\(\int \frac {-3+x^4}{(1+x^4) \sqrt [4]{x+x^5}} \, dx\) [148]

Optimal result

Integrand size = 22, antiderivative size = 18 \[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=-\frac {4 \left (x+x^5\right )^{3/4}}{1+x^4} \]

[Out]

-4*(x^5+x)^(3/4)/(x^4+1)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2081, 460} \[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=-\frac {4 x}{\sqrt [4]{x^5+x}} \]

[In]

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

[Out]

(-4*x)/(x + x^5)^(1/4)

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 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 {-3+x^4}{\sqrt [4]{x} \left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x+x^5}} \\ & = -\frac {4 x}{\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 = 10.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.33 \[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\frac {4 \sqrt [4]{1+x^4} \left (-19 x \operatorname {Hypergeometric2F1}\left (\frac {3}{16},\frac {5}{4},\frac {19}{16},-x^4\right )+x^5 \operatorname {Hypergeometric2F1}\left (\frac {19}{16},\frac {5}{4},\frac {35}{16},-x^4\right )\right )}{19 \sqrt [4]{x+x^5}} \]

[In]

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

[Out]

(4*(1 + x^4)^(1/4)*(-19*x*Hypergeometric2F1[3/16, 5/4, 19/16, -x^4] + x^5*Hypergeometric2F1[19/16, 5/4, 35/16,
-x^4]))/(19*(x + x^5)^(1/4))

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61

method	result	size
gosper	\(-\frac {4 x}{\left (x^{5}+x \right )^{\frac {1}{4}}}\)	\(11\)
risch	\(-\frac {4 x}{{\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}}\)	\(13\)
pseudoelliptic	\(-\frac {4 x}{{\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}}\)	\(13\)
trager	\(-\frac {4 \left (x^{5}+x \right )^{\frac {3}{4}}}{x^{4}+1}\)	\(17\)
meijerg	\(-4 x^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [\frac {3}{16}, \frac {5}{4}\right ], \left [\frac {19}{16}\right ], -x^{4}\right )+\frac {4 x^{\frac {19}{4}} \operatorname {hypergeom}\left (\left [\frac {19}{16}, \frac {5}{4}\right ], \left [\frac {35}{16}\right ], -x^{4}\right )}{19}\)	\(34\)

[In]

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

[Out]

-4*x/(x^5+x)^(1/4)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=-\frac {4 \, {\left (x^{5} + x\right )}^{\frac {3}{4}}}{x^{4} + 1} \]

[In]

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

[Out]

-4*(x^5 + x)^(3/4)/(x^4 + 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.96 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=-\frac {4\,{\left (x^5+x\right )}^{3/4}}{x^4+1} \]

[In]

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

[Out]

-(4*(x + x^5)^(3/4))/(x^4 + 1)