\(\int (1+3 x^2) \sqrt [3]{x+x^3} \, dx\) [25]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [A] (verification not implemented)
   Sympy [B] (verification not implemented)
   Maxima [A] (verification not implemented)
   Giac [A] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 17, antiderivative size = 13 \[ \int \left (1+3 x^2\right ) \sqrt [3]{x+x^3} \, dx=\frac {3}{4} \left (x+x^3\right )^{4/3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1602} \[ \int \left (1+3 x^2\right ) \sqrt [3]{x+x^3} \, dx=\frac {3}{4} \left (x^3+x\right )^{4/3} \]

[In]

Int[(1 + 3*x^2)*(x + x^3)^(1/3),x]

[Out]

(3*(x + x^3)^(4/3))/4

Rule 1602

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*x^(p - q +
 1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

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

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.46 \[ \int \left (1+3 x^2\right ) \sqrt [3]{x+x^3} \, dx=\frac {3}{4} x \left (1+x^2\right ) \sqrt [3]{x+x^3} \]

[In]

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

[Out]

(3*x*(1 + x^2)*(x + x^3)^(1/3))/4

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77

method result size
derivativedivides \(\frac {3 \left (x^{3}+x \right )^{\frac {4}{3}}}{4}\) \(10\)
default \(\frac {3 \left (x^{3}+x \right )^{\frac {4}{3}}}{4}\) \(10\)
gosper \(\frac {3 \left (x^{2}+1\right ) x \left (x^{3}+x \right )^{\frac {1}{3}}}{4}\) \(16\)
trager \(\frac {3 \left (x^{2}+1\right ) x \left (x^{3}+x \right )^{\frac {1}{3}}}{4}\) \(16\)
risch \(\frac {3 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}} \left (x^{2}+1\right ) x}{4}\) \(18\)
pseudoelliptic \(\frac {3 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}} \left (x^{2}+1\right ) x}{4}\) \(18\)
meijerg \(\frac {3 x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{2}\right )}{4}+\frac {9 x^{\frac {10}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], -x^{2}\right )}{10}\) \(34\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \left (1+3 x^2\right ) \sqrt [3]{x+x^3} \, dx=\frac {3}{4} \, {\left (x^{3} + x\right )}^{\frac {4}{3}} \]

[In]

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

[Out]

3/4*(x^3 + x)^(4/3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (10) = 20\).

Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.08 \[ \int \left (1+3 x^2\right ) \sqrt [3]{x+x^3} \, dx=\frac {3 x^{3} \sqrt [3]{x^{3} + x}}{4} + \frac {3 x \sqrt [3]{x^{3} + x}}{4} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \left (1+3 x^2\right ) \sqrt [3]{x+x^3} \, dx=\frac {3}{4} \, {\left (x^{3} + x\right )}^{\frac {4}{3}} \]

[In]

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

[Out]

3/4*(x^3 + x)^(4/3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \left (1+3 x^2\right ) \sqrt [3]{x+x^3} \, dx=\frac {3}{4} \, {\left (x^{3} + x\right )}^{\frac {4}{3}} \]

[In]

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

[Out]

3/4*(x^3 + x)^(4/3)

Mupad [B] (verification not implemented)

Time = 4.88 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \left (1+3 x^2\right ) \sqrt [3]{x+x^3} \, dx=\frac {3\,{\left (x^3+x\right )}^{4/3}}{4} \]

[In]

int((3*x^2 + 1)*(x + x^3)^(1/3),x)

[Out]

(3*(x + x^3)^(4/3))/4