3.7.3 \(\int (2-x^2) \sqrt [4]{6 x-x^3} \, dx\) [603]

Optimal. Leaf size=17 \[ \frac {4}{15} \left (6 x-x^3\right )^{5/4} \]

[Out]

4/15*(-x^3+6*x)^(5/4)

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Rubi [A]
time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1602} \begin {gather*} \frac {4}{15} \left (6 x-x^3\right )^{5/4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 - x^2)*(6*x - x^3)^(1/4),x]

[Out]

(4*(6*x - x^3)^(5/4))/15

Rule 1602

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*x^(p - q +
1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \left (2-x^2\right ) \sqrt [4]{6 x-x^3} \, dx &=\frac {4}{15} \left (6 x-x^3\right )^{5/4}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 2 in optimal.
time = 10.04, size = 72, normalized size = 4.24 \begin {gather*} -\frac {4 \sqrt [4]{-x \left (-6+x^2\right )} \left (-26 x \, _2F_1\left (-\frac {1}{4},\frac {5}{8};\frac {13}{8};\frac {x^2}{6}\right )+5 x^3 \, _2F_1\left (-\frac {1}{4},\frac {13}{8};\frac {21}{8};\frac {x^2}{6}\right )\right )}{65 \sqrt [4]{1-\frac {x^2}{6}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 - x^2)*(6*x - x^3)^(1/4),x]

[Out]

(-4*(-(x*(-6 + x^2)))^(1/4)*(-26*x*Hypergeometric2F1[-1/4, 5/8, 13/8, x^2/6] + 5*x^3*Hypergeometric2F1[-1/4, 1
3/8, 21/8, x^2/6]))/(65*(1 - x^2/6)^(1/4))

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Maple [A]
time = 0.32, size = 14, normalized size = 0.82

method	result	size
default	\(\frac {4 \left (-x^{3}+6 x \right )^{\frac {5}{4}}}{15}\)	\(14\)
risch	\(-\frac {4 \left (-x \left (x^{2}-6\right )\right )^{\frac {1}{4}} x \left (x^{2}-6\right )}{15}\)	\(19\)
gosper	\(-\frac {4 \left (-x^{3}+6 x \right )^{\frac {1}{4}} x \left (x^{2}-6\right )}{15}\)	\(20\)
trager	\(-\frac {4 \left (-x^{3}+6 x \right )^{\frac {1}{4}} x \left (x^{2}-6\right )}{15}\)	\(20\)
meijerg	\(\frac {8 \,6^{\frac {1}{4}} x^{\frac {5}{4}} \hypergeom \left (\left [-\frac {1}{4}, \frac {5}{8}\right ], \left [\frac {13}{8}\right ], \frac {x^{2}}{6}\right )}{5}-\frac {4 \,6^{\frac {1}{4}} x^{\frac {13}{4}} \hypergeom \left (\left [-\frac {1}{4}, \frac {13}{8}\right ], \left [\frac {21}{8}\right ], \frac {x^{2}}{6}\right )}{13}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+2)*(-x^3+6*x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

4/15*(-x^3+6*x)^(5/4)

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Maxima [A]
time = 0.28, size = 13, normalized size = 0.76 \begin {gather*} \frac {4}{15} \, {\left (-x^{3} + 6 \, x\right )}^{\frac {5}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+2)*(-x^3+6*x)^(1/4),x, algorithm="maxima")

[Out]

4/15*(-x^3 + 6*x)^(5/4)

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Fricas [A]
time = 0.35, size = 20, normalized size = 1.18 \begin {gather*} -\frac {4}{15} \, {\left (x^{3} - 6 \, x\right )} {\left (-x^{3} + 6 \, x\right )}^{\frac {1}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+2)*(-x^3+6*x)^(1/4),x, algorithm="fricas")

[Out]

-4/15*(x^3 - 6*x)*(-x^3 + 6*x)^(1/4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).
time = 0.09, size = 31, normalized size = 1.82 \begin {gather*} - \frac {4 x^{3} \sqrt [4]{- x^{3} + 6 x}}{15} + \frac {8 x \sqrt [4]{- x^{3} + 6 x}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2+2)*(-x**3+6*x)**(1/4),x)

[Out]

-4*x**3*(-x**3 + 6*x)**(1/4)/15 + 8*x*(-x**3 + 6*x)**(1/4)/5

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Giac [A]
time = 3.33, size = 13, normalized size = 0.76 \begin {gather*} \frac {4}{15} \, {\left (-x^{3} + 6 \, x\right )}^{\frac {5}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+2)*(-x^3+6*x)^(1/4),x, algorithm="giac")

[Out]

4/15*(-x^3 + 6*x)^(5/4)

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Mupad [B]
time = 3.16, size = 19, normalized size = 1.12 \begin {gather*} -\frac {4\,x\,\left (x^2-6\right )\,{\left (6\,x-x^3\right )}^{1/4}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2 - 2)*(6*x - x^3)^(1/4),x)

[Out]

-(4*x*(x^2 - 6)*(6*x - x^3)^(1/4))/15

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