3.3.94 \(\int \frac {(-1+x^3-x^5-2 x^7)^{2/3} (1-x^3+x^5+2 x^7) (-3+2 x^5+8 x^7)}{x^9} \, dx\)

Optimal. Leaf size=26 \[ \frac {3 \left (-2 x^7-x^5+x^3-1\right )^{8/3}}{8 x^8} \]

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Rubi [A]  time = 0.32, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {6688, 1590} \begin {gather*} \frac {3 \left (-2 x^7-x^5+x^3-1\right )^{8/3}}{8 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-1 + x^3 - x^5 - 2*x^7)^(2/3)*(1 - x^3 + x^5 + 2*x^7)*(-3 + 2*x^5 + 8*x^7))/x^9,x]

[Out]

(3*(-1 + x^3 - x^5 - 2*x^7)^(8/3))/(8*x^8)

Rule 1590

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[(Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*Rr^(n + 1))/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x
, r]), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp
, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rule 6688

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

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3-x^5-2 x^7\right )^{2/3} \left (1-x^3+x^5+2 x^7\right ) \left (-3+2 x^5+8 x^7\right )}{x^9} \, dx &=\int \frac {\left (3-2 x^5-8 x^7\right ) \left (-1+x^3-x^5-2 x^7\right )^{5/3}}{x^9} \, dx\\ &=\frac {3 \left (-1+x^3-x^5-2 x^7\right )^{8/3}}{8 x^8}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 26, normalized size = 1.00 \begin {gather*} \frac {3 \left (-2 x^7-x^5+x^3-1\right )^{8/3}}{8 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^3 - x^5 - 2*x^7)^(2/3)*(1 - x^3 + x^5 + 2*x^7)*(-3 + 2*x^5 + 8*x^7))/x^9,x]

[Out]

(3*(-1 + x^3 - x^5 - 2*x^7)^(8/3))/(8*x^8)

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IntegrateAlgebraic [A]  time = 0.13, size = 26, normalized size = 1.00 \begin {gather*} \frac {3 \left (-1+x^3-x^5-2 x^7\right )^{8/3}}{8 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^3 - x^5 - 2*x^7)^(2/3)*(1 - x^3 + x^5 + 2*x^7)*(-3 + 2*x^5 + 8*x^7))/x^9,x]

[Out]

(3*(-1 + x^3 - x^5 - 2*x^7)^(8/3))/(8*x^8)

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fricas [B]  time = 0.46, size = 62, normalized size = 2.38 \begin {gather*} \frac {3 \, {\left (4 \, x^{14} + 4 \, x^{12} - 3 \, x^{10} - 2 \, x^{8} + 4 \, x^{7} + x^{6} + 2 \, x^{5} - 2 \, x^{3} + 1\right )} {\left (-2 \, x^{7} - x^{5} + x^{3} - 1\right )}^{\frac {2}{3}}}{8 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^7-x^5+x^3-1)^(2/3)*(2*x^7+x^5-x^3+1)*(8*x^7+2*x^5-3)/x^9,x, algorithm="fricas")

[Out]

3/8*(4*x^14 + 4*x^12 - 3*x^10 - 2*x^8 + 4*x^7 + x^6 + 2*x^5 - 2*x^3 + 1)*(-2*x^7 - x^5 + x^3 - 1)^(2/3)/x^8

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (8 \, x^{7} + 2 \, x^{5} - 3\right )} {\left (2 \, x^{7} + x^{5} - x^{3} + 1\right )} {\left (-2 \, x^{7} - x^{5} + x^{3} - 1\right )}^{\frac {2}{3}}}{x^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^7-x^5+x^3-1)^(2/3)*(2*x^7+x^5-x^3+1)*(8*x^7+2*x^5-3)/x^9,x, algorithm="giac")

[Out]

integrate((8*x^7 + 2*x^5 - 3)*(2*x^7 + x^5 - x^3 + 1)*(-2*x^7 - x^5 + x^3 - 1)^(2/3)/x^9, x)

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maple [A]  time = 0.12, size = 23, normalized size = 0.88

method	result	size
gosper	\(\frac {3 \left (-2 x^{7}-x^{5}+x^{3}-1\right )^{\frac {8}{3}}}{8 x^{8}}\)	\(23\)
trager	\(\frac {3 \left (4 x^{14}+4 x^{12}-3 x^{10}-2 x^{8}+4 x^{7}+x^{6}+2 x^{5}-2 x^{3}+1\right ) \left (-2 x^{7}-x^{5}+x^{3}-1\right )^{\frac {2}{3}}}{8 x^{8}}\)	\(63\)
risch	\(-\frac {3 \left (8 x^{21}+12 x^{19}-6 x^{17}-11 x^{15}+12 x^{14}+3 x^{13}+12 x^{12}+3 x^{11}-9 x^{10}-x^{9}-6 x^{8}+6 x^{7}+3 x^{6}+3 x^{5}-3 x^{3}+1\right )}{8 x^{8} \left (-2 x^{7}-x^{5}+x^{3}-1\right )^{\frac {1}{3}}}\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^7-x^5+x^3-1)^(2/3)*(2*x^7+x^5-x^3+1)*(8*x^7+2*x^5-3)/x^9,x,method=_RETURNVERBOSE)

[Out]

3/8*(-2*x^7-x^5+x^3-1)^(8/3)/x^8

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maxima [B]  time = 0.67, size = 62, normalized size = 2.38 \begin {gather*} \frac {3 \, {\left (4 \, x^{14} + 4 \, x^{12} - 3 \, x^{10} - 2 \, x^{8} + 4 \, x^{7} + x^{6} + 2 \, x^{5} - 2 \, x^{3} + 1\right )} {\left (-2 \, x^{7} - x^{5} + x^{3} - 1\right )}^{\frac {2}{3}}}{8 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^7-x^5+x^3-1)^(2/3)*(2*x^7+x^5-x^3+1)*(8*x^7+2*x^5-3)/x^9,x, algorithm="maxima")

[Out]

3/8*(4*x^14 + 4*x^12 - 3*x^10 - 2*x^8 + 4*x^7 + x^6 + 2*x^5 - 2*x^3 + 1)*(-2*x^7 - x^5 + x^3 - 1)^(2/3)/x^8

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mupad [B]  time = 1.52, size = 147, normalized size = 5.65 \begin {gather*} \frac {3\,{\left (-2\,x^7-x^5+x^3-1\right )}^{2/3}}{2\,x}+\frac {3\,{\left (-2\,x^7-x^5+x^3-1\right )}^{2/3}}{8\,x^2}+\frac {3\,{\left (-2\,x^7-x^5+x^3-1\right )}^{2/3}}{4\,x^3}-\frac {3\,{\left (-2\,x^7-x^5+x^3-1\right )}^{2/3}}{4\,x^5}+\frac {3\,{\left (-2\,x^7-x^5+x^3-1\right )}^{2/3}}{8\,x^8}-\left (-\frac {3\,x^6}{2}-\frac {3\,x^4}{2}+\frac {9\,x^2}{8}+\frac {3}{4}\right )\,{\left (-2\,x^7-x^5+x^3-1\right )}^{2/3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x^5 + 8*x^7 - 3)*(x^3 - x^5 - 2*x^7 - 1)^(5/3))/x^9,x)

[Out]

(3*(x^3 - x^5 - 2*x^7 - 1)^(2/3))/(2*x) + (3*(x^3 - x^5 - 2*x^7 - 1)^(2/3))/(8*x^2) + (3*(x^3 - x^5 - 2*x^7 -
1)^(2/3))/(4*x^3) - (3*(x^3 - x^5 - 2*x^7 - 1)^(2/3))/(4*x^5) + (3*(x^3 - x^5 - 2*x^7 - 1)^(2/3))/(8*x^8) - ((
9*x^2)/8 - (3*x^4)/2 - (3*x^6)/2 + 3/4)*(x^3 - x^5 - 2*x^7 - 1)^(2/3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (8 x^{7} + 2 x^{5} - 3\right ) \left (- 2 x^{7} - x^{5} + x^{3} - 1\right )^{\frac {2}{3}} \left (2 x^{7} + x^{5} - x^{3} + 1\right )}{x^{9}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**7-x**5+x**3-1)**(2/3)*(2*x**7+x**5-x**3+1)*(8*x**7+2*x**5-3)/x**9,x)

[Out]

Integral((8*x**7 + 2*x**5 - 3)*(-2*x**7 - x**5 + x**3 - 1)**(2/3)*(2*x**7 + x**5 - x**3 + 1)/x**9, x)

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