\(\int \frac {-4+x^5}{x^2 (1+x^5)^{3/4}} \, dx\) [56]

Optimal result

Integrand size = 18, antiderivative size = 14 \[ \int \frac {-4+x^5}{x^2 \left (1+x^5\right )^{3/4}} \, dx=\frac {4 \sqrt [4]{1+x^5}}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {460} \[ \int \frac {-4+x^5}{x^2 \left (1+x^5\right )^{3/4}} \, dx=\frac {4 \sqrt [4]{x^5+1}}{x} \]

[In]

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

[Out]

(4*(1 + x^5)^(1/4))/x

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]

Rubi steps \begin{align*} \text {integral}& = \frac {4 \sqrt [4]{1+x^5}}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {-4+x^5}{x^2 \left (1+x^5\right )^{3/4}} \, dx=\frac {4 \sqrt [4]{1+x^5}}{x} \]

[In]

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

[Out]

(4*(1 + x^5)^(1/4))/x

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
trager	\(\frac {4 \left (x^{5}+1\right )^{\frac {1}{4}}}{x}\)	\(13\)
risch	\(\frac {4 \left (x^{5}+1\right )^{\frac {1}{4}}}{x}\)	\(13\)
pseudoelliptic	\(\frac {4 \left (x^{5}+1\right )^{\frac {1}{4}}}{x}\)	\(13\)
gosper	\(\frac {4 \left (x^{4}-x^{3}+x^{2}-x +1\right ) \left (1+x \right )}{x \left (x^{5}+1\right )^{\frac {3}{4}}}\)	\(32\)
meijerg	\(\frac {x^{4} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {4}{5}\right ], \left [\frac {9}{5}\right ], -x^{5}\right )}{4}+\frac {4 \operatorname {hypergeom}\left (\left [-\frac {1}{5}, \frac {3}{4}\right ], \left [\frac {4}{5}\right ], -x^{5}\right )}{x}\)	\(34\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-4+x^5}{x^2 \left (1+x^5\right )^{3/4}} \, dx=\frac {4 \, {\left (x^{5} + 1\right )}^{\frac {1}{4}}}{x} \]

[In]

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

[Out]

4*(x^5 + 1)^(1/4)/x

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.78 (sec) , antiderivative size = 63, normalized size of antiderivative = 4.50 \[ \int \frac {-4+x^5}{x^2 \left (1+x^5\right )^{3/4}} \, dx=\frac {x^{4} \Gamma \left (\frac {4}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {4}{5} \\ \frac {9}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 \Gamma \left (\frac {9}{5}\right )} - \frac {4 \Gamma \left (- \frac {1}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{5}, \frac {3}{4} \\ \frac {4}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 x \Gamma \left (\frac {4}{5}\right )} \]

[In]

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

[Out]

x**4*gamma(4/5)*hyper((3/4, 4/5), (9/5,), x**5*exp_polar(I*pi))/(5*gamma(9/5)) - 4*gamma(-1/5)*hyper((-1/5, 3/
4), (4/5,), x**5*exp_polar(I*pi))/(5*x*gamma(4/5))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (12) = 24\).

Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.00 \[ \int \frac {-4+x^5}{x^2 \left (1+x^5\right )^{3/4}} \, dx=\frac {4 \, {\left (x^{4} - x^{3} + x^{2} - x + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}}}{x} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.84 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-4+x^5}{x^2 \left (1+x^5\right )^{3/4}} \, dx=\frac {4\,{\left (x^5+1\right )}^{1/4}}{x} \]

[In]

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

[Out]

(4*(x^5 + 1)^(1/4))/x