\(\int \frac {x^2 (-2+x^4)}{\sqrt [3]{-x+x^5} (-1+x^4+x^8)} \, dx\) [1466]

Optimal result

Integrand size = 30, antiderivative size = 103 \[ \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx=\frac {1}{4} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-x+x^5}}{-2 x^3+\sqrt [3]{-x+x^5}}\right )+\frac {1}{4} \log \left (x^3+\sqrt [3]{-x+x^5}\right )-\frac {1}{8} \log \left (x^6-x^3 \sqrt [3]{-x+x^5}+\left (-x+x^5\right )^{2/3}\right ) \]

[Out]

1/4*3^(1/2)*arctan(3^(1/2)*(x^5-x)^(1/3)/(-2*x^3+(x^5-x)^(1/3)))+1/4*ln(x^3+(x^5-x)^(1/3))-1/8*ln(x^6-x^3*(x^5
-x)^(1/3)+(x^5-x)^(2/3))

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.43 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2081, 6860, 477, 476, 525, 524} \[ \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx=\frac {3 \sqrt [3]{1-x^4} x^3 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^4,-\frac {2 x^4}{1-\sqrt {5}}\right )}{8 \sqrt [3]{x^5-x}}+\frac {3 \sqrt [3]{1-x^4} x^3 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^4,-\frac {2 x^4}{1+\sqrt {5}}\right )}{8 \sqrt [3]{x^5-x}} \]

[In]

Int[(x^2*(-2 + x^4))/((-x + x^5)^(1/3)*(-1 + x^4 + x^8)),x]

[Out]

(3*x^3*(1 - x^4)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^4, (-2*x^4)/(1 - Sqrt[5])])/(8*(-x + x^5)^(1/3)) + (3*x^3*
(1 - x^4)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^4, (-2*x^4)/(1 + Sqrt[5])])/(8*(-x + x^5)^(1/3))

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 524

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)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], 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 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[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 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*} \text {integral}& = \frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \frac {x^{5/3} \left (-2+x^4\right )}{\sqrt [3]{-1+x^4} \left (-1+x^4+x^8\right )} \, dx}{\sqrt [3]{-x+x^5}} \\ & = \frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \left (\frac {\left (1-\sqrt {5}\right ) x^{5/3}}{\sqrt [3]{-1+x^4} \left (1-\sqrt {5}+2 x^4\right )}+\frac {\left (1+\sqrt {5}\right ) x^{5/3}}{\sqrt [3]{-1+x^4} \left (1+\sqrt {5}+2 x^4\right )}\right ) \, dx}{\sqrt [3]{-x+x^5}} \\ & = \frac {\left (\left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \frac {x^{5/3}}{\sqrt [3]{-1+x^4} \left (1-\sqrt {5}+2 x^4\right )} \, dx}{\sqrt [3]{-x+x^5}}+\frac {\left (\left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \frac {x^{5/3}}{\sqrt [3]{-1+x^4} \left (1+\sqrt {5}+2 x^4\right )} \, dx}{\sqrt [3]{-x+x^5}} \\ & = \frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \text {Subst}\left (\int \frac {x^7}{\sqrt [3]{-1+x^{12}} \left (1-\sqrt {5}+2 x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \text {Subst}\left (\int \frac {x^7}{\sqrt [3]{-1+x^{12}} \left (1+\sqrt {5}+2 x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}} \\ & = \frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^3} \left (1-\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^3} \left (1+\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}} \\ & = \frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^3} \left (1-\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^3} \left (1+\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}} \\ & = \frac {3 x^3 \sqrt [3]{1-x^4} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^4,-\frac {2 x^4}{1-\sqrt {5}}\right )}{8 \sqrt [3]{-x+x^5}}+\frac {3 x^3 \sqrt [3]{1-x^4} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^4,-\frac {2 x^4}{1+\sqrt {5}}\right )}{8 \sqrt [3]{-x+x^5}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.45 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx=\frac {1}{4} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-x+x^5}}{-2 x^3+\sqrt [3]{-x+x^5}}\right )+\frac {1}{4} \log \left (x^3+\sqrt [3]{-x+x^5}\right )-\frac {1}{8} \log \left (x^6-x^3 \sqrt [3]{-x+x^5}+\left (-x+x^5\right )^{2/3}\right ) \]

[In]

Integrate[(x^2*(-2 + x^4))/((-x + x^5)^(1/3)*(-1 + x^4 + x^8)),x]

[Out]

(Sqrt[3]*ArcTan[(Sqrt[3]*(-x + x^5)^(1/3))/(-2*x^3 + (-x + x^5)^(1/3))])/4 + Log[x^3 + (-x + x^5)^(1/3)]/4 - L
og[x^6 - x^3*(-x + x^5)^(1/3) + (-x + x^5)^(2/3)]/8

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 27.87 (sec) , antiderivative size = 626, normalized size of antiderivative = 6.08

method	result	size
trager	\(\text {Expression too large to display}\)	\(626\)

[In]

int(x^2*(x^4-2)/(x^5-x)^(1/3)/(x^8+x^4-1),x,method=_RETURNVERBOSE)

[Out]

RootOf(16*_Z^2+4*_Z+1)*ln(-(1027936778409280454041232991842640*RootOf(16*_Z^2+4*_Z+1)^2*x^8-401507123405306026
5622960199240704*RootOf(16*_Z^2+4*_Z+1)*x^8-457129439550601836183569143942165*x^8+6458608904506033300021952070
673408*RootOf(16*_Z^2+4*_Z+1)*(x^5-x)^(1/3)*x^5-17543454351518386415637043060781056*RootOf(16*_Z^2+4*_Z+1)^2*x
^4+1589389345365026959344463317732671*(x^5-x)^(1/3)*x^5-6458608904506033300021952070673408*RootOf(16*_Z^2+4*_Z
+1)*(x^5-x)^(2/3)*x^2-2199310112028943683020577141723220*RootOf(16*_Z^2+4*_Z+1)*x^4-15893893453650269593444633
17732671*(x^5-x)^(2/3)*x^2-28452039806054056193998079498475*x^4+17543454351518386415637043060781056*RootOf(16*
_Z^2+4*_Z+1)^2+2199310112028943683020577141723220*RootOf(16*_Z^2+4*_Z+1)+28452039806054056193998079498475)/(x^
8+x^4-1))-1/4*ln(-(1027936778409280454041232991842640*RootOf(16*_Z^2+4*_Z+1)^2*x^8+452903962325770049264357669
5162024*RootOf(16*_Z^2+4*_Z+1)*x^8+610884417613243258599747967858176*x^8-6458608904506033300021952070673408*Ro
otOf(16*_Z^2+4*_Z+1)*(x^5-x)^(1/3)*x^5-17543454351518386415637043060781056*RootOf(16*_Z^2+4*_Z+1)^2*x^4-252628
80761481365661024699935681*(x^5-x)^(1/3)*x^5+6458608904506033300021952070673408*RootOf(16*_Z^2+4*_Z+1)*(x^5-x)
^(2/3)*x^2-6572417063730249524797944388667308*RootOf(16*_Z^2+4*_Z+1)*x^4+25262880761481365661024699935681*(x^5
-x)^(2/3)*x^2-575090408768717286416168985366486*x^4+17543454351518386415637043060781056*RootOf(16*_Z^2+4*_Z+1)
^2+6572417063730249524797944388667308*RootOf(16*_Z^2+4*_Z+1)+575090408768717286416168985366486)/(x^8+x^4-1))-l
n(-(1027936778409280454041232991842640*RootOf(16*_Z^2+4*_Z+1)^2*x^8+4529039623257700492643576695162024*RootOf(
16*_Z^2+4*_Z+1)*x^8+610884417613243258599747967858176*x^8-6458608904506033300021952070673408*RootOf(16*_Z^2+4*
_Z+1)*(x^5-x)^(1/3)*x^5-17543454351518386415637043060781056*RootOf(16*_Z^2+4*_Z+1)^2*x^4-252628807614813656610
24699935681*(x^5-x)^(1/3)*x^5+6458608904506033300021952070673408*RootOf(16*_Z^2+4*_Z+1)*(x^5-x)^(2/3)*x^2-6572
417063730249524797944388667308*RootOf(16*_Z^2+4*_Z+1)*x^4+25262880761481365661024699935681*(x^5-x)^(2/3)*x^2-5
75090408768717286416168985366486*x^4+17543454351518386415637043060781056*RootOf(16*_Z^2+4*_Z+1)^2+657241706373
0249524797944388667308*RootOf(16*_Z^2+4*_Z+1)+575090408768717286416168985366486)/(x^8+x^4-1))*RootOf(16*_Z^2+4
*_Z+1)

Fricas [A] (verification not implemented)

none

Time = 2.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.09 \[ \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx=-\frac {1}{4} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{8} + 2 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {1}{3}} x^{5} + 4 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {2}{3}} x^{2}}{x^{8} - 8 \, x^{4} + 8}\right ) + \frac {1}{8} \, \log \left (\frac {x^{8} + 3 \, {\left (x^{5} - x\right )}^{\frac {1}{3}} x^{5} + x^{4} + 3 \, {\left (x^{5} - x\right )}^{\frac {2}{3}} x^{2} - 1}{x^{8} + x^{4} - 1}\right ) \]

[In]

integrate(x^2*(x^4-2)/(x^5-x)^(1/3)/(x^8+x^4-1),x, algorithm="fricas")

[Out]

-1/4*sqrt(3)*arctan((sqrt(3)*x^8 + 2*sqrt(3)*(x^5 - x)^(1/3)*x^5 + 4*sqrt(3)*(x^5 - x)^(2/3)*x^2)/(x^8 - 8*x^4
+ 8)) + 1/8*log((x^8 + 3*(x^5 - x)^(1/3)*x^5 + x^4 + 3*(x^5 - x)^(2/3)*x^2 - 1)/(x^8 + x^4 - 1))

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx=\text {Timed out} \]

[In]

integrate(x**2*(x**4-2)/(x**5-x)**(1/3)/(x**8+x**4-1),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx=\int { \frac {{\left (x^{4} - 2\right )} x^{2}}{{\left (x^{8} + x^{4} - 1\right )} {\left (x^{5} - x\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate(x^2*(x^4-2)/(x^5-x)^(1/3)/(x^8+x^4-1),x, algorithm="maxima")

[Out]

integrate((x^4 - 2)*x^2/((x^8 + x^4 - 1)*(x^5 - x)^(1/3)), x)

Giac [F]

\[ \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx=\int { \frac {{\left (x^{4} - 2\right )} x^{2}}{{\left (x^{8} + x^{4} - 1\right )} {\left (x^{5} - x\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate(x^2*(x^4-2)/(x^5-x)^(1/3)/(x^8+x^4-1),x, algorithm="giac")

[Out]

integrate((x^4 - 2)*x^2/((x^8 + x^4 - 1)*(x^5 - x)^(1/3)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx=\int \frac {x^2\,\left (x^4-2\right )}{{\left (x^5-x\right )}^{1/3}\,\left (x^8+x^4-1\right )} \,d x \]

[In]

int((x^2*(x^4 - 2))/((x^5 - x)^(1/3)*(x^4 + x^8 - 1)),x)

[Out]

int((x^2*(x^4 - 2))/((x^5 - x)^(1/3)*(x^4 + x^8 - 1)), x)