3.1094 \(\int \frac{x^3}{(c+a^2 c x^2)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=227 \[ \frac{44}{3} \text{Unintegrable}\left (\frac{x^3}{\left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right )+8 a^2 \text{Unintegrable}\left (\frac{x^5}{\left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right )+\frac{8 \sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^4 c \sqrt{a^2 c x^2+c}}-\frac{8 x^4}{3 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}-\frac{2 x^3}{3 a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}} \]

[Out]

(-2*x^3)/(3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) - (4*x^2)/(a^2*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])
- (8*x^4)/(3*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (8*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*S
qrt[ArcTan[a*x]]])/(a^4*c*Sqrt[c + a^2*c*x^2]) + (44*Unintegrable[x^3/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]
), x])/3 + 8*a^2*Unintegrable[x^5/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]

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Rubi [A]  time = 0.927643, 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 = {} \[ \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-2*x^3)/(3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) - (4*x^2)/(a^2*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])
- (8*x^4)/(3*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (8*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*S
qrt[ArcTan[a*x]]])/(a^4*c*Sqrt[c + a^2*c*x^2]) + (44*Defer[Int][x^3/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]),
x])/3 + 8*a^2*Defer[Int][x^5/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]

Rubi steps

\begin{align*} \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac{2 x^3}{3 a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}+\frac{2 \int \frac{x^2}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx}{a}+\frac{1}{3} (4 a) \int \frac{x^4}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 x^3}{3 a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}-\frac{8 x^4}{3 c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+4 \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{32}{3} \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{x}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a^2}+\left (8 a^2\right ) \int \frac{x^5}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx\\ &=-\frac{2 x^3}{3 a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}-\frac{8 x^4}{3 c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+4 \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{32}{3} \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\left (8 a^2\right ) \int \frac{x^5}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{\left (8 \sqrt{1+a^2 x^2}\right ) \int \frac{x}{\left (1+a^2 x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a^2 c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2 x^3}{3 a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}-\frac{8 x^4}{3 c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+4 \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{32}{3} \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\left (8 a^2\right ) \int \frac{x^5}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{\left (8 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2 x^3}{3 a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}-\frac{8 x^4}{3 c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+4 \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{32}{3} \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\left (8 a^2\right ) \int \frac{x^5}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{\left (16 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{a^4 c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2 x^3}{3 a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}-\frac{8 x^4}{3 c \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}+\frac{8 \sqrt{2 \pi } \sqrt{1+a^2 x^2} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^4 c \sqrt{c+a^2 c x^2}}+4 \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{32}{3} \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\left (8 a^2\right ) \int \frac{x^5}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx\\ \end{align*}

Mathematica [A]  time = 2.57426, size = 0, normalized size = 0. \[ \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 3.161, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \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^3/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x)

[Out]

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

[Out]

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