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

Optimal. Leaf size=34 \[ \frac{x^{a+1} \left (3 x^a+2 x^{2 a}+6\right )^{\frac{1}{a}+1}}{6 (a+1)} \]

[Out]

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

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Rubi [A]  time = 0.0436392, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {1594, 1747} \[ \frac{x^{a+1} \left (3 x^a+2 x^{2 a}+6\right )^{\frac{1}{a}+1}}{6 (a+1)} \]

Antiderivative was successfully verified.

[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 1594

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 1747

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*} \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 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]  time = 0.117796, size = 33, normalized size = 0.97 \[ \frac{x^{a+1} \left (3 x^a+2 x^{2 a}+6\right )^{\frac{1}{a}+1}}{6 a+6} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.039, size = 44, normalized size = 1.3 \begin{align*}{\frac{x{x}^{a} \left ( 6+3\,{x}^{a}+2\, \left ({x}^{a} \right ) ^{2} \right ) \sqrt [a]{6+3\,{x}^{a}+2\, \left ({x}^{a} \right ) ^{2}}}{6+6\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [A]  time = 1.13808, size = 65, normalized size = 1.91 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 2.1829, size = 109, normalized size = 3.21 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)