3.2131 \(\int \left (a+b \sqrt{x}\right )^3 x \, dx\)

Optimal. Leaf size=44 \[ \frac{a^3 x^2}{2}+\frac{6}{5} a^2 b x^{5/2}+a b^2 x^3+\frac{2}{7} b^3 x^{7/2} \]

[Out]

(a^3*x^2)/2 + (6*a^2*b*x^(5/2))/5 + a*b^2*x^3 + (2*b^3*x^(7/2))/7

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Rubi [A]  time = 0.0640526, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{a^3 x^2}{2}+\frac{6}{5} a^2 b x^{5/2}+a b^2 x^3+\frac{2}{7} b^3 x^{7/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^3*x,x]

[Out]

(a^3*x^2)/2 + (6*a^2*b*x^(5/2))/5 + a*b^2*x^3 + (2*b^3*x^(7/2))/7

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Rubi in Sympy [A]  time = 9.17054, size = 41, normalized size = 0.93 \[ \frac{a^{3} x^{2}}{2} + \frac{6 a^{2} b x^{\frac{5}{2}}}{5} + a b^{2} x^{3} + \frac{2 b^{3} x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x**2/2 + 6*a**2*b*x**(5/2)/5 + a*b**2*x**3 + 2*b**3*x**(7/2)/7

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Mathematica [A]  time = 0.0109396, size = 44, normalized size = 1. \[ \frac{a^3 x^2}{2}+\frac{6}{5} a^2 b x^{5/2}+a b^2 x^3+\frac{2}{7} b^3 x^{7/2} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*x^2)/2 + (6*a^2*b*x^(5/2))/5 + a*b^2*x^3 + (2*b^3*x^(7/2))/7

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Maple [A]  time = 0.002, size = 35, normalized size = 0.8 \[{\frac{{x}^{2}{a}^{3}}{2}}+{\frac{6\,{a}^{2}b}{5}{x}^{{\frac{5}{2}}}}+a{b}^{2}{x}^{3}+{\frac{2\,{b}^{3}}{7}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(a+b*x^(1/2))^3,x)

[Out]

1/2*x^2*a^3+6/5*a^2*b*x^(5/2)+a*b^2*x^3+2/7*b^3*x^(7/2)

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Maxima [A]  time = 1.42754, size = 86, normalized size = 1.95 \[ \frac{2 \,{\left (b \sqrt{x} + a\right )}^{7}}{7 \, b^{4}} - \frac{{\left (b \sqrt{x} + a\right )}^{6} a}{b^{4}} + \frac{6 \,{\left (b \sqrt{x} + a\right )}^{5} a^{2}}{5 \, b^{4}} - \frac{{\left (b \sqrt{x} + a\right )}^{4} a^{3}}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.232691, size = 54, normalized size = 1.23 \[ a b^{2} x^{3} + \frac{1}{2} \, a^{3} x^{2} + \frac{2}{35} \,{\left (5 \, b^{3} x^{3} + 21 \, a^{2} b x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*b^2*x^3 + 1/2*a^3*x^2 + 2/35*(5*b^3*x^3 + 21*a^2*b*x^2)*sqrt(x)

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Sympy [A]  time = 1.43852, size = 41, normalized size = 0.93 \[ \frac{a^{3} x^{2}}{2} + \frac{6 a^{2} b x^{\frac{5}{2}}}{5} + a b^{2} x^{3} + \frac{2 b^{3} x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x**2/2 + 6*a**2*b*x**(5/2)/5 + a*b**2*x**3 + 2*b**3*x**(7/2)/7

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GIAC/XCAS [A]  time = 0.217415, size = 46, normalized size = 1.05 \[ \frac{2}{7} \, b^{3} x^{\frac{7}{2}} + a b^{2} x^{3} + \frac{6}{5} \, a^{2} b x^{\frac{5}{2}} + \frac{1}{2} \, a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*b^3*x^(7/2) + a*b^2*x^3 + 6/5*a^2*b*x^(5/2) + 1/2*a^3*x^2