3.263 \(\int x^3 \sqrt{2 r x-x^2} \, dx\)

Optimal. Leaf size=113 \[ \frac{7}{4} r^5 \tan ^{-1}\left (\frac{x}{\sqrt{2 r x-x^2}}\right )-\frac{7}{8} r^3 (r-x) \sqrt{2 r x-x^2}-\frac{7}{12} r^2 \left (2 r x-x^2\right )^{3/2}-\frac{7}{20} r x \left (2 r x-x^2\right )^{3/2}-\frac{1}{5} x^2 \left (2 r x-x^2\right )^{3/2} \]

[Out]

(-7*r^3*(r - x)*Sqrt[2*r*x - x^2])/8 - (7*r^2*(2*r*x - x^2)^(3/2))/12 - (7*r*x*(
2*r*x - x^2)^(3/2))/20 - (x^2*(2*r*x - x^2)^(3/2))/5 + (7*r^5*ArcTan[x/Sqrt[2*r*
x - x^2]])/4

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Rubi [A]  time = 0.105325, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{7}{4} r^5 \tan ^{-1}\left (\frac{x}{\sqrt{2 r x-x^2}}\right )-\frac{7}{8} r^3 (r-x) \sqrt{2 r x-x^2}-\frac{7}{12} r^2 \left (2 r x-x^2\right )^{3/2}-\frac{7}{20} r x \left (2 r x-x^2\right )^{3/2}-\frac{1}{5} x^2 \left (2 r x-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x^3*Sqrt[2*r*x - x^2],x]

[Out]

(-7*r^3*(r - x)*Sqrt[2*r*x - x^2])/8 - (7*r^2*(2*r*x - x^2)^(3/2))/12 - (7*r*x*(
2*r*x - x^2)^(3/2))/20 - (x^2*(2*r*x - x^2)^(3/2))/5 + (7*r^5*ArcTan[x/Sqrt[2*r*
x - x^2]])/4

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Rubi in Sympy [A]  time = 5.77736, size = 100, normalized size = 0.88 \[ \frac{7 r^{5} \operatorname{atan}{\left (\frac{x}{\sqrt{2 r x - x^{2}}} \right )}}{4} - \frac{7 r^{3} \left (2 r - 2 x\right ) \sqrt{2 r x - x^{2}}}{16} - \frac{7 r^{2} \left (2 r x - x^{2}\right )^{\frac{3}{2}}}{12} - \frac{7 r x \left (2 r x - x^{2}\right )^{\frac{3}{2}}}{20} - \frac{x^{2} \left (2 r x - x^{2}\right )^{\frac{3}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(2*r*x-x**2)**(1/2),x)

[Out]

7*r**5*atan(x/sqrt(2*r*x - x**2))/4 - 7*r**3*(2*r - 2*x)*sqrt(2*r*x - x**2)/16 -
7*r**2*(2*r*x - x**2)**(3/2)/12 - 7*r*x*(2*r*x - x**2)**(3/2)/20 - x**2*(2*r*x
- x**2)**(3/2)/5

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Mathematica [A]  time = 0.0963619, size = 82, normalized size = 0.73 \[ \frac{1}{120} \sqrt{-x (x-2 r)} \left (-\frac{210 r^5 \log \left (\sqrt{x-2 r}+\sqrt{x}\right )}{\sqrt{x} \sqrt{x-2 r}}-105 r^4-35 r^3 x-14 r^2 x^2-6 r x^3+24 x^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*Sqrt[2*r*x - x^2],x]

[Out]

(Sqrt[-(x*(-2*r + x))]*(-105*r^4 - 35*r^3*x - 14*r^2*x^2 - 6*r*x^3 + 24*x^4 - (2
10*r^5*Log[Sqrt[x] + Sqrt[-2*r + x]])/(Sqrt[x]*Sqrt[-2*r + x])))/120

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Maple [A]  time = 0.007, size = 111, normalized size = 1. \[ -{\frac{{x}^{2}}{5} \left ( 2\,rx-{x}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{7\,rx}{20} \left ( 2\,rx-{x}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{r}^{2}}{12} \left ( 2\,rx-{x}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{r}^{3}x}{8}\sqrt{2\,rx-{x}^{2}}}-{\frac{7\,{r}^{4}}{8}\sqrt{2\,rx-{x}^{2}}}+{\frac{7\,{r}^{5}}{8}\arctan \left ({(x-r){\frac{1}{\sqrt{2\,rx-{x}^{2}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(2*r*x-x^2)^(1/2),x)

[Out]

-1/5*x^2*(2*r*x-x^2)^(3/2)-7/20*r*x*(2*r*x-x^2)^(3/2)-7/12*r^2*(2*r*x-x^2)^(3/2)
+7/8*r^3*(2*r*x-x^2)^(1/2)*x-7/8*(2*r*x-x^2)^(1/2)*r^4+7/8*r^5*arctan((x-r)/(2*r
*x-x^2)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(2*r*x - x^2)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261184, size = 92, normalized size = 0.81 \[ -\frac{7}{4} \, r^{5} \arctan \left (\frac{\sqrt{2 \, r x - x^{2}}}{x}\right ) - \frac{1}{120} \,{\left (105 \, r^{4} + 35 \, r^{3} x + 14 \, r^{2} x^{2} + 6 \, r x^{3} - 24 \, x^{4}\right )} \sqrt{2 \, r x - x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(2*r*x - x^2)*x^3,x, algorithm="fricas")

[Out]

-7/4*r^5*arctan(sqrt(2*r*x - x^2)/x) - 1/120*(105*r^4 + 35*r^3*x + 14*r^2*x^2 +
6*r*x^3 - 24*x^4)*sqrt(2*r*x - x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{3} \sqrt{- x \left (- 2 r + x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(2*r*x-x**2)**(1/2),x)

[Out]

Integral(x**3*sqrt(-x*(-2*r + x)), x)

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GIAC/XCAS [A]  time = 0.208697, size = 85, normalized size = 0.75 \[ -\frac{7}{8} \, r^{5} \arcsin \left (\frac{r - x}{r}\right ){\rm sign}\left (r\right ) - \frac{1}{120} \,{\left (105 \, r^{4} +{\left (35 \, r^{3} + 2 \,{\left (7 \, r^{2} + 3 \,{\left (r - 4 \, x\right )} x\right )} x\right )} x\right )} \sqrt{2 \, r x - x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(2*r*x - x^2)*x^3,x, algorithm="giac")

[Out]

-7/8*r^5*arcsin((r - x)/r)*sign(r) - 1/120*(105*r^4 + (35*r^3 + 2*(7*r^2 + 3*(r
- 4*x)*x)*x)*x)*sqrt(2*r*x - x^2)