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

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

[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 = 0.0596816, 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 \[ \frac{x^{a+1} \left (2 x^{2 a}+3 x^a+6\right )^{\frac{1}{a}+1}}{6 (a+1)} \]

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))

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Rubi in Sympy [A]  time = 9.29847, size = 27, normalized size = 0.79 \[ \frac{x^{a + 1} \left (2 x^{2 a} + 3 x^{a} + 6\right )^{1 + \frac{1}{a}}}{6 \left (a + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((6+3*x**a+2*x**(2*a))**(1/a)*(x**a+x**(2*a)+x**(3*a)),x)

[Out]

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

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Mathematica [A]  time = 0.0564303, size = 33, normalized size = 0.97 \[ \frac{x^{a+1} \left (2 x^{2 a}+3 x^a+6\right )^{\frac{1}{a}+1}}{6 a+6} \]

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 = 0.066, size = 44, normalized size = 1.3 \[{\frac{x{x}^{a} \left ( 6+3\,{x}^{a}+2\, \left ({x}^{a} \right ) ^{2} \right ) \sqrt [a]{6+3\,{x}^{a}+2\, \left ({x}^{a} \right ) ^{2}}}{6+6\,a}} \]

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 = 1.76613, size = 65, normalized size = 1.91 \[ \frac{{\left (2 \, x x^{3 \, a} + 3 \, x x^{2 \, a} + 6 \, x x^{a}\right )}{\left (2 \, x^{2 \, a} + 3 \, x^{a} + 6\right )}^{\left (\frac{1}{a}\right )}}{6 \,{\left (a + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^(2*a) + 3*x^a + 6)^(1/a)*(x^(3*a) + x^(2*a) + x^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 = 0.223727, size = 65, normalized size = 1.91 \[ \frac{{\left (2 \, x x^{3 \, a} + 3 \, x x^{2 \, a} + 6 \, x x^{a}\right )}{\left (2 \, x^{2 \, a} + 3 \, x^{a} + 6\right )}^{\left (\frac{1}{a}\right )}}{6 \,{\left (a + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^(2*a) + 3*x^a + 6)^(1/a)*(x^(3*a) + x^(2*a) + x^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. \[ \text{Timed out} \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (2 \, x^{2 \, a} + 3 \, x^{a} + 6\right )}^{\left (\frac{1}{a}\right )}{\left (x^{3 \, a} + x^{2 \, a} + x^{a}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^(2*a) + 3*x^a + 6)^(1/a)*(x^(3*a) + x^(2*a) + x^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)