3.18.64 \(\int \frac {\sqrt [3]{x+2 x^3} (-1+x^4)}{x^4 (2-x^2+x^4)} \, dx\)

Optimal. Leaf size=119 \[ \frac {1}{8} \text {RootSum}\left [2 \text {$\#$1}^6-9 \text {$\#$1}^3+11\& ,\frac {-\text {$\#$1}^3 \log \left (\sqrt [3]{2 x^3+x}-\text {$\#$1} x\right )+\text {$\#$1}^3 \log (x)-11 \log \left (\sqrt [3]{2 x^3+x}-\text {$\#$1} x\right )+11 \log (x)}{4 \text {$\#$1}^5-9 \text {$\#$1}^2}\& \right ]+\frac {3 \sqrt [3]{2 x^3+x} \left (4 x^2+1\right )}{16 x^3} \]

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Rubi [C]  time = 1.87, antiderivative size = 504, normalized size of antiderivative = 4.24, number of steps used = 12, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2056, 6728, 264, 466, 465, 510} \begin {gather*} \frac {3 \left (-5 \sqrt {7}+7 i\right ) \sqrt [3]{2 x^3+x} \left (-\left (\left (-6 \left (2+i \sqrt {7}\right ) x^2-3 i \sqrt {7}+5\right ) x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 \left (2 i-\sqrt {7}\right ) x^2}{\left (i-\sqrt {7}\right ) \left (2 x^2+1\right )}\right )\right )+3 \left (2 \left (2+i \sqrt {7}\right ) x^2-3 i \sqrt {7}+5\right ) x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 \left (2 i-\sqrt {7}\right ) x^2}{\left (i-\sqrt {7}\right ) \left (2 x^2+1\right )}\right )+2 \left (2 x^2+1\right ) \left (-3 \left (1+i \sqrt {7}\right ) x^2-i \sqrt {7}+3\right )\right )}{224 \left (-\sqrt {7}+5 i\right ) x^3 \left (2 x^2+1\right )}+\frac {3 \left (5 \sqrt {7}+7 i\right ) \sqrt [3]{2 x^3+x} \left (-\left (\left (-6 \left (2-i \sqrt {7}\right ) x^2+3 i \sqrt {7}+5\right ) x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 \left (2 i+\sqrt {7}\right ) x^2}{\left (i+\sqrt {7}\right ) \left (2 x^2+1\right )}\right )\right )+3 \left (2 \left (2-i \sqrt {7}\right ) x^2+3 i \sqrt {7}+5\right ) x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 \left (2 i+\sqrt {7}\right ) x^2}{\left (i+\sqrt {7}\right ) \left (2 x^2+1\right )}\right )+2 \left (2 x^2+1\right ) \left (-3 \left (1-i \sqrt {7}\right ) x^2+i \sqrt {7}+3\right )\right )}{224 \left (\sqrt {7}+5 i\right ) x^3 \left (2 x^2+1\right )}-\frac {3 \sqrt [3]{2 x^3+x} \left (2 x^2+1\right )}{8 x^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(1 + 2*x^2)*(x + 2*x^3)^(1/3))/(8*x^3) + (3*(7*I - 5*Sqrt[7])*(x + 2*x^3)^(1/3)*(2*(1 + 2*x^2)*(3 - I*Sqrt
[7] - 3*(1 + I*Sqrt[7])*x^2) - x^2*(5 - (3*I)*Sqrt[7] - 6*(2 + I*Sqrt[7])*x^2)*Hypergeometric2F1[2/3, 1, 5/3,
(2*(2*I - Sqrt[7])*x^2)/((I - Sqrt[7])*(1 + 2*x^2))] + 3*x^2*(5 - (3*I)*Sqrt[7] + 2*(2 + I*Sqrt[7])*x^2)*Hyper
geometric2F1[2/3, 2, 5/3, (2*(2*I - Sqrt[7])*x^2)/((I - Sqrt[7])*(1 + 2*x^2))]))/(224*(5*I - Sqrt[7])*x^3*(1 +
2*x^2)) + (3*(7*I + 5*Sqrt[7])*(x + 2*x^3)^(1/3)*(2*(1 + 2*x^2)*(3 + I*Sqrt[7] - 3*(1 - I*Sqrt[7])*x^2) - x^2
*(5 + (3*I)*Sqrt[7] - 6*(2 - I*Sqrt[7])*x^2)*Hypergeometric2F1[2/3, 1, 5/3, (2*(2*I + Sqrt[7])*x^2)/((I + Sqrt
[7])*(1 + 2*x^2))] + 3*x^2*(5 + (3*I)*Sqrt[7] + 2*(2 - I*Sqrt[7])*x^2)*Hypergeometric2F1[2/3, 2, 5/3, (2*(2*I
+ Sqrt[7])*x^2)/((I + Sqrt[7])*(1 + 2*x^2))]))/(224*(5*I + Sqrt[7])*x^3*(1 + 2*x^2))

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 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 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 {\sqrt [3]{x+2 x^3} \left (-1+x^4\right )}{x^4 \left (2-x^2+x^4\right )} \, dx &=\frac {\sqrt [3]{x+2 x^3} \int \frac {\sqrt [3]{1+2 x^2} \left (-1+x^4\right )}{x^{11/3} \left (2-x^2+x^4\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ &=\frac {\sqrt [3]{x+2 x^3} \int \left (\frac {\sqrt [3]{1+2 x^2}}{x^{11/3}}-\frac {\left (3-x^2\right ) \sqrt [3]{1+2 x^2}}{x^{11/3} \left (2-x^2+x^4\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ &=\frac {\sqrt [3]{x+2 x^3} \int \frac {\sqrt [3]{1+2 x^2}}{x^{11/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}-\frac {\sqrt [3]{x+2 x^3} \int \frac {\left (3-x^2\right ) \sqrt [3]{1+2 x^2}}{x^{11/3} \left (2-x^2+x^4\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ &=-\frac {3 \left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{8 x^3}-\frac {\sqrt [3]{x+2 x^3} \int \left (\frac {\left (-1-\frac {5 i}{\sqrt {7}}\right ) \sqrt [3]{1+2 x^2}}{x^{11/3} \left (-1-i \sqrt {7}+2 x^2\right )}+\frac {\left (-1+\frac {5 i}{\sqrt {7}}\right ) \sqrt [3]{1+2 x^2}}{x^{11/3} \left (-1+i \sqrt {7}+2 x^2\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ &=-\frac {3 \left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{8 x^3}-\frac {\left (\left (-7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \int \frac {\sqrt [3]{1+2 x^2}}{x^{11/3} \left (-1+i \sqrt {7}+2 x^2\right )} \, dx}{7 \sqrt [3]{x} \sqrt [3]{1+2 x^2}}+\frac {\left (\left (7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \int \frac {\sqrt [3]{1+2 x^2}}{x^{11/3} \left (-1-i \sqrt {7}+2 x^2\right )} \, dx}{7 \sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ &=-\frac {3 \left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{8 x^3}-\frac {\left (3 \left (-7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6}}{x^9 \left (-1+i \sqrt {7}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{7 \sqrt [3]{x} \sqrt [3]{1+2 x^2}}+\frac {\left (3 \left (7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6}}{x^9 \left (-1-i \sqrt {7}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{7 \sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ &=-\frac {3 \left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{8 x^3}-\frac {\left (3 \left (-7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^3}}{x^5 \left (-1+i \sqrt {7}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{14 \sqrt [3]{x} \sqrt [3]{1+2 x^2}}+\frac {\left (3 \left (7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^3}}{x^5 \left (-1-i \sqrt {7}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{14 \sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ &=-\frac {3 \left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{8 x^3}+\frac {3 \left (7 i-5 \sqrt {7}\right ) \sqrt [3]{x+2 x^3} \left (2 \left (1+2 x^2\right ) \left (3-i \sqrt {7}-3 \left (1+i \sqrt {7}\right ) x^2\right )-x^2 \left (5-3 i \sqrt {7}-6 \left (2+i \sqrt {7}\right ) x^2\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 \left (2 i-\sqrt {7}\right ) x^2}{\left (i-\sqrt {7}\right ) \left (1+2 x^2\right )}\right )+3 x^2 \left (5-3 i \sqrt {7}+2 \left (2+i \sqrt {7}\right ) x^2\right ) \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 \left (2 i-\sqrt {7}\right ) x^2}{\left (i-\sqrt {7}\right ) \left (1+2 x^2\right )}\right )\right )}{224 \left (5 i-\sqrt {7}\right ) x^3 \left (1+2 x^2\right )}+\frac {3 \left (7 i+5 \sqrt {7}\right ) \sqrt [3]{x+2 x^3} \left (2 \left (1+2 x^2\right ) \left (3+i \sqrt {7}-3 \left (1-i \sqrt {7}\right ) x^2\right )-x^2 \left (5+3 i \sqrt {7}-6 \left (2-i \sqrt {7}\right ) x^2\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 \left (2 i+\sqrt {7}\right ) x^2}{\left (i+\sqrt {7}\right ) \left (1+2 x^2\right )}\right )+3 x^2 \left (5+3 i \sqrt {7}+2 \left (2-i \sqrt {7}\right ) x^2\right ) \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 \left (2 i+\sqrt {7}\right ) x^2}{\left (i+\sqrt {7}\right ) \left (1+2 x^2\right )}\right )\right )}{224 \left (5 i+\sqrt {7}\right ) x^3 \left (1+2 x^2\right )}\\ \end {align*}

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Mathematica [C]  time = 0.65, size = 453, normalized size = 3.81 \begin {gather*} \frac {3 \sqrt [3]{2 x^3+x} \left (\frac {\left (-5 \sqrt {7}+7 i\right ) \left (\left (6 \left (2+i \sqrt {7}\right ) x^2+3 i \sqrt {7}-5\right ) x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 \left (-2 i+\sqrt {7}\right ) x^2}{\left (-i+\sqrt {7}\right ) \left (2 x^2+1\right )}\right )+3 \left (\left (4+2 i \sqrt {7}\right ) x^2-3 i \sqrt {7}+5\right ) x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 \left (-2 i+\sqrt {7}\right ) x^2}{\left (-i+\sqrt {7}\right ) \left (2 x^2+1\right )}\right )+2 \left (2 x^2+1\right ) \left (\left (-3-3 i \sqrt {7}\right ) x^2-i \sqrt {7}+3\right )\right )}{-\sqrt {7}+5 i}+\frac {\left (5 \sqrt {7}+7 i\right ) \left (-\left (\left (6 i \left (\sqrt {7}+2 i\right ) x^2+3 i \sqrt {7}+5\right ) x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 \left (2 i+\sqrt {7}\right ) x^2}{\left (i+\sqrt {7}\right ) \left (2 x^2+1\right )}\right )\right )+3 \left (\left (4-2 i \sqrt {7}\right ) x^2+3 i \sqrt {7}+5\right ) x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 \left (2 i+\sqrt {7}\right ) x^2}{\left (i+\sqrt {7}\right ) \left (2 x^2+1\right )}\right )+2 \left (2 x^2+1\right ) \left (3 i \left (\sqrt {7}+i\right ) x^2+i \sqrt {7}+3\right )\right )}{\sqrt {7}+5 i}-28 \left (2 x^2+1\right )^2\right )}{224 x^3 \left (2 x^2+1\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(3*(x + 2*x^3)^(1/3)*(-28*(1 + 2*x^2)^2 + ((7*I - 5*Sqrt[7])*(2*(1 + 2*x^2)*(3 - I*Sqrt[7] + (-3 - (3*I)*Sqrt[
7])*x^2) + x^2*(-5 + (3*I)*Sqrt[7] + 6*(2 + I*Sqrt[7])*x^2)*Hypergeometric2F1[2/3, 1, 5/3, (2*(-2*I + Sqrt[7])
*x^2)/((-I + Sqrt[7])*(1 + 2*x^2))] + 3*x^2*(5 - (3*I)*Sqrt[7] + (4 + (2*I)*Sqrt[7])*x^2)*Hypergeometric2F1[2/
3, 2, 5/3, (2*(-2*I + Sqrt[7])*x^2)/((-I + Sqrt[7])*(1 + 2*x^2))]))/(5*I - Sqrt[7]) + ((7*I + 5*Sqrt[7])*(2*(1
+ 2*x^2)*(3 + I*Sqrt[7] + (3*I)*(I + Sqrt[7])*x^2) - x^2*(5 + (3*I)*Sqrt[7] + (6*I)*(2*I + Sqrt[7])*x^2)*Hype
rgeometric2F1[2/3, 1, 5/3, (2*(2*I + Sqrt[7])*x^2)/((I + Sqrt[7])*(1 + 2*x^2))] + 3*x^2*(5 + (3*I)*Sqrt[7] + (
4 - (2*I)*Sqrt[7])*x^2)*Hypergeometric2F1[2/3, 2, 5/3, (2*(2*I + Sqrt[7])*x^2)/((I + Sqrt[7])*(1 + 2*x^2))]))/
(5*I + Sqrt[7])))/(224*x^3*(1 + 2*x^2))

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IntegrateAlgebraic [A]  time = 0.28, size = 119, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+4 x^2\right ) \sqrt [3]{x+2 x^3}}{16 x^3}+\frac {1}{8} \text {RootSum}\left [11-9 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {11 \log (x)-11 \log \left (\sqrt [3]{x+2 x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{x+2 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-9 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(1 + 4*x^2)*(x + 2*x^3)^(1/3))/(16*x^3) + RootSum[11 - 9*#1^3 + 2*#1^6 & , (11*Log[x] - 11*Log[(x + 2*x^3)^
(1/3) - x*#1] + Log[x]*#1^3 - Log[(x + 2*x^3)^(1/3) - x*#1]*#1^3)/(-9*#1^2 + 4*#1^5) & ]/8

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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((2*x^3+x)^(1/3)*(x^4-1)/x^4/(x^4-x^2+2),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^{4} - 1\right )} {\left (2 \, x^{3} + x\right )}^{\frac {1}{3}}}{{\left (x^{4} - x^{2} + 2\right )} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 257.41, size = 12390, normalized size = 104.12

method	result	size
trager	\(\text {Expression too large to display}\)	\(12390\)
risch	\(\text {Expression too large to display}\)	\(13837\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^3+x)^(1/3)*(x^4-1)/x^4/(x^4-x^2+2),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*} \int \frac {{\left (x^{4} - 1\right )} {\left (2 \, x^{3} + x\right )}^{\frac {1}{3}}}{{\left (x^{4} - x^{2} + 2\right )} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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