3.2132 \(\int \left (a+b \sqrt{x}\right )^3 \, dx\)

Optimal. Leaf size=38 \[ \frac{2 \left (a+b \sqrt{x}\right )^5}{5 b^2}-\frac{a \left (a+b \sqrt{x}\right )^4}{2 b^2} \]

[Out]

-(a*(a + b*Sqrt[x])^4)/(2*b^2) + (2*(a + b*Sqrt[x])^5)/(5*b^2)

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Rubi [A]  time = 0.0451122, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 \left (a+b \sqrt{x}\right )^5}{5 b^2}-\frac{a \left (a+b \sqrt{x}\right )^4}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^3,x]

[Out]

-(a*(a + b*Sqrt[x])^4)/(2*b^2) + (2*(a + b*Sqrt[x])^5)/(5*b^2)

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Rubi in Sympy [A]  time = 6.77902, size = 32, normalized size = 0.84 \[ - \frac{a \left (a + b \sqrt{x}\right )^{4}}{2 b^{2}} + \frac{2 \left (a + b \sqrt{x}\right )^{5}}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**3,x)

[Out]

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

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Mathematica [A]  time = 0.00854131, size = 40, normalized size = 1.05 \[ a^3 x+2 a^2 b x^{3/2}+\frac{3}{2} a b^2 x^2+\frac{2}{5} b^3 x^{5/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^3,x]

[Out]

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

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Maple [A]  time = 0.002, size = 33, normalized size = 0.9 \[{\frac{2\,{b}^{3}}{5}{x}^{{\frac{5}{2}}}}+{\frac{3\,a{b}^{2}{x}^{2}}{2}}+2\,{a}^{2}b{x}^{3/2}+{a}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^3,x)

[Out]

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

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Maxima [A]  time = 1.42096, size = 43, normalized size = 1.13 \[ \frac{2}{5} \, b^{3} x^{\frac{5}{2}} + \frac{3}{2} \, a b^{2} x^{2} + 2 \, a^{2} b x^{\frac{3}{2}} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.232683, size = 47, normalized size = 1.24 \[ \frac{3}{2} \, a b^{2} x^{2} + a^{3} x + \frac{2}{5} \,{\left (b^{3} x^{2} + 5 \, a^{2} b x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.21469, size = 39, normalized size = 1.03 \[ a^{3} x + 2 a^{2} b x^{\frac{3}{2}} + \frac{3 a b^{2} x^{2}}{2} + \frac{2 b^{3} x^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**3,x)

[Out]

a**3*x + 2*a**2*b*x**(3/2) + 3*a*b**2*x**2/2 + 2*b**3*x**(5/2)/5

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GIAC/XCAS [A]  time = 0.215377, size = 43, normalized size = 1.13 \[ \frac{2}{5} \, b^{3} x^{\frac{5}{2}} + \frac{3}{2} \, a b^{2} x^{2} + 2 \, a^{2} b x^{\frac{3}{2}} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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