3.2173 \(\int \frac{\left (a+b \sqrt{x}\right )^{15}}{x} \, dx\)

Optimal. Leaf size=205 \[ a^{15} \log (x)+30 a^{14} b \sqrt{x}+105 a^{13} b^2 x+\frac{910}{3} a^{12} b^3 x^{3/2}+\frac{1365}{2} a^{11} b^4 x^2+\frac{6006}{5} a^{10} b^5 x^{5/2}+\frac{5005}{3} a^9 b^6 x^3+\frac{12870}{7} a^8 b^7 x^{7/2}+\frac{6435}{4} a^7 b^8 x^4+\frac{10010}{9} a^6 b^9 x^{9/2}+\frac{3003}{5} a^5 b^{10} x^5+\frac{2730}{11} a^4 b^{11} x^{11/2}+\frac{455}{6} a^3 b^{12} x^6+\frac{210}{13} a^2 b^{13} x^{13/2}+\frac{15}{7} a b^{14} x^7+\frac{2}{15} b^{15} x^{15/2} \]

[Out]

30*a^14*b*Sqrt[x] + 105*a^13*b^2*x + (910*a^12*b^3*x^(3/2))/3 + (1365*a^11*b^4*x
^2)/2 + (6006*a^10*b^5*x^(5/2))/5 + (5005*a^9*b^6*x^3)/3 + (12870*a^8*b^7*x^(7/2
))/7 + (6435*a^7*b^8*x^4)/4 + (10010*a^6*b^9*x^(9/2))/9 + (3003*a^5*b^10*x^5)/5
+ (2730*a^4*b^11*x^(11/2))/11 + (455*a^3*b^12*x^6)/6 + (210*a^2*b^13*x^(13/2))/1
3 + (15*a*b^14*x^7)/7 + (2*b^15*x^(15/2))/15 + a^15*Log[x]

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Rubi [A]  time = 0.255, antiderivative size = 205, 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 \[ a^{15} \log (x)+30 a^{14} b \sqrt{x}+105 a^{13} b^2 x+\frac{910}{3} a^{12} b^3 x^{3/2}+\frac{1365}{2} a^{11} b^4 x^2+\frac{6006}{5} a^{10} b^5 x^{5/2}+\frac{5005}{3} a^9 b^6 x^3+\frac{12870}{7} a^8 b^7 x^{7/2}+\frac{6435}{4} a^7 b^8 x^4+\frac{10010}{9} a^6 b^9 x^{9/2}+\frac{3003}{5} a^5 b^{10} x^5+\frac{2730}{11} a^4 b^{11} x^{11/2}+\frac{455}{6} a^3 b^{12} x^6+\frac{210}{13} a^2 b^{13} x^{13/2}+\frac{15}{7} a b^{14} x^7+\frac{2}{15} b^{15} x^{15/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^15/x,x]

[Out]

30*a^14*b*Sqrt[x] + 105*a^13*b^2*x + (910*a^12*b^3*x^(3/2))/3 + (1365*a^11*b^4*x
^2)/2 + (6006*a^10*b^5*x^(5/2))/5 + (5005*a^9*b^6*x^3)/3 + (12870*a^8*b^7*x^(7/2
))/7 + (6435*a^7*b^8*x^4)/4 + (10010*a^6*b^9*x^(9/2))/9 + (3003*a^5*b^10*x^5)/5
+ (2730*a^4*b^11*x^(11/2))/11 + (455*a^3*b^12*x^6)/6 + (210*a^2*b^13*x^(13/2))/1
3 + (15*a*b^14*x^7)/7 + (2*b^15*x^(15/2))/15 + a^15*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 a^{15} \log{\left (\sqrt{x} \right )} + 30 a^{14} b \sqrt{x} + 210 a^{13} b^{2} \int ^{\sqrt{x}} x\, dx + \frac{910 a^{12} b^{3} x^{\frac{3}{2}}}{3} + \frac{1365 a^{11} b^{4} x^{2}}{2} + \frac{6006 a^{10} b^{5} x^{\frac{5}{2}}}{5} + \frac{5005 a^{9} b^{6} x^{3}}{3} + \frac{12870 a^{8} b^{7} x^{\frac{7}{2}}}{7} + \frac{6435 a^{7} b^{8} x^{4}}{4} + \frac{10010 a^{6} b^{9} x^{\frac{9}{2}}}{9} + \frac{3003 a^{5} b^{10} x^{5}}{5} + \frac{2730 a^{4} b^{11} x^{\frac{11}{2}}}{11} + \frac{455 a^{3} b^{12} x^{6}}{6} + \frac{210 a^{2} b^{13} x^{\frac{13}{2}}}{13} + \frac{15 a b^{14} x^{7}}{7} + \frac{2 b^{15} x^{\frac{15}{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**15*log(sqrt(x)) + 30*a**14*b*sqrt(x) + 210*a**13*b**2*Integral(x, (x, sqrt(
x))) + 910*a**12*b**3*x**(3/2)/3 + 1365*a**11*b**4*x**2/2 + 6006*a**10*b**5*x**(
5/2)/5 + 5005*a**9*b**6*x**3/3 + 12870*a**8*b**7*x**(7/2)/7 + 6435*a**7*b**8*x**
4/4 + 10010*a**6*b**9*x**(9/2)/9 + 3003*a**5*b**10*x**5/5 + 2730*a**4*b**11*x**(
11/2)/11 + 455*a**3*b**12*x**6/6 + 210*a**2*b**13*x**(13/2)/13 + 15*a*b**14*x**7
/7 + 2*b**15*x**(15/2)/15

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Mathematica [A]  time = 0.044582, size = 205, normalized size = 1. \[ a^{15} \log (x)+30 a^{14} b \sqrt{x}+105 a^{13} b^2 x+\frac{910}{3} a^{12} b^3 x^{3/2}+\frac{1365}{2} a^{11} b^4 x^2+\frac{6006}{5} a^{10} b^5 x^{5/2}+\frac{5005}{3} a^9 b^6 x^3+\frac{12870}{7} a^8 b^7 x^{7/2}+\frac{6435}{4} a^7 b^8 x^4+\frac{10010}{9} a^6 b^9 x^{9/2}+\frac{3003}{5} a^5 b^{10} x^5+\frac{2730}{11} a^4 b^{11} x^{11/2}+\frac{455}{6} a^3 b^{12} x^6+\frac{210}{13} a^2 b^{13} x^{13/2}+\frac{15}{7} a b^{14} x^7+\frac{2}{15} b^{15} x^{15/2} \]

Antiderivative was successfully verified.

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

[Out]

30*a^14*b*Sqrt[x] + 105*a^13*b^2*x + (910*a^12*b^3*x^(3/2))/3 + (1365*a^11*b^4*x
^2)/2 + (6006*a^10*b^5*x^(5/2))/5 + (5005*a^9*b^6*x^3)/3 + (12870*a^8*b^7*x^(7/2
))/7 + (6435*a^7*b^8*x^4)/4 + (10010*a^6*b^9*x^(9/2))/9 + (3003*a^5*b^10*x^5)/5
+ (2730*a^4*b^11*x^(11/2))/11 + (455*a^3*b^12*x^6)/6 + (210*a^2*b^13*x^(13/2))/1
3 + (15*a*b^14*x^7)/7 + (2*b^15*x^(15/2))/15 + a^15*Log[x]

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Maple [A]  time = 0.006, size = 164, normalized size = 0.8 \[ 105\,{a}^{13}{b}^{2}x+{\frac{910\,{a}^{12}{b}^{3}}{3}{x}^{{\frac{3}{2}}}}+{\frac{1365\,{a}^{11}{b}^{4}{x}^{2}}{2}}+{\frac{6006\,{a}^{10}{b}^{5}}{5}{x}^{{\frac{5}{2}}}}+{\frac{5005\,{a}^{9}{b}^{6}{x}^{3}}{3}}+{\frac{12870\,{a}^{8}{b}^{7}}{7}{x}^{{\frac{7}{2}}}}+{\frac{6435\,{a}^{7}{b}^{8}{x}^{4}}{4}}+{\frac{10010\,{a}^{6}{b}^{9}}{9}{x}^{{\frac{9}{2}}}}+{\frac{3003\,{a}^{5}{b}^{10}{x}^{5}}{5}}+{\frac{2730\,{a}^{4}{b}^{11}}{11}{x}^{{\frac{11}{2}}}}+{\frac{455\,{a}^{3}{b}^{12}{x}^{6}}{6}}+{\frac{210\,{a}^{2}{b}^{13}}{13}{x}^{{\frac{13}{2}}}}+{\frac{15\,a{b}^{14}{x}^{7}}{7}}+{\frac{2\,{b}^{15}}{15}{x}^{{\frac{15}{2}}}}+{a}^{15}\ln \left ( x \right ) +30\,{a}^{14}b\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^15/x,x)

[Out]

105*a^13*b^2*x+910/3*a^12*b^3*x^(3/2)+1365/2*a^11*b^4*x^2+6006/5*a^10*b^5*x^(5/2
)+5005/3*a^9*b^6*x^3+12870/7*a^8*b^7*x^(7/2)+6435/4*a^7*b^8*x^4+10010/9*a^6*b^9*
x^(9/2)+3003/5*a^5*b^10*x^5+2730/11*a^4*b^11*x^(11/2)+455/6*a^3*b^12*x^6+210/13*
a^2*b^13*x^(13/2)+15/7*a*b^14*x^7+2/15*b^15*x^(15/2)+a^15*ln(x)+30*a^14*b*x^(1/2
)

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Maxima [A]  time = 1.42025, size = 220, normalized size = 1.07 \[ \frac{2}{15} \, b^{15} x^{\frac{15}{2}} + \frac{15}{7} \, a b^{14} x^{7} + \frac{210}{13} \, a^{2} b^{13} x^{\frac{13}{2}} + \frac{455}{6} \, a^{3} b^{12} x^{6} + \frac{2730}{11} \, a^{4} b^{11} x^{\frac{11}{2}} + \frac{3003}{5} \, a^{5} b^{10} x^{5} + \frac{10010}{9} \, a^{6} b^{9} x^{\frac{9}{2}} + \frac{6435}{4} \, a^{7} b^{8} x^{4} + \frac{12870}{7} \, a^{8} b^{7} x^{\frac{7}{2}} + \frac{5005}{3} \, a^{9} b^{6} x^{3} + \frac{6006}{5} \, a^{10} b^{5} x^{\frac{5}{2}} + \frac{1365}{2} \, a^{11} b^{4} x^{2} + \frac{910}{3} \, a^{12} b^{3} x^{\frac{3}{2}} + 105 \, a^{13} b^{2} x + a^{15} \log \left (x\right ) + 30 \, a^{14} b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*b^15*x^(15/2) + 15/7*a*b^14*x^7 + 210/13*a^2*b^13*x^(13/2) + 455/6*a^3*b^12
*x^6 + 2730/11*a^4*b^11*x^(11/2) + 3003/5*a^5*b^10*x^5 + 10010/9*a^6*b^9*x^(9/2)
+ 6435/4*a^7*b^8*x^4 + 12870/7*a^8*b^7*x^(7/2) + 5005/3*a^9*b^6*x^3 + 6006/5*a^
10*b^5*x^(5/2) + 1365/2*a^11*b^4*x^2 + 910/3*a^12*b^3*x^(3/2) + 105*a^13*b^2*x +
a^15*log(x) + 30*a^14*b*sqrt(x)

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Fricas [A]  time = 0.240013, size = 225, normalized size = 1.1 \[ \frac{15}{7} \, a b^{14} x^{7} + \frac{455}{6} \, a^{3} b^{12} x^{6} + \frac{3003}{5} \, a^{5} b^{10} x^{5} + \frac{6435}{4} \, a^{7} b^{8} x^{4} + \frac{5005}{3} \, a^{9} b^{6} x^{3} + \frac{1365}{2} \, a^{11} b^{4} x^{2} + 105 \, a^{13} b^{2} x + 2 \, a^{15} \log \left (\sqrt{x}\right ) + \frac{2}{45045} \,{\left (3003 \, b^{15} x^{7} + 363825 \, a^{2} b^{13} x^{6} + 5589675 \, a^{4} b^{11} x^{5} + 25050025 \, a^{6} b^{9} x^{4} + 41409225 \, a^{8} b^{7} x^{3} + 27054027 \, a^{10} b^{5} x^{2} + 6831825 \, a^{12} b^{3} x + 675675 \, a^{14} b\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15/7*a*b^14*x^7 + 455/6*a^3*b^12*x^6 + 3003/5*a^5*b^10*x^5 + 6435/4*a^7*b^8*x^4
+ 5005/3*a^9*b^6*x^3 + 1365/2*a^11*b^4*x^2 + 105*a^13*b^2*x + 2*a^15*log(sqrt(x)
) + 2/45045*(3003*b^15*x^7 + 363825*a^2*b^13*x^6 + 5589675*a^4*b^11*x^5 + 250500
25*a^6*b^9*x^4 + 41409225*a^8*b^7*x^3 + 27054027*a^10*b^5*x^2 + 6831825*a^12*b^3
*x + 675675*a^14*b)*sqrt(x)

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Sympy [A]  time = 19.0353, size = 211, normalized size = 1.03 \[ a^{15} \log{\left (x \right )} + 30 a^{14} b \sqrt{x} + 105 a^{13} b^{2} x + \frac{910 a^{12} b^{3} x^{\frac{3}{2}}}{3} + \frac{1365 a^{11} b^{4} x^{2}}{2} + \frac{6006 a^{10} b^{5} x^{\frac{5}{2}}}{5} + \frac{5005 a^{9} b^{6} x^{3}}{3} + \frac{12870 a^{8} b^{7} x^{\frac{7}{2}}}{7} + \frac{6435 a^{7} b^{8} x^{4}}{4} + \frac{10010 a^{6} b^{9} x^{\frac{9}{2}}}{9} + \frac{3003 a^{5} b^{10} x^{5}}{5} + \frac{2730 a^{4} b^{11} x^{\frac{11}{2}}}{11} + \frac{455 a^{3} b^{12} x^{6}}{6} + \frac{210 a^{2} b^{13} x^{\frac{13}{2}}}{13} + \frac{15 a b^{14} x^{7}}{7} + \frac{2 b^{15} x^{\frac{15}{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**15*log(x) + 30*a**14*b*sqrt(x) + 105*a**13*b**2*x + 910*a**12*b**3*x**(3/2)/3
+ 1365*a**11*b**4*x**2/2 + 6006*a**10*b**5*x**(5/2)/5 + 5005*a**9*b**6*x**3/3 +
12870*a**8*b**7*x**(7/2)/7 + 6435*a**7*b**8*x**4/4 + 10010*a**6*b**9*x**(9/2)/9
+ 3003*a**5*b**10*x**5/5 + 2730*a**4*b**11*x**(11/2)/11 + 455*a**3*b**12*x**6/6
+ 210*a**2*b**13*x**(13/2)/13 + 15*a*b**14*x**7/7 + 2*b**15*x**(15/2)/15

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GIAC/XCAS [A]  time = 0.218695, size = 221, normalized size = 1.08 \[ \frac{2}{15} \, b^{15} x^{\frac{15}{2}} + \frac{15}{7} \, a b^{14} x^{7} + \frac{210}{13} \, a^{2} b^{13} x^{\frac{13}{2}} + \frac{455}{6} \, a^{3} b^{12} x^{6} + \frac{2730}{11} \, a^{4} b^{11} x^{\frac{11}{2}} + \frac{3003}{5} \, a^{5} b^{10} x^{5} + \frac{10010}{9} \, a^{6} b^{9} x^{\frac{9}{2}} + \frac{6435}{4} \, a^{7} b^{8} x^{4} + \frac{12870}{7} \, a^{8} b^{7} x^{\frac{7}{2}} + \frac{5005}{3} \, a^{9} b^{6} x^{3} + \frac{6006}{5} \, a^{10} b^{5} x^{\frac{5}{2}} + \frac{1365}{2} \, a^{11} b^{4} x^{2} + \frac{910}{3} \, a^{12} b^{3} x^{\frac{3}{2}} + 105 \, a^{13} b^{2} x + a^{15}{\rm ln}\left ({\left | x \right |}\right ) + 30 \, a^{14} b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*b^15*x^(15/2) + 15/7*a*b^14*x^7 + 210/13*a^2*b^13*x^(13/2) + 455/6*a^3*b^12
*x^6 + 2730/11*a^4*b^11*x^(11/2) + 3003/5*a^5*b^10*x^5 + 10010/9*a^6*b^9*x^(9/2)
+ 6435/4*a^7*b^8*x^4 + 12870/7*a^8*b^7*x^(7/2) + 5005/3*a^9*b^6*x^3 + 6006/5*a^
10*b^5*x^(5/2) + 1365/2*a^11*b^4*x^2 + 910/3*a^12*b^3*x^(3/2) + 105*a^13*b^2*x +
a^15*ln(abs(x)) + 30*a^14*b*sqrt(x)