\(\int \frac {1}{(1+x^2) \sqrt [3]{x+x^3}} \, dx\) [168]

Optimal result

Integrand size = 17, antiderivative size = 20 \[ \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{x+x^3}} \, dx=\frac {3 \left (x+x^3\right )^{2/3}}{2 \left (1+x^2\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2081, 270} \[ \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{x+x^3}} \, dx=\frac {3 x}{2 \sqrt [3]{x^3+x}} \]

[In]

Int[1/((1 + x^2)*(x + x^3)^(1/3)),x]

[Out]

(3*x)/(2*(x + x^3)^(1/3))

Rule 270

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

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

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{x+x^3}} \, dx=\frac {3 x}{2 \sqrt [3]{x+x^3}} \]

[In]

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

[Out]

(3*x)/(2*(x + x^3)^(1/3))

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.55

method	result	size
gosper	\(\frac {3 x}{2 \left (x^{3}+x \right )^{\frac {1}{3}}}\)	\(11\)
meijerg	\(\frac {3 x^{\frac {2}{3}}}{2 \left (x^{2}+1\right )^{\frac {1}{3}}}\)	\(13\)
risch	\(\frac {3 x}{2 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}}}\)	\(13\)
pseudoelliptic	\(\frac {3 x}{2 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}}}\)	\(13\)
trager	\(\frac {3 \left (x^{3}+x \right )^{\frac {2}{3}}}{2 \left (x^{2}+1\right )}\)	\(17\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{x+x^3}} \, dx=\frac {3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}}}{2 \, {\left (x^{2} + 1\right )}} \]

[In]

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

[Out]

3/2*(x^3 + x)^(2/3)/(x^2 + 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-3/4*(x^3 + x)/((x^(7/3) + x^(1/3))*(x^2 + 1)^(1/3)) + integrate(3/2*(x^2 + 1)^(2/3)/(x^(13/3) + 2*x^(7/3) + x
^(1/3)), x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{x+x^3}} \, dx=\frac {3}{2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}} \]

[In]

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

[Out]

3/2/(1/x^2 + 1)^(1/3)

Mupad [B] (verification not implemented)

Time = 5.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{x+x^3}} \, dx=\frac {3\,{\left (x^3+x\right )}^{2/3}}{2\,\left (x^2+1\right )} \]

[In]

int(1/((x^2 + 1)*(x + x^3)^(1/3)),x)

[Out]

(3*(x + x^3)^(2/3))/(2*(x^2 + 1))