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

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

[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 = 0.0427148, 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, Rules used = {670, 640, 612, 620, 203} \[ -\frac{7}{8} r^3 (r-x) \sqrt{2 r x-x^2}-\frac{7}{12} r^2 \left (2 r x-x^2\right )^{3/2}+\frac{7}{4} r^5 \tan ^{-1}\left (\frac{x}{\sqrt{2 r x-x^2}}\right )-\frac{7}{20} r x \left (2 r x-x^2\right )^{3/2}-\frac{1}{5} x^2 \left (2 r x-x^2\right )^{3/2} \]

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

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[((m + p)*(2*c*d - b*e))/(c*(m + 2*p + 1)), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int x^3 \sqrt{2 r x-x^2} \, dx &=-\frac{1}{5} x^2 \left (2 r x-x^2\right )^{3/2}+\frac{1}{5} (7 r) \int x^2 \sqrt{2 r x-x^2} \, dx\\ &=-\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}+\frac{1}{4} \left (7 r^2\right ) \int x \sqrt{2 r x-x^2} \, dx\\ &=-\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}+\frac{1}{4} \left (7 r^3\right ) \int \sqrt{2 r x-x^2} \, dx\\ &=-\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}+\frac{1}{8} \left (7 r^5\right ) \int \frac{1}{\sqrt{2 r x-x^2}} \, dx\\ &=-\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}+\frac{1}{4} \left (7 r^5\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\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}+\frac{7}{4} r^5 \tan ^{-1}\left (\frac{x}{\sqrt{2 r x-x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.114559, size = 88, normalized size = 0.78 \[ \frac{1}{120} \sqrt{-x (x-2 r)} \left (-14 r^2 x^2-35 r^3 x+\frac{210 r^{9/2} \sin ^{-1}\left (\frac{\sqrt{x}}{\sqrt{2} \sqrt{r}}\right )}{\sqrt{x} \sqrt{2-\frac{x}{r}}}-105 r^4-6 r x^3+24 x^4\right ) \]

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 + (210*r^(9/2)*ArcSin[Sqrt[x]/(Sqr
t[2]*Sqrt[r])])/(Sqrt[x]*Sqrt[2 - x/r])))/120

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Maple [A]  time = 0.005, size = 111, normalized size = 1. \begin{align*} -{\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 ) } \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(2*r*x-x^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.03297, size = 158, normalized size = 1.4 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(2*r*x-x^2)^(1/2),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. \begin{align*} \int x^{3} \sqrt{- x \left (- 2 r + x\right )}\, dx \end{align*}

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 [A]  time = 1.06861, size = 85, normalized size = 0.75 \begin{align*} -\frac{7}{8} \, r^{5} \arcsin \left (\frac{r - x}{r}\right ) \mathrm{sgn}\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(2*r*x-x^2)^(1/2),x, algorithm="giac")

[Out]

-7/8*r^5*arcsin((r - x)/r)*sgn(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)