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

Optimal. Leaf size=19 \[ 3 a \sqrt [3]{x}+\frac{3}{4} b x^{4/3} \]

[Out]

3*a*x^(1/3) + (3*b*x^(4/3))/4

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Rubi [A]  time = 0.0133091, antiderivative size = 19, 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 \[ 3 a \sqrt [3]{x}+\frac{3}{4} b x^{4/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/x^(2/3),x]

[Out]

3*a*x^(1/3) + (3*b*x^(4/3))/4

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Rubi in Sympy [A]  time = 2.33952, size = 17, normalized size = 0.89 \[ 3 a \sqrt [3]{x} + \frac{3 b x^{\frac{4}{3}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a*x**(1/3) + 3*b*x**(4/3)/4

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Mathematica [A]  time = 0.00468135, size = 16, normalized size = 0.84 \[ \frac{3}{4} \sqrt [3]{x} (4 a+b x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/x^(2/3),x]

[Out]

(3*x^(1/3)*(4*a + b*x))/4

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Maple [A]  time = 0.004, size = 13, normalized size = 0.7 \[{\frac{3\,bx+12\,a}{4}\sqrt [3]{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/x^(2/3),x)

[Out]

3/4*x^(1/3)*(b*x+4*a)

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Maxima [A]  time = 1.34718, size = 18, normalized size = 0.95 \[ \frac{3}{4} \, b x^{\frac{4}{3}} + 3 \, a x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*b*x^(4/3) + 3*a*x^(1/3)

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Fricas [A]  time = 0.208549, size = 16, normalized size = 0.84 \[ \frac{3}{4} \,{\left (b x + 4 \, a\right )} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*(b*x + 4*a)*x^(1/3)

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Sympy [A]  time = 1.71138, size = 17, normalized size = 0.89 \[ 3 a \sqrt [3]{x} + \frac{3 b x^{\frac{4}{3}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/x**(2/3),x)

[Out]

3*a*x**(1/3) + 3*b*x**(4/3)/4

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GIAC/XCAS [A]  time = 0.204884, size = 18, normalized size = 0.95 \[ \frac{3}{4} \, b x^{\frac{4}{3}} + 3 \, a x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*b*x^(4/3) + 3*a*x^(1/3)