\(\int (6+3 x^a+2 x^{2 a})^{\frac {1}{a}} (x^a+x^{2 a}+x^{3 a}) \, dx\) [32]

Optimal result

Integrand size = 33, antiderivative size = 34 \[ \int \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \left (x^a+x^{2 a}+x^{3 a}\right ) \, dx=\frac {x^{1+a} \left (6+3 x^a+2 x^{2 a}\right )^{1+\frac {1}{a}}}{6 (1+a)} \]

[Out]

1/6*x^(1+a)*(6+3*x^a+2*x^(2*a))^(1+1/a)/(1+a)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1608, 1761} \[ \int \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \left (x^a+x^{2 a}+x^{3 a}\right ) \, dx=\frac {x^{a+1} \left (2 x^{2 a}+3 x^a+6\right )^{\frac {1}{a}+1}}{6 (a+1)} \]

[In]

Int[(6 + 3*x^a + 2*x^(2*a))^a^(-1)*(x^a + x^(2*a) + x^(3*a)),x]

[Out]

(x^(1 + a)*(6 + 3*x^a + 2*x^(2*a))^(1 + a^(-1)))/(6*(1 + a))

Rule 1608

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1761

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_.) + (f_.)*(x
_)^(n2_.)), x_Symbol] :> Simp[d*(g*x)^(m + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(a*g*(m + 1))), x] /; FreeQ[{a,
b, c, d, e, f, g, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1), 0] && EqQ[a*f*(m
+ 1) - c*d*(m + 2*n*(p + 1) + 1), 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \int x^a \left (1+x^a+x^{2 a}\right ) \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \, dx \\ & = \frac {x^{1+a} \left (6+3 x^a+2 x^{2 a}\right )^{1+\frac {1}{a}}}{6 (1+a)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \left (x^a+x^{2 a}+x^{3 a}\right ) \, dx=\frac {x^{1+a} \left (6+3 x^a+2 x^{2 a}\right )^{1+\frac {1}{a}}}{6+6 a} \]

[In]

Integrate[(6 + 3*x^a + 2*x^(2*a))^a^(-1)*(x^a + x^(2*a) + x^(3*a)),x]

[Out]

(x^(1 + a)*(6 + 3*x^a + 2*x^(2*a))^(1 + a^(-1)))/(6 + 6*a)

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.29

method	result	size
risch	\(\frac {x \,x^{a} \left (6+3 x^{a}+2 x^{2 a}\right ) \left (6+3 x^{a}+2 x^{2 a}\right )^{\frac {1}{a}}}{6+6 a}\)	\(44\)

[In]

int((6+3*x^a+2*x^(2*a))^(1/a)*(x^a+x^(2*a)+x^(3*a)),x,method=_RETURNVERBOSE)

[Out]

1/6*x*x^a*(6+3*x^a+2*(x^a)^2)/(1+a)*(6+3*x^a+2*(x^a)^2)^(1/a)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.41 \[ \int \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \left (x^a+x^{2 a}+x^{3 a}\right ) \, dx=\frac {{\left (2 \, x x^{3 \, a} + 3 \, x x^{2 \, a} + 6 \, x x^{a}\right )} {\left (2 \, x^{2 \, a} + 3 \, x^{a} + 6\right )}^{\left (\frac {1}{a}\right )}}{6 \, {\left (a + 1\right )}} \]

[In]

integrate((6+3*x^a+2*x^(2*a))^(1/a)*(x^a+x^(2*a)+x^(3*a)),x, algorithm="fricas")

[Out]

1/6*(2*x*x^(3*a) + 3*x*x^(2*a) + 6*x*x^a)*(2*x^(2*a) + 3*x^a + 6)^(1/a)/(a + 1)

Sympy [F(-1)]

Timed out. \[ \int \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \left (x^a+x^{2 a}+x^{3 a}\right ) \, dx=\text {Timed out} \]

[In]

integrate((6+3*x**a+2*x**(2*a))**(1/a)*(x**a+x**(2*a)+x**(3*a)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.41 \[ \int \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \left (x^a+x^{2 a}+x^{3 a}\right ) \, dx=\frac {{\left (2 \, x x^{3 \, a} + 3 \, x x^{2 \, a} + 6 \, x x^{a}\right )} {\left (2 \, x^{2 \, a} + 3 \, x^{a} + 6\right )}^{\left (\frac {1}{a}\right )}}{6 \, {\left (a + 1\right )}} \]

[In]

integrate((6+3*x^a+2*x^(2*a))^(1/a)*(x^a+x^(2*a)+x^(3*a)),x, algorithm="maxima")

[Out]

1/6*(2*x*x^(3*a) + 3*x*x^(2*a) + 6*x*x^a)*(2*x^(2*a) + 3*x^a + 6)^(1/a)/(a + 1)

Giac [F]

\[ \int \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \left (x^a+x^{2 a}+x^{3 a}\right ) \, dx=\int { {\left (2 \, x^{2 \, a} + 3 \, x^{a} + 6\right )}^{\left (\frac {1}{a}\right )} {\left (x^{3 \, a} + x^{2 \, a} + x^{a}\right )} \,d x } \]

[In]

integrate((6+3*x^a+2*x^(2*a))^(1/a)*(x^a+x^(2*a)+x^(3*a)),x, algorithm="giac")

[Out]

integrate((2*x^(2*a) + 3*x^a + 6)^(1/a)*(x^(3*a) + x^(2*a) + x^a), x)

Mupad [F(-1)]

Timed out. \[ \int \left (6+3 x^a+2 x^{2 a}\right )^{\frac {1}{a}} \left (x^a+x^{2 a}+x^{3 a}\right ) \, dx=\int \left (x^a+x^{2\,a}+x^{3\,a}\right )\,{\left (3\,x^a+2\,x^{2\,a}+6\right )}^{1/a} \,d x \]

[In]

int((x^a + x^(2*a) + x^(3*a))*(3*x^a + 2*x^(2*a) + 6)^(1/a),x)

[Out]

int((x^a + x^(2*a) + x^(3*a))*(3*x^a + 2*x^(2*a) + 6)^(1/a), x)