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

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

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Rubi [B]  time = 1.96, antiderivative size = 357, normalized size of antiderivative = 3.08, number of steps used = 11, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2056, 6728, 466, 465, 511, 510} \begin {gather*} \frac {3 \left (4-\sqrt {2}\right ) \sqrt [3]{x^3-2 x} \left (-3 \left (\left (4-3 \sqrt {2}\right ) x^2+2 \left (2-\sqrt {2}\right )\right ) x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {\left (2-\sqrt {2}\right ) x^2}{2-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 )+2 \left (2-x^2\right ) \left (2-3 \left (1-\sqrt {2}\right ) x^2\right )\right )}{256 x^3 \left (2-x^2\right )}+\frac {3 \left (4+\sqrt {2}\right ) \sqrt [3]{x^3-2 x} \left (-3 \left (\left (4+3 \sqrt {2}\right ) x^2+2 \left (2+\sqrt {2}\right )\right ) x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {\left (2+\sqrt {2}\right ) x^2}{2-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 )+2 \left (2-x^2\right ) \left (2-3 \left (1+\sqrt {2}\right ) x^2\right )\right )}{256 x^3 \left (2-x^2\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[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)*(2*(2 - x^2)*(2 - 3*(1 - Sqrt[2])*x^2) + (2*(2 - Sqrt[2])*x^2 - 3*(4 - 3*S
qrt[2])*x^4)*Hypergeometric2F1[2/3, 1, 5/3, -(((2 - Sqrt[2])*x^2)/(2 - x^2))] - 3*x^2*(2*(2 - Sqrt[2]) + (4 -
3*Sqrt[2])*x^2)*Hypergeometric2F1[2/3, 2, 5/3, -(((2 - Sqrt[2])*x^2)/(2 - x^2))]))/(256*x^3*(2 - x^2)) + (3*(4
+ Sqrt[2])*(-2*x + x^3)^(1/3)*(2*(2 - x^2)*(2 - 3*(1 + Sqrt[2])*x^2) + (2*(2 + Sqrt[2])*x^2 - 3*(4 + 3*Sqrt[2
])*x^4)*Hypergeometric2F1[2/3, 1, 5/3, -(((2 + Sqrt[2])*x^2)/(2 - x^2))] - 3*x^2*(2*(2 + Sqrt[2]) + (4 + 3*Sqr
t[2])*x^2)*Hypergeometric2F1[2/3, 2, 5/3, -(((2 + Sqrt[2])*x^2)/(2 - x^2))]))/(256*x^3*(2 - x^2))

Rule 465

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 466

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 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 511

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

Rule 2056

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 6728

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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 [B]  time = 1.13, size = 318, normalized size = 2.74 \begin {gather*} \frac {3 \left (-\frac {\left (3+\sqrt {2}\right ) \left (\left (3 \left (3 \sqrt {2}-4\right ) x^2-2 \sqrt {2}+4\right ) x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {\left (-2+\sqrt {2}\right ) x^2}{x^2-2}\right )+3 \left (\left (3 \sqrt {2}-4\right ) x^2+2 \left (\sqrt {2}-2\right )\right ) x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {\left (-2+\sqrt {2}\right ) x^2}{x^2-2}\right )-2 \left (x^2-2\right ) \left (3 \left (\sqrt {2}-1\right ) x^2+2\right )\right )}{2+\sqrt {2}}-\left (1+\frac {1}{\sqrt {2}}\right ) \left (3-\sqrt {2}\right ) \left (-3 \left (\left (4+3 \sqrt {2}\right ) x^2+2 \left (2+\sqrt {2}\right )\right ) x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {\left (2+\sqrt {2}\right ) x^2}{x^2-2}\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}{x^2-2}\right )+2 \left (2-x^2\right ) \left (2-3 \left (1+\sqrt {2}\right ) x^2\right )\right )\right )}{128 x^2 \left (x \left (x^2-2\right )\right )^{2/3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(3*(-(((3 + Sqrt[2])*(-2*(-2 + x^2)*(2 + 3*(-1 + Sqrt[2])*x^2) + x^2*(4 - 2*Sqrt[2] + 3*(-4 + 3*Sqrt[2])*x^2)*
Hypergeometric2F1[2/3, 1, 5/3, -(((-2 + Sqrt[2])*x^2)/(-2 + x^2))] + 3*x^2*(2*(-2 + Sqrt[2]) + (-4 + 3*Sqrt[2]
)*x^2)*Hypergeometric2F1[2/3, 2, 5/3, -(((-2 + Sqrt[2])*x^2)/(-2 + x^2))]))/(2 + Sqrt[2])) - (1 + 1/Sqrt[2])*(
3 - Sqrt[2])*(2*(2 - x^2)*(2 - 3*(1 + Sqrt[2])*x^2) + (2*(2 + Sqrt[2])*x^2 - 3*(4 + 3*Sqrt[2])*x^4)*Hypergeome
tric2F1[2/3, 1, 5/3, ((2 + Sqrt[2])*x^2)/(-2 + x^2)] - 3*x^2*(2*(2 + Sqrt[2]) + (4 + 3*Sqrt[2])*x^2)*Hypergeom
etric2F1[2/3, 2, 5/3, ((2 + Sqrt[2])*x^2)/(-2 + x^2)])))/(128*x^2*(x*(-2 + x^2))^(2/3))

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IntegrateAlgebraic [A]  time = 0.29, size = 116, normalized size = 1.00 \begin {gather*} -\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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(-2 + 7*x^2)*(-2*x + x^3)^(1/3))/(16*x^3) + RootSum[2 - 4*#1^3 + #1^6 & , (-6*Log[x] + 6*Log[(-2*x + x^3)^
(1/3) - x*#1] + 11*Log[x]*#1^3 - 11*Log[(-2*x + x^3)^(1/3) - x*#1]*#1^3)/(-2*#1^2 + #1^5) & ]/16

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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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - 2 \, x\right )}^{\frac {1}{3}} {\left (x^{2} + 4\right )}}{{\left (x^{4} - 4 \, x^{2} - 4\right )} x^{4}}\,{d x} \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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maple [B]  time = 265.77, size = 7672, normalized size = 66.14

method	result	size
trager	\(\text {Expression too large to display}\)	\(7672\)
risch	\(\text {Expression too large to display}\)	\(16927\)

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,method=_RETURNVERBOSE)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {3 \, {\left (36 \, x^{7} - 36 \, x^{5} + 5 \, {\left (3 \, x^{5} + 2 \, x^{3} - 16 \, x\right )} x^{2} - 8 \, x^{3} - 128 \, x\right )} {\left (x^{2} - 2\right )}^{\frac {1}{3}}}{1120 \, {\left (x^{\frac {23}{3}} - 4 \, x^{\frac {17}{3}} - 4 \, x^{\frac {11}{3}}\right )}} + \int \frac {3 \, {\left (36 \, x^{6} - 6 \, x^{4} + {\left (18 \, x^{6} + 27 \, x^{4} + 26 \, x^{2} - 304\right )} x^{2} + 12 \, x^{2} - 288\right )} {\left (x^{2} - 2\right )}^{\frac {1}{3}}}{70 \, {\left (x^{\frac {35}{3}} - 8 \, x^{\frac {29}{3}} + 8 \, x^{\frac {23}{3}} + 32 \, x^{\frac {17}{3}} + 16 \, x^{\frac {11}{3}}\right )}}\,{d x} \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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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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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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