3.6.44 \(\int \frac {-1+x^3}{x^6 \sqrt [3]{x^2+x^3}} \, dx\)

Optimal. Leaf size=43 \[ -\frac {3 \left (x^3+x^2\right )^{2/3} \left (19071 x^5-12714 x^4+10595 x^3-3600 x^2+3300 x-3080\right )}{52360 x^7} \]

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Rubi [B]  time = 0.32, antiderivative size = 109, normalized size of antiderivative = 2.53, number of steps used = 11, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2052, 2016, 2014} \begin {gather*} -\frac {57213 \left (x^3+x^2\right )^{2/3}}{52360 x^2}+\frac {19071 \left (x^3+x^2\right )^{2/3}}{26180 x^3}+\frac {3 \left (x^3+x^2\right )^{2/3}}{17 x^7}-\frac {45 \left (x^3+x^2\right )^{2/3}}{238 x^6}+\frac {270 \left (x^3+x^2\right )^{2/3}}{1309 x^5}-\frac {6357 \left (x^3+x^2\right )^{2/3}}{10472 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x^3)/(x^6*(x^2 + x^3)^(1/3)),x]

[Out]

(3*(x^2 + x^3)^(2/3))/(17*x^7) - (45*(x^2 + x^3)^(2/3))/(238*x^6) + (270*(x^2 + x^3)^(2/3))/(1309*x^5) - (6357
*(x^2 + x^3)^(2/3))/(10472*x^4) + (19071*(x^2 + x^3)^(2/3))/(26180*x^3) - (57213*(x^2 + x^3)^(2/3))/(52360*x^2
)

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2016

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rule 2052

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(c*x)
^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !In
tegerQ[p] && NeQ[n, j]

Rubi steps

\begin {align*} \int \frac {-1+x^3}{x^6 \sqrt [3]{x^2+x^3}} \, dx &=\int \left (-\frac {1}{x^6 \sqrt [3]{x^2+x^3}}+\frac {1}{x^3 \sqrt [3]{x^2+x^3}}\right ) \, dx\\ &=-\int \frac {1}{x^6 \sqrt [3]{x^2+x^3}} \, dx+\int \frac {1}{x^3 \sqrt [3]{x^2+x^3}} \, dx\\ &=\frac {3 \left (x^2+x^3\right )^{2/3}}{17 x^7}-\frac {3 \left (x^2+x^3\right )^{2/3}}{8 x^4}-\frac {3}{4} \int \frac {1}{x^2 \sqrt [3]{x^2+x^3}} \, dx+\frac {15}{17} \int \frac {1}{x^5 \sqrt [3]{x^2+x^3}} \, dx\\ &=\frac {3 \left (x^2+x^3\right )^{2/3}}{17 x^7}-\frac {45 \left (x^2+x^3\right )^{2/3}}{238 x^6}-\frac {3 \left (x^2+x^3\right )^{2/3}}{8 x^4}+\frac {9 \left (x^2+x^3\right )^{2/3}}{20 x^3}+\frac {9}{20} \int \frac {1}{x \sqrt [3]{x^2+x^3}} \, dx-\frac {90}{119} \int \frac {1}{x^4 \sqrt [3]{x^2+x^3}} \, dx\\ &=\frac {3 \left (x^2+x^3\right )^{2/3}}{17 x^7}-\frac {45 \left (x^2+x^3\right )^{2/3}}{238 x^6}+\frac {270 \left (x^2+x^3\right )^{2/3}}{1309 x^5}-\frac {3 \left (x^2+x^3\right )^{2/3}}{8 x^4}+\frac {9 \left (x^2+x^3\right )^{2/3}}{20 x^3}-\frac {27 \left (x^2+x^3\right )^{2/3}}{40 x^2}+\frac {810 \int \frac {1}{x^3 \sqrt [3]{x^2+x^3}} \, dx}{1309}\\ &=\frac {3 \left (x^2+x^3\right )^{2/3}}{17 x^7}-\frac {45 \left (x^2+x^3\right )^{2/3}}{238 x^6}+\frac {270 \left (x^2+x^3\right )^{2/3}}{1309 x^5}-\frac {6357 \left (x^2+x^3\right )^{2/3}}{10472 x^4}+\frac {9 \left (x^2+x^3\right )^{2/3}}{20 x^3}-\frac {27 \left (x^2+x^3\right )^{2/3}}{40 x^2}-\frac {1215 \int \frac {1}{x^2 \sqrt [3]{x^2+x^3}} \, dx}{2618}\\ &=\frac {3 \left (x^2+x^3\right )^{2/3}}{17 x^7}-\frac {45 \left (x^2+x^3\right )^{2/3}}{238 x^6}+\frac {270 \left (x^2+x^3\right )^{2/3}}{1309 x^5}-\frac {6357 \left (x^2+x^3\right )^{2/3}}{10472 x^4}+\frac {19071 \left (x^2+x^3\right )^{2/3}}{26180 x^3}-\frac {27 \left (x^2+x^3\right )^{2/3}}{40 x^2}+\frac {729 \int \frac {1}{x \sqrt [3]{x^2+x^3}} \, dx}{2618}\\ &=\frac {3 \left (x^2+x^3\right )^{2/3}}{17 x^7}-\frac {45 \left (x^2+x^3\right )^{2/3}}{238 x^6}+\frac {270 \left (x^2+x^3\right )^{2/3}}{1309 x^5}-\frac {6357 \left (x^2+x^3\right )^{2/3}}{10472 x^4}+\frac {19071 \left (x^2+x^3\right )^{2/3}}{26180 x^3}-\frac {57213 \left (x^2+x^3\right )^{2/3}}{52360 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 43, normalized size = 1.00 \begin {gather*} -\frac {3 \left (x^2 (x+1)\right )^{2/3} \left (19071 x^5-12714 x^4+10595 x^3-3600 x^2+3300 x-3080\right )}{52360 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(x^2*(1 + x))^(2/3)*(-3080 + 3300*x - 3600*x^2 + 10595*x^3 - 12714*x^4 + 19071*x^5))/(52360*x^7)

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IntegrateAlgebraic [A]  time = 0.28, size = 43, normalized size = 1.00 \begin {gather*} -\frac {3 \left (x^2+x^3\right )^{2/3} \left (-3080+3300 x-3600 x^2+10595 x^3-12714 x^4+19071 x^5\right )}{52360 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x^3)/(x^6*(x^2 + x^3)^(1/3)),x]

[Out]

(-3*(x^2 + x^3)^(2/3)*(-3080 + 3300*x - 3600*x^2 + 10595*x^3 - 12714*x^4 + 19071*x^5))/(52360*x^7)

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fricas [A]  time = 0.59, size = 39, normalized size = 0.91 \begin {gather*} -\frac {3 \, {\left (19071 \, x^{5} - 12714 \, x^{4} + 10595 \, x^{3} - 3600 \, x^{2} + 3300 \, x - 3080\right )} {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{52360 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/52360*(19071*x^5 - 12714*x^4 + 10595*x^3 - 3600*x^2 + 3300*x - 3080)*(x^3 + x^2)^(2/3)/x^7

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giac [A]  time = 0.45, size = 55, normalized size = 1.28 \begin {gather*} \frac {3}{17} \, {\left (\frac {1}{x} + 1\right )}^{\frac {17}{3}} - \frac {15}{14} \, {\left (\frac {1}{x} + 1\right )}^{\frac {14}{3}} + \frac {30}{11} \, {\left (\frac {1}{x} + 1\right )}^{\frac {11}{3}} - \frac {33}{8} \, {\left (\frac {1}{x} + 1\right )}^{\frac {8}{3}} + \frac {21}{5} \, {\left (\frac {1}{x} + 1\right )}^{\frac {5}{3}} - 3 \, {\left (\frac {1}{x} + 1\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/17*(1/x + 1)^(17/3) - 15/14*(1/x + 1)^(14/3) + 30/11*(1/x + 1)^(11/3) - 33/8*(1/x + 1)^(8/3) + 21/5*(1/x + 1
)^(5/3) - 3*(1/x + 1)^(2/3)

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maple [C]  time = 0.07, size = 36, normalized size = 0.84

method	result	size
meijerg	\(\frac {3 \hypergeom \left (\left [-\frac {17}{3}, \frac {1}{3}\right ], \left [-\frac {14}{3}\right ], -x \right )}{17 x^{\frac {17}{3}}}-\frac {3 \left (\frac {9}{5} x^{2}-\frac {6}{5} x +1\right ) \left (1+x \right )^{\frac {2}{3}}}{8 x^{\frac {8}{3}}}\)	\(36\)
trager	\(-\frac {3 \left (x^{3}+x^{2}\right )^{\frac {2}{3}} \left (19071 x^{5}-12714 x^{4}+10595 x^{3}-3600 x^{2}+3300 x -3080\right )}{52360 x^{7}}\)	\(40\)
gosper	\(-\frac {3 \left (19071 x^{5}-12714 x^{4}+10595 x^{3}-3600 x^{2}+3300 x -3080\right ) \left (1+x \right )}{52360 \left (x^{3}+x^{2}\right )^{\frac {1}{3}} x^{5}}\)	\(43\)
risch	\(-\frac {3 \left (19071 x^{6}+6357 x^{5}-2119 x^{4}+6995 x^{3}-300 x^{2}+220 x -3080\right )}{52360 x^{5} \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/17*hypergeom([-17/3,1/3],[-14/3],-x)/x^(17/3)-3/8*(9/5*x^2-6/5*x+1)*(1+x)^(2/3)/x^(8/3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} - 1}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.30, size = 85, normalized size = 1.98 \begin {gather*} \frac {19071\,{\left (x^3+x^2\right )}^{2/3}}{26180\,x^3}-\frac {57213\,{\left (x^3+x^2\right )}^{2/3}}{52360\,x^2}-\frac {6357\,{\left (x^3+x^2\right )}^{2/3}}{10472\,x^4}+\frac {270\,{\left (x^3+x^2\right )}^{2/3}}{1309\,x^5}-\frac {45\,{\left (x^3+x^2\right )}^{2/3}}{238\,x^6}+\frac {3\,{\left (x^3+x^2\right )}^{2/3}}{17\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3 - 1)/(x^6*(x^2 + x^3)^(1/3)),x)

[Out]

(19071*(x^2 + x^3)^(2/3))/(26180*x^3) - (57213*(x^2 + x^3)^(2/3))/(52360*x^2) - (6357*(x^2 + x^3)^(2/3))/(1047
2*x^4) + (270*(x^2 + x^3)^(2/3))/(1309*x^5) - (45*(x^2 + x^3)^(2/3))/(238*x^6) + (3*(x^2 + x^3)^(2/3))/(17*x^7
)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x^{2} + x + 1\right )}{x^{6} \sqrt [3]{x^{2} \left (x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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