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

Optimal. Leaf size=213 \[ -\frac{c \text{Unintegrable}\left (\frac{a^2 c x^2+c}{\sqrt{\tan ^{-1}(a x)}},x\right )}{64 a}-\frac{c^2 \text{Unintegrable}\left (\frac{1}{\sqrt{\tan ^{-1}(a x)}},x\right )}{24 a}-\frac{2 c^2 \text{Unintegrable}\left (\tan ^{-1}(a x)^{3/2},x\right )}{9 a}+\frac{c^2 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^{5/2}}{6 a^2}-\frac{c^2 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}}{12 a}+\frac{c^2 \left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)}}{32 a^2}-\frac{c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}{9 a}+\frac{c^2 \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}}{12 a^2} \]

[Out]

(c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(12*a^2) + (c^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(32*a^2) - (c^2*x*(1
+ a^2*x^2)*ArcTan[a*x]^(3/2))/(9*a) - (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(12*a) + (c^2*(1 + a^2*x^2)^3*
ArcTan[a*x]^(5/2))/(6*a^2) - (c^2*Unintegrable[1/Sqrt[ArcTan[a*x]], x])/(24*a) - (c*Unintegrable[(c + a^2*c*x^
2)/Sqrt[ArcTan[a*x]], x])/(64*a) - (2*c^2*Unintegrable[ArcTan[a*x]^(3/2), x])/(9*a)

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Rubi [A]  time = 0.113981, 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 = {} \[ \int x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(12*a^2) + (c^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(32*a^2) - (c^2*x*(1
+ a^2*x^2)*ArcTan[a*x]^(3/2))/(9*a) - (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(12*a) + (c^2*(1 + a^2*x^2)^3*
ArcTan[a*x]^(5/2))/(6*a^2) - (c^2*Defer[Int][1/Sqrt[ArcTan[a*x]], x])/(24*a) - (c*Defer[Int][(c + a^2*c*x^2)/S
qrt[ArcTan[a*x]], x])/(64*a) - (2*c^2*Defer[Int][ArcTan[a*x]^(3/2), x])/(9*a)

Rubi steps

\begin{align*} \int x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2} \, dx &=\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{5/2}}{6 a^2}-\frac{5 \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2} \, dx}{12 a}\\ &=\frac{c^2 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}{32 a^2}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{12 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{5/2}}{6 a^2}-\frac{c \int \frac{c+a^2 c x^2}{\sqrt{\tan ^{-1}(a x)}} \, dx}{64 a}-\frac{c \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2} \, dx}{3 a}\\ &=\frac{c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}{12 a^2}+\frac{c^2 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}{32 a^2}-\frac{c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}{9 a}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{12 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{5/2}}{6 a^2}-\frac{c \int \frac{c+a^2 c x^2}{\sqrt{\tan ^{-1}(a x)}} \, dx}{64 a}-\frac{c^2 \int \frac{1}{\sqrt{\tan ^{-1}(a x)}} \, dx}{24 a}-\frac{\left (2 c^2\right ) \int \tan ^{-1}(a x)^{3/2} \, dx}{9 a}\\ \end{align*}

Mathematica [A]  time = 1.42827, size = 0, normalized size = 0. \[ \int x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.532, size = 0, normalized size = 0. \begin{align*} \int x \left ({a}^{2}c{x}^{2}+c \right ) ^{2} \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)^2*arctan(a*x)^(5/2),x)

[Out]

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

[Out]

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