\(\int \frac {3+x^4}{x^3 \sqrt [4]{-x+x^5}} \, dx\) [146]

Optimal result

Integrand size = 20, antiderivative size = 18 \[ \int \frac {3+x^4}{x^3 \sqrt [4]{-x+x^5}} \, dx=\frac {4 \left (-x+x^5\right )^{3/4}}{3 x^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1604} \[ \int \frac {3+x^4}{x^3 \sqrt [4]{-x+x^5}} \, dx=\frac {4 \left (x^5-x\right )^{3/4}}{3 x^3} \]

[In]

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

[Out]

(4*(-x + x^5)^(3/4))/(3*x^3)

Rule 1604

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*(Rr^(n + 1)/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x,
r])), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp
, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {4 \left (-x+x^5\right )^{3/4}}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {3+x^4}{x^3 \sqrt [4]{-x+x^5}} \, dx=\frac {4 \left (x \left (-1+x^4\right )\right )^{3/4}}{3 x^3} \]

[In]

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

[Out]

(4*(x*(-1 + x^4))^(3/4))/(3*x^3)

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83

method	result	size
trager	\(\frac {4 \left (x^{5}-x \right )^{\frac {3}{4}}}{3 x^{3}}\)	\(15\)
pseudoelliptic	\(\frac {4 \left (x^{5}-x \right )^{\frac {3}{4}}}{3 x^{3}}\)	\(15\)
risch	\(\frac {\frac {4 x^{4}}{3}-\frac {4}{3}}{x^{2} {\left (x \left (x^{4}-1\right )\right )}^{\frac {1}{4}}}\)	\(20\)
gosper	\(\frac {4 \left (x^{2}+1\right ) \left (x -1\right ) \left (1+x \right )}{3 x^{2} \left (x^{5}-x \right )^{\frac {1}{4}}}\)	\(26\)
meijerg	\(-\frac {4 {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [-\frac {9}{16}, \frac {1}{4}\right ], \left [\frac {7}{16}\right ], x^{4}\right )}{3 \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}} x^{\frac {9}{4}}}+\frac {4 {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} x^{\frac {7}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {7}{16}\right ], \left [\frac {23}{16}\right ], x^{4}\right )}{7 \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}\)	\(66\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {3+x^4}{x^3 \sqrt [4]{-x+x^5}} \, dx=\frac {4 \, {\left (x^{5} - x\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]

[In]

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

[Out]

4/3*(x^5 - x)^(3/4)/x^3

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {3+x^4}{x^3 \sqrt [4]{-x+x^5}} \, dx=\frac {4\,{\left (x^5-x\right )}^{3/4}}{3\,x^3} \]

[In]

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

[Out]

(4*(x^5 - x)^(3/4))/(3*x^3)