3.2140 \(\int \left (a+b \sqrt{x}\right )^5 x^3 \, dx\)

Optimal. Leaf size=73 \[ \frac{a^5 x^4}{4}+\frac{10}{9} a^4 b x^{9/2}+2 a^3 b^2 x^5+\frac{20}{11} a^2 b^3 x^{11/2}+\frac{5}{6} a b^4 x^6+\frac{2}{13} b^5 x^{13/2} \]

[Out]

(a^5*x^4)/4 + (10*a^4*b*x^(9/2))/9 + 2*a^3*b^2*x^5 + (20*a^2*b^3*x^(11/2))/11 +
(5*a*b^4*x^6)/6 + (2*b^5*x^(13/2))/13

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Rubi [A]  time = 0.103858, antiderivative size = 73, 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 \[ \frac{a^5 x^4}{4}+\frac{10}{9} a^4 b x^{9/2}+2 a^3 b^2 x^5+\frac{20}{11} a^2 b^3 x^{11/2}+\frac{5}{6} a b^4 x^6+\frac{2}{13} b^5 x^{13/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^5*x^3,x]

[Out]

(a^5*x^4)/4 + (10*a^4*b*x^(9/2))/9 + 2*a^3*b^2*x^5 + (20*a^2*b^3*x^(11/2))/11 +
(5*a*b^4*x^6)/6 + (2*b^5*x^(13/2))/13

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Rubi in Sympy [A]  time = 16.011, size = 71, normalized size = 0.97 \[ \frac{a^{5} x^{4}}{4} + \frac{10 a^{4} b x^{\frac{9}{2}}}{9} + 2 a^{3} b^{2} x^{5} + \frac{20 a^{2} b^{3} x^{\frac{11}{2}}}{11} + \frac{5 a b^{4} x^{6}}{6} + \frac{2 b^{5} x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(a+b*x**(1/2))**5,x)

[Out]

a**5*x**4/4 + 10*a**4*b*x**(9/2)/9 + 2*a**3*b**2*x**5 + 20*a**2*b**3*x**(11/2)/1
1 + 5*a*b**4*x**6/6 + 2*b**5*x**(13/2)/13

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Mathematica [A]  time = 0.013598, size = 73, normalized size = 1. \[ \frac{a^5 x^4}{4}+\frac{10}{9} a^4 b x^{9/2}+2 a^3 b^2 x^5+\frac{20}{11} a^2 b^3 x^{11/2}+\frac{5}{6} a b^4 x^6+\frac{2}{13} b^5 x^{13/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^5*x^3,x]

[Out]

(a^5*x^4)/4 + (10*a^4*b*x^(9/2))/9 + 2*a^3*b^2*x^5 + (20*a^2*b^3*x^(11/2))/11 +
(5*a*b^4*x^6)/6 + (2*b^5*x^(13/2))/13

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Maple [A]  time = 0.003, size = 58, normalized size = 0.8 \[{\frac{{a}^{5}{x}^{4}}{4}}+{\frac{10\,{a}^{4}b}{9}{x}^{{\frac{9}{2}}}}+2\,{a}^{3}{b}^{2}{x}^{5}+{\frac{20\,{a}^{2}{b}^{3}}{11}{x}^{{\frac{11}{2}}}}+{\frac{5\,a{b}^{4}{x}^{6}}{6}}+{\frac{2\,{b}^{5}}{13}{x}^{{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(a+b*x^(1/2))^5,x)

[Out]

1/4*a^5*x^4+10/9*a^4*b*x^(9/2)+2*a^3*b^2*x^5+20/11*a^2*b^3*x^(11/2)+5/6*a*b^4*x^
6+2/13*b^5*x^(13/2)

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Maxima [A]  time = 1.44719, size = 178, normalized size = 2.44 \[ \frac{2 \,{\left (b \sqrt{x} + a\right )}^{13}}{13 \, b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{12} a}{6 \, b^{8}} + \frac{42 \,{\left (b \sqrt{x} + a\right )}^{11} a^{2}}{11 \, b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{10} a^{3}}{b^{8}} + \frac{70 \,{\left (b \sqrt{x} + a\right )}^{9} a^{4}}{9 \, b^{8}} - \frac{21 \,{\left (b \sqrt{x} + a\right )}^{8} a^{5}}{4 \, b^{8}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{7} a^{6}}{b^{8}} - \frac{{\left (b \sqrt{x} + a\right )}^{6} a^{7}}{3 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5*x^3,x, algorithm="maxima")

[Out]

2/13*(b*sqrt(x) + a)^13/b^8 - 7/6*(b*sqrt(x) + a)^12*a/b^8 + 42/11*(b*sqrt(x) +
a)^11*a^2/b^8 - 7*(b*sqrt(x) + a)^10*a^3/b^8 + 70/9*(b*sqrt(x) + a)^9*a^4/b^8 -
21/4*(b*sqrt(x) + a)^8*a^5/b^8 + 2*(b*sqrt(x) + a)^7*a^6/b^8 - 1/3*(b*sqrt(x) +
a)^6*a^7/b^8

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Fricas [A]  time = 0.232221, size = 85, normalized size = 1.16 \[ \frac{5}{6} \, a b^{4} x^{6} + 2 \, a^{3} b^{2} x^{5} + \frac{1}{4} \, a^{5} x^{4} + \frac{2}{1287} \,{\left (99 \, b^{5} x^{6} + 1170 \, a^{2} b^{3} x^{5} + 715 \, a^{4} b x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5*x^3,x, algorithm="fricas")

[Out]

5/6*a*b^4*x^6 + 2*a^3*b^2*x^5 + 1/4*a^5*x^4 + 2/1287*(99*b^5*x^6 + 1170*a^2*b^3*
x^5 + 715*a^4*b*x^4)*sqrt(x)

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Sympy [A]  time = 3.41764, size = 71, normalized size = 0.97 \[ \frac{a^{5} x^{4}}{4} + \frac{10 a^{4} b x^{\frac{9}{2}}}{9} + 2 a^{3} b^{2} x^{5} + \frac{20 a^{2} b^{3} x^{\frac{11}{2}}}{11} + \frac{5 a b^{4} x^{6}}{6} + \frac{2 b^{5} x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(a+b*x**(1/2))**5,x)

[Out]

a**5*x**4/4 + 10*a**4*b*x**(9/2)/9 + 2*a**3*b**2*x**5 + 20*a**2*b**3*x**(11/2)/1
1 + 5*a*b**4*x**6/6 + 2*b**5*x**(13/2)/13

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GIAC/XCAS [A]  time = 0.215554, size = 77, normalized size = 1.05 \[ \frac{2}{13} \, b^{5} x^{\frac{13}{2}} + \frac{5}{6} \, a b^{4} x^{6} + \frac{20}{11} \, a^{2} b^{3} x^{\frac{11}{2}} + 2 \, a^{3} b^{2} x^{5} + \frac{10}{9} \, a^{4} b x^{\frac{9}{2}} + \frac{1}{4} \, a^{5} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5*x^3,x, algorithm="giac")

[Out]

2/13*b^5*x^(13/2) + 5/6*a*b^4*x^6 + 20/11*a^2*b^3*x^(11/2) + 2*a^3*b^2*x^5 + 10/
9*a^4*b*x^(9/2) + 1/4*a^5*x^4