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

Optimal. Leaf size=197 \[ -\frac{5 \text{Unintegrable}\left (\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right )}{16 a^3}+\frac{25 \text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{a^2 c x^2+c}},x\right )}{12 a^3}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{3 a^2 c}-\frac{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{3 a^4 c}-\frac{5 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{12 a^3 c}+\frac{5 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{8 a^4 c} \]

[Out]

(5*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(8*a^4*c) - (5*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(12*a^3*c) -
(2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*a^4*c) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*a^2*c) -
(5*Unintegrable[1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/(16*a^3) + (25*Unintegrable[ArcTan[a*x]^(3/2)/
Sqrt[c + a^2*c*x^2], x])/(12*a^3)

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Rubi [A]  time = 0.472049, 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 = {} \[ \int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(5*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(8*a^4*c) - (5*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(12*a^3*c) -
(2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*a^4*c) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*a^2*c) -
(5*Defer[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/(16*a^3) + (25*Defer[Int][ArcTan[a*x]^(3/2)/Sqrt
[c + a^2*c*x^2], x])/(12*a^3)

Rubi steps

\begin{align*} \int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx &=\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 a^2 c}-\frac{2 \int \frac{x \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx}{3 a^2}-\frac{5 \int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx}{6 a}\\ &=-\frac{5 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{12 a^3 c}-\frac{2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 a^4 c}+\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 a^2 c}+\frac{5 \int \frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx}{12 a^3}+\frac{5 \int \frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx}{3 a^3}+\frac{5 \int \frac{x \sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx}{8 a^2}\\ &=\frac{5 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}{8 a^4 c}-\frac{5 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{12 a^3 c}-\frac{2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 a^4 c}+\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 a^2 c}-\frac{5 \int \frac{1}{\sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx}{16 a^3}+\frac{5 \int \frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx}{12 a^3}+\frac{5 \int \frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx}{3 a^3}\\ \end{align*}

Mathematica [A]  time = 3.85488, size = 0, normalized size = 0. \[ \int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 4.998, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x)

[Out]

int(x^3*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),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^3*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),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^3*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),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**3*atan(a*x)**(5/2)/(a**2*c*x**2+c)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

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