\(\int \frac {4+x^5}{x^2 (-1+x^5)^{3/4}} \, dx\) [57]

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 = 0.40 (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 = 0.84 (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	\(\frac {4 \left (x^{4}+x^{3}+x^{2}+x +1\right ) \left (x -1\right )}{x \left (x^{5}-1\right )^{\frac {3}{4}}}\)	\(28\)
meijerg	\(\frac {{\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} x^{4} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {4}{5}\right ], \left [\frac {9}{5}\right ], x^{5}\right )}{4 \operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}}}-\frac {4 {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{5}, \frac {3}{4}\right ], \left [\frac {4}{5}\right ], x^{5}\right )}{\operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}} x}\)	\(66\)

[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 = 0.25 (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 = 0.81 (sec) , antiderivative size = 68, normalized size of antiderivative = 4.86 \[ \int \frac {4+x^5}{x^2 \left (-1+x^5\right )^{3/4}} \, dx=\frac {x^{4} e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {4}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {4}{5} \\ \frac {9}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 \Gamma \left (\frac {9}{5}\right )} - \frac {4 e^{\frac {i \pi }{4}} \Gamma \left (- \frac {1}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{5}, \frac {3}{4} \\ \frac {4}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x \Gamma \left (\frac {4}{5}\right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.71 \[ \int \frac {4+x^5}{x^2 \left (-1+x^5\right )^{3/4}} \, dx=\frac {4 \, {\left (x^{4} + x^{3} + x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}}}{x} \]

[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 = 4.82 (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