3.2115 \(\int (a+b \sqrt {x}) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ a x+\frac {2}{3} b x^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Sqrt[x],x]

[Out]

a*x + (2*b*x^(3/2))/3

Rubi steps

\begin {align*} \int \left (a+b \sqrt {x}\right ) \, dx &=a x+\frac {2}{3} b x^{3/2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \[ a x+\frac {2}{3} b x^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Sqrt[x],x]

[Out]

a*x + (2*b*x^(3/2))/3

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fricas [A]  time = 1.20, size = 10, normalized size = 0.71 \[ \frac {2}{3} \, b x^{\frac {3}{2}} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*x^(1/2),x, algorithm="fricas")

[Out]

2/3*b*x^(3/2) + a*x

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giac [A]  time = 0.16, size = 10, normalized size = 0.71 \[ \frac {2}{3} \, b x^{\frac {3}{2}} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*x^(1/2),x, algorithm="giac")

[Out]

2/3*b*x^(3/2) + a*x

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maple [A]  time = 0.00, size = 11, normalized size = 0.79 \[ \frac {2 b \,x^{\frac {3}{2}}}{3}+a x \]

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 = 0.86, size = 10, normalized size = 0.71 \[ \frac {2}{3} \, b x^{\frac {3}{2}} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*x^(1/2),x, algorithm="maxima")

[Out]

2/3*b*x^(3/2) + a*x

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mupad [B]  time = 0.02, size = 10, normalized size = 0.71 \[ a\,x+\frac {2\,b\,x^{3/2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*x^(1/2),x)

[Out]

a*x + (2*b*x^(3/2))/3

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sympy [A]  time = 0.06, 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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