\(\int \frac {(-3+x^4) (1-2 x^3+x^4) (1-x^3+x^4)}{x^6 (1+x^4) \sqrt [4]{x+x^5}} \, dx\) [612]

Optimal result

Integrand size = 45, antiderivative size = 48 \[ \int \frac {\left (-3+x^4\right ) \left (1-2 x^3+x^4\right ) \left (1-x^3+x^4\right )}{x^6 \left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\frac {4 \left (x+x^5\right )^{3/4} \left (1-7 x^3+2 x^4-14 x^6-7 x^7+x^8\right )}{7 x^6 \left (1+x^4\right )} \]

[Out]

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 2 in optimal.

Time = 0.38 (sec) , antiderivative size = 208, normalized size of antiderivative = 4.33, number of steps used = 18, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2081, 1847, 1599, 1492, 460, 1849, 1600, 1858, 371} \[ \int \frac {\left (-3+x^4\right ) \left (1-2 x^3+x^4\right ) \left (1-x^3+x^4\right )}{x^6 \left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\frac {8 \sqrt [4]{x^4+1} x^5 \operatorname {Hypergeometric2F1}\left (\frac {19}{16},\frac {5}{4},\frac {35}{16},-x^4\right )}{19 \sqrt [4]{x^5+x}}-\frac {8 \sqrt [4]{x^4+1} x \operatorname {Hypergeometric2F1}\left (\frac {3}{16},\frac {5}{4},\frac {19}{16},-x^4\right )}{\sqrt [4]{x^5+x}}+\frac {4 \sqrt [4]{x^4+1} x^7 \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {27}{16},\frac {43}{16},-x^4\right )}{27 \sqrt [4]{x^5+x}}+\frac {4 \sqrt [4]{x^4+1} x^3 \operatorname {Hypergeometric2F1}\left (\frac {11}{16},\frac {5}{4},\frac {27}{16},-x^4\right )}{7 \sqrt [4]{x^5+x}}+\frac {8}{7 \sqrt [4]{x^5+x} x}+\frac {4}{7 \sqrt [4]{x^5+x} x^5}-\frac {4 \left (x^4+1\right )}{\sqrt [4]{x^5+x} x^2} \]

[In]

Int[((-3 + x^4)*(1 - 2*x^3 + x^4)*(1 - x^3 + x^4))/(x^6*(1 + x^4)*(x + x^5)^(1/4)),x]

[Out]

4/(7*x^5*(x + x^5)^(1/4)) + 8/(7*x*(x + x^5)^(1/4)) - (4*(1 + x^4))/(x^2*(x + x^5)^(1/4)) - (8*x*(1 + x^4)^(1/
4)*Hypergeometric2F1[3/16, 5/4, 19/16, -x^4])/(x + x^5)^(1/4) + (4*x^3*(1 + x^4)^(1/4)*Hypergeometric2F1[11/16
, 5/4, 27/16, -x^4])/(7*(x + x^5)^(1/4)) + (8*x^5*(1 + x^4)^(1/4)*Hypergeometric2F1[19/16, 5/4, 35/16, -x^4])/
(19*(x + x^5)^(1/4)) + (4*x^7*(1 + x^4)^(1/4)*Hypergeometric2F1[5/4, 27/16, 43/16, -x^4])/(27*(x + x^5)^(1/4))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 460

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

Rule 1492

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x_Sym
bol] :> Int[(f*x)^m*(d + e*x^n)^(q + p)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && Eq
Q[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rule 1599

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

Rule 1600

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 1847

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[
Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {
j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] &&  !PolyQ[Pq, x^(n/2)]

Rule 1849

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{Pq0 = Coeff[Pq, x, 0]}, Simp[Pq0
*(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] + Dist[1/(2*a*c*(m + 1)), Int[(c*x)^(m + 1)*ExpandToSum
[2*a*(m + 1)*((Pq - Pq0)/x) - 2*b*Pq0*(m + n*(p + 1) + 1)*x^(n - 1), x]*(a + b*x^n)^p, x], x] /; NeQ[Pq0, 0]]
/; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[m, -1] && LeQ[n - 1, Expon[Pq, x]]

Rule 1858

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IGtQ[m, 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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {\left (-3+x^4\right ) \left (1-2 x^3+x^4\right ) \left (1-x^3+x^4\right )}{x^{25/4} \left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x+x^5}} \\ & = \frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \left (\frac {9 x^2+6 x^6-3 x^{10}}{x^{21/4} \left (1+x^4\right )^{5/4}}+\frac {-3-5 x^4-6 x^6-x^8+2 x^{10}+x^{12}}{x^{25/4} \left (1+x^4\right )^{5/4}}\right ) \, dx}{\sqrt [4]{x+x^5}} \\ & = \frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {9 x^2+6 x^6-3 x^{10}}{x^{21/4} \left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x+x^5}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {-3-5 x^4-6 x^6-x^8+2 x^{10}+x^{12}}{x^{25/4} \left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x+x^5}} \\ & = \frac {4}{7 x^5 \sqrt [4]{x+x^5}}-\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {15 x^3+63 x^5+\frac {21 x^7}{2}-21 x^9-\frac {21 x^{11}}{2}}{x^{21/4} \left (1+x^4\right )^{5/4}} \, dx}{21 \sqrt [4]{x+x^5}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {9+6 x^4-3 x^8}{x^{13/4} \left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x+x^5}} \\ & = \frac {4}{7 x^5 \sqrt [4]{x+x^5}}-\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {15 x^2+63 x^4+\frac {21 x^6}{2}-21 x^8-\frac {21 x^{10}}{2}}{x^{17/4} \left (1+x^4\right )^{5/4}} \, dx}{21 \sqrt [4]{x+x^5}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {9-3 x^4}{x^{13/4} \sqrt [4]{1+x^4}} \, dx}{\sqrt [4]{x+x^5}} \\ & = \frac {4}{7 x^5 \sqrt [4]{x+x^5}}-\frac {4 \left (1+x^4\right )}{x^2 \sqrt [4]{x+x^5}}-\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {15 x+63 x^3+\frac {21 x^5}{2}-21 x^7-\frac {21 x^9}{2}}{x^{13/4} \left (1+x^4\right )^{5/4}} \, dx}{21 \sqrt [4]{x+x^5}} \\ & = \frac {4}{7 x^5 \sqrt [4]{x+x^5}}-\frac {4 \left (1+x^4\right )}{x^2 \sqrt [4]{x+x^5}}-\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {15+63 x^2+\frac {21 x^4}{2}-21 x^6-\frac {21 x^8}{2}}{x^{9/4} \left (1+x^4\right )^{5/4}} \, dx}{21 \sqrt [4]{x+x^5}} \\ & = \frac {4}{7 x^5 \sqrt [4]{x+x^5}}+\frac {8}{7 x \sqrt [4]{x+x^5}}-\frac {4 \left (1+x^4\right )}{x^2 \sqrt [4]{x+x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {-\frac {315 x}{2}+\frac {165 x^3}{4}+\frac {105 x^5}{2}+\frac {105 x^7}{4}}{x^{5/4} \left (1+x^4\right )^{5/4}} \, dx}{105 \sqrt [4]{x+x^5}} \\ & = \frac {4}{7 x^5 \sqrt [4]{x+x^5}}+\frac {8}{7 x \sqrt [4]{x+x^5}}-\frac {4 \left (1+x^4\right )}{x^2 \sqrt [4]{x+x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {-\frac {315}{2}+\frac {165 x^2}{4}+\frac {105 x^4}{2}+\frac {105 x^6}{4}}{\sqrt [4]{x} \left (1+x^4\right )^{5/4}} \, dx}{105 \sqrt [4]{x+x^5}} \\ & = \frac {4}{7 x^5 \sqrt [4]{x+x^5}}+\frac {8}{7 x \sqrt [4]{x+x^5}}-\frac {4 \left (1+x^4\right )}{x^2 \sqrt [4]{x+x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \left (-\frac {315}{2 \sqrt [4]{x} \left (1+x^4\right )^{5/4}}+\frac {165 x^{7/4}}{4 \left (1+x^4\right )^{5/4}}+\frac {105 x^{15/4}}{2 \left (1+x^4\right )^{5/4}}+\frac {105 x^{23/4}}{4 \left (1+x^4\right )^{5/4}}\right ) \, dx}{105 \sqrt [4]{x+x^5}} \\ & = \frac {4}{7 x^5 \sqrt [4]{x+x^5}}+\frac {8}{7 x \sqrt [4]{x+x^5}}-\frac {4 \left (1+x^4\right )}{x^2 \sqrt [4]{x+x^5}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{23/4}}{\left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x+x^5}}+\frac {\left (11 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{7/4}}{\left (1+x^4\right )^{5/4}} \, dx}{7 \sqrt [4]{x+x^5}}+\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{15/4}}{\left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x+x^5}}-\frac {\left (6 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x+x^5}} \\ & = \frac {4}{7 x^5 \sqrt [4]{x+x^5}}+\frac {8}{7 x \sqrt [4]{x+x^5}}-\frac {4 \left (1+x^4\right )}{x^2 \sqrt [4]{x+x^5}}-\frac {8 x \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {3}{16},\frac {5}{4},\frac {19}{16},-x^4\right )}{\sqrt [4]{x+x^5}}+\frac {4 x^3 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {11}{16},\frac {5}{4},\frac {27}{16},-x^4\right )}{7 \sqrt [4]{x+x^5}}+\frac {8 x^5 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {19}{16},\frac {5}{4},\frac {35}{16},-x^4\right )}{19 \sqrt [4]{x+x^5}}+\frac {4 x^7 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {27}{16},\frac {43}{16},-x^4\right )}{27 \sqrt [4]{x+x^5}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 2 in optimal.

Time = 10.13 (sec) , antiderivative size = 203, normalized size of antiderivative = 4.23 \[ \int \frac {\left (-3+x^4\right ) \left (1-2 x^3+x^4\right ) \left (1-x^3+x^4\right )}{x^6 \left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\frac {4 \sqrt [4]{1+x^4} \left (129789 \operatorname {Hypergeometric2F1}\left (-\frac {21}{16},\frac {5}{4},-\frac {5}{16},-x^4\right )+x^3 \left (-908523 \operatorname {Hypergeometric2F1}\left (-\frac {9}{16},\frac {5}{4},\frac {7}{16},-x^4\right )+908523 x \operatorname {Hypergeometric2F1}\left (-\frac {5}{16},\frac {5}{4},\frac {11}{16},-x^4\right )-1817046 x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{16},\frac {5}{4},\frac {19}{16},-x^4\right )+778734 x^4 \operatorname {Hypergeometric2F1}\left (\frac {7}{16},\frac {5}{4},\frac {23}{16},-x^4\right )-82593 x^5 \operatorname {Hypergeometric2F1}\left (\frac {11}{16},\frac {5}{4},\frac {27}{16},-x^4\right )+95634 x^7 \operatorname {Hypergeometric2F1}\left (\frac {19}{16},\frac {5}{4},\frac {35}{16},-x^4\right )-118503 x^8 \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {23}{16},\frac {39}{16},-x^4\right )+33649 x^9 \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {27}{16},\frac {43}{16},-x^4\right )\right )\right )}{908523 x^5 \sqrt [4]{x+x^5}} \]

[In]

Integrate[((-3 + x^4)*(1 - 2*x^3 + x^4)*(1 - x^3 + x^4))/(x^6*(1 + x^4)*(x + x^5)^(1/4)),x]

[Out]

(4*(1 + x^4)^(1/4)*(129789*Hypergeometric2F1[-21/16, 5/4, -5/16, -x^4] + x^3*(-908523*Hypergeometric2F1[-9/16,
5/4, 7/16, -x^4] + 908523*x*Hypergeometric2F1[-5/16, 5/4, 11/16, -x^4] - 1817046*x^3*Hypergeometric2F1[3/16,
5/4, 19/16, -x^4] + 778734*x^4*Hypergeometric2F1[7/16, 5/4, 23/16, -x^4] - 82593*x^5*Hypergeometric2F1[11/16,
5/4, 27/16, -x^4] + 95634*x^7*Hypergeometric2F1[19/16, 5/4, 35/16, -x^4] - 118503*x^8*Hypergeometric2F1[5/4, 2
3/16, 39/16, -x^4] + 33649*x^9*Hypergeometric2F1[5/4, 27/16, 43/16, -x^4])))/(908523*x^5*(x + x^5)^(1/4))

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.79

method	result	size
gosper	\(\frac {\frac {4}{7} x^{8}+\frac {8}{7} x^{4}+\frac {4}{7}-8 x^{6}-4 x^{7}-4 x^{3}}{\left (x^{5}+x \right )^{\frac {1}{4}} x^{5}}\)	\(38\)
risch	\(\frac {\frac {4}{7} x^{8}+\frac {8}{7} x^{4}+\frac {4}{7}-8 x^{6}-4 x^{7}-4 x^{3}}{x^{5} {\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}}\)	\(40\)
pseudoelliptic	\(\frac {\frac {4}{7} x^{8}+\frac {8}{7} x^{4}+\frac {4}{7}-8 x^{6}-4 x^{7}-4 x^{3}}{x^{5} {\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}}\)	\(40\)
trager	\(\frac {4 \left (x^{5}+x \right )^{\frac {3}{4}} \left (x^{8}-7 x^{7}-14 x^{6}+2 x^{4}-7 x^{3}+1\right )}{7 x^{6} \left (x^{4}+1\right )}\)	\(45\)
meijerg	\(\frac {4 \operatorname {hypergeom}\left (\left [-\frac {21}{16}, \frac {5}{4}\right ], \left [-\frac {5}{16}\right ], -x^{4}\right )}{7 x^{\frac {21}{4}}}+\frac {4 \operatorname {hypergeom}\left (\left [-\frac {5}{16}, \frac {5}{4}\right ], \left [\frac {11}{16}\right ], -x^{4}\right )}{x^{\frac {5}{4}}}-\frac {4 \operatorname {hypergeom}\left (\left [-\frac {9}{16}, \frac {5}{4}\right ], \left [\frac {7}{16}\right ], -x^{4}\right )}{x^{\frac {9}{4}}}-\frac {4 x^{\frac {11}{4}} \operatorname {hypergeom}\left (\left [\frac {11}{16}, \frac {5}{4}\right ], \left [\frac {27}{16}\right ], -x^{4}\right )}{11}+\frac {24 x^{\frac {7}{4}} \operatorname {hypergeom}\left (\left [\frac {7}{16}, \frac {5}{4}\right ], \left [\frac {23}{16}\right ], -x^{4}\right )}{7}-8 x^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [\frac {3}{16}, \frac {5}{4}\right ], \left [\frac {19}{16}\right ], -x^{4}\right )+\frac {4 x^{\frac {27}{4}} \operatorname {hypergeom}\left (\left [\frac {5}{4}, \frac {27}{16}\right ], \left [\frac {43}{16}\right ], -x^{4}\right )}{27}-\frac {12 x^{\frac {23}{4}} \operatorname {hypergeom}\left (\left [\frac {5}{4}, \frac {23}{16}\right ], \left [\frac {39}{16}\right ], -x^{4}\right )}{23}+\frac {8 x^{\frac {19}{4}} \operatorname {hypergeom}\left (\left [\frac {19}{16}, \frac {5}{4}\right ], \left [\frac {35}{16}\right ], -x^{4}\right )}{19}\)	\(146\)

[In]

int((x^4-3)*(x^4-2*x^3+1)*(x^4-x^3+1)/x^6/(x^4+1)/(x^5+x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.90 \[ \int \frac {\left (-3+x^4\right ) \left (1-2 x^3+x^4\right ) \left (1-x^3+x^4\right )}{x^6 \left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\frac {4 \, {\left (x^{8} - 7 \, x^{7} - 14 \, x^{6} + 2 \, x^{4} - 7 \, x^{3} + 1\right )} {\left (x^{5} + x\right )}^{\frac {3}{4}}}{7 \, {\left (x^{10} + x^{6}\right )}} \]

[In]

integrate((x^4-3)*(x^4-2*x^3+1)*(x^4-x^3+1)/x^6/(x^4+1)/(x^5+x)^(1/4),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\left (-3+x^4\right ) \left (1-2 x^3+x^4\right ) \left (1-x^3+x^4\right )}{x^6 \left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int \frac {\left (x - 1\right ) \left (x^{4} - 3\right ) \left (x^{4} - x^{3} + 1\right ) \left (x^{3} - x^{2} - x - 1\right )}{x^{6} \sqrt [4]{x \left (x^{4} + 1\right )} \left (x^{4} + 1\right )}\, dx \]

[In]

integrate((x**4-3)*(x**4-2*x**3+1)*(x**4-x**3+1)/x**6/(x**4+1)/(x**5+x)**(1/4),x)

[Out]

Integral((x - 1)*(x**4 - 3)*(x**4 - x**3 + 1)*(x**3 - x**2 - x - 1)/(x**6*(x*(x**4 + 1))**(1/4)*(x**4 + 1)), x
)

Maxima [F]

\[ \int \frac {\left (-3+x^4\right ) \left (1-2 x^3+x^4\right ) \left (1-x^3+x^4\right )}{x^6 \left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int { \frac {{\left (x^{4} - x^{3} + 1\right )} {\left (x^{4} - 2 \, x^{3} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )} x^{6}} \,d x } \]

[In]

integrate((x^4-3)*(x^4-2*x^3+1)*(x^4-x^3+1)/x^6/(x^4+1)/(x^5+x)^(1/4),x, algorithm="maxima")

[Out]

integrate((x^4 - x^3 + 1)*(x^4 - 2*x^3 + 1)*(x^4 - 3)/((x^5 + x)^(1/4)*(x^4 + 1)*x^6), x)

Giac [F]

\[ \int \frac {\left (-3+x^4\right ) \left (1-2 x^3+x^4\right ) \left (1-x^3+x^4\right )}{x^6 \left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int { \frac {{\left (x^{4} - x^{3} + 1\right )} {\left (x^{4} - 2 \, x^{3} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )} x^{6}} \,d x } \]

[In]

integrate((x^4-3)*(x^4-2*x^3+1)*(x^4-x^3+1)/x^6/(x^4+1)/(x^5+x)^(1/4),x, algorithm="giac")

[Out]

integrate((x^4 - x^3 + 1)*(x^4 - 2*x^3 + 1)*(x^4 - 3)/((x^5 + x)^(1/4)*(x^4 + 1)*x^6), x)

Mupad [B] (verification not implemented)

Time = 5.78 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-3+x^4\right ) \left (1-2 x^3+x^4\right ) \left (1-x^3+x^4\right )}{x^6 \left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\frac {4\,{\left (x^5+x\right )}^{3/4}}{7\,x^2}-\frac {8\,{\left (x^5+x\right )}^{3/4}}{x^4+1}-\frac {4\,{\left (x^5+x\right )}^{3/4}}{x^3}+\frac {4\,{\left (x^5+x\right )}^{3/4}}{7\,x^6} \]

[In]

int(((x^4 - 3)*(x^4 - x^3 + 1)*(x^4 - 2*x^3 + 1))/(x^6*(x^4 + 1)*(x + x^5)^(1/4)),x)

[Out]

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