}+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{5/2}+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{5/2}+\frac{1}{56} (5 c) \int \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)} \, dx+\frac{1}{56} \left (9 c^2\right ) \int \left (c+a^2 c x^2\right ) \sqrt{\tan ^{-1}(a x)} \, dx+\frac{1}{7} \left (3 c^3\right ) \int \sqrt{\tan ^{-1}(a x)} \, dx+\frac{1}{35} \left (16 c^3\right ) \int \tan ^{-1}(a x)^{5/2} \, dx\\ \end{align*}

Mathematica [A]  time = 2.1729, size = 0, normalized size = 0. \[ \int \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2),x]

[Out]

Integrate[(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2), x]

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Maple [A]  time = 0.592, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}c{x}^{2}+c \right ) ^{3} \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)^3*arctan(a*x)^(5/2),x)

[Out]

int((a^2*c*x^2+c)^3*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)^3*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)^3*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)**3*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 )}^{3} \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)^3*arctan(a*x)^(5/2),x, algorithm="giac")

[Out]

integrate((a^2*c*x^2 + c)^3*arctan(a*x)^(5/2), x)