3.2129 \(\int \left (a+b \sqrt{x}\right )^3 x^3 \, dx\)

Optimal. Leaf size=47 \[ \frac{a^3 x^4}{4}+\frac{2}{3} a^2 b x^{9/2}+\frac{3}{5} a b^2 x^5+\frac{2}{11} b^3 x^{11/2} \]

[Out]

(a^3*x^4)/4 + (2*a^2*b*x^(9/2))/3 + (3*a*b^2*x^5)/5 + (2*b^3*x^(11/2))/11

_______________________________________________________________________________________

Rubi [A]  time = 0.0739327, 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 \[ \frac{a^3 x^4}{4}+\frac{2}{3} a^2 b x^{9/2}+\frac{3}{5} a b^2 x^5+\frac{2}{11} b^3 x^{11/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^3*x^3,x]

[Out]

(a^3*x^4)/4 + (2*a^2*b*x^(9/2))/3 + (3*a*b^2*x^5)/5 + (2*b^3*x^(11/2))/11

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.7863, size = 44, normalized size = 0.94 \[ \frac{a^{3} x^{4}}{4} + \frac{2 a^{2} b x^{\frac{9}{2}}}{3} + \frac{3 a b^{2} x^{5}}{5} + \frac{2 b^{3} x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x**4/4 + 2*a**2*b*x**(9/2)/3 + 3*a*b**2*x**5/5 + 2*b**3*x**(11/2)/11

_______________________________________________________________________________________

Mathematica [A]  time = 0.0110557, size = 47, normalized size = 1. \[ \frac{a^3 x^4}{4}+\frac{2}{3} a^2 b x^{9/2}+\frac{3}{5} a b^2 x^5+\frac{2}{11} b^3 x^{11/2} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*x^4)/4 + (2*a^2*b*x^(9/2))/3 + (3*a*b^2*x^5)/5 + (2*b^3*x^(11/2))/11

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 36, normalized size = 0.8 \[{\frac{{a}^{3}{x}^{4}}{4}}+{\frac{2\,{a}^{2}b}{3}{x}^{{\frac{9}{2}}}}+{\frac{3\,a{b}^{2}{x}^{5}}{5}}+{\frac{2\,{b}^{3}}{11}{x}^{{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(a+b*x^(1/2))^3,x)

[Out]

1/4*a^3*x^4+2/3*a^2*b*x^(9/2)+3/5*a*b^2*x^5+2/11*b^3*x^(11/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.44976, size = 178, normalized size = 3.79 \[ \frac{2 \,{\left (b \sqrt{x} + a\right )}^{11}}{11 \, b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{10} a}{5 \, b^{8}} + \frac{14 \,{\left (b \sqrt{x} + a\right )}^{9} a^{2}}{3 \, b^{8}} - \frac{35 \,{\left (b \sqrt{x} + a\right )}^{8} a^{3}}{4 \, b^{8}} + \frac{10 \,{\left (b \sqrt{x} + a\right )}^{7} a^{4}}{b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{6} a^{5}}{b^{8}} + \frac{14 \,{\left (b \sqrt{x} + a\right )}^{5} a^{6}}{5 \, b^{8}} - \frac{{\left (b \sqrt{x} + a\right )}^{4} a^{7}}{2 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*(b*sqrt(x) + a)^11/b^8 - 7/5*(b*sqrt(x) + a)^10*a/b^8 + 14/3*(b*sqrt(x) + a
)^9*a^2/b^8 - 35/4*(b*sqrt(x) + a)^8*a^3/b^8 + 10*(b*sqrt(x) + a)^7*a^4/b^8 - 7*
(b*sqrt(x) + a)^6*a^5/b^8 + 14/5*(b*sqrt(x) + a)^5*a^6/b^8 - 1/2*(b*sqrt(x) + a)
^4*a^7/b^8

_______________________________________________________________________________________

Fricas [A]  time = 0.244725, size = 55, normalized size = 1.17 \[ \frac{3}{5} \, a b^{2} x^{5} + \frac{1}{4} \, a^{3} x^{4} + \frac{2}{33} \,{\left (3 \, b^{3} x^{5} + 11 \, a^{2} b x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/5*a*b^2*x^5 + 1/4*a^3*x^4 + 2/33*(3*b^3*x^5 + 11*a^2*b*x^4)*sqrt(x)

_______________________________________________________________________________________

Sympy [A]  time = 2.42545, size = 44, normalized size = 0.94 \[ \frac{a^{3} x^{4}}{4} + \frac{2 a^{2} b x^{\frac{9}{2}}}{3} + \frac{3 a b^{2} x^{5}}{5} + \frac{2 b^{3} x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x**4/4 + 2*a**2*b*x**(9/2)/3 + 3*a*b**2*x**5/5 + 2*b**3*x**(11/2)/11

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212665, size = 47, normalized size = 1. \[ \frac{2}{11} \, b^{3} x^{\frac{11}{2}} + \frac{3}{5} \, a b^{2} x^{5} + \frac{2}{3} \, a^{2} b x^{\frac{9}{2}} + \frac{1}{4} \, a^{3} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*b^3*x^(11/2) + 3/5*a*b^2*x^5 + 2/3*a^2*b*x^(9/2) + 1/4*a^3*x^4