3.2154 \(\int \left (a+b \sqrt{x}\right )^{10} x \, dx\)

Optimal. Leaf size=80 \[ -\frac{2 a^3 \left (a+b \sqrt{x}\right )^{11}}{11 b^4}+\frac{a^2 \left (a+b \sqrt{x}\right )^{12}}{2 b^4}+\frac{\left (a+b \sqrt{x}\right )^{14}}{7 b^4}-\frac{6 a \left (a+b \sqrt{x}\right )^{13}}{13 b^4} \]

[Out]

(-2*a^3*(a + b*Sqrt[x])^11)/(11*b^4) + (a^2*(a + b*Sqrt[x])^12)/(2*b^4) - (6*a*(
a + b*Sqrt[x])^13)/(13*b^4) + (a + b*Sqrt[x])^14/(7*b^4)

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Rubi [A]  time = 0.132747, antiderivative size = 80, 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 \[ -\frac{2 a^3 \left (a+b \sqrt{x}\right )^{11}}{11 b^4}+\frac{a^2 \left (a+b \sqrt{x}\right )^{12}}{2 b^4}+\frac{\left (a+b \sqrt{x}\right )^{14}}{7 b^4}-\frac{6 a \left (a+b \sqrt{x}\right )^{13}}{13 b^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^10*x,x]

[Out]

(-2*a^3*(a + b*Sqrt[x])^11)/(11*b^4) + (a^2*(a + b*Sqrt[x])^12)/(2*b^4) - (6*a*(
a + b*Sqrt[x])^13)/(13*b^4) + (a + b*Sqrt[x])^14/(7*b^4)

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Rubi in Sympy [A]  time = 21.7596, size = 71, normalized size = 0.89 \[ - \frac{2 a^{3} \left (a + b \sqrt{x}\right )^{11}}{11 b^{4}} + \frac{a^{2} \left (a + b \sqrt{x}\right )^{12}}{2 b^{4}} - \frac{6 a \left (a + b \sqrt{x}\right )^{13}}{13 b^{4}} + \frac{\left (a + b \sqrt{x}\right )^{14}}{7 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(a+b*x**(1/2))**10,x)

[Out]

-2*a**3*(a + b*sqrt(x))**11/(11*b**4) + a**2*(a + b*sqrt(x))**12/(2*b**4) - 6*a*
(a + b*sqrt(x))**13/(13*b**4) + (a + b*sqrt(x))**14/(7*b**4)

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Mathematica [A]  time = 0.0217947, size = 136, normalized size = 1.7 \[ \frac{a^{10} x^2}{2}+4 a^9 b x^{5/2}+15 a^8 b^2 x^3+\frac{240}{7} a^7 b^3 x^{7/2}+\frac{105}{2} a^6 b^4 x^4+56 a^5 b^5 x^{9/2}+42 a^4 b^6 x^5+\frac{240}{11} a^3 b^7 x^{11/2}+\frac{15}{2} a^2 b^8 x^6+\frac{20}{13} a b^9 x^{13/2}+\frac{b^{10} x^7}{7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^10*x,x]

[Out]

(a^10*x^2)/2 + 4*a^9*b*x^(5/2) + 15*a^8*b^2*x^3 + (240*a^7*b^3*x^(7/2))/7 + (105
*a^6*b^4*x^4)/2 + 56*a^5*b^5*x^(9/2) + 42*a^4*b^6*x^5 + (240*a^3*b^7*x^(11/2))/1
1 + (15*a^2*b^8*x^6)/2 + (20*a*b^9*x^(13/2))/13 + (b^10*x^7)/7

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Maple [A]  time = 0.004, size = 113, normalized size = 1.4 \[{\frac{{x}^{7}{b}^{10}}{7}}+{\frac{20\,a{b}^{9}}{13}{x}^{{\frac{13}{2}}}}+{\frac{15\,{x}^{6}{a}^{2}{b}^{8}}{2}}+{\frac{240\,{a}^{3}{b}^{7}}{11}{x}^{{\frac{11}{2}}}}+42\,{x}^{5}{a}^{4}{b}^{6}+56\,{x}^{9/2}{a}^{5}{b}^{5}+{\frac{105\,{x}^{4}{a}^{6}{b}^{4}}{2}}+{\frac{240\,{a}^{7}{b}^{3}}{7}{x}^{{\frac{7}{2}}}}+15\,{x}^{3}{a}^{8}{b}^{2}+4\,{x}^{5/2}{a}^{9}b+{\frac{{x}^{2}{a}^{10}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(a+b*x^(1/2))^10,x)

[Out]

1/7*x^7*b^10+20/13*x^(13/2)*a*b^9+15/2*x^6*a^2*b^8+240/11*x^(11/2)*a^3*b^7+42*x^
5*a^4*b^6+56*x^(9/2)*a^5*b^5+105/2*x^4*a^6*b^4+240/7*x^(7/2)*a^7*b^3+15*x^3*a^8*
b^2+4*x^(5/2)*a^9*b+1/2*x^2*a^10

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Maxima [A]  time = 1.44178, size = 86, normalized size = 1.08 \[ \frac{{\left (b \sqrt{x} + a\right )}^{14}}{7 \, b^{4}} - \frac{6 \,{\left (b \sqrt{x} + a\right )}^{13} a}{13 \, b^{4}} + \frac{{\left (b \sqrt{x} + a\right )}^{12} a^{2}}{2 \, b^{4}} - \frac{2 \,{\left (b \sqrt{x} + a\right )}^{11} a^{3}}{11 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10*x,x, algorithm="maxima")

[Out]

1/7*(b*sqrt(x) + a)^14/b^4 - 6/13*(b*sqrt(x) + a)^13*a/b^4 + 1/2*(b*sqrt(x) + a)
^12*a^2/b^4 - 2/11*(b*sqrt(x) + a)^11*a^3/b^4

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Fricas [A]  time = 0.233335, size = 159, normalized size = 1.99 \[ \frac{1}{7} \, b^{10} x^{7} + \frac{15}{2} \, a^{2} b^{8} x^{6} + 42 \, a^{4} b^{6} x^{5} + \frac{105}{2} \, a^{6} b^{4} x^{4} + 15 \, a^{8} b^{2} x^{3} + \frac{1}{2} \, a^{10} x^{2} + \frac{4}{1001} \,{\left (385 \, a b^{9} x^{6} + 5460 \, a^{3} b^{7} x^{5} + 14014 \, a^{5} b^{5} x^{4} + 8580 \, a^{7} b^{3} x^{3} + 1001 \, a^{9} b x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10*x,x, algorithm="fricas")

[Out]

1/7*b^10*x^7 + 15/2*a^2*b^8*x^6 + 42*a^4*b^6*x^5 + 105/2*a^6*b^4*x^4 + 15*a^8*b^
2*x^3 + 1/2*a^10*x^2 + 4/1001*(385*a*b^9*x^6 + 5460*a^3*b^7*x^5 + 14014*a^5*b^5*
x^4 + 8580*a^7*b^3*x^3 + 1001*a^9*b*x^2)*sqrt(x)

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Sympy [A]  time = 3.49441, size = 136, normalized size = 1.7 \[ \frac{a^{10} x^{2}}{2} + 4 a^{9} b x^{\frac{5}{2}} + 15 a^{8} b^{2} x^{3} + \frac{240 a^{7} b^{3} x^{\frac{7}{2}}}{7} + \frac{105 a^{6} b^{4} x^{4}}{2} + 56 a^{5} b^{5} x^{\frac{9}{2}} + 42 a^{4} b^{6} x^{5} + \frac{240 a^{3} b^{7} x^{\frac{11}{2}}}{11} + \frac{15 a^{2} b^{8} x^{6}}{2} + \frac{20 a b^{9} x^{\frac{13}{2}}}{13} + \frac{b^{10} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(a+b*x**(1/2))**10,x)

[Out]

a**10*x**2/2 + 4*a**9*b*x**(5/2) + 15*a**8*b**2*x**3 + 240*a**7*b**3*x**(7/2)/7
+ 105*a**6*b**4*x**4/2 + 56*a**5*b**5*x**(9/2) + 42*a**4*b**6*x**5 + 240*a**3*b*
*7*x**(11/2)/11 + 15*a**2*b**8*x**6/2 + 20*a*b**9*x**(13/2)/13 + b**10*x**7/7

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GIAC/XCAS [A]  time = 0.216736, size = 151, normalized size = 1.89 \[ \frac{1}{7} \, b^{10} x^{7} + \frac{20}{13} \, a b^{9} x^{\frac{13}{2}} + \frac{15}{2} \, a^{2} b^{8} x^{6} + \frac{240}{11} \, a^{3} b^{7} x^{\frac{11}{2}} + 42 \, a^{4} b^{6} x^{5} + 56 \, a^{5} b^{5} x^{\frac{9}{2}} + \frac{105}{2} \, a^{6} b^{4} x^{4} + \frac{240}{7} \, a^{7} b^{3} x^{\frac{7}{2}} + 15 \, a^{8} b^{2} x^{3} + 4 \, a^{9} b x^{\frac{5}{2}} + \frac{1}{2} \, a^{10} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10*x,x, algorithm="giac")

[Out]

1/7*b^10*x^7 + 20/13*a*b^9*x^(13/2) + 15/2*a^2*b^8*x^6 + 240/11*a^3*b^7*x^(11/2)
+ 42*a^4*b^6*x^5 + 56*a^5*b^5*x^(9/2) + 105/2*a^6*b^4*x^4 + 240/7*a^7*b^3*x^(7/
2) + 15*a^8*b^2*x^3 + 4*a^9*b*x^(5/2) + 1/2*a^10*x^2