3.4.94 \(\int \frac {(-4+x^3) (-1+x^3)^{2/3}}{x^{12}} \, dx\)

Optimal. Leaf size=33 \[ \frac {\left (x^3-1\right )^{2/3} \left (-39 x^9-26 x^6-95 x^3+160\right )}{440 x^{11}} \]

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Rubi [A]  time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.48, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {453, 271, 264} \begin {gather*} -\frac {4 \left (x^3-1\right )^{5/3}}{11 x^{11}}-\frac {13 \left (x^3-1\right )^{5/3}}{88 x^8}-\frac {39 \left (x^3-1\right )^{5/3}}{440 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-4 + x^3)*(-1 + x^3)^(2/3))/x^12,x]

[Out]

(-4*(-1 + x^3)^(5/3))/(11*x^11) - (13*(-1 + x^3)^(5/3))/(88*x^8) - (39*(-1 + x^3)^(5/3))/(440*x^5)

Rule 264

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

Rule 271

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

Rule 453

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

Rubi steps

\begin {align*} \int \frac {\left (-4+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^{12}} \, dx &=-\frac {4 \left (-1+x^3\right )^{5/3}}{11 x^{11}}-\frac {13}{11} \int \frac {\left (-1+x^3\right )^{2/3}}{x^9} \, dx\\ &=-\frac {4 \left (-1+x^3\right )^{5/3}}{11 x^{11}}-\frac {13 \left (-1+x^3\right )^{5/3}}{88 x^8}-\frac {39}{88} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx\\ &=-\frac {4 \left (-1+x^3\right )^{5/3}}{11 x^{11}}-\frac {13 \left (-1+x^3\right )^{5/3}}{88 x^8}-\frac {39 \left (-1+x^3\right )^{5/3}}{440 x^5}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 1.00 \begin {gather*} -\frac {\left (x^3-1\right )^{2/3} \left (39 x^9+26 x^6+95 x^3-160\right )}{440 x^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-4 + x^3)*(-1 + x^3)^(2/3))/x^12,x]

[Out]

-1/440*((-1 + x^3)^(2/3)*(-160 + 95*x^3 + 26*x^6 + 39*x^9))/x^11

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IntegrateAlgebraic [A]  time = 0.09, size = 33, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (160-95 x^3-26 x^6-39 x^9\right )}{440 x^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-4 + x^3)*(-1 + x^3)^(2/3))/x^12,x]

[Out]

((-1 + x^3)^(2/3)*(160 - 95*x^3 - 26*x^6 - 39*x^9))/(440*x^11)

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fricas [A]  time = 0.46, size = 29, normalized size = 0.88 \begin {gather*} -\frac {{\left (39 \, x^{9} + 26 \, x^{6} + 95 \, x^{3} - 160\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{440 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-4)*(x^3-1)^(2/3)/x^12,x, algorithm="fricas")

[Out]

-1/440*(39*x^9 + 26*x^6 + 95*x^3 - 160)*(x^3 - 1)^(2/3)/x^11

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} - 4\right )}}{x^{12}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-4)*(x^3-1)^(2/3)/x^12,x, algorithm="giac")

[Out]

integrate((x^3 - 1)^(2/3)*(x^3 - 4)/x^12, x)

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maple [A]  time = 0.10, size = 30, normalized size = 0.91

method	result	size
trager	\(-\frac {\left (39 x^{9}+26 x^{6}+95 x^{3}-160\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{440 x^{11}}\)	\(30\)
gosper	\(-\frac {\left (x^{2}+x +1\right ) \left (-1+x \right ) \left (39 x^{6}+65 x^{3}+160\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{440 x^{11}}\)	\(34\)
risch	\(-\frac {39 x^{12}-13 x^{9}+69 x^{6}-255 x^{3}+160}{440 x^{11} \left (x^{3}-1\right )^{\frac {1}{3}}}\)	\(35\)
meijerg	\(\frac {4 \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-\frac {9}{20} x^{9}-\frac {3}{10} x^{6}-\frac {1}{4} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{11 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{11}}-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-\frac {3}{5} x^{6}-\frac {2}{5} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{8 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{8}}\)	\(95\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-4)*(x^3-1)^(2/3)/x^12,x,method=_RETURNVERBOSE)

[Out]

-1/440*(39*x^9+26*x^6+95*x^3-160)/x^11*(x^3-1)^(2/3)

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maxima [A]  time = 0.40, size = 37, normalized size = 1.12 \begin {gather*} -\frac {3 \, {\left (x^{3} - 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} + \frac {7 \, {\left (x^{3} - 1\right )}^{\frac {8}{3}}}{8 \, x^{8}} - \frac {4 \, {\left (x^{3} - 1\right )}^{\frac {11}{3}}}{11 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-4)*(x^3-1)^(2/3)/x^12,x, algorithm="maxima")

[Out]

-3/5*(x^3 - 1)^(5/3)/x^5 + 7/8*(x^3 - 1)^(8/3)/x^8 - 4/11*(x^3 - 1)^(11/3)/x^11

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mupad [B]  time = 0.37, size = 49, normalized size = 1.48 \begin {gather*} \frac {4\,{\left (x^3-1\right )}^{2/3}}{11\,x^{11}}-\frac {13\,{\left (x^3-1\right )}^{2/3}}{220\,x^5}-\frac {19\,{\left (x^3-1\right )}^{2/3}}{88\,x^8}-\frac {39\,{\left (x^3-1\right )}^{2/3}}{440\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 1)^(2/3)*(x^3 - 4))/x^12,x)

[Out]

(4*(x^3 - 1)^(2/3))/(11*x^11) - (13*(x^3 - 1)^(2/3))/(220*x^5) - (19*(x^3 - 1)^(2/3))/(88*x^8) - (39*(x^3 - 1)
^(2/3))/(440*x^2)

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sympy [C]  time = 3.34, size = 563, normalized size = 17.06 \begin {gather*} \begin {cases} \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {2 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {5 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {3 x^{6} \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {x^{3} \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} + \frac {5 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{9} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{6} \Gamma \left (- \frac {2}{3}\right )} - \frac {7 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases} - 4 \left (\begin {cases} \frac {2 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {11}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {4 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {11}{3}\right )}{9 x^{3} \Gamma \left (- \frac {2}{3}\right )} + \frac {10 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {11}{3}\right )}{27 x^{6} \Gamma \left (- \frac {2}{3}\right )} - \frac {40 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {11}{3}\right )}{27 x^{9} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {2 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} - \frac {4 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {10 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{27 x^{6} \Gamma \left (- \frac {2}{3}\right )} + \frac {40 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{27 x^{9} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-4)*(x**3-1)**(2/3)/x**12,x)

[Out]

Piecewise(((-1 + x**(-3))**(2/3)*exp(2*I*pi/3)*gamma(-8/3)/(3*gamma(-2/3)) + 2*(-1 + x**(-3))**(2/3)*exp(2*I*p
i/3)*gamma(-8/3)/(9*x**3*gamma(-2/3)) - 5*(-1 + x**(-3))**(2/3)*exp(2*I*pi/3)*gamma(-8/3)/(9*x**6*gamma(-2/3))
, 1/Abs(x**3) > 1), (3*x**6*(1 - 1/x**3)**(2/3)*gamma(-8/3)/(9*x**6*gamma(-2/3) - 9*x**3*gamma(-2/3)) - x**3*(
1 - 1/x**3)**(2/3)*gamma(-8/3)/(9*x**6*gamma(-2/3) - 9*x**3*gamma(-2/3)) + 5*(1 - 1/x**3)**(2/3)*gamma(-8/3)/(
9*x**9*gamma(-2/3) - 9*x**6*gamma(-2/3)) - 7*(1 - 1/x**3)**(2/3)*gamma(-8/3)/(9*x**6*gamma(-2/3) - 9*x**3*gamm
a(-2/3)), True)) - 4*Piecewise((2*(-1 + x**(-3))**(2/3)*exp(-I*pi/3)*gamma(-11/3)/(3*gamma(-2/3)) + 4*(-1 + x*
*(-3))**(2/3)*exp(-I*pi/3)*gamma(-11/3)/(9*x**3*gamma(-2/3)) + 10*(-1 + x**(-3))**(2/3)*exp(-I*pi/3)*gamma(-11
/3)/(27*x**6*gamma(-2/3)) - 40*(-1 + x**(-3))**(2/3)*exp(-I*pi/3)*gamma(-11/3)/(27*x**9*gamma(-2/3)), 1/Abs(x*
*3) > 1), (-2*(1 - 1/x**3)**(2/3)*gamma(-11/3)/(3*gamma(-2/3)) - 4*(1 - 1/x**3)**(2/3)*gamma(-11/3)/(9*x**3*ga
mma(-2/3)) - 10*(1 - 1/x**3)**(2/3)*gamma(-11/3)/(27*x**6*gamma(-2/3)) + 40*(1 - 1/x**3)**(2/3)*gamma(-11/3)/(
27*x**9*gamma(-2/3)), True))

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