3.2171 \(\int \left (a+b \sqrt{x}\right )^{15} x \, dx\)

Optimal. Leaf size=80 \[ -\frac{a^3 \left (a+b \sqrt{x}\right )^{16}}{8 b^4}+\frac{6 a^2 \left (a+b \sqrt{x}\right )^{17}}{17 b^4}+\frac{2 \left (a+b \sqrt{x}\right )^{19}}{19 b^4}-\frac{a \left (a+b \sqrt{x}\right )^{18}}{3 b^4} \]

[Out]

-(a^3*(a + b*Sqrt[x])^16)/(8*b^4) + (6*a^2*(a + b*Sqrt[x])^17)/(17*b^4) - (a*(a
+ b*Sqrt[x])^18)/(3*b^4) + (2*(a + b*Sqrt[x])^19)/(19*b^4)

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Rubi [A]  time = 0.185615, 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 \[ -\frac{a^3 \left (a+b \sqrt{x}\right )^{16}}{8 b^4}+\frac{6 a^2 \left (a+b \sqrt{x}\right )^{17}}{17 b^4}+\frac{2 \left (a+b \sqrt{x}\right )^{19}}{19 b^4}-\frac{a \left (a+b \sqrt{x}\right )^{18}}{3 b^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^15*x,x]

[Out]

-(a^3*(a + b*Sqrt[x])^16)/(8*b^4) + (6*a^2*(a + b*Sqrt[x])^17)/(17*b^4) - (a*(a
+ b*Sqrt[x])^18)/(3*b^4) + (2*(a + b*Sqrt[x])^19)/(19*b^4)

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Rubi in Sympy [A]  time = 33.4942, size = 71, normalized size = 0.89 \[ - \frac{a^{3} \left (a + b \sqrt{x}\right )^{16}}{8 b^{4}} + \frac{6 a^{2} \left (a + b \sqrt{x}\right )^{17}}{17 b^{4}} - \frac{a \left (a + b \sqrt{x}\right )^{18}}{3 b^{4}} + \frac{2 \left (a + b \sqrt{x}\right )^{19}}{19 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(a+b*x**(1/2))**15,x)

[Out]

-a**3*(a + b*sqrt(x))**16/(8*b**4) + 6*a**2*(a + b*sqrt(x))**17/(17*b**4) - a*(a
+ b*sqrt(x))**18/(3*b**4) + 2*(a + b*sqrt(x))**19/(19*b**4)

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Mathematica [B]  time = 0.0315292, size = 199, normalized size = 2.49 \[ \frac{a^{15} x^2}{2}+6 a^{14} b x^{5/2}+35 a^{13} b^2 x^3+130 a^{12} b^3 x^{7/2}+\frac{1365}{4} a^{11} b^4 x^4+\frac{2002}{3} a^{10} b^5 x^{9/2}+1001 a^9 b^6 x^5+1170 a^8 b^7 x^{11/2}+\frac{2145}{2} a^7 b^8 x^6+770 a^6 b^9 x^{13/2}+429 a^5 b^{10} x^7+182 a^4 b^{11} x^{15/2}+\frac{455}{8} a^3 b^{12} x^8+\frac{210}{17} a^2 b^{13} x^{17/2}+\frac{5}{3} a b^{14} x^9+\frac{2}{19} b^{15} x^{19/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^15*x,x]

[Out]

(a^15*x^2)/2 + 6*a^14*b*x^(5/2) + 35*a^13*b^2*x^3 + 130*a^12*b^3*x^(7/2) + (1365
*a^11*b^4*x^4)/4 + (2002*a^10*b^5*x^(9/2))/3 + 1001*a^9*b^6*x^5 + 1170*a^8*b^7*x
^(11/2) + (2145*a^7*b^8*x^6)/2 + 770*a^6*b^9*x^(13/2) + 429*a^5*b^10*x^7 + 182*a
^4*b^11*x^(15/2) + (455*a^3*b^12*x^8)/8 + (210*a^2*b^13*x^(17/2))/17 + (5*a*b^14
*x^9)/3 + (2*b^15*x^(19/2))/19

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Maple [B]  time = 0.005, size = 168, normalized size = 2.1 \[{\frac{2\,{b}^{15}}{19}{x}^{{\frac{19}{2}}}}+{\frac{5\,{x}^{9}a{b}^{14}}{3}}+{\frac{210\,{a}^{2}{b}^{13}}{17}{x}^{{\frac{17}{2}}}}+{\frac{455\,{x}^{8}{a}^{3}{b}^{12}}{8}}+182\,{x}^{15/2}{a}^{4}{b}^{11}+429\,{x}^{7}{a}^{5}{b}^{10}+770\,{x}^{13/2}{a}^{6}{b}^{9}+{\frac{2145\,{x}^{6}{a}^{7}{b}^{8}}{2}}+1170\,{x}^{11/2}{a}^{8}{b}^{7}+1001\,{x}^{5}{a}^{9}{b}^{6}+{\frac{2002\,{a}^{10}{b}^{5}}{3}{x}^{{\frac{9}{2}}}}+{\frac{1365\,{x}^{4}{a}^{11}{b}^{4}}{4}}+130\,{x}^{7/2}{a}^{12}{b}^{3}+35\,{x}^{3}{a}^{13}{b}^{2}+6\,{x}^{5/2}{a}^{14}b+{\frac{{x}^{2}{a}^{15}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(a+b*x^(1/2))^15,x)

[Out]

2/19*x^(19/2)*b^15+5/3*x^9*a*b^14+210/17*x^(17/2)*a^2*b^13+455/8*x^8*a^3*b^12+18
2*x^(15/2)*a^4*b^11+429*x^7*a^5*b^10+770*x^(13/2)*a^6*b^9+2145/2*x^6*a^7*b^8+117
0*x^(11/2)*a^8*b^7+1001*x^5*a^9*b^6+2002/3*x^(9/2)*a^10*b^5+1365/4*x^4*a^11*b^4+
130*x^(7/2)*a^12*b^3+35*x^3*a^13*b^2+6*x^(5/2)*a^14*b+1/2*x^2*a^15

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Maxima [A]  time = 1.43665, size = 86, normalized size = 1.08 \[ \frac{2 \,{\left (b \sqrt{x} + a\right )}^{19}}{19 \, b^{4}} - \frac{{\left (b \sqrt{x} + a\right )}^{18} a}{3 \, b^{4}} + \frac{6 \,{\left (b \sqrt{x} + a\right )}^{17} a^{2}}{17 \, b^{4}} - \frac{{\left (b \sqrt{x} + a\right )}^{16} a^{3}}{8 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15*x,x, algorithm="maxima")

[Out]

2/19*(b*sqrt(x) + a)^19/b^4 - 1/3*(b*sqrt(x) + a)^18*a/b^4 + 6/17*(b*sqrt(x) + a
)^17*a^2/b^4 - 1/8*(b*sqrt(x) + a)^16*a^3/b^4

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Fricas [A]  time = 0.234355, size = 234, normalized size = 2.92 \[ \frac{5}{3} \, a b^{14} x^{9} + \frac{455}{8} \, a^{3} b^{12} x^{8} + 429 \, a^{5} b^{10} x^{7} + \frac{2145}{2} \, a^{7} b^{8} x^{6} + 1001 \, a^{9} b^{6} x^{5} + \frac{1365}{4} \, a^{11} b^{4} x^{4} + 35 \, a^{13} b^{2} x^{3} + \frac{1}{2} \, a^{15} x^{2} + \frac{2}{969} \,{\left (51 \, b^{15} x^{9} + 5985 \, a^{2} b^{13} x^{8} + 88179 \, a^{4} b^{11} x^{7} + 373065 \, a^{6} b^{9} x^{6} + 566865 \, a^{8} b^{7} x^{5} + 323323 \, a^{10} b^{5} x^{4} + 62985 \, a^{12} b^{3} x^{3} + 2907 \, a^{14} b x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15*x,x, algorithm="fricas")

[Out]

5/3*a*b^14*x^9 + 455/8*a^3*b^12*x^8 + 429*a^5*b^10*x^7 + 2145/2*a^7*b^8*x^6 + 10
01*a^9*b^6*x^5 + 1365/4*a^11*b^4*x^4 + 35*a^13*b^2*x^3 + 1/2*a^15*x^2 + 2/969*(5
1*b^15*x^9 + 5985*a^2*b^13*x^8 + 88179*a^4*b^11*x^7 + 373065*a^6*b^9*x^6 + 56686
5*a^8*b^7*x^5 + 323323*a^10*b^5*x^4 + 62985*a^12*b^3*x^3 + 2907*a^14*b*x^2)*sqrt
(x)

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Sympy [A]  time = 8.44242, size = 204, normalized size = 2.55 \[ \frac{a^{15} x^{2}}{2} + 6 a^{14} b x^{\frac{5}{2}} + 35 a^{13} b^{2} x^{3} + 130 a^{12} b^{3} x^{\frac{7}{2}} + \frac{1365 a^{11} b^{4} x^{4}}{4} + \frac{2002 a^{10} b^{5} x^{\frac{9}{2}}}{3} + 1001 a^{9} b^{6} x^{5} + 1170 a^{8} b^{7} x^{\frac{11}{2}} + \frac{2145 a^{7} b^{8} x^{6}}{2} + 770 a^{6} b^{9} x^{\frac{13}{2}} + 429 a^{5} b^{10} x^{7} + 182 a^{4} b^{11} x^{\frac{15}{2}} + \frac{455 a^{3} b^{12} x^{8}}{8} + \frac{210 a^{2} b^{13} x^{\frac{17}{2}}}{17} + \frac{5 a b^{14} x^{9}}{3} + \frac{2 b^{15} x^{\frac{19}{2}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(a+b*x**(1/2))**15,x)

[Out]

a**15*x**2/2 + 6*a**14*b*x**(5/2) + 35*a**13*b**2*x**3 + 130*a**12*b**3*x**(7/2)
+ 1365*a**11*b**4*x**4/4 + 2002*a**10*b**5*x**(9/2)/3 + 1001*a**9*b**6*x**5 + 1
170*a**8*b**7*x**(11/2) + 2145*a**7*b**8*x**6/2 + 770*a**6*b**9*x**(13/2) + 429*
a**5*b**10*x**7 + 182*a**4*b**11*x**(15/2) + 455*a**3*b**12*x**8/8 + 210*a**2*b*
*13*x**(17/2)/17 + 5*a*b**14*x**9/3 + 2*b**15*x**(19/2)/19

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GIAC/XCAS [A]  time = 0.218277, size = 225, normalized size = 2.81 \[ \frac{2}{19} \, b^{15} x^{\frac{19}{2}} + \frac{5}{3} \, a b^{14} x^{9} + \frac{210}{17} \, a^{2} b^{13} x^{\frac{17}{2}} + \frac{455}{8} \, a^{3} b^{12} x^{8} + 182 \, a^{4} b^{11} x^{\frac{15}{2}} + 429 \, a^{5} b^{10} x^{7} + 770 \, a^{6} b^{9} x^{\frac{13}{2}} + \frac{2145}{2} \, a^{7} b^{8} x^{6} + 1170 \, a^{8} b^{7} x^{\frac{11}{2}} + 1001 \, a^{9} b^{6} x^{5} + \frac{2002}{3} \, a^{10} b^{5} x^{\frac{9}{2}} + \frac{1365}{4} \, a^{11} b^{4} x^{4} + 130 \, a^{12} b^{3} x^{\frac{7}{2}} + 35 \, a^{13} b^{2} x^{3} + 6 \, a^{14} b x^{\frac{5}{2}} + \frac{1}{2} \, a^{15} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15*x,x, algorithm="giac")

[Out]

2/19*b^15*x^(19/2) + 5/3*a*b^14*x^9 + 210/17*a^2*b^13*x^(17/2) + 455/8*a^3*b^12*
x^8 + 182*a^4*b^11*x^(15/2) + 429*a^5*b^10*x^7 + 770*a^6*b^9*x^(13/2) + 2145/2*a
^7*b^8*x^6 + 1170*a^8*b^7*x^(11/2) + 1001*a^9*b^6*x^5 + 2002/3*a^10*b^5*x^(9/2)
+ 1365/4*a^11*b^4*x^4 + 130*a^12*b^3*x^(7/2) + 35*a^13*b^2*x^3 + 6*a^14*b*x^(5/2
) + 1/2*a^15*x^2