3.2172 \(\int \left (a+b \sqrt{x}\right )^{15} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 \left (a+b \sqrt{x}\right )^{17}}{17 b^2}-\frac{a \left (a+b \sqrt{x}\right )^{16}}{8 b^2} \]

[Out]

-(a*(a + b*Sqrt[x])^16)/(8*b^2) + (2*(a + b*Sqrt[x])^17)/(17*b^2)

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Rubi [A]  time = 0.0523329, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 \left (a+b \sqrt{x}\right )^{17}}{17 b^2}-\frac{a \left (a+b \sqrt{x}\right )^{16}}{8 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^15,x]

[Out]

-(a*(a + b*Sqrt[x])^16)/(8*b^2) + (2*(a + b*Sqrt[x])^17)/(17*b^2)

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Rubi in Sympy [A]  time = 24.6897, size = 32, normalized size = 0.84 \[ - \frac{a \left (a + b \sqrt{x}\right )^{16}}{8 b^{2}} + \frac{2 \left (a + b \sqrt{x}\right )^{17}}{17 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*(a + b*sqrt(x))**16/(8*b**2) + 2*(a + b*sqrt(x))**17/(17*b**2)

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Mathematica [B]  time = 0.0267295, size = 190, normalized size = 5. \[ a^{15} x+10 a^{14} b x^{3/2}+\frac{105}{2} a^{13} b^2 x^2+182 a^{12} b^3 x^{5/2}+455 a^{11} b^4 x^3+858 a^{10} b^5 x^{7/2}+\frac{5005}{4} a^9 b^6 x^4+1430 a^8 b^7 x^{9/2}+1287 a^7 b^8 x^5+910 a^6 b^9 x^{11/2}+\frac{1001}{2} a^5 b^{10} x^6+210 a^4 b^{11} x^{13/2}+65 a^3 b^{12} x^7+14 a^2 b^{13} x^{15/2}+\frac{15}{8} a b^{14} x^8+\frac{2}{17} b^{15} x^{17/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^15,x]

[Out]

a^15*x + 10*a^14*b*x^(3/2) + (105*a^13*b^2*x^2)/2 + 182*a^12*b^3*x^(5/2) + 455*a
^11*b^4*x^3 + 858*a^10*b^5*x^(7/2) + (5005*a^9*b^6*x^4)/4 + 1430*a^8*b^7*x^(9/2)
+ 1287*a^7*b^8*x^5 + 910*a^6*b^9*x^(11/2) + (1001*a^5*b^10*x^6)/2 + 210*a^4*b^1
1*x^(13/2) + 65*a^3*b^12*x^7 + 14*a^2*b^13*x^(15/2) + (15*a*b^14*x^8)/8 + (2*b^1
5*x^(17/2))/17

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Maple [B]  time = 0.005, size = 165, normalized size = 4.3 \[{\frac{2\,{b}^{15}}{17}{x}^{{\frac{17}{2}}}}+{\frac{15\,{x}^{8}a{b}^{14}}{8}}+14\,{x}^{15/2}{a}^{2}{b}^{13}+65\,{x}^{7}{a}^{3}{b}^{12}+210\,{x}^{13/2}{a}^{4}{b}^{11}+{\frac{1001\,{x}^{6}{a}^{5}{b}^{10}}{2}}+910\,{x}^{11/2}{a}^{6}{b}^{9}+1287\,{x}^{5}{a}^{7}{b}^{8}+1430\,{x}^{9/2}{a}^{8}{b}^{7}+{\frac{5005\,{x}^{4}{a}^{9}{b}^{6}}{4}}+858\,{x}^{7/2}{a}^{10}{b}^{5}+455\,{x}^{3}{a}^{11}{b}^{4}+182\,{x}^{5/2}{a}^{12}{b}^{3}+{\frac{105\,{x}^{2}{a}^{13}{b}^{2}}{2}}+10\,{x}^{3/2}{a}^{14}b+x{a}^{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^15,x)

[Out]

2/17*x^(17/2)*b^15+15/8*x^8*a*b^14+14*x^(15/2)*a^2*b^13+65*x^7*a^3*b^12+210*x^(1
3/2)*a^4*b^11+1001/2*x^6*a^5*b^10+910*x^(11/2)*a^6*b^9+1287*x^5*a^7*b^8+1430*x^(
9/2)*a^8*b^7+5005/4*x^4*a^9*b^6+858*x^(7/2)*a^10*b^5+455*x^3*a^11*b^4+182*x^(5/2
)*a^12*b^3+105/2*x^2*a^13*b^2+10*x^(3/2)*a^14*b+x*a^15

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Maxima [A]  time = 1.41542, size = 41, normalized size = 1.08 \[ \frac{2 \,{\left (b \sqrt{x} + a\right )}^{17}}{17 \, b^{2}} - \frac{{\left (b \sqrt{x} + a\right )}^{16} a}{8 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/17*(b*sqrt(x) + a)^17/b^2 - 1/8*(b*sqrt(x) + a)^16*a/b^2

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Fricas [A]  time = 0.231075, size = 225, normalized size = 5.92 \[ \frac{15}{8} \, a b^{14} x^{8} + 65 \, a^{3} b^{12} x^{7} + \frac{1001}{2} \, a^{5} b^{10} x^{6} + 1287 \, a^{7} b^{8} x^{5} + \frac{5005}{4} \, a^{9} b^{6} x^{4} + 455 \, a^{11} b^{4} x^{3} + \frac{105}{2} \, a^{13} b^{2} x^{2} + a^{15} x + \frac{2}{17} \,{\left (b^{15} x^{8} + 119 \, a^{2} b^{13} x^{7} + 1785 \, a^{4} b^{11} x^{6} + 7735 \, a^{6} b^{9} x^{5} + 12155 \, a^{8} b^{7} x^{4} + 7293 \, a^{10} b^{5} x^{3} + 1547 \, a^{12} b^{3} x^{2} + 85 \, a^{14} b x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15/8*a*b^14*x^8 + 65*a^3*b^12*x^7 + 1001/2*a^5*b^10*x^6 + 1287*a^7*b^8*x^5 + 500
5/4*a^9*b^6*x^4 + 455*a^11*b^4*x^3 + 105/2*a^13*b^2*x^2 + a^15*x + 2/17*(b^15*x^
8 + 119*a^2*b^13*x^7 + 1785*a^4*b^11*x^6 + 7735*a^6*b^9*x^5 + 12155*a^8*b^7*x^4
+ 7293*a^10*b^5*x^3 + 1547*a^12*b^3*x^2 + 85*a^14*b*x)*sqrt(x)

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Sympy [A]  time = 6.78041, size = 197, normalized size = 5.18 \[ a^{15} x + 10 a^{14} b x^{\frac{3}{2}} + \frac{105 a^{13} b^{2} x^{2}}{2} + 182 a^{12} b^{3} x^{\frac{5}{2}} + 455 a^{11} b^{4} x^{3} + 858 a^{10} b^{5} x^{\frac{7}{2}} + \frac{5005 a^{9} b^{6} x^{4}}{4} + 1430 a^{8} b^{7} x^{\frac{9}{2}} + 1287 a^{7} b^{8} x^{5} + 910 a^{6} b^{9} x^{\frac{11}{2}} + \frac{1001 a^{5} b^{10} x^{6}}{2} + 210 a^{4} b^{11} x^{\frac{13}{2}} + 65 a^{3} b^{12} x^{7} + 14 a^{2} b^{13} x^{\frac{15}{2}} + \frac{15 a b^{14} x^{8}}{8} + \frac{2 b^{15} x^{\frac{17}{2}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**15,x)

[Out]

a**15*x + 10*a**14*b*x**(3/2) + 105*a**13*b**2*x**2/2 + 182*a**12*b**3*x**(5/2)
+ 455*a**11*b**4*x**3 + 858*a**10*b**5*x**(7/2) + 5005*a**9*b**6*x**4/4 + 1430*a
**8*b**7*x**(9/2) + 1287*a**7*b**8*x**5 + 910*a**6*b**9*x**(11/2) + 1001*a**5*b*
*10*x**6/2 + 210*a**4*b**11*x**(13/2) + 65*a**3*b**12*x**7 + 14*a**2*b**13*x**(1
5/2) + 15*a*b**14*x**8/8 + 2*b**15*x**(17/2)/17

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GIAC/XCAS [A]  time = 0.221683, size = 221, normalized size = 5.82 \[ \frac{2}{17} \, b^{15} x^{\frac{17}{2}} + \frac{15}{8} \, a b^{14} x^{8} + 14 \, a^{2} b^{13} x^{\frac{15}{2}} + 65 \, a^{3} b^{12} x^{7} + 210 \, a^{4} b^{11} x^{\frac{13}{2}} + \frac{1001}{2} \, a^{5} b^{10} x^{6} + 910 \, a^{6} b^{9} x^{\frac{11}{2}} + 1287 \, a^{7} b^{8} x^{5} + 1430 \, a^{8} b^{7} x^{\frac{9}{2}} + \frac{5005}{4} \, a^{9} b^{6} x^{4} + 858 \, a^{10} b^{5} x^{\frac{7}{2}} + 455 \, a^{11} b^{4} x^{3} + 182 \, a^{12} b^{3} x^{\frac{5}{2}} + \frac{105}{2} \, a^{13} b^{2} x^{2} + 10 \, a^{14} b x^{\frac{3}{2}} + a^{15} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/17*b^15*x^(17/2) + 15/8*a*b^14*x^8 + 14*a^2*b^13*x^(15/2) + 65*a^3*b^12*x^7 +
210*a^4*b^11*x^(13/2) + 1001/2*a^5*b^10*x^6 + 910*a^6*b^9*x^(11/2) + 1287*a^7*b^
8*x^5 + 1430*a^8*b^7*x^(9/2) + 5005/4*a^9*b^6*x^4 + 858*a^10*b^5*x^(7/2) + 455*a
^11*b^4*x^3 + 182*a^12*b^3*x^(5/2) + 105/2*a^13*b^2*x^2 + 10*a^14*b*x^(3/2) + a^
15*x