\(\int \frac {2+x}{(1+x) \sqrt [4]{x^2+x^3}} \, dx\) [52]

Optimal result

Integrand size = 20, antiderivative size = 14 \[ \int \frac {2+x}{(1+x) \sqrt [4]{x^2+x^3}} \, dx=\frac {4 x}{\sqrt [4]{x^2 (1+x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2081, 75} \[ \int \frac {2+x}{(1+x) \sqrt [4]{x^2+x^3}} \, dx=\frac {4 x}{\sqrt [4]{x^3+x^2}} \]

[In]

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

[Out]

(4*x)/(x^2 + x^3)^(1/4)

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 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 {x} \sqrt [4]{1+x}\right ) \int \frac {2+x}{\sqrt {x} (1+x)^{5/4}} \, dx}{\sqrt [4]{x^2+x^3}} \\ & = \frac {4 x}{\sqrt [4]{x^2+x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.99 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {2+x}{(1+x) \sqrt [4]{x^2+x^3}} \, dx=\frac {4 x}{\sqrt [4]{x^2 (1+x)}} \]

[In]

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

[Out]

(4*x)/(x^2*(1 + x))^(1/4)

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
gosper	\(\frac {4 x}{\left (x^{3}+x^{2}\right )^{\frac {1}{4}}}\)	\(13\)
risch	\(\frac {4 x}{\left (x^{2} \left (1+x \right )\right )^{\frac {1}{4}}}\)	\(13\)
trager	\(\frac {4 \left (x^{3}+x^{2}\right )^{\frac {3}{4}}}{\left (1+x \right ) x}\)	\(20\)
meijerg	\(4 \sqrt {x}\, \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -x \right )+\frac {2 x^{\frac {3}{2}} \operatorname {hypergeom}\left (\left [\frac {5}{4}, \frac {3}{2}\right ], \left [\frac {5}{2}\right ], -x \right )}{3}\)	\(30\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int \frac {2+x}{(1+x) \sqrt [4]{x^2+x^3}} \, dx=\frac {4 \, {\left (x^{3} + x^{2}\right )}^{\frac {3}{4}}}{x^{2} + x} \]

[In]

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

[Out]

4*(x^3 + x^2)^(3/4)/(x^2 + x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79 \[ \int \frac {2+x}{(1+x) \sqrt [4]{x^2+x^3}} \, dx=\frac {4}{{\left (\frac {1}{x} + \frac {1}{x^{2}}\right )}^{\frac {1}{4}}} \]

[In]

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

[Out]

4/(1/x + 1/x^2)^(1/4)

Mupad [B] (verification not implemented)

Time = 4.91 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int \frac {2+x}{(1+x) \sqrt [4]{x^2+x^3}} \, dx=\frac {4\,{\left (x^3+x^2\right )}^{3/4}}{x\,\left (x+1\right )} \]

[In]

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

[Out]

(4*(x^2 + x^3)^(3/4))/(x*(x + 1))