3.2130 \(\int \left (a+b \sqrt{x}\right )^3 x^2 \, dx\)

Optimal. Leaf size=47 \[ \frac{a^3 x^3}{3}+\frac{6}{7} a^2 b x^{7/2}+\frac{3}{4} a b^2 x^4+\frac{2}{9} b^3 x^{9/2} \]

[Out]

(a^3*x^3)/3 + (6*a^2*b*x^(7/2))/7 + (3*a*b^2*x^4)/4 + (2*b^3*x^(9/2))/9

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Rubi [A]  time = 0.0702033, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{a^3 x^3}{3}+\frac{6}{7} a^2 b x^{7/2}+\frac{3}{4} a b^2 x^4+\frac{2}{9} b^3 x^{9/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^3*x^2,x]

[Out]

(a^3*x^3)/3 + (6*a^2*b*x^(7/2))/7 + (3*a*b^2*x^4)/4 + (2*b^3*x^(9/2))/9

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Rubi in Sympy [A]  time = 10.005, size = 44, normalized size = 0.94 \[ \frac{a^{3} x^{3}}{3} + \frac{6 a^{2} b x^{\frac{7}{2}}}{7} + \frac{3 a b^{2} x^{4}}{4} + \frac{2 b^{3} x^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x**3/3 + 6*a**2*b*x**(7/2)/7 + 3*a*b**2*x**4/4 + 2*b**3*x**(9/2)/9

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Mathematica [A]  time = 0.0111514, size = 47, normalized size = 1. \[ \frac{a^3 x^3}{3}+\frac{6}{7} a^2 b x^{7/2}+\frac{3}{4} a b^2 x^4+\frac{2}{9} b^3 x^{9/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^3*x^2,x]

[Out]

(a^3*x^3)/3 + (6*a^2*b*x^(7/2))/7 + (3*a*b^2*x^4)/4 + (2*b^3*x^(9/2))/9

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Maple [A]  time = 0.003, size = 36, normalized size = 0.8 \[{\frac{{a}^{3}{x}^{3}}{3}}+{\frac{6\,{a}^{2}b}{7}{x}^{{\frac{7}{2}}}}+{\frac{3\,a{b}^{2}{x}^{4}}{4}}+{\frac{2\,{b}^{3}}{9}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(a+b*x^(1/2))^3,x)

[Out]

1/3*a^3*x^3+6/7*a^2*b*x^(7/2)+3/4*a*b^2*x^4+2/9*b^3*x^(9/2)

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Maxima [A]  time = 1.42609, size = 132, normalized size = 2.81 \[ \frac{2 \,{\left (b \sqrt{x} + a\right )}^{9}}{9 \, b^{6}} - \frac{5 \,{\left (b \sqrt{x} + a\right )}^{8} a}{4 \, b^{6}} + \frac{20 \,{\left (b \sqrt{x} + a\right )}^{7} a^{2}}{7 \, b^{6}} - \frac{10 \,{\left (b \sqrt{x} + a\right )}^{6} a^{3}}{3 \, b^{6}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{5} a^{4}}{b^{6}} - \frac{{\left (b \sqrt{x} + a\right )}^{4} a^{5}}{2 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(b*sqrt(x) + a)^9/b^6 - 5/4*(b*sqrt(x) + a)^8*a/b^6 + 20/7*(b*sqrt(x) + a)^7
*a^2/b^6 - 10/3*(b*sqrt(x) + a)^6*a^3/b^6 + 2*(b*sqrt(x) + a)^5*a^4/b^6 - 1/2*(b
*sqrt(x) + a)^4*a^5/b^6

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Fricas [A]  time = 0.233188, size = 55, normalized size = 1.17 \[ \frac{3}{4} \, a b^{2} x^{4} + \frac{1}{3} \, a^{3} x^{3} + \frac{2}{63} \,{\left (7 \, b^{3} x^{4} + 27 \, a^{2} b x^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*a*b^2*x^4 + 1/3*a^3*x^3 + 2/63*(7*b^3*x^4 + 27*a^2*b*x^3)*sqrt(x)

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Sympy [A]  time = 1.83807, size = 44, normalized size = 0.94 \[ \frac{a^{3} x^{3}}{3} + \frac{6 a^{2} b x^{\frac{7}{2}}}{7} + \frac{3 a b^{2} x^{4}}{4} + \frac{2 b^{3} x^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x**3/3 + 6*a**2*b*x**(7/2)/7 + 3*a*b**2*x**4/4 + 2*b**3*x**(9/2)/9

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GIAC/XCAS [A]  time = 0.214679, size = 47, normalized size = 1. \[ \frac{2}{9} \, b^{3} x^{\frac{9}{2}} + \frac{3}{4} \, a b^{2} x^{4} + \frac{6}{7} \, a^{2} b x^{\frac{7}{2}} + \frac{1}{3} \, a^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*b^3*x^(9/2) + 3/4*a*b^2*x^4 + 6/7*a^2*b*x^(7/2) + 1/3*a^3*x^3