\(\int \frac {4+x^3}{x^4 \sqrt [4]{1+x^3}} \, dx\) [75]

Optimal result

Integrand size = 18, antiderivative size = 16 \[ \int \frac {4+x^3}{x^4 \sqrt [4]{1+x^3}} \, dx=-\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {457, 75} \[ \int \frac {4+x^3}{x^4 \sqrt [4]{1+x^3}} \, dx=-\frac {4 \left (x^3+1\right )^{3/4}}{3 x^3} \]

[In]

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

[Out]

(-4*(1 + x^3)^(3/4))/(3*x^3)

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {4+x}{x^2 \sqrt [4]{1+x}} \, dx,x,x^3\right ) \\ & = -\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {4+x^3}{x^4 \sqrt [4]{1+x^3}} \, dx=-\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3} \]

[In]

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

[Out]

(-4*(1 + x^3)^(3/4))/(3*x^3)

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
trager	\(-\frac {4 \left (x^{3}+1\right )^{\frac {3}{4}}}{3 x^{3}}\)	\(13\)
risch	\(-\frac {4 \left (x^{3}+1\right )^{\frac {3}{4}}}{3 x^{3}}\)	\(13\)
pseudoelliptic	\(-\frac {4 \left (x^{3}+1\right )^{\frac {3}{4}}}{3 x^{3}}\)	\(13\)
gosper	\(-\frac {4 \left (1+x \right ) \left (x^{2}-x +1\right )}{3 x^{3} \left (x^{3}+1\right )^{\frac {1}{4}}}\)	\(24\)
meijerg	\(\frac {2 \sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) \left (\frac {5 \pi \sqrt {2}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {9}{4}\right ], \left [2, 3\right ], -x^{3}\right )}{32 \Gamma \left (\frac {3}{4}\right )}-\frac {\left (3-3 \ln \left (2\right )-\frac {\pi }{2}+3 \ln \left (x \right )\right ) \pi \sqrt {2}}{4 \Gamma \left (\frac {3}{4}\right )}-\frac {\pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right ) x^{3}}\right )}{3 \pi }+\frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) \left (-\frac {\pi \sqrt {2}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], -x^{3}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \left (2\right )-\frac {\pi }{2}+3 \ln \left (x \right )\right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{6 \pi }\)	\(133\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {4+x^3}{x^4 \sqrt [4]{1+x^3}} \, dx=-\frac {4 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]

[In]

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

[Out]

-4/3*(x^3 + 1)^(3/4)/x^3

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (15) = 30\).

Time = 8.53 (sec) , antiderivative size = 49, normalized size of antiderivative = 3.06 \[ \int \frac {4+x^3}{x^4 \sqrt [4]{1+x^3}} \, dx=- \frac {2 \sqrt [4]{x^{3} + 1}}{3 \left (\sqrt {x^{3} + 1} + 1\right )} - \frac {1}{3 \left (\sqrt [4]{x^{3} + 1} + 1\right )} - \frac {1}{3 \left (\sqrt [4]{x^{3} + 1} - 1\right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {4+x^3}{x^4 \sqrt [4]{1+x^3}} \, dx=-\frac {4 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]

[In]

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

[Out]

-4/3*(x^3 + 1)^(3/4)/x^3

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {4+x^3}{x^4 \sqrt [4]{1+x^3}} \, dx=-\frac {4 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]

[In]

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

[Out]

-4/3*(x^3 + 1)^(3/4)/x^3

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {4+x^3}{x^4 \sqrt [4]{1+x^3}} \, dx=-\frac {4\,{\left (x^3+1\right )}^{3/4}}{3\,x^3} \]

[In]

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

[Out]

-(4*(x^3 + 1)^(3/4))/(3*x^3)