3.2188 \(\int \frac{\left (a+b \sqrt{x}\right )^{15}}{x^{17}} \, dx\)

Optimal. Leaf size=207 \[ -\frac{a^{15}}{16 x^{16}}-\frac{30 a^{14} b}{31 x^{31/2}}-\frac{7 a^{13} b^2}{x^{15}}-\frac{910 a^{12} b^3}{29 x^{29/2}}-\frac{195 a^{11} b^4}{2 x^{14}}-\frac{2002 a^{10} b^5}{9 x^{27/2}}-\frac{385 a^9 b^6}{x^{13}}-\frac{2574 a^8 b^7}{5 x^{25/2}}-\frac{2145 a^7 b^8}{4 x^{12}}-\frac{10010 a^6 b^9}{23 x^{23/2}}-\frac{273 a^5 b^{10}}{x^{11}}-\frac{130 a^4 b^{11}}{x^{21/2}}-\frac{91 a^3 b^{12}}{2 x^{10}}-\frac{210 a^2 b^{13}}{19 x^{19/2}}-\frac{5 a b^{14}}{3 x^9}-\frac{2 b^{15}}{17 x^{17/2}} \]

[Out]

-a^15/(16*x^16) - (30*a^14*b)/(31*x^(31/2)) - (7*a^13*b^2)/x^15 - (910*a^12*b^3)
/(29*x^(29/2)) - (195*a^11*b^4)/(2*x^14) - (2002*a^10*b^5)/(9*x^(27/2)) - (385*a
^9*b^6)/x^13 - (2574*a^8*b^7)/(5*x^(25/2)) - (2145*a^7*b^8)/(4*x^12) - (10010*a^
6*b^9)/(23*x^(23/2)) - (273*a^5*b^10)/x^11 - (130*a^4*b^11)/x^(21/2) - (91*a^3*b
^12)/(2*x^10) - (210*a^2*b^13)/(19*x^(19/2)) - (5*a*b^14)/(3*x^9) - (2*b^15)/(17
*x^(17/2))

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Rubi [A]  time = 0.314122, antiderivative size = 207, 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 \[ -\frac{a^{15}}{16 x^{16}}-\frac{30 a^{14} b}{31 x^{31/2}}-\frac{7 a^{13} b^2}{x^{15}}-\frac{910 a^{12} b^3}{29 x^{29/2}}-\frac{195 a^{11} b^4}{2 x^{14}}-\frac{2002 a^{10} b^5}{9 x^{27/2}}-\frac{385 a^9 b^6}{x^{13}}-\frac{2574 a^8 b^7}{5 x^{25/2}}-\frac{2145 a^7 b^8}{4 x^{12}}-\frac{10010 a^6 b^9}{23 x^{23/2}}-\frac{273 a^5 b^{10}}{x^{11}}-\frac{130 a^4 b^{11}}{x^{21/2}}-\frac{91 a^3 b^{12}}{2 x^{10}}-\frac{210 a^2 b^{13}}{19 x^{19/2}}-\frac{5 a b^{14}}{3 x^9}-\frac{2 b^{15}}{17 x^{17/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^15/x^17,x]

[Out]

-a^15/(16*x^16) - (30*a^14*b)/(31*x^(31/2)) - (7*a^13*b^2)/x^15 - (910*a^12*b^3)
/(29*x^(29/2)) - (195*a^11*b^4)/(2*x^14) - (2002*a^10*b^5)/(9*x^(27/2)) - (385*a
^9*b^6)/x^13 - (2574*a^8*b^7)/(5*x^(25/2)) - (2145*a^7*b^8)/(4*x^12) - (10010*a^
6*b^9)/(23*x^(23/2)) - (273*a^5*b^10)/x^11 - (130*a^4*b^11)/x^(21/2) - (91*a^3*b
^12)/(2*x^10) - (210*a^2*b^13)/(19*x^(19/2)) - (5*a*b^14)/(3*x^9) - (2*b^15)/(17
*x^(17/2))

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Rubi in Sympy [A]  time = 53.764, size = 212, normalized size = 1.02 \[ - \frac{a^{15}}{16 x^{16}} - \frac{30 a^{14} b}{31 x^{\frac{31}{2}}} - \frac{7 a^{13} b^{2}}{x^{15}} - \frac{910 a^{12} b^{3}}{29 x^{\frac{29}{2}}} - \frac{195 a^{11} b^{4}}{2 x^{14}} - \frac{2002 a^{10} b^{5}}{9 x^{\frac{27}{2}}} - \frac{385 a^{9} b^{6}}{x^{13}} - \frac{2574 a^{8} b^{7}}{5 x^{\frac{25}{2}}} - \frac{2145 a^{7} b^{8}}{4 x^{12}} - \frac{10010 a^{6} b^{9}}{23 x^{\frac{23}{2}}} - \frac{273 a^{5} b^{10}}{x^{11}} - \frac{130 a^{4} b^{11}}{x^{\frac{21}{2}}} - \frac{91 a^{3} b^{12}}{2 x^{10}} - \frac{210 a^{2} b^{13}}{19 x^{\frac{19}{2}}} - \frac{5 a b^{14}}{3 x^{9}} - \frac{2 b^{15}}{17 x^{\frac{17}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**15/x**17,x)

[Out]

-a**15/(16*x**16) - 30*a**14*b/(31*x**(31/2)) - 7*a**13*b**2/x**15 - 910*a**12*b
**3/(29*x**(29/2)) - 195*a**11*b**4/(2*x**14) - 2002*a**10*b**5/(9*x**(27/2)) -
385*a**9*b**6/x**13 - 2574*a**8*b**7/(5*x**(25/2)) - 2145*a**7*b**8/(4*x**12) -
10010*a**6*b**9/(23*x**(23/2)) - 273*a**5*b**10/x**11 - 130*a**4*b**11/x**(21/2)
- 91*a**3*b**12/(2*x**10) - 210*a**2*b**13/(19*x**(19/2)) - 5*a*b**14/(3*x**9)
- 2*b**15/(17*x**(17/2))

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Mathematica [A]  time = 0.0713114, size = 207, normalized size = 1. \[ -\frac{a^{15}}{16 x^{16}}-\frac{30 a^{14} b}{31 x^{31/2}}-\frac{7 a^{13} b^2}{x^{15}}-\frac{910 a^{12} b^3}{29 x^{29/2}}-\frac{195 a^{11} b^4}{2 x^{14}}-\frac{2002 a^{10} b^5}{9 x^{27/2}}-\frac{385 a^9 b^6}{x^{13}}-\frac{2574 a^8 b^7}{5 x^{25/2}}-\frac{2145 a^7 b^8}{4 x^{12}}-\frac{10010 a^6 b^9}{23 x^{23/2}}-\frac{273 a^5 b^{10}}{x^{11}}-\frac{130 a^4 b^{11}}{x^{21/2}}-\frac{91 a^3 b^{12}}{2 x^{10}}-\frac{210 a^2 b^{13}}{19 x^{19/2}}-\frac{5 a b^{14}}{3 x^9}-\frac{2 b^{15}}{17 x^{17/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^15/x^17,x]

[Out]

-a^15/(16*x^16) - (30*a^14*b)/(31*x^(31/2)) - (7*a^13*b^2)/x^15 - (910*a^12*b^3)
/(29*x^(29/2)) - (195*a^11*b^4)/(2*x^14) - (2002*a^10*b^5)/(9*x^(27/2)) - (385*a
^9*b^6)/x^13 - (2574*a^8*b^7)/(5*x^(25/2)) - (2145*a^7*b^8)/(4*x^12) - (10010*a^
6*b^9)/(23*x^(23/2)) - (273*a^5*b^10)/x^11 - (130*a^4*b^11)/x^(21/2) - (91*a^3*b
^12)/(2*x^10) - (210*a^2*b^13)/(19*x^(19/2)) - (5*a*b^14)/(3*x^9) - (2*b^15)/(17
*x^(17/2))

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Maple [A]  time = 0.006, size = 168, normalized size = 0.8 \[ -{\frac{{a}^{15}}{16\,{x}^{16}}}-{\frac{30\,{a}^{14}b}{31}{x}^{-{\frac{31}{2}}}}-7\,{\frac{{a}^{13}{b}^{2}}{{x}^{15}}}-{\frac{910\,{a}^{12}{b}^{3}}{29}{x}^{-{\frac{29}{2}}}}-{\frac{195\,{a}^{11}{b}^{4}}{2\,{x}^{14}}}-{\frac{2002\,{a}^{10}{b}^{5}}{9}{x}^{-{\frac{27}{2}}}}-385\,{\frac{{a}^{9}{b}^{6}}{{x}^{13}}}-{\frac{2574\,{a}^{8}{b}^{7}}{5}{x}^{-{\frac{25}{2}}}}-{\frac{2145\,{a}^{7}{b}^{8}}{4\,{x}^{12}}}-{\frac{10010\,{a}^{6}{b}^{9}}{23}{x}^{-{\frac{23}{2}}}}-273\,{\frac{{a}^{5}{b}^{10}}{{x}^{11}}}-130\,{\frac{{a}^{4}{b}^{11}}{{x}^{21/2}}}-{\frac{91\,{a}^{3}{b}^{12}}{2\,{x}^{10}}}-{\frac{210\,{a}^{2}{b}^{13}}{19}{x}^{-{\frac{19}{2}}}}-{\frac{5\,a{b}^{14}}{3\,{x}^{9}}}-{\frac{2\,{b}^{15}}{17}{x}^{-{\frac{17}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^15/x^17,x)

[Out]

-1/16*a^15/x^16-30/31*a^14*b/x^(31/2)-7*a^13*b^2/x^15-910/29*a^12*b^3/x^(29/2)-1
95/2*a^11*b^4/x^14-2002/9*a^10*b^5/x^(27/2)-385*a^9*b^6/x^13-2574/5*a^8*b^7/x^(2
5/2)-2145/4*a^7*b^8/x^12-10010/23*a^6*b^9/x^(23/2)-273*a^5*b^10/x^11-130*a^4*b^1
1/x^(21/2)-91/2*a^3*b^12/x^10-210/19*a^2*b^13/x^(19/2)-5/3*a*b^14/x^9-2/17*b^15/
x^(17/2)

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Maxima [A]  time = 1.44707, size = 225, normalized size = 1.09 \[ -\frac{565722720 \, b^{15} x^{\frac{15}{2}} + 8014405200 \, a b^{14} x^{7} + 53148160800 \, a^{2} b^{13} x^{\frac{13}{2}} + 218793261960 \, a^{3} b^{12} x^{6} + 625123605600 \, a^{4} b^{11} x^{\frac{11}{2}} + 1312759571760 \, a^{5} b^{10} x^{5} + 2092805114400 \, a^{6} b^{9} x^{\frac{9}{2}} + 2578634873100 \, a^{7} b^{8} x^{4} + 2475489478176 \, a^{8} b^{7} x^{\frac{7}{2}} + 1851327601200 \, a^{9} b^{6} x^{3} + 1069655947360 \, a^{10} b^{5} x^{\frac{5}{2}} + 468842704200 \, a^{11} b^{4} x^{2} + 150891904800 \, a^{12} b^{3} x^{\frac{3}{2}} + 33660501840 \, a^{13} b^{2} x + 4653525600 \, a^{14} b \sqrt{x} + 300540195 \, a^{15}}{4808643120 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^17,x, algorithm="maxima")

[Out]

-1/4808643120*(565722720*b^15*x^(15/2) + 8014405200*a*b^14*x^7 + 53148160800*a^2
*b^13*x^(13/2) + 218793261960*a^3*b^12*x^6 + 625123605600*a^4*b^11*x^(11/2) + 13
12759571760*a^5*b^10*x^5 + 2092805114400*a^6*b^9*x^(9/2) + 2578634873100*a^7*b^8
*x^4 + 2475489478176*a^8*b^7*x^(7/2) + 1851327601200*a^9*b^6*x^3 + 1069655947360
*a^10*b^5*x^(5/2) + 468842704200*a^11*b^4*x^2 + 150891904800*a^12*b^3*x^(3/2) +
33660501840*a^13*b^2*x + 4653525600*a^14*b*sqrt(x) + 300540195*a^15)/x^16

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Fricas [A]  time = 0.240377, size = 227, normalized size = 1.1 \[ -\frac{8014405200 \, a b^{14} x^{7} + 218793261960 \, a^{3} b^{12} x^{6} + 1312759571760 \, a^{5} b^{10} x^{5} + 2578634873100 \, a^{7} b^{8} x^{4} + 1851327601200 \, a^{9} b^{6} x^{3} + 468842704200 \, a^{11} b^{4} x^{2} + 33660501840 \, a^{13} b^{2} x + 300540195 \, a^{15} + 32 \,{\left (17678835 \, b^{15} x^{7} + 1660880025 \, a^{2} b^{13} x^{6} + 19535112675 \, a^{4} b^{11} x^{5} + 65400159825 \, a^{6} b^{9} x^{4} + 77359046193 \, a^{8} b^{7} x^{3} + 33426748355 \, a^{10} b^{5} x^{2} + 4715372025 \, a^{12} b^{3} x + 145422675 \, a^{14} b\right )} \sqrt{x}}{4808643120 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^17,x, algorithm="fricas")

[Out]

-1/4808643120*(8014405200*a*b^14*x^7 + 218793261960*a^3*b^12*x^6 + 1312759571760
*a^5*b^10*x^5 + 2578634873100*a^7*b^8*x^4 + 1851327601200*a^9*b^6*x^3 + 46884270
4200*a^11*b^4*x^2 + 33660501840*a^13*b^2*x + 300540195*a^15 + 32*(17678835*b^15*
x^7 + 1660880025*a^2*b^13*x^6 + 19535112675*a^4*b^11*x^5 + 65400159825*a^6*b^9*x
^4 + 77359046193*a^8*b^7*x^3 + 33426748355*a^10*b^5*x^2 + 4715372025*a^12*b^3*x
+ 145422675*a^14*b)*sqrt(x))/x^16

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**15/x**17,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221718, size = 225, normalized size = 1.09 \[ -\frac{565722720 \, b^{15} x^{\frac{15}{2}} + 8014405200 \, a b^{14} x^{7} + 53148160800 \, a^{2} b^{13} x^{\frac{13}{2}} + 218793261960 \, a^{3} b^{12} x^{6} + 625123605600 \, a^{4} b^{11} x^{\frac{11}{2}} + 1312759571760 \, a^{5} b^{10} x^{5} + 2092805114400 \, a^{6} b^{9} x^{\frac{9}{2}} + 2578634873100 \, a^{7} b^{8} x^{4} + 2475489478176 \, a^{8} b^{7} x^{\frac{7}{2}} + 1851327601200 \, a^{9} b^{6} x^{3} + 1069655947360 \, a^{10} b^{5} x^{\frac{5}{2}} + 468842704200 \, a^{11} b^{4} x^{2} + 150891904800 \, a^{12} b^{3} x^{\frac{3}{2}} + 33660501840 \, a^{13} b^{2} x + 4653525600 \, a^{14} b \sqrt{x} + 300540195 \, a^{15}}{4808643120 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^17,x, algorithm="giac")

[Out]

-1/4808643120*(565722720*b^15*x^(15/2) + 8014405200*a*b^14*x^7 + 53148160800*a^2
*b^13*x^(13/2) + 218793261960*a^3*b^12*x^6 + 625123605600*a^4*b^11*x^(11/2) + 13
12759571760*a^5*b^10*x^5 + 2092805114400*a^6*b^9*x^(9/2) + 2578634873100*a^7*b^8
*x^4 + 2475489478176*a^8*b^7*x^(7/2) + 1851327601200*a^9*b^6*x^3 + 1069655947360
*a^10*b^5*x^(5/2) + 468842704200*a^11*b^4*x^2 + 150891904800*a^12*b^3*x^(3/2) +
33660501840*a^13*b^2*x + 4653525600*a^14*b*sqrt(x) + 300540195*a^15)/x^16