3.11.8 \(\int \sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}} \, dx\)

Optimal. Leaf size=76 \[ \frac {\left (210 x^4-7 x^3-3 x^2+681 x-229\right ) \sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}}{910 (x+1) \left (x^2-x+1\right )} \]

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Rubi [A]  time = 0.06, antiderivative size = 139, normalized size of antiderivative = 1.83, number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {6688, 6719, 1850} \begin {gather*} \frac {\sqrt [3]{(1-3 x) \left (x^3+1\right )^3} (1-3 x)^4}{351 \left (x^3+1\right )}-\frac {\sqrt [3]{(1-3 x) \left (x^3+1\right )^3} (1-3 x)^3}{90 \left (x^3+1\right )}+\frac {\sqrt [3]{(1-3 x) \left (x^3+1\right )^3} (1-3 x)^2}{63 \left (x^3+1\right )}-\frac {7 \sqrt [3]{(1-3 x) \left (x^3+1\right )^3} (1-3 x)}{27 \left (x^3+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3),x]

[Out]

(-7*(1 - 3*x)*((1 - 3*x)*(1 + x^3)^3)^(1/3))/(27*(1 + x^3)) + ((1 - 3*x)^2*((1 - 3*x)*(1 + x^3)^3)^(1/3))/(63*
(1 + x^3)) - ((1 - 3*x)^3*((1 - 3*x)*(1 + x^3)^3)^(1/3))/(90*(1 + x^3)) + ((1 - 3*x)^4*((1 - 3*x)*(1 + x^3)^3)
^(1/3))/(351*(1 + x^3))

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps

\begin {align*} \int \sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}} \, dx &=\int \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \, dx\\ &=\frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \int \sqrt [3]{-1+3 x} \left (1+x^3\right ) \, dx}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}\\ &=\frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \int \left (\frac {28}{27} \sqrt [3]{-1+3 x}+\frac {1}{9} (-1+3 x)^{4/3}+\frac {1}{9} (-1+3 x)^{7/3}+\frac {1}{27} (-1+3 x)^{10/3}\right ) \, dx}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}\\ &=-\frac {7 (1-3 x) \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}{27 \left (1+x^3\right )}+\frac {(1-3 x)^2 \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}{63 \left (1+x^3\right )}-\frac {(1-3 x)^3 \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}{90 \left (1+x^3\right )}+\frac {(1-3 x)^4 \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}{351 \left (1+x^3\right )}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 44, normalized size = 0.58 \begin {gather*} -\frac {\left (-\left ((3 x-1) \left (x^3+1\right )^3\right )\right )^{4/3} \left (70 x^3+21 x^2+6 x+229\right )}{910 \left (x^3+1\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3),x]

[Out]

-1/910*((-((-1 + 3*x)*(1 + x^3)^3))^(4/3)*(229 + 6*x + 21*x^2 + 70*x^3))/(1 + x^3)^4

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IntegrateAlgebraic [A]  time = 0.10, size = 71, normalized size = 0.93 \begin {gather*} \frac {\left (-229-6 x-21 x^2-70 x^3\right ) \left (1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}\right )^{4/3}}{910 (1+x)^4 \left (1-x+x^2\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3),x]

[Out]

((-229 - 6*x - 21*x^2 - 70*x^3)*(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(4/3))/(910*(1 + x)^4
*(1 - x + x^2)^4)

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fricas [A]  time = 0.43, size = 64, normalized size = 0.84 \begin {gather*} \frac {{\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 1\right )}^{\frac {1}{3}} {\left (210 \, x^{4} - 7 \, x^{3} - 3 \, x^{2} + 681 \, x - 229\right )}}{910 \, {\left (x^{3} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3),x, algorithm="fricas")

[Out]

1/910*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3)*(210*x^4 - 7*x^3 - 3*x^2 + 681*x - 229)/
(x^3 + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 1\right )}^{\frac {1}{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3),x, algorithm="giac")

[Out]

integrate((-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3), x)

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maple [A]  time = 0.05, size = 46, normalized size = 0.61

method	result	size
risch	\(\frac {\left (-\left (-1+3 x \right ) \left (x^{3}+1\right )^{3}\right )^{\frac {1}{3}} \left (210 x^{4}-7 x^{3}-3 x^{2}+681 x -229\right )}{910 x^{3}+910}\)	\(46\)
trager	\(\frac {\left (210 x^{4}-7 x^{3}-3 x^{2}+681 x -229\right ) \left (-3 x^{10}+x^{9}-9 x^{7}+3 x^{6}-9 x^{4}+3 x^{3}-3 x +1\right )^{\frac {1}{3}}}{910 x^{3}+910}\)	\(65\)
gosper	\(\frac {\left (-1+3 x \right ) \left (70 x^{3}+21 x^{2}+6 x +229\right ) \left (-3 x^{10}+x^{9}-9 x^{7}+3 x^{6}-9 x^{4}+3 x^{3}-3 x +1\right )^{\frac {1}{3}}}{910 \left (1+x \right ) \left (x^{2}-x +1\right )}\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/910*(-(-1+3*x)*(x^3+1)^3)^(1/3)/(x^3+1)*(210*x^4-7*x^3-3*x^2+681*x-229)

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maxima [A]  time = 0.43, size = 29, normalized size = 0.38 \begin {gather*} -\frac {1}{910} \, {\left (210 \, x^{4} - 7 \, x^{3} - 3 \, x^{2} + 681 \, x - 229\right )} {\left (3 \, x - 1\right )}^{\frac {1}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3),x, algorithm="maxima")

[Out]

-1/910*(210*x^4 - 7*x^3 - 3*x^2 + 681*x - 229)*(3*x - 1)^(1/3)

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mupad [B]  time = 0.78, size = 64, normalized size = 0.84 \begin {gather*} -\frac {\left (-\frac {3\,x^4}{13}+\frac {x^3}{130}+\frac {3\,x^2}{910}-\frac {681\,x}{910}+\frac {229}{910}\right )\,{\left (-3\,x^{10}+x^9-9\,x^7+3\,x^6-9\,x^4+3\,x^3-3\,x+1\right )}^{1/3}}{x^3+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^3 - 3*x - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10 + 1)^(1/3),x)

[Out]

-(((3*x^2)/910 - (681*x)/910 + x^3/130 - (3*x^4)/13 + 229/910)*(3*x^3 - 3*x - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*
x^10 + 1)^(1/3))/(x^3 + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{- 3 x^{10} + x^{9} - 9 x^{7} + 3 x^{6} - 9 x^{4} + 3 x^{3} - 3 x + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x**10+x**9-9*x**7+3*x**6-9*x**4+3*x**3-3*x+1)**(1/3),x)

[Out]

Integral((-3*x**10 + x**9 - 9*x**7 + 3*x**6 - 9*x**4 + 3*x**3 - 3*x + 1)**(1/3), x)

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