3.652 \(\int x^{2/3} (a+b x) \, dx\)

Optimal. Leaf size=21 \[ \frac{3}{5} a x^{5/3}+\frac{3}{8} b x^{8/3} \]

[Out]

(3*a*x^(5/3))/5 + (3*b*x^(8/3))/8

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Rubi [A]  time = 0.0133689, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{3}{5} a x^{5/3}+\frac{3}{8} b x^{8/3} \]

Antiderivative was successfully verified.

[In]  Int[x^(2/3)*(a + b*x),x]

[Out]

(3*a*x^(5/3))/5 + (3*b*x^(8/3))/8

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Rubi in Sympy [A]  time = 2.42239, size = 19, normalized size = 0.9 \[ \frac{3 a x^{\frac{5}{3}}}{5} + \frac{3 b x^{\frac{8}{3}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a*x**(5/3)/5 + 3*b*x**(8/3)/8

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Mathematica [A]  time = 0.00510021, size = 17, normalized size = 0.81 \[ \frac{3}{40} x^{5/3} (8 a+5 b x) \]

Antiderivative was successfully verified.

[In]  Integrate[x^(2/3)*(a + b*x),x]

[Out]

(3*x^(5/3)*(8*a + 5*b*x))/40

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Maple [A]  time = 0.005, size = 14, normalized size = 0.7 \[{\frac{15\,bx+24\,a}{40}{x}^{{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(2/3)*(b*x+a),x)

[Out]

3/40*x^(5/3)*(5*b*x+8*a)

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Maxima [A]  time = 1.34303, size = 18, normalized size = 0.86 \[ \frac{3}{8} \, b x^{\frac{8}{3}} + \frac{3}{5} \, a x^{\frac{5}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*x^(2/3),x, algorithm="maxima")

[Out]

3/8*b*x^(8/3) + 3/5*a*x^(5/3)

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Fricas [A]  time = 0.203833, size = 22, normalized size = 1.05 \[ \frac{3}{40} \,{\left (5 \, b x^{2} + 8 \, a x\right )} x^{\frac{2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*x^(2/3),x, algorithm="fricas")

[Out]

3/40*(5*b*x^2 + 8*a*x)*x^(2/3)

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Sympy [A]  time = 1.82854, size = 19, normalized size = 0.9 \[ \frac{3 a x^{\frac{5}{3}}}{5} + \frac{3 b x^{\frac{8}{3}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(2/3)*(b*x+a),x)

[Out]

3*a*x**(5/3)/5 + 3*b*x**(8/3)/8

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GIAC/XCAS [A]  time = 0.207311, size = 18, normalized size = 0.86 \[ \frac{3}{8} \, b x^{\frac{8}{3}} + \frac{3}{5} \, a x^{\frac{5}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*x^(2/3),x, algorithm="giac")

[Out]

3/8*b*x^(8/3) + 3/5*a*x^(5/3)