3.3.98 \(\int x^5 \sqrt {1-2 x^3} \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{45} \sqrt {1-2 x^3} \left (6 x^6-x^3-1\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \begin {gather*} \frac {1}{30} \left (1-2 x^3\right )^{5/2}-\frac {1}{18} \left (1-2 x^3\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*Sqrt[1 - 2*x^3],x]

[Out]

-1/18*(1 - 2*x^3)^(3/2) + (1 - 2*x^3)^(5/2)/30

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int x^5 \sqrt {1-2 x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \sqrt {1-2 x} x \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {1}{2} \sqrt {1-2 x}-\frac {1}{2} (1-2 x)^{3/2}\right ) \, dx,x,x^3\right )\\ &=-\frac {1}{18} \left (1-2 x^3\right )^{3/2}+\frac {1}{30} \left (1-2 x^3\right )^{5/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 0.81 \begin {gather*} -\frac {1}{45} \left (1-2 x^3\right )^{3/2} \left (3 x^3+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*Sqrt[1 - 2*x^3],x]

[Out]

-1/45*((1 - 2*x^3)^(3/2)*(1 + 3*x^3))

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IntegrateAlgebraic [A]  time = 0.02, size = 22, normalized size = 0.81 \begin {gather*} \frac {1}{45} \left (-1-3 x^3\right ) \left (1-2 x^3\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^5*Sqrt[1 - 2*x^3],x]

[Out]

((-1 - 3*x^3)*(1 - 2*x^3)^(3/2))/45

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fricas [A]  time = 0.45, size = 23, normalized size = 0.85 \begin {gather*} \frac {1}{45} \, {\left (6 \, x^{6} - x^{3} - 1\right )} \sqrt {-2 \, x^{3} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-2*x^3+1)^(1/2),x, algorithm="fricas")

[Out]

1/45*(6*x^6 - x^3 - 1)*sqrt(-2*x^3 + 1)

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giac [A]  time = 0.31, size = 32, normalized size = 1.19 \begin {gather*} \frac {1}{30} \, {\left (2 \, x^{3} - 1\right )}^{2} \sqrt {-2 \, x^{3} + 1} - \frac {1}{18} \, {\left (-2 \, x^{3} + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(2*x^3 - 1)^2*sqrt(-2*x^3 + 1) - 1/18*(-2*x^3 + 1)^(3/2)

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maple [A]  time = 0.13, size = 19, normalized size = 0.70

method	result	size
gosper	\(-\frac {\left (3 x^{3}+1\right ) \left (-2 x^{3}+1\right )^{\frac {3}{2}}}{45}\)	\(19\)
trager	\(\left (\frac {2}{15} x^{6}-\frac {1}{45} x^{3}-\frac {1}{45}\right ) \sqrt {-2 x^{3}+1}\)	\(23\)
risch	\(-\frac {\left (6 x^{6}-x^{3}-1\right ) \left (2 x^{3}-1\right )}{45 \sqrt {-2 x^{3}+1}}\)	\(31\)
meijerg	\(-\frac {-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (-2 x^{3}+1\right )^{\frac {3}{2}} \left (6 x^{3}+2\right )}{15}}{24 \sqrt {\pi }}\)	\(33\)
default	\(\frac {2 x^{6} \sqrt {-2 x^{3}+1}}{15}-\frac {x^{3} \sqrt {-2 x^{3}+1}}{45}-\frac {\sqrt {-2 x^{3}+1}}{45}\)	\(41\)
elliptic	\(\frac {2 x^{6} \sqrt {-2 x^{3}+1}}{15}-\frac {x^{3} \sqrt {-2 x^{3}+1}}{45}-\frac {\sqrt {-2 x^{3}+1}}{45}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(-2*x^3+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/45*(3*x^3+1)*(-2*x^3+1)^(3/2)

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maxima [A]  time = 0.32, size = 23, normalized size = 0.85 \begin {gather*} \frac {1}{30} \, {\left (-2 \, x^{3} + 1\right )}^{\frac {5}{2}} - \frac {1}{18} \, {\left (-2 \, x^{3} + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-2*x^3+1)^(1/2),x, algorithm="maxima")

[Out]

1/30*(-2*x^3 + 1)^(5/2) - 1/18*(-2*x^3 + 1)^(3/2)

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mupad [B]  time = 0.15, size = 34, normalized size = 1.26 \begin {gather*} -\frac {\frac {5\,{\left (2\,x^3-1\right )}^2}{2}+\frac {3\,{\left (2\,x^3-1\right )}^3}{2}}{45\,\sqrt {1-2\,x^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(1 - 2*x^3)^(1/2),x)

[Out]

-((5*(2*x^3 - 1)^2)/2 + (3*(2*x^3 - 1)^3)/2)/(45*(1 - 2*x^3)^(1/2))

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sympy [B]  time = 0.29, size = 42, normalized size = 1.56 \begin {gather*} \frac {2 x^{6} \sqrt {1 - 2 x^{3}}}{15} - \frac {x^{3} \sqrt {1 - 2 x^{3}}}{45} - \frac {\sqrt {1 - 2 x^{3}}}{45} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(-2*x**3+1)**(1/2),x)

[Out]

2*x**6*sqrt(1 - 2*x**3)/15 - x**3*sqrt(1 - 2*x**3)/45 - sqrt(1 - 2*x**3)/45

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