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

Optimal. Leaf size=256 \[ -\frac{9 c^2 \text{Unintegrable}\left (\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right )}{64 a}-\frac{3 c^2 \text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{a^2 c x^2+c}},x\right )}{16 a}-\frac{c \text{Unintegrable}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{\tan ^{-1}(a x)}},x\right )}{32 a}+\frac{\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^{5/2}}{5 a^2 c}-\frac{x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{8 a}-\frac{3 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{16 a}+\frac{\left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}{16 a^2}+\frac{9 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{32 a^2} \]

[Out]

(9*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(32*a^2) + ((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(16*a^2) - (3
*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(16*a) - (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(8*a) + ((c +
a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2))/(5*a^2*c) - (9*c^2*Unintegrable[1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])
, x])/(64*a) - (c*Unintegrable[Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x])/(32*a) - (3*c^2*Unintegrable[ArcTan[
a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x])/(16*a)

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Rubi [A]  time = 0.272472, 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 = {} \[ \int x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(9*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(32*a^2) + ((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(16*a^2) - (3
*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(16*a) - (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(8*a) + ((c +
a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2))/(5*a^2*c) - (9*c^2*Defer[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]),
x])/(64*a) - (c*Defer[Int][Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x])/(32*a) - (3*c^2*Defer[Int][ArcTan[a*x]^(
3/2)/Sqrt[c + a^2*c*x^2], x])/(16*a)

Rubi steps

\begin{align*} \int x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2} \, dx &=\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2}}{5 a^2 c}-\frac{\int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2} \, dx}{2 a}\\ &=\frac{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}{16 a^2}-\frac{x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{8 a}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2}}{5 a^2 c}-\frac{c \int \frac{\sqrt{c+a^2 c x^2}}{\sqrt{\tan ^{-1}(a x)}} \, dx}{32 a}-\frac{(3 c) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2} \, dx}{8 a}\\ &=\frac{9 c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}{32 a^2}+\frac{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}{16 a^2}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{16 a}-\frac{x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{8 a}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2}}{5 a^2 c}-\frac{c \int \frac{\sqrt{c+a^2 c x^2}}{\sqrt{\tan ^{-1}(a x)}} \, dx}{32 a}-\frac{\left (9 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx}{64 a}-\frac{\left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx}{16 a}\\ \end{align*}

Mathematica [A]  time = 2.55925, size = 0, normalized size = 0. \[ \int x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.796, size = 0, normalized size = 0. \begin{align*} \int x \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{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)^(3/2)*arctan(a*x)^(5/2),x)

[Out]

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

[Out]

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