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

Optimal. Leaf size=305 \[ -\frac{9 c^2 \text{Unintegrable}\left (\frac{a^2 c x^2+c}{\sqrt{\tan ^{-1}(a x)}},x\right )}{896 a}-\frac{5 c \text{Unintegrable}\left (\frac{\left (a^2 c x^2+c\right )^2}{\sqrt{\tan ^{-1}(a x)}},x\right )}{896 a}-\frac{3 c^3 \text{Unintegrable}\left (\frac{1}{\sqrt{\tan ^{-1}(a x)}},x\right )}{112 a}-\frac{c^3 \text{Unintegrable}\left (\tan ^{-1}(a x)^{3/2},x\right )}{7 a}+\frac{c^3 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x)^{5/2}}{8 a^2}-\frac{5 c^3 x \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^{3/2}}{112 a}+\frac{5 c^3 \left (a^2 x^2+1\right )^3 \sqrt{\tan ^{-1}(a x)}}{448 a^2}-\frac{3 c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}}{56 a}+\frac{9 c^3 \left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)}}{448 a^2}-\frac{c^3 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}{14 a}+\frac{3 c^3 \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}}{56 a^2} \]

[Out]

(3*c^3*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(56*a^2) + (9*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(448*a^2) + (5*c^
3*(1 + a^2*x^2)^3*Sqrt[ArcTan[a*x]])/(448*a^2) - (c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(14*a) - (3*c^3*x*(1
+ a^2*x^2)^2*ArcTan[a*x]^(3/2))/(56*a) - (5*c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^(3/2))/(112*a) + (c^3*(1 + a^2*x
^2)^4*ArcTan[a*x]^(5/2))/(8*a^2) - (3*c^3*Unintegrable[1/Sqrt[ArcTan[a*x]], x])/(112*a) - (9*c^2*Unintegrable[
(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x])/(896*a) - (5*c*Unintegrable[(c + a^2*c*x^2)^2/Sqrt[ArcTan[a*x]], x])/(8
96*a) - (c^3*Unintegrable[ArcTan[a*x]^(3/2), x])/(7*a)

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Rubi [A]  time = 0.187492, 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 = {} \[ \int x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(3*c^3*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(56*a^2) + (9*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(448*a^2) + (5*c^
3*(1 + a^2*x^2)^3*Sqrt[ArcTan[a*x]])/(448*a^2) - (c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(14*a) - (3*c^3*x*(1
+ a^2*x^2)^2*ArcTan[a*x]^(3/2))/(56*a) - (5*c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^(3/2))/(112*a) + (c^3*(1 + a^2*x
^2)^4*ArcTan[a*x]^(5/2))/(8*a^2) - (3*c^3*Defer[Int][1/Sqrt[ArcTan[a*x]], x])/(112*a) - (9*c^2*Defer[Int][(c +
a^2*c*x^2)/Sqrt[ArcTan[a*x]], x])/(896*a) - (5*c*Defer[Int][(c + a^2*c*x^2)^2/Sqrt[ArcTan[a*x]], x])/(896*a)
- (c^3*Defer[Int][ArcTan[a*x]^(3/2), x])/(7*a)

Rubi steps

\begin{align*} \int x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2} \, dx &=\frac{c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^{5/2}}{8 a^2}-\frac{5 \int \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2} \, dx}{16 a}\\ &=\frac{5 c^3 \left (1+a^2 x^2\right )^3 \sqrt{\tan ^{-1}(a x)}}{448 a^2}-\frac{5 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{3/2}}{112 a}+\frac{c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^{5/2}}{8 a^2}-\frac{(5 c) \int \frac{\left (c+a^2 c x^2\right )^2}{\sqrt{\tan ^{-1}(a x)}} \, dx}{896 a}-\frac{(15 c) \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2} \, dx}{56 a}\\ &=\frac{9 c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}{448 a^2}+\frac{5 c^3 \left (1+a^2 x^2\right )^3 \sqrt{\tan ^{-1}(a x)}}{448 a^2}-\frac{3 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{56 a}-\frac{5 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{3/2}}{112 a}+\frac{c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^{5/2}}{8 a^2}-\frac{(5 c) \int \frac{\left (c+a^2 c x^2\right )^2}{\sqrt{\tan ^{-1}(a x)}} \, dx}{896 a}-\frac{\left (9 c^2\right ) \int \frac{c+a^2 c x^2}{\sqrt{\tan ^{-1}(a x)}} \, dx}{896 a}-\frac{\left (3 c^2\right ) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2} \, dx}{14 a}\\ &=\frac{3 c^3 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}{56 a^2}+\frac{9 c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}{448 a^2}+\frac{5 c^3 \left (1+a^2 x^2\right )^3 \sqrt{\tan ^{-1}(a x)}}{448 a^2}-\frac{c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}{14 a}-\frac{3 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{56 a}-\frac{5 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{3/2}}{112 a}+\frac{c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^{5/2}}{8 a^2}-\frac{(5 c) \int \frac{\left (c+a^2 c x^2\right )^2}{\sqrt{\tan ^{-1}(a x)}} \, dx}{896 a}-\frac{\left (9 c^2\right ) \int \frac{c+a^2 c x^2}{\sqrt{\tan ^{-1}(a x)}} \, dx}{896 a}-\frac{\left (3 c^3\right ) \int \frac{1}{\sqrt{\tan ^{-1}(a x)}} \, dx}{112 a}-\frac{c^3 \int \tan ^{-1}(a x)^{3/2} \, dx}{7 a}\\ \end{align*}

Mathematica [A]  time = 1.50749, size = 0, normalized size = 0. \[ \int x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.76, size = 0, normalized size = 0. \begin{align*} \int x \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(x*(a^2*c*x^2+c)^3*arctan(a*x)^(5/2),x)

[Out]

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

[Out]

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