3.2152 \(\int \left (a+b \sqrt{x}\right )^{10} x^3 \, dx\)

Optimal. Leaf size=162 \[ -\frac{2 a^7 \left (a+b \sqrt{x}\right )^{11}}{11 b^8}+\frac{7 a^6 \left (a+b \sqrt{x}\right )^{12}}{6 b^8}-\frac{42 a^5 \left (a+b \sqrt{x}\right )^{13}}{13 b^8}+\frac{5 a^4 \left (a+b \sqrt{x}\right )^{14}}{b^8}-\frac{14 a^3 \left (a+b \sqrt{x}\right )^{15}}{3 b^8}+\frac{21 a^2 \left (a+b \sqrt{x}\right )^{16}}{8 b^8}+\frac{\left (a+b \sqrt{x}\right )^{18}}{9 b^8}-\frac{14 a \left (a+b \sqrt{x}\right )^{17}}{17 b^8} \]

[Out]

(-2*a^7*(a + b*Sqrt[x])^11)/(11*b^8) + (7*a^6*(a + b*Sqrt[x])^12)/(6*b^8) - (42*
a^5*(a + b*Sqrt[x])^13)/(13*b^8) + (5*a^4*(a + b*Sqrt[x])^14)/b^8 - (14*a^3*(a +
b*Sqrt[x])^15)/(3*b^8) + (21*a^2*(a + b*Sqrt[x])^16)/(8*b^8) - (14*a*(a + b*Sqr
t[x])^17)/(17*b^8) + (a + b*Sqrt[x])^18/(9*b^8)

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Rubi [A]  time = 0.217695, antiderivative size = 162, 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 \[ -\frac{2 a^7 \left (a+b \sqrt{x}\right )^{11}}{11 b^8}+\frac{7 a^6 \left (a+b \sqrt{x}\right )^{12}}{6 b^8}-\frac{42 a^5 \left (a+b \sqrt{x}\right )^{13}}{13 b^8}+\frac{5 a^4 \left (a+b \sqrt{x}\right )^{14}}{b^8}-\frac{14 a^3 \left (a+b \sqrt{x}\right )^{15}}{3 b^8}+\frac{21 a^2 \left (a+b \sqrt{x}\right )^{16}}{8 b^8}+\frac{\left (a+b \sqrt{x}\right )^{18}}{9 b^8}-\frac{14 a \left (a+b \sqrt{x}\right )^{17}}{17 b^8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^10*x^3,x]

[Out]

(-2*a^7*(a + b*Sqrt[x])^11)/(11*b^8) + (7*a^6*(a + b*Sqrt[x])^12)/(6*b^8) - (42*
a^5*(a + b*Sqrt[x])^13)/(13*b^8) + (5*a^4*(a + b*Sqrt[x])^14)/b^8 - (14*a^3*(a +
b*Sqrt[x])^15)/(3*b^8) + (21*a^2*(a + b*Sqrt[x])^16)/(8*b^8) - (14*a*(a + b*Sqr
t[x])^17)/(17*b^8) + (a + b*Sqrt[x])^18/(9*b^8)

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Rubi in Sympy [A]  time = 32.8326, size = 136, normalized size = 0.84 \[ \frac{a^{10} x^{4}}{4} + \frac{20 a^{9} b x^{\frac{9}{2}}}{9} + 9 a^{8} b^{2} x^{5} + \frac{240 a^{7} b^{3} x^{\frac{11}{2}}}{11} + 35 a^{6} b^{4} x^{6} + \frac{504 a^{5} b^{5} x^{\frac{13}{2}}}{13} + 30 a^{4} b^{6} x^{7} + 16 a^{3} b^{7} x^{\frac{15}{2}} + \frac{45 a^{2} b^{8} x^{8}}{8} + \frac{20 a b^{9} x^{\frac{17}{2}}}{17} + \frac{b^{10} x^{9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(a+b*x**(1/2))**10,x)

[Out]

a**10*x**4/4 + 20*a**9*b*x**(9/2)/9 + 9*a**8*b**2*x**5 + 240*a**7*b**3*x**(11/2)
/11 + 35*a**6*b**4*x**6 + 504*a**5*b**5*x**(13/2)/13 + 30*a**4*b**6*x**7 + 16*a*
*3*b**7*x**(15/2) + 45*a**2*b**8*x**8/8 + 20*a*b**9*x**(17/2)/17 + b**10*x**9/9

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Mathematica [A]  time = 0.0207013, size = 136, normalized size = 0.84 \[ \frac{a^{10} x^4}{4}+\frac{20}{9} a^9 b x^{9/2}+9 a^8 b^2 x^5+\frac{240}{11} a^7 b^3 x^{11/2}+35 a^6 b^4 x^6+\frac{504}{13} a^5 b^5 x^{13/2}+30 a^4 b^6 x^7+16 a^3 b^7 x^{15/2}+\frac{45}{8} a^2 b^8 x^8+\frac{20}{17} a b^9 x^{17/2}+\frac{b^{10} x^9}{9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^10*x^3,x]

[Out]

(a^10*x^4)/4 + (20*a^9*b*x^(9/2))/9 + 9*a^8*b^2*x^5 + (240*a^7*b^3*x^(11/2))/11
+ 35*a^6*b^4*x^6 + (504*a^5*b^5*x^(13/2))/13 + 30*a^4*b^6*x^7 + 16*a^3*b^7*x^(15
/2) + (45*a^2*b^8*x^8)/8 + (20*a*b^9*x^(17/2))/17 + (b^10*x^9)/9

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Maple [A]  time = 0.004, size = 113, normalized size = 0.7 \[{\frac{{x}^{9}{b}^{10}}{9}}+{\frac{20\,a{b}^{9}}{17}{x}^{{\frac{17}{2}}}}+{\frac{45\,{x}^{8}{a}^{2}{b}^{8}}{8}}+16\,{x}^{15/2}{a}^{3}{b}^{7}+30\,{a}^{4}{b}^{6}{x}^{7}+{\frac{504\,{a}^{5}{b}^{5}}{13}{x}^{{\frac{13}{2}}}}+35\,{x}^{6}{a}^{6}{b}^{4}+{\frac{240\,{a}^{7}{b}^{3}}{11}{x}^{{\frac{11}{2}}}}+9\,{x}^{5}{a}^{8}{b}^{2}+{\frac{20\,{a}^{9}b}{9}{x}^{{\frac{9}{2}}}}+{\frac{{a}^{10}{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(a+b*x^(1/2))^10,x)

[Out]

1/9*x^9*b^10+20/17*x^(17/2)*a*b^9+45/8*x^8*a^2*b^8+16*x^(15/2)*a^3*b^7+30*a^4*b^
6*x^7+504/13*x^(13/2)*a^5*b^5+35*x^6*a^6*b^4+240/11*x^(11/2)*a^7*b^3+9*x^5*a^8*b
^2+20/9*x^(9/2)*a^9*b+1/4*a^10*x^4

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Maxima [A]  time = 1.43828, size = 178, normalized size = 1.1 \[ \frac{{\left (b \sqrt{x} + a\right )}^{18}}{9 \, b^{8}} - \frac{14 \,{\left (b \sqrt{x} + a\right )}^{17} a}{17 \, b^{8}} + \frac{21 \,{\left (b \sqrt{x} + a\right )}^{16} a^{2}}{8 \, b^{8}} - \frac{14 \,{\left (b \sqrt{x} + a\right )}^{15} a^{3}}{3 \, b^{8}} + \frac{5 \,{\left (b \sqrt{x} + a\right )}^{14} a^{4}}{b^{8}} - \frac{42 \,{\left (b \sqrt{x} + a\right )}^{13} a^{5}}{13 \, b^{8}} + \frac{7 \,{\left (b \sqrt{x} + a\right )}^{12} a^{6}}{6 \, b^{8}} - \frac{2 \,{\left (b \sqrt{x} + a\right )}^{11} a^{7}}{11 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10*x^3,x, algorithm="maxima")

[Out]

1/9*(b*sqrt(x) + a)^18/b^8 - 14/17*(b*sqrt(x) + a)^17*a/b^8 + 21/8*(b*sqrt(x) +
a)^16*a^2/b^8 - 14/3*(b*sqrt(x) + a)^15*a^3/b^8 + 5*(b*sqrt(x) + a)^14*a^4/b^8 -
42/13*(b*sqrt(x) + a)^13*a^5/b^8 + 7/6*(b*sqrt(x) + a)^12*a^6/b^8 - 2/11*(b*sqr
t(x) + a)^11*a^7/b^8

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Fricas [A]  time = 0.235762, size = 159, normalized size = 0.98 \[ \frac{1}{9} \, b^{10} x^{9} + \frac{45}{8} \, a^{2} b^{8} x^{8} + 30 \, a^{4} b^{6} x^{7} + 35 \, a^{6} b^{4} x^{6} + 9 \, a^{8} b^{2} x^{5} + \frac{1}{4} \, a^{10} x^{4} + \frac{4}{21879} \,{\left (6435 \, a b^{9} x^{8} + 87516 \, a^{3} b^{7} x^{7} + 212058 \, a^{5} b^{5} x^{6} + 119340 \, a^{7} b^{3} x^{5} + 12155 \, a^{9} b x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10*x^3,x, algorithm="fricas")

[Out]

1/9*b^10*x^9 + 45/8*a^2*b^8*x^8 + 30*a^4*b^6*x^7 + 35*a^6*b^4*x^6 + 9*a^8*b^2*x^
5 + 1/4*a^10*x^4 + 4/21879*(6435*a*b^9*x^8 + 87516*a^3*b^7*x^7 + 212058*a^5*b^5*
x^6 + 119340*a^7*b^3*x^5 + 12155*a^9*b*x^4)*sqrt(x)

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Sympy [A]  time = 17.7314, size = 136, normalized size = 0.84 \[ \frac{a^{10} x^{4}}{4} + \frac{20 a^{9} b x^{\frac{9}{2}}}{9} + 9 a^{8} b^{2} x^{5} + \frac{240 a^{7} b^{3} x^{\frac{11}{2}}}{11} + 35 a^{6} b^{4} x^{6} + \frac{504 a^{5} b^{5} x^{\frac{13}{2}}}{13} + 30 a^{4} b^{6} x^{7} + 16 a^{3} b^{7} x^{\frac{15}{2}} + \frac{45 a^{2} b^{8} x^{8}}{8} + \frac{20 a b^{9} x^{\frac{17}{2}}}{17} + \frac{b^{10} x^{9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(a+b*x**(1/2))**10,x)

[Out]

a**10*x**4/4 + 20*a**9*b*x**(9/2)/9 + 9*a**8*b**2*x**5 + 240*a**7*b**3*x**(11/2)
/11 + 35*a**6*b**4*x**6 + 504*a**5*b**5*x**(13/2)/13 + 30*a**4*b**6*x**7 + 16*a*
*3*b**7*x**(15/2) + 45*a**2*b**8*x**8/8 + 20*a*b**9*x**(17/2)/17 + b**10*x**9/9

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GIAC/XCAS [A]  time = 0.218264, size = 151, normalized size = 0.93 \[ \frac{1}{9} \, b^{10} x^{9} + \frac{20}{17} \, a b^{9} x^{\frac{17}{2}} + \frac{45}{8} \, a^{2} b^{8} x^{8} + 16 \, a^{3} b^{7} x^{\frac{15}{2}} + 30 \, a^{4} b^{6} x^{7} + \frac{504}{13} \, a^{5} b^{5} x^{\frac{13}{2}} + 35 \, a^{6} b^{4} x^{6} + \frac{240}{11} \, a^{7} b^{3} x^{\frac{11}{2}} + 9 \, a^{8} b^{2} x^{5} + \frac{20}{9} \, a^{9} b x^{\frac{9}{2}} + \frac{1}{4} \, a^{10} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10*x^3,x, algorithm="giac")

[Out]

1/9*b^10*x^9 + 20/17*a*b^9*x^(17/2) + 45/8*a^2*b^8*x^8 + 16*a^3*b^7*x^(15/2) + 3
0*a^4*b^6*x^7 + 504/13*a^5*b^5*x^(13/2) + 35*a^6*b^4*x^6 + 240/11*a^7*b^3*x^(11/
2) + 9*a^8*b^2*x^5 + 20/9*a^9*b*x^(9/2) + 1/4*a^10*x^4