3.2141 \(\int \left (a+b \sqrt{x}\right )^5 x^2 \, dx\)

Optimal. Leaf size=72 \[ \frac{a^5 x^3}{3}+\frac{10}{7} a^4 b x^{7/2}+\frac{5}{2} a^3 b^2 x^4+\frac{20}{9} a^2 b^3 x^{9/2}+a b^4 x^5+\frac{2}{11} b^5 x^{11/2} \]

[Out]

(a^5*x^3)/3 + (10*a^4*b*x^(7/2))/7 + (5*a^3*b^2*x^4)/2 + (20*a^2*b^3*x^(9/2))/9
+ a*b^4*x^5 + (2*b^5*x^(11/2))/11

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Rubi [A]  time = 0.100692, antiderivative size = 72, 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 \[ \frac{a^5 x^3}{3}+\frac{10}{7} a^4 b x^{7/2}+\frac{5}{2} a^3 b^2 x^4+\frac{20}{9} a^2 b^3 x^{9/2}+a b^4 x^5+\frac{2}{11} b^5 x^{11/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^5*x^2,x]

[Out]

(a^5*x^3)/3 + (10*a^4*b*x^(7/2))/7 + (5*a^3*b^2*x^4)/2 + (20*a^2*b^3*x^(9/2))/9
+ a*b^4*x^5 + (2*b^5*x^(11/2))/11

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Rubi in Sympy [A]  time = 15.1954, size = 70, normalized size = 0.97 \[ \frac{a^{5} x^{3}}{3} + \frac{10 a^{4} b x^{\frac{7}{2}}}{7} + \frac{5 a^{3} b^{2} x^{4}}{2} + \frac{20 a^{2} b^{3} x^{\frac{9}{2}}}{9} + a b^{4} x^{5} + \frac{2 b^{5} x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(a+b*x**(1/2))**5,x)

[Out]

a**5*x**3/3 + 10*a**4*b*x**(7/2)/7 + 5*a**3*b**2*x**4/2 + 20*a**2*b**3*x**(9/2)/
9 + a*b**4*x**5 + 2*b**5*x**(11/2)/11

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Mathematica [A]  time = 0.0135334, size = 72, normalized size = 1. \[ \frac{a^5 x^3}{3}+\frac{10}{7} a^4 b x^{7/2}+\frac{5}{2} a^3 b^2 x^4+\frac{20}{9} a^2 b^3 x^{9/2}+a b^4 x^5+\frac{2}{11} b^5 x^{11/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^5*x^2,x]

[Out]

(a^5*x^3)/3 + (10*a^4*b*x^(7/2))/7 + (5*a^3*b^2*x^4)/2 + (20*a^2*b^3*x^(9/2))/9
+ a*b^4*x^5 + (2*b^5*x^(11/2))/11

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Maple [A]  time = 0.003, size = 57, normalized size = 0.8 \[{\frac{{a}^{5}{x}^{3}}{3}}+{\frac{10\,{a}^{4}b}{7}{x}^{{\frac{7}{2}}}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{4}}{2}}+{\frac{20\,{a}^{2}{b}^{3}}{9}{x}^{{\frac{9}{2}}}}+a{b}^{4}{x}^{5}+{\frac{2\,{b}^{5}}{11}{x}^{{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(a+b*x^(1/2))^5,x)

[Out]

1/3*a^5*x^3+10/7*a^4*b*x^(7/2)+5/2*a^3*b^2*x^4+20/9*a^2*b^3*x^(9/2)+a*b^4*x^5+2/
11*b^5*x^(11/2)

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Maxima [A]  time = 1.43546, size = 132, normalized size = 1.83 \[ \frac{2 \,{\left (b \sqrt{x} + a\right )}^{11}}{11 \, b^{6}} - \frac{{\left (b \sqrt{x} + a\right )}^{10} a}{b^{6}} + \frac{20 \,{\left (b \sqrt{x} + a\right )}^{9} a^{2}}{9 \, b^{6}} - \frac{5 \,{\left (b \sqrt{x} + a\right )}^{8} a^{3}}{2 \, b^{6}} + \frac{10 \,{\left (b \sqrt{x} + a\right )}^{7} a^{4}}{7 \, b^{6}} - \frac{{\left (b \sqrt{x} + a\right )}^{6} a^{5}}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5*x^2,x, algorithm="maxima")

[Out]

2/11*(b*sqrt(x) + a)^11/b^6 - (b*sqrt(x) + a)^10*a/b^6 + 20/9*(b*sqrt(x) + a)^9*
a^2/b^6 - 5/2*(b*sqrt(x) + a)^8*a^3/b^6 + 10/7*(b*sqrt(x) + a)^7*a^4/b^6 - 1/3*(
b*sqrt(x) + a)^6*a^5/b^6

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Fricas [A]  time = 0.235365, size = 84, normalized size = 1.17 \[ a b^{4} x^{5} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{1}{3} \, a^{5} x^{3} + \frac{2}{693} \,{\left (63 \, b^{5} x^{5} + 770 \, a^{2} b^{3} x^{4} + 495 \, a^{4} b x^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5*x^2,x, algorithm="fricas")

[Out]

a*b^4*x^5 + 5/2*a^3*b^2*x^4 + 1/3*a^5*x^3 + 2/693*(63*b^5*x^5 + 770*a^2*b^3*x^4
+ 495*a^4*b*x^3)*sqrt(x)

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Sympy [A]  time = 2.48461, size = 70, normalized size = 0.97 \[ \frac{a^{5} x^{3}}{3} + \frac{10 a^{4} b x^{\frac{7}{2}}}{7} + \frac{5 a^{3} b^{2} x^{4}}{2} + \frac{20 a^{2} b^{3} x^{\frac{9}{2}}}{9} + a b^{4} x^{5} + \frac{2 b^{5} x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(a+b*x**(1/2))**5,x)

[Out]

a**5*x**3/3 + 10*a**4*b*x**(7/2)/7 + 5*a**3*b**2*x**4/2 + 20*a**2*b**3*x**(9/2)/
9 + a*b**4*x**5 + 2*b**5*x**(11/2)/11

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GIAC/XCAS [A]  time = 0.222323, size = 76, normalized size = 1.06 \[ \frac{2}{11} \, b^{5} x^{\frac{11}{2}} + a b^{4} x^{5} + \frac{20}{9} \, a^{2} b^{3} x^{\frac{9}{2}} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{10}{7} \, a^{4} b x^{\frac{7}{2}} + \frac{1}{3} \, a^{5} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5*x^2,x, algorithm="giac")

[Out]

2/11*b^5*x^(11/2) + a*b^4*x^5 + 20/9*a^2*b^3*x^(9/2) + 5/2*a^3*b^2*x^4 + 10/7*a^
4*b*x^(7/2) + 1/3*a^5*x^3