3.2180 \(\int \frac{\left (a+b \sqrt{x}\right )^{15}}{x^9} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b \sqrt{x}\right )^{16}}{8 a x^8} \]

[Out]

-(a + b*Sqrt[x])^16/(8*a*x^8)

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Rubi [A]  time = 0.0164378, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{\left (a+b \sqrt{x}\right )^{16}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^15/x^9,x]

[Out]

-(a + b*Sqrt[x])^16/(8*a*x^8)

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Rubi in Sympy [A]  time = 2.79467, size = 17, normalized size = 0.81 \[ - \frac{\left (a + b \sqrt{x}\right )^{16}}{8 a x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**15/x**9,x)

[Out]

-(a + b*sqrt(x))**16/(8*a*x**8)

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Mathematica [B]  time = 0.0538755, size = 183, normalized size = 8.71 \[ -\frac{a^{15}+16 a^{14} b \sqrt{x}+120 a^{13} b^2 x+560 a^{12} b^3 x^{3/2}+1820 a^{11} b^4 x^2+4368 a^{10} b^5 x^{5/2}+8008 a^9 b^6 x^3+11440 a^8 b^7 x^{7/2}+12870 a^7 b^8 x^4+11440 a^6 b^9 x^{9/2}+8008 a^5 b^{10} x^5+4368 a^4 b^{11} x^{11/2}+1820 a^3 b^{12} x^6+560 a^2 b^{13} x^{13/2}+120 a b^{14} x^7+16 b^{15} x^{15/2}}{8 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^15/x^9,x]

[Out]

-(a^15 + 16*a^14*b*Sqrt[x] + 120*a^13*b^2*x + 560*a^12*b^3*x^(3/2) + 1820*a^11*b
^4*x^2 + 4368*a^10*b^5*x^(5/2) + 8008*a^9*b^6*x^3 + 11440*a^8*b^7*x^(7/2) + 1287
0*a^7*b^8*x^4 + 11440*a^6*b^9*x^(9/2) + 8008*a^5*b^10*x^5 + 4368*a^4*b^11*x^(11/
2) + 1820*a^3*b^12*x^6 + 560*a^2*b^13*x^(13/2) + 120*a*b^14*x^7 + 16*b^15*x^(15/
2))/(8*x^8)

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Maple [B]  time = 0.006, size = 168, normalized size = 8. \[ -2\,{\frac{{b}^{15}}{\sqrt{x}}}-15\,{\frac{a{b}^{14}}{x}}-70\,{\frac{{a}^{2}{b}^{13}}{{x}^{3/2}}}-{\frac{455\,{a}^{3}{b}^{12}}{2\,{x}^{2}}}-546\,{\frac{{a}^{4}{b}^{11}}{{x}^{5/2}}}-1001\,{\frac{{a}^{5}{b}^{10}}{{x}^{3}}}-1430\,{\frac{{a}^{6}{b}^{9}}{{x}^{7/2}}}-{\frac{6435\,{a}^{7}{b}^{8}}{4\,{x}^{4}}}-1430\,{\frac{{a}^{8}{b}^{7}}{{x}^{9/2}}}-1001\,{\frac{{a}^{9}{b}^{6}}{{x}^{5}}}-546\,{\frac{{a}^{10}{b}^{5}}{{x}^{11/2}}}-{\frac{455\,{a}^{11}{b}^{4}}{2\,{x}^{6}}}-70\,{\frac{{a}^{12}{b}^{3}}{{x}^{13/2}}}-15\,{\frac{{a}^{13}{b}^{2}}{{x}^{7}}}-2\,{\frac{{a}^{14}b}{{x}^{15/2}}}-{\frac{{a}^{15}}{8\,{x}^{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^15/x^9,x)

[Out]

-2*b^15/x^(1/2)-15*a*b^14/x-70*a^2*b^13/x^(3/2)-455/2*a^3*b^12/x^2-546*a^4*b^11/
x^(5/2)-1001*a^5*b^10/x^3-1430*a^6*b^9/x^(7/2)-6435/4*a^7*b^8/x^4-1430*a^8*b^7/x
^(9/2)-1001*a^9*b^6/x^5-546*a^10*b^5/x^(11/2)-455/2*a^11*b^4/x^6-70*a^12*b^3/x^(
13/2)-15*a^13*b^2/x^7-2*a^14*b/x^(15/2)-1/8*a^15/x^8

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Maxima [A]  time = 1.43426, size = 223, normalized size = 10.62 \[ -\frac{16 \, b^{15} x^{\frac{15}{2}} + 120 \, a b^{14} x^{7} + 560 \, a^{2} b^{13} x^{\frac{13}{2}} + 1820 \, a^{3} b^{12} x^{6} + 4368 \, a^{4} b^{11} x^{\frac{11}{2}} + 8008 \, a^{5} b^{10} x^{5} + 11440 \, a^{6} b^{9} x^{\frac{9}{2}} + 12870 \, a^{7} b^{8} x^{4} + 11440 \, a^{8} b^{7} x^{\frac{7}{2}} + 8008 \, a^{9} b^{6} x^{3} + 4368 \, a^{10} b^{5} x^{\frac{5}{2}} + 1820 \, a^{11} b^{4} x^{2} + 560 \, a^{12} b^{3} x^{\frac{3}{2}} + 120 \, a^{13} b^{2} x + 16 \, a^{14} b \sqrt{x} + a^{15}}{8 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^9,x, algorithm="maxima")

[Out]

-1/8*(16*b^15*x^(15/2) + 120*a*b^14*x^7 + 560*a^2*b^13*x^(13/2) + 1820*a^3*b^12*
x^6 + 4368*a^4*b^11*x^(11/2) + 8008*a^5*b^10*x^5 + 11440*a^6*b^9*x^(9/2) + 12870
*a^7*b^8*x^4 + 11440*a^8*b^7*x^(7/2) + 8008*a^9*b^6*x^3 + 4368*a^10*b^5*x^(5/2)
+ 1820*a^11*b^4*x^2 + 560*a^12*b^3*x^(3/2) + 120*a^13*b^2*x + 16*a^14*b*sqrt(x)
+ a^15)/x^8

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Fricas [A]  time = 0.239793, size = 221, normalized size = 10.52 \[ -\frac{120 \, a b^{14} x^{7} + 1820 \, a^{3} b^{12} x^{6} + 8008 \, a^{5} b^{10} x^{5} + 12870 \, a^{7} b^{8} x^{4} + 8008 \, a^{9} b^{6} x^{3} + 1820 \, a^{11} b^{4} x^{2} + 120 \, a^{13} b^{2} x + a^{15} + 16 \,{\left (b^{15} x^{7} + 35 \, a^{2} b^{13} x^{6} + 273 \, a^{4} b^{11} x^{5} + 715 \, a^{6} b^{9} x^{4} + 715 \, a^{8} b^{7} x^{3} + 273 \, a^{10} b^{5} x^{2} + 35 \, a^{12} b^{3} x + a^{14} b\right )} \sqrt{x}}{8 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^9,x, algorithm="fricas")

[Out]

-1/8*(120*a*b^14*x^7 + 1820*a^3*b^12*x^6 + 8008*a^5*b^10*x^5 + 12870*a^7*b^8*x^4
+ 8008*a^9*b^6*x^3 + 1820*a^11*b^4*x^2 + 120*a^13*b^2*x + a^15 + 16*(b^15*x^7 +
35*a^2*b^13*x^6 + 273*a^4*b^11*x^5 + 715*a^6*b^9*x^4 + 715*a^8*b^7*x^3 + 273*a^
10*b^5*x^2 + 35*a^12*b^3*x + a^14*b)*sqrt(x))/x^8

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Sympy [A]  time = 27.7204, size = 197, normalized size = 9.38 \[ - \frac{a^{15}}{8 x^{8}} - \frac{2 a^{14} b}{x^{\frac{15}{2}}} - \frac{15 a^{13} b^{2}}{x^{7}} - \frac{70 a^{12} b^{3}}{x^{\frac{13}{2}}} - \frac{455 a^{11} b^{4}}{2 x^{6}} - \frac{546 a^{10} b^{5}}{x^{\frac{11}{2}}} - \frac{1001 a^{9} b^{6}}{x^{5}} - \frac{1430 a^{8} b^{7}}{x^{\frac{9}{2}}} - \frac{6435 a^{7} b^{8}}{4 x^{4}} - \frac{1430 a^{6} b^{9}}{x^{\frac{7}{2}}} - \frac{1001 a^{5} b^{10}}{x^{3}} - \frac{546 a^{4} b^{11}}{x^{\frac{5}{2}}} - \frac{455 a^{3} b^{12}}{2 x^{2}} - \frac{70 a^{2} b^{13}}{x^{\frac{3}{2}}} - \frac{15 a b^{14}}{x} - \frac{2 b^{15}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**15/x**9,x)

[Out]

-a**15/(8*x**8) - 2*a**14*b/x**(15/2) - 15*a**13*b**2/x**7 - 70*a**12*b**3/x**(1
3/2) - 455*a**11*b**4/(2*x**6) - 546*a**10*b**5/x**(11/2) - 1001*a**9*b**6/x**5
- 1430*a**8*b**7/x**(9/2) - 6435*a**7*b**8/(4*x**4) - 1430*a**6*b**9/x**(7/2) -
1001*a**5*b**10/x**3 - 546*a**4*b**11/x**(5/2) - 455*a**3*b**12/(2*x**2) - 70*a*
*2*b**13/x**(3/2) - 15*a*b**14/x - 2*b**15/sqrt(x)

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GIAC/XCAS [A]  time = 0.22247, size = 223, normalized size = 10.62 \[ -\frac{16 \, b^{15} x^{\frac{15}{2}} + 120 \, a b^{14} x^{7} + 560 \, a^{2} b^{13} x^{\frac{13}{2}} + 1820 \, a^{3} b^{12} x^{6} + 4368 \, a^{4} b^{11} x^{\frac{11}{2}} + 8008 \, a^{5} b^{10} x^{5} + 11440 \, a^{6} b^{9} x^{\frac{9}{2}} + 12870 \, a^{7} b^{8} x^{4} + 11440 \, a^{8} b^{7} x^{\frac{7}{2}} + 8008 \, a^{9} b^{6} x^{3} + 4368 \, a^{10} b^{5} x^{\frac{5}{2}} + 1820 \, a^{11} b^{4} x^{2} + 560 \, a^{12} b^{3} x^{\frac{3}{2}} + 120 \, a^{13} b^{2} x + 16 \, a^{14} b \sqrt{x} + a^{15}}{8 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^9,x, algorithm="giac")

[Out]

-1/8*(16*b^15*x^(15/2) + 120*a*b^14*x^7 + 560*a^2*b^13*x^(13/2) + 1820*a^3*b^12*
x^6 + 4368*a^4*b^11*x^(11/2) + 8008*a^5*b^10*x^5 + 11440*a^6*b^9*x^(9/2) + 12870
*a^7*b^8*x^4 + 11440*a^8*b^7*x^(7/2) + 8008*a^9*b^6*x^3 + 4368*a^10*b^5*x^(5/2)
+ 1820*a^11*b^4*x^2 + 560*a^12*b^3*x^(3/2) + 120*a^13*b^2*x + 16*a^14*b*sqrt(x)
+ a^15)/x^8