3.2112 \(\int \left (a+b \sqrt{x}\right ) x \, dx\)

Optimal. Leaf size=19 \[ \frac{a x^2}{2}+\frac{2}{5} b x^{5/2} \]

[Out]

(a*x^2)/2 + (2*b*x^(5/2))/5

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Rubi [A]  time = 0.0156251, 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 \[ \frac{a x^2}{2}+\frac{2}{5} b x^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])*x,x]

[Out]

(a*x^2)/2 + (2*b*x^(5/2))/5

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a \int x\, dx + \frac{2 b x^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*Integral(x, x) + 2*b*x**(5/2)/5

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Mathematica [A]  time = 0.00517189, size = 19, normalized size = 1. \[ \frac{a x^2}{2}+\frac{2}{5} b x^{5/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])*x,x]

[Out]

(a*x^2)/2 + (2*b*x^(5/2))/5

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Maple [A]  time = 0.002, size = 14, normalized size = 0.7 \[{\frac{a{x}^{2}}{2}}+{\frac{2\,b}{5}{x}^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(a+b*x^(1/2)),x)

[Out]

1/2*a*x^2+2/5*b*x^(5/2)

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Maxima [A]  time = 1.43598, size = 86, normalized size = 4.53 \[ \frac{2 \,{\left (b \sqrt{x} + a\right )}^{5}}{5 \, b^{4}} - \frac{3 \,{\left (b \sqrt{x} + a\right )}^{4} a}{2 \, b^{4}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{3} a^{2}}{b^{4}} - \frac{{\left (b \sqrt{x} + a\right )}^{2} a^{3}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(b*sqrt(x) + a)^5/b^4 - 3/2*(b*sqrt(x) + a)^4*a/b^4 + 2*(b*sqrt(x) + a)^3*a^
2/b^4 - (b*sqrt(x) + a)^2*a^3/b^4

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Fricas [A]  time = 0.23097, size = 18, normalized size = 0.95 \[ \frac{2}{5} \, b x^{\frac{5}{2}} + \frac{1}{2} \, a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*b*x^(5/2) + 1/2*a*x^2

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Sympy [A]  time = 1.18036, size = 15, normalized size = 0.79 \[ \frac{a x^{2}}{2} + \frac{2 b x^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(a+b*x**(1/2)),x)

[Out]

a*x**2/2 + 2*b*x**(5/2)/5

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GIAC/XCAS [A]  time = 0.213773, size = 18, normalized size = 0.95 \[ \frac{2}{5} \, b x^{\frac{5}{2}} + \frac{1}{2} \, a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*b*x^(5/2) + 1/2*a*x^2