3.18.17 \(\int \frac {(4+x^2) \sqrt [3]{-2 x+x^3}}{x^4 (-4-4 x^2+x^4)} \, dx\) [1717]

Optimal. Leaf size=116 \[ -\frac {3 \left (-2+7 x^2\right ) \sqrt [3]{-2 x+x^3}}{16 x^3}+\frac {1}{16} \text {RootSum}\left [2-4 \text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {-6 \log (x)+6 \log \left (\sqrt [3]{-2 x+x^3}-x \text {$\#$1}\right )+11 \log (x) \text {$\#$1}^3-11 \log \left (\sqrt [3]{-2 x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-2 \text {$\#$1}^2+\text {$\#$1}^5}\& \right ] \]

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(561\) vs. \(2(116)=232\).
time = 0.89, antiderivative size = 561, normalized size of antiderivative = 4.84, number of steps used = 15, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2081, 6860, 477, 476, 486, 597, 12, 503} \begin {gather*} \frac {\sqrt {3} \sqrt [3]{\frac {1}{2} \left (674 \sqrt {2}-953\right )} \sqrt [3]{x^3-2 x} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{2-\sqrt {2}} x^{2/3}}{\sqrt [3]{x^2-2}}+1}{\sqrt {3}}\right )}{16 \sqrt [3]{x} \sqrt [3]{x^2-2}}-\frac {\sqrt {3} \sqrt [3]{\frac {1}{2} \left (953+674 \sqrt {2}\right )} \sqrt [3]{x^3-2 x} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{2+\sqrt {2}} x^{2/3}}{\sqrt [3]{x^2-2}}+1}{\sqrt {3}}\right )}{16 \sqrt [3]{x} \sqrt [3]{x^2-2}}-\frac {21 \left (4+3 \sqrt {2}\right ) \sqrt [3]{x^3-2 x}}{128 x}-\frac {21 \left (4-3 \sqrt {2}\right ) \sqrt [3]{x^3-2 x}}{128 x}+\frac {3 \left (4+\sqrt {2}\right ) \sqrt [3]{x^3-2 x}}{64 x^3}+\frac {3 \left (4-\sqrt {2}\right ) \sqrt [3]{x^3-2 x}}{64 x^3}-\frac {\sqrt [3]{\frac {1}{2} \left (674 \sqrt {2}-953\right )} \sqrt [3]{x^3-2 x} \log \left (2 \left (1+\sqrt {2}\right )-x^2\right )}{32 \sqrt [3]{x} \sqrt [3]{x^2-2}}+\frac {\sqrt [3]{\frac {1}{2} \left (953+674 \sqrt {2}\right )} \sqrt [3]{x^3-2 x} \log \left (x^2-2 \left (1-\sqrt {2}\right )\right )}{32 \sqrt [3]{x} \sqrt [3]{x^2-2}}+\frac {3 \sqrt [3]{\frac {1}{2} \left (674 \sqrt {2}-953\right )} \sqrt [3]{x^3-2 x} \log \left (\sqrt [3]{2-\sqrt {2}} x^{2/3}-\sqrt [3]{x^2-2}\right )}{32 \sqrt [3]{x} \sqrt [3]{x^2-2}}-\frac {3 \sqrt [3]{\frac {1}{2} \left (953+674 \sqrt {2}\right )} \sqrt [3]{x^3-2 x} \log \left (\sqrt [3]{2+\sqrt {2}} x^{2/3}-\sqrt [3]{x^2-2}\right )}{32 \sqrt [3]{x} \sqrt [3]{x^2-2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(4 - Sqrt[2])*(-2*x + x^3)^(1/3))/(64*x^3) + (3*(4 + Sqrt[2])*(-2*x + x^3)^(1/3))/(64*x^3) - (21*(4 - 3*Sqr
t[2])*(-2*x + x^3)^(1/3))/(128*x) - (21*(4 + 3*Sqrt[2])*(-2*x + x^3)^(1/3))/(128*x) + (Sqrt[3]*((-953 + 674*Sq
rt[2])/2)^(1/3)*(-2*x + x^3)^(1/3)*ArcTan[(1 + (2*(2 - Sqrt[2])^(1/3)*x^(2/3))/(-2 + x^2)^(1/3))/Sqrt[3]])/(16
*x^(1/3)*(-2 + x^2)^(1/3)) - (Sqrt[3]*((953 + 674*Sqrt[2])/2)^(1/3)*(-2*x + x^3)^(1/3)*ArcTan[(1 + (2*(2 + Sqr
t[2])^(1/3)*x^(2/3))/(-2 + x^2)^(1/3))/Sqrt[3]])/(16*x^(1/3)*(-2 + x^2)^(1/3)) - (((-953 + 674*Sqrt[2])/2)^(1/
3)*(-2*x + x^3)^(1/3)*Log[2*(1 + Sqrt[2]) - x^2])/(32*x^(1/3)*(-2 + x^2)^(1/3)) + (((953 + 674*Sqrt[2])/2)^(1/
3)*(-2*x + x^3)^(1/3)*Log[-2*(1 - Sqrt[2]) + x^2])/(32*x^(1/3)*(-2 + x^2)^(1/3)) + (3*((-953 + 674*Sqrt[2])/2)
^(1/3)*(-2*x + x^3)^(1/3)*Log[(2 - Sqrt[2])^(1/3)*x^(2/3) - (-2 + x^2)^(1/3)])/(32*x^(1/3)*(-2 + x^2)^(1/3)) -
(3*((953 + 674*Sqrt[2])/2)^(1/3)*(-2*x + x^3)^(1/3)*Log[(2 + Sqrt[2])^(1/3)*x^(2/3) - (-2 + x^2)^(1/3)])/(32*
x^(1/3)*(-2 + x^2)^(1/3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 476

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 486

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

Rule 503

Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Si
mp[-ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q^2), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/
(2*c*q^2), x] + Simp[Log[c + d*x^3]/(6*c*q^2), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\left (4+x^2\right ) \sqrt [3]{-2 x+x^3}}{x^4 \left (-4-4 x^2+x^4\right )} \, dx &=\frac {\sqrt [3]{-2 x+x^3} \int \frac {\sqrt [3]{-2+x^2} \left (4+x^2\right )}{x^{11/3} \left (-4-4 x^2+x^4\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+x^2}}\\ &=\frac {\sqrt [3]{-2 x+x^3} \int \left (\frac {\left (1+\frac {3}{\sqrt {2}}\right ) \sqrt [3]{-2+x^2}}{x^{11/3} \left (-4-4 \sqrt {2}+2 x^2\right )}+\frac {\left (1-\frac {3}{\sqrt {2}}\right ) \sqrt [3]{-2+x^2}}{x^{11/3} \left (-4+4 \sqrt {2}+2 x^2\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-2+x^2}}\\ &=\frac {\left (\left (2-3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \int \frac {\sqrt [3]{-2+x^2}}{x^{11/3} \left (-4+4 \sqrt {2}+2 x^2\right )} \, dx}{2 \sqrt [3]{x} \sqrt [3]{-2+x^2}}+\frac {\left (\left (2+3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \int \frac {\sqrt [3]{-2+x^2}}{x^{11/3} \left (-4-4 \sqrt {2}+2 x^2\right )} \, dx}{2 \sqrt [3]{x} \sqrt [3]{-2+x^2}}\\ &=\frac {\left (3 \left (2-3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-2+x^6}}{x^9 \left (-4+4 \sqrt {2}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{-2+x^2}}+\frac {\left (3 \left (2+3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-2+x^6}}{x^9 \left (-4-4 \sqrt {2}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{-2+x^2}}\\ &=\frac {\left (3 \left (2-3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-2+x^3}}{x^5 \left (-4+4 \sqrt {2}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{x} \sqrt [3]{-2+x^2}}+\frac {\left (3 \left (2+3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-2+x^3}}{x^5 \left (-4-4 \sqrt {2}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{x} \sqrt [3]{-2+x^2}}\\ &=\frac {\left (3 \left (2-3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1-\frac {x^3}{2}}}{x^5 \left (-4+4 \sqrt {2}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2\ 2^{2/3} \sqrt [3]{x} \sqrt [3]{2-x^2}}+\frac {\left (3 \left (2+3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1-\frac {x^3}{2}}}{x^5 \left (-4-4 \sqrt {2}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2\ 2^{2/3} \sqrt [3]{x} \sqrt [3]{2-x^2}}\\ &=\frac {3 \left (4-\sqrt {2}\right ) \sqrt [3]{-2 x+x^3} \left (2 \left (2-x^2\right ) \left (2-3 \left (1-\sqrt {2}\right ) x^2\right )+\left (2 \left (2-\sqrt {2}\right ) x^2-3 \left (4-3 \sqrt {2}\right ) x^4\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {\left (2-\sqrt {2}\right ) x^2}{2-x^2}\right )-3 x^2 \left (2 \left (2-\sqrt {2}\right )+\left (4-3 \sqrt {2}\right ) x^2\right ) \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {\left (2-\sqrt {2}\right ) x^2}{2-x^2}\right )\right )}{256 x^3 \left (2-x^2\right )}+\frac {3 \left (4+\sqrt {2}\right ) \sqrt [3]{-2 x+x^3} \left (2 \left (2-x^2\right ) \left (2-3 \left (1+\sqrt {2}\right ) x^2\right )+\left (2 \left (2+\sqrt {2}\right ) x^2-3 \left (4+3 \sqrt {2}\right ) x^4\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {\left (2+\sqrt {2}\right ) x^2}{2-x^2}\right )-3 x^2 \left (2 \left (2+\sqrt {2}\right )+\left (4+3 \sqrt {2}\right ) x^2\right ) \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {\left (2+\sqrt {2}\right ) x^2}{2-x^2}\right )\right )}{256 x^3 \left (2-x^2\right )}\\ \end {align*}

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Mathematica [A]
time = 2.77, size = 142, normalized size = 1.22 \begin {gather*} \frac {\sqrt [3]{x \left (-2+x^2\right )} \left (9 \left (2-7 x^2\right ) \sqrt [3]{-2+x^2}+x^{8/3} \text {RootSum}\left [2-4 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-12 \log (x)+18 \log \left (\sqrt [3]{-2+x^2}-x^{2/3} \text {$\#$1}\right )+22 \log (x) \text {$\#$1}^3-33 \log \left (\sqrt [3]{-2+x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}^3}{-2 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ]\right )}{48 x^3 \sqrt [3]{-2+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((x*(-2 + x^2))^(1/3)*(9*(2 - 7*x^2)*(-2 + x^2)^(1/3) + x^(8/3)*RootSum[2 - 4*#1^3 + #1^6 & , (-12*Log[x] + 18
*Log[(-2 + x^2)^(1/3) - x^(2/3)*#1] + 22*Log[x]*#1^3 - 33*Log[(-2 + x^2)^(1/3) - x^(2/3)*#1]*#1^3)/(-2*#1^2 +
#1^5) & ]))/(48*x^3*(-2 + x^2)^(1/3))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 142.34, size = 14280, normalized size = 123.10 \[\text {output too large to display}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/1120*(36*x^7 - 36*x^5 + 5*(3*x^5 + 2*x^3 - 16*x)*x^2 - 8*x^3 - 128*x)*(x^2 - 2)^(1/3)/(x^(23/3) - 4*x^(17/3)
- 4*x^(11/3)) + integrate(3/70*(36*x^6 - 6*x^4 + (18*x^6 + 27*x^4 + 26*x^2 - 304)*x^2 + 12*x^2 - 288)*(x^2 -
2)^(1/3)/(x^(35/3) - 8*x^(29/3) + 8*x^(23/3) + 32*x^(17/3) + 16*x^(11/3)), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x \left (x^{2} - 2\right )} \left (x^{2} + 4\right )}{x^{4} \left (x^{4} - 4 x^{2} - 4\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x*(x**2 - 2))**(1/3)*(x**2 + 4)/(x**4*(x**4 - 4*x**2 - 4)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (x^2+4\right )\,{\left (x^3-2\,x\right )}^{1/3}}{x^4\,\left (-x^4+4\,x^2+4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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