3.2113 \(\int \left (a+b \sqrt{x}\right ) \, dx\)

Optimal. Leaf size=14 \[ a x+\frac{2}{3} b x^{3/2} \]

[Out]

a*x + (2*b*x^(3/2))/3

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Rubi [A]  time = 0.0109322, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ a x+\frac{2}{3} b x^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[a + b*Sqrt[x],x]

[Out]

a*x + (2*b*x^(3/2))/3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 b x^{\frac{3}{2}}}{3} + \int a\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(a+b*x**(1/2),x)

[Out]

2*b*x**(3/2)/3 + Integral(a, x)

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Mathematica [A]  time = 0.00305744, size = 14, normalized size = 1. \[ a x+\frac{2}{3} b x^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[a + b*Sqrt[x],x]

[Out]

a*x + (2*b*x^(3/2))/3

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Maple [A]  time = 0.002, size = 11, normalized size = 0.8 \[ ax+{\frac{2\,b}{3}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(a+b*x^(1/2),x)

[Out]

a*x+2/3*b*x^(3/2)

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Maxima [A]  time = 1.42894, size = 14, normalized size = 1. \[ \frac{2}{3} \, b x^{\frac{3}{2}} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(b*sqrt(x) + a,x, algorithm="maxima")

[Out]

2/3*b*x^(3/2) + a*x

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Fricas [A]  time = 0.241529, size = 14, normalized size = 1. \[ \frac{2}{3} \, b x^{\frac{3}{2}} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(b*sqrt(x) + a,x, algorithm="fricas")

[Out]

2/3*b*x^(3/2) + a*x

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Sympy [A]  time = 0.063559, size = 12, normalized size = 0.86 \[ a x + \frac{2 b x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a+b*x**(1/2),x)

[Out]

a*x + 2*b*x**(3/2)/3

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GIAC/XCAS [A]  time = 0.214009, size = 14, normalized size = 1. \[ \frac{2}{3} \, b x^{\frac{3}{2}} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(b*sqrt(x) + a,x, algorithm="giac")

[Out]

2/3*b*x^(3/2) + a*x