3.2174 \(\int \frac{\left (a+b \sqrt{x}\right )^{15}}{x^2} \, dx\)

Optimal. Leaf size=192 \[ -\frac{a^{15}}{x}-\frac{30 a^{14} b}{\sqrt{x}}+105 a^{13} b^2 \log (x)+910 a^{12} b^3 \sqrt{x}+1365 a^{11} b^4 x+2002 a^{10} b^5 x^{3/2}+\frac{5005}{2} a^9 b^6 x^2+2574 a^8 b^7 x^{5/2}+2145 a^7 b^8 x^3+1430 a^6 b^9 x^{7/2}+\frac{3003}{4} a^5 b^{10} x^4+\frac{910}{3} a^4 b^{11} x^{9/2}+91 a^3 b^{12} x^5+\frac{210}{11} a^2 b^{13} x^{11/2}+\frac{5}{2} a b^{14} x^6+\frac{2}{13} b^{15} x^{13/2} \]

[Out]

-(a^15/x) - (30*a^14*b)/Sqrt[x] + 910*a^12*b^3*Sqrt[x] + 1365*a^11*b^4*x + 2002*
a^10*b^5*x^(3/2) + (5005*a^9*b^6*x^2)/2 + 2574*a^8*b^7*x^(5/2) + 2145*a^7*b^8*x^
3 + 1430*a^6*b^9*x^(7/2) + (3003*a^5*b^10*x^4)/4 + (910*a^4*b^11*x^(9/2))/3 + 91
*a^3*b^12*x^5 + (210*a^2*b^13*x^(11/2))/11 + (5*a*b^14*x^6)/2 + (2*b^15*x^(13/2)
)/13 + 105*a^13*b^2*Log[x]

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Rubi [A]  time = 0.315878, antiderivative size = 192, 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 \[ -\frac{a^{15}}{x}-\frac{30 a^{14} b}{\sqrt{x}}+105 a^{13} b^2 \log (x)+910 a^{12} b^3 \sqrt{x}+1365 a^{11} b^4 x+2002 a^{10} b^5 x^{3/2}+\frac{5005}{2} a^9 b^6 x^2+2574 a^8 b^7 x^{5/2}+2145 a^7 b^8 x^3+1430 a^6 b^9 x^{7/2}+\frac{3003}{4} a^5 b^{10} x^4+\frac{910}{3} a^4 b^{11} x^{9/2}+91 a^3 b^{12} x^5+\frac{210}{11} a^2 b^{13} x^{11/2}+\frac{5}{2} a b^{14} x^6+\frac{2}{13} b^{15} x^{13/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^15/x^2,x]

[Out]

-(a^15/x) - (30*a^14*b)/Sqrt[x] + 910*a^12*b^3*Sqrt[x] + 1365*a^11*b^4*x + 2002*
a^10*b^5*x^(3/2) + (5005*a^9*b^6*x^2)/2 + 2574*a^8*b^7*x^(5/2) + 2145*a^7*b^8*x^
3 + 1430*a^6*b^9*x^(7/2) + (3003*a^5*b^10*x^4)/4 + (910*a^4*b^11*x^(9/2))/3 + 91
*a^3*b^12*x^5 + (210*a^2*b^13*x^(11/2))/11 + (5*a*b^14*x^6)/2 + (2*b^15*x^(13/2)
)/13 + 105*a^13*b^2*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{15}}{x} - \frac{30 a^{14} b}{\sqrt{x}} + 210 a^{13} b^{2} \log{\left (\sqrt{x} \right )} + 910 a^{12} b^{3} \sqrt{x} + 2730 a^{11} b^{4} \int ^{\sqrt{x}} x\, dx + 2002 a^{10} b^{5} x^{\frac{3}{2}} + \frac{5005 a^{9} b^{6} x^{2}}{2} + 2574 a^{8} b^{7} x^{\frac{5}{2}} + 2145 a^{7} b^{8} x^{3} + 1430 a^{6} b^{9} x^{\frac{7}{2}} + \frac{3003 a^{5} b^{10} x^{4}}{4} + \frac{910 a^{4} b^{11} x^{\frac{9}{2}}}{3} + 91 a^{3} b^{12} x^{5} + \frac{210 a^{2} b^{13} x^{\frac{11}{2}}}{11} + \frac{5 a b^{14} x^{6}}{2} + \frac{2 b^{15} x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**15/x - 30*a**14*b/sqrt(x) + 210*a**13*b**2*log(sqrt(x)) + 910*a**12*b**3*sqr
t(x) + 2730*a**11*b**4*Integral(x, (x, sqrt(x))) + 2002*a**10*b**5*x**(3/2) + 50
05*a**9*b**6*x**2/2 + 2574*a**8*b**7*x**(5/2) + 2145*a**7*b**8*x**3 + 1430*a**6*
b**9*x**(7/2) + 3003*a**5*b**10*x**4/4 + 910*a**4*b**11*x**(9/2)/3 + 91*a**3*b**
12*x**5 + 210*a**2*b**13*x**(11/2)/11 + 5*a*b**14*x**6/2 + 2*b**15*x**(13/2)/13

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Mathematica [A]  time = 0.0875781, size = 192, normalized size = 1. \[ -\frac{a^{15}}{x}-\frac{30 a^{14} b}{\sqrt{x}}+105 a^{13} b^2 \log (x)+910 a^{12} b^3 \sqrt{x}+1365 a^{11} b^4 x+2002 a^{10} b^5 x^{3/2}+\frac{5005}{2} a^9 b^6 x^2+2574 a^8 b^7 x^{5/2}+2145 a^7 b^8 x^3+1430 a^6 b^9 x^{7/2}+\frac{3003}{4} a^5 b^{10} x^4+\frac{910}{3} a^4 b^{11} x^{9/2}+91 a^3 b^{12} x^5+\frac{210}{11} a^2 b^{13} x^{11/2}+\frac{5}{2} a b^{14} x^6+\frac{2}{13} b^{15} x^{13/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^15/x^2,x]

[Out]

-(a^15/x) - (30*a^14*b)/Sqrt[x] + 910*a^12*b^3*Sqrt[x] + 1365*a^11*b^4*x + 2002*
a^10*b^5*x^(3/2) + (5005*a^9*b^6*x^2)/2 + 2574*a^8*b^7*x^(5/2) + 2145*a^7*b^8*x^
3 + 1430*a^6*b^9*x^(7/2) + (3003*a^5*b^10*x^4)/4 + (910*a^4*b^11*x^(9/2))/3 + 91
*a^3*b^12*x^5 + (210*a^2*b^13*x^(11/2))/11 + (5*a*b^14*x^6)/2 + (2*b^15*x^(13/2)
)/13 + 105*a^13*b^2*Log[x]

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Maple [A]  time = 0.005, size = 165, normalized size = 0.9 \[ -{\frac{{a}^{15}}{x}}+1365\,{a}^{11}{b}^{4}x+2002\,{a}^{10}{b}^{5}{x}^{3/2}+{\frac{5005\,{a}^{9}{b}^{6}{x}^{2}}{2}}+2574\,{a}^{8}{b}^{7}{x}^{5/2}+2145\,{a}^{7}{b}^{8}{x}^{3}+1430\,{a}^{6}{b}^{9}{x}^{7/2}+{\frac{3003\,{a}^{5}{b}^{10}{x}^{4}}{4}}+{\frac{910\,{a}^{4}{b}^{11}}{3}{x}^{{\frac{9}{2}}}}+91\,{a}^{3}{b}^{12}{x}^{5}+{\frac{210\,{a}^{2}{b}^{13}}{11}{x}^{{\frac{11}{2}}}}+{\frac{5\,a{b}^{14}{x}^{6}}{2}}+{\frac{2\,{b}^{15}}{13}{x}^{{\frac{13}{2}}}}+105\,{a}^{13}{b}^{2}\ln \left ( x \right ) -30\,{\frac{{a}^{14}b}{\sqrt{x}}}+910\,{a}^{12}{b}^{3}\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^15/x^2,x)

[Out]

-a^15/x+1365*a^11*b^4*x+2002*a^10*b^5*x^(3/2)+5005/2*a^9*b^6*x^2+2574*a^8*b^7*x^
(5/2)+2145*a^7*b^8*x^3+1430*a^6*b^9*x^(7/2)+3003/4*a^5*b^10*x^4+910/3*a^4*b^11*x
^(9/2)+91*a^3*b^12*x^5+210/11*a^2*b^13*x^(11/2)+5/2*a*b^14*x^6+2/13*b^15*x^(13/2
)+105*a^13*b^2*ln(x)-30*a^14*b/x^(1/2)+910*a^12*b^3*x^(1/2)

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Maxima [A]  time = 1.43246, size = 223, normalized size = 1.16 \[ \frac{2}{13} \, b^{15} x^{\frac{13}{2}} + \frac{5}{2} \, a b^{14} x^{6} + \frac{210}{11} \, a^{2} b^{13} x^{\frac{11}{2}} + 91 \, a^{3} b^{12} x^{5} + \frac{910}{3} \, a^{4} b^{11} x^{\frac{9}{2}} + \frac{3003}{4} \, a^{5} b^{10} x^{4} + 1430 \, a^{6} b^{9} x^{\frac{7}{2}} + 2145 \, a^{7} b^{8} x^{3} + 2574 \, a^{8} b^{7} x^{\frac{5}{2}} + \frac{5005}{2} \, a^{9} b^{6} x^{2} + 2002 \, a^{10} b^{5} x^{\frac{3}{2}} + 1365 \, a^{11} b^{4} x + 105 \, a^{13} b^{2} \log \left (x\right ) + 910 \, a^{12} b^{3} \sqrt{x} - \frac{30 \, a^{14} b \sqrt{x} + a^{15}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^2,x, algorithm="maxima")

[Out]

2/13*b^15*x^(13/2) + 5/2*a*b^14*x^6 + 210/11*a^2*b^13*x^(11/2) + 91*a^3*b^12*x^5
+ 910/3*a^4*b^11*x^(9/2) + 3003/4*a^5*b^10*x^4 + 1430*a^6*b^9*x^(7/2) + 2145*a^
7*b^8*x^3 + 2574*a^8*b^7*x^(5/2) + 5005/2*a^9*b^6*x^2 + 2002*a^10*b^5*x^(3/2) +
1365*a^11*b^4*x + 105*a^13*b^2*log(x) + 910*a^12*b^3*sqrt(x) - (30*a^14*b*sqrt(x
) + a^15)/x

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Fricas [A]  time = 0.240229, size = 232, normalized size = 1.21 \[ \frac{4290 \, a b^{14} x^{7} + 156156 \, a^{3} b^{12} x^{6} + 1288287 \, a^{5} b^{10} x^{5} + 3680820 \, a^{7} b^{8} x^{4} + 4294290 \, a^{9} b^{6} x^{3} + 2342340 \, a^{11} b^{4} x^{2} + 360360 \, a^{13} b^{2} x \log \left (\sqrt{x}\right ) - 1716 \, a^{15} + 8 \,{\left (33 \, b^{15} x^{7} + 4095 \, a^{2} b^{13} x^{6} + 65065 \, a^{4} b^{11} x^{5} + 306735 \, a^{6} b^{9} x^{4} + 552123 \, a^{8} b^{7} x^{3} + 429429 \, a^{10} b^{5} x^{2} + 195195 \, a^{12} b^{3} x - 6435 \, a^{14} b\right )} \sqrt{x}}{1716 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^2,x, algorithm="fricas")

[Out]

1/1716*(4290*a*b^14*x^7 + 156156*a^3*b^12*x^6 + 1288287*a^5*b^10*x^5 + 3680820*a
^7*b^8*x^4 + 4294290*a^9*b^6*x^3 + 2342340*a^11*b^4*x^2 + 360360*a^13*b^2*x*log(
sqrt(x)) - 1716*a^15 + 8*(33*b^15*x^7 + 4095*a^2*b^13*x^6 + 65065*a^4*b^11*x^5 +
306735*a^6*b^9*x^4 + 552123*a^8*b^7*x^3 + 429429*a^10*b^5*x^2 + 195195*a^12*b^3
*x - 6435*a^14*b)*sqrt(x))/x

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Sympy [A]  time = 18.8701, size = 197, normalized size = 1.03 \[ - \frac{a^{15}}{x} - \frac{30 a^{14} b}{\sqrt{x}} + 105 a^{13} b^{2} \log{\left (x \right )} + 910 a^{12} b^{3} \sqrt{x} + 1365 a^{11} b^{4} x + 2002 a^{10} b^{5} x^{\frac{3}{2}} + \frac{5005 a^{9} b^{6} x^{2}}{2} + 2574 a^{8} b^{7} x^{\frac{5}{2}} + 2145 a^{7} b^{8} x^{3} + 1430 a^{6} b^{9} x^{\frac{7}{2}} + \frac{3003 a^{5} b^{10} x^{4}}{4} + \frac{910 a^{4} b^{11} x^{\frac{9}{2}}}{3} + 91 a^{3} b^{12} x^{5} + \frac{210 a^{2} b^{13} x^{\frac{11}{2}}}{11} + \frac{5 a b^{14} x^{6}}{2} + \frac{2 b^{15} x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**15/x - 30*a**14*b/sqrt(x) + 105*a**13*b**2*log(x) + 910*a**12*b**3*sqrt(x) +
1365*a**11*b**4*x + 2002*a**10*b**5*x**(3/2) + 5005*a**9*b**6*x**2/2 + 2574*a**
8*b**7*x**(5/2) + 2145*a**7*b**8*x**3 + 1430*a**6*b**9*x**(7/2) + 3003*a**5*b**1
0*x**4/4 + 910*a**4*b**11*x**(9/2)/3 + 91*a**3*b**12*x**5 + 210*a**2*b**13*x**(1
1/2)/11 + 5*a*b**14*x**6/2 + 2*b**15*x**(13/2)/13

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GIAC/XCAS [A]  time = 0.221688, size = 224, normalized size = 1.17 \[ \frac{2}{13} \, b^{15} x^{\frac{13}{2}} + \frac{5}{2} \, a b^{14} x^{6} + \frac{210}{11} \, a^{2} b^{13} x^{\frac{11}{2}} + 91 \, a^{3} b^{12} x^{5} + \frac{910}{3} \, a^{4} b^{11} x^{\frac{9}{2}} + \frac{3003}{4} \, a^{5} b^{10} x^{4} + 1430 \, a^{6} b^{9} x^{\frac{7}{2}} + 2145 \, a^{7} b^{8} x^{3} + 2574 \, a^{8} b^{7} x^{\frac{5}{2}} + \frac{5005}{2} \, a^{9} b^{6} x^{2} + 2002 \, a^{10} b^{5} x^{\frac{3}{2}} + 1365 \, a^{11} b^{4} x + 105 \, a^{13} b^{2}{\rm ln}\left ({\left | x \right |}\right ) + 910 \, a^{12} b^{3} \sqrt{x} - \frac{30 \, a^{14} b \sqrt{x} + a^{15}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^2,x, algorithm="giac")

[Out]

2/13*b^15*x^(13/2) + 5/2*a*b^14*x^6 + 210/11*a^2*b^13*x^(11/2) + 91*a^3*b^12*x^5
+ 910/3*a^4*b^11*x^(9/2) + 3003/4*a^5*b^10*x^4 + 1430*a^6*b^9*x^(7/2) + 2145*a^
7*b^8*x^3 + 2574*a^8*b^7*x^(5/2) + 5005/2*a^9*b^6*x^2 + 2002*a^10*b^5*x^(3/2) +
1365*a^11*b^4*x + 105*a^13*b^2*ln(abs(x)) + 910*a^12*b^3*sqrt(x) - (30*a^14*b*sq
rt(x) + a^15)/x