3.2175 \(\int \frac{\left (a+b \sqrt{x}\right )^{15}}{x^3} \, dx\)

Optimal. Leaf size=190 \[ -\frac{a^{15}}{2 x^2}-\frac{10 a^{14} b}{x^{3/2}}-\frac{105 a^{13} b^2}{x}-\frac{910 a^{12} b^3}{\sqrt{x}}+1365 a^{11} b^4 \log (x)+6006 a^{10} b^5 \sqrt{x}+5005 a^9 b^6 x+4290 a^8 b^7 x^{3/2}+\frac{6435}{2} a^7 b^8 x^2+2002 a^6 b^9 x^{5/2}+1001 a^5 b^{10} x^3+390 a^4 b^{11} x^{7/2}+\frac{455}{4} a^3 b^{12} x^4+\frac{70}{3} a^2 b^{13} x^{9/2}+3 a b^{14} x^5+\frac{2}{11} b^{15} x^{11/2} \]

[Out]

-a^15/(2*x^2) - (10*a^14*b)/x^(3/2) - (105*a^13*b^2)/x - (910*a^12*b^3)/Sqrt[x]
+ 6006*a^10*b^5*Sqrt[x] + 5005*a^9*b^6*x + 4290*a^8*b^7*x^(3/2) + (6435*a^7*b^8*
x^2)/2 + 2002*a^6*b^9*x^(5/2) + 1001*a^5*b^10*x^3 + 390*a^4*b^11*x^(7/2) + (455*
a^3*b^12*x^4)/4 + (70*a^2*b^13*x^(9/2))/3 + 3*a*b^14*x^5 + (2*b^15*x^(11/2))/11
+ 1365*a^11*b^4*Log[x]

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Rubi [A]  time = 0.310178, antiderivative size = 190, 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 \[ -\frac{a^{15}}{2 x^2}-\frac{10 a^{14} b}{x^{3/2}}-\frac{105 a^{13} b^2}{x}-\frac{910 a^{12} b^3}{\sqrt{x}}+1365 a^{11} b^4 \log (x)+6006 a^{10} b^5 \sqrt{x}+5005 a^9 b^6 x+4290 a^8 b^7 x^{3/2}+\frac{6435}{2} a^7 b^8 x^2+2002 a^6 b^9 x^{5/2}+1001 a^5 b^{10} x^3+390 a^4 b^{11} x^{7/2}+\frac{455}{4} a^3 b^{12} x^4+\frac{70}{3} a^2 b^{13} x^{9/2}+3 a b^{14} x^5+\frac{2}{11} b^{15} x^{11/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^15/x^3,x]

[Out]

-a^15/(2*x^2) - (10*a^14*b)/x^(3/2) - (105*a^13*b^2)/x - (910*a^12*b^3)/Sqrt[x]
+ 6006*a^10*b^5*Sqrt[x] + 5005*a^9*b^6*x + 4290*a^8*b^7*x^(3/2) + (6435*a^7*b^8*
x^2)/2 + 2002*a^6*b^9*x^(5/2) + 1001*a^5*b^10*x^3 + 390*a^4*b^11*x^(7/2) + (455*
a^3*b^12*x^4)/4 + (70*a^2*b^13*x^(9/2))/3 + 3*a*b^14*x^5 + (2*b^15*x^(11/2))/11
+ 1365*a^11*b^4*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{15}}{2 x^{2}} - \frac{10 a^{14} b}{x^{\frac{3}{2}}} - \frac{105 a^{13} b^{2}}{x} - \frac{910 a^{12} b^{3}}{\sqrt{x}} + 2730 a^{11} b^{4} \log{\left (\sqrt{x} \right )} + 6006 a^{10} b^{5} \sqrt{x} + 10010 a^{9} b^{6} \int ^{\sqrt{x}} x\, dx + 4290 a^{8} b^{7} x^{\frac{3}{2}} + \frac{6435 a^{7} b^{8} x^{2}}{2} + 2002 a^{6} b^{9} x^{\frac{5}{2}} + 1001 a^{5} b^{10} x^{3} + 390 a^{4} b^{11} x^{\frac{7}{2}} + \frac{455 a^{3} b^{12} x^{4}}{4} + \frac{70 a^{2} b^{13} x^{\frac{9}{2}}}{3} + 3 a b^{14} x^{5} + \frac{2 b^{15} x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**15/x**3,x)

[Out]

-a**15/(2*x**2) - 10*a**14*b/x**(3/2) - 105*a**13*b**2/x - 910*a**12*b**3/sqrt(x
) + 2730*a**11*b**4*log(sqrt(x)) + 6006*a**10*b**5*sqrt(x) + 10010*a**9*b**6*Int
egral(x, (x, sqrt(x))) + 4290*a**8*b**7*x**(3/2) + 6435*a**7*b**8*x**2/2 + 2002*
a**6*b**9*x**(5/2) + 1001*a**5*b**10*x**3 + 390*a**4*b**11*x**(7/2) + 455*a**3*b
**12*x**4/4 + 70*a**2*b**13*x**(9/2)/3 + 3*a*b**14*x**5 + 2*b**15*x**(11/2)/11

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Mathematica [A]  time = 0.107152, size = 190, normalized size = 1. \[ -\frac{a^{15}}{2 x^2}-\frac{10 a^{14} b}{x^{3/2}}-\frac{105 a^{13} b^2}{x}-\frac{910 a^{12} b^3}{\sqrt{x}}+1365 a^{11} b^4 \log (x)+6006 a^{10} b^5 \sqrt{x}+5005 a^9 b^6 x+4290 a^8 b^7 x^{3/2}+\frac{6435}{2} a^7 b^8 x^2+2002 a^6 b^9 x^{5/2}+1001 a^5 b^{10} x^3+390 a^4 b^{11} x^{7/2}+\frac{455}{4} a^3 b^{12} x^4+\frac{70}{3} a^2 b^{13} x^{9/2}+3 a b^{14} x^5+\frac{2}{11} b^{15} x^{11/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^15/x^3,x]

[Out]

-a^15/(2*x^2) - (10*a^14*b)/x^(3/2) - (105*a^13*b^2)/x - (910*a^12*b^3)/Sqrt[x]
+ 6006*a^10*b^5*Sqrt[x] + 5005*a^9*b^6*x + 4290*a^8*b^7*x^(3/2) + (6435*a^7*b^8*
x^2)/2 + 2002*a^6*b^9*x^(5/2) + 1001*a^5*b^10*x^3 + 390*a^4*b^11*x^(7/2) + (455*
a^3*b^12*x^4)/4 + (70*a^2*b^13*x^(9/2))/3 + 3*a*b^14*x^5 + (2*b^15*x^(11/2))/11
+ 1365*a^11*b^4*Log[x]

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Maple [A]  time = 0.008, size = 165, normalized size = 0.9 \[ -{\frac{{a}^{15}}{2\,{x}^{2}}}-10\,{\frac{{a}^{14}b}{{x}^{3/2}}}-105\,{\frac{{a}^{13}{b}^{2}}{x}}+5005\,{a}^{9}{b}^{6}x+4290\,{a}^{8}{b}^{7}{x}^{3/2}+{\frac{6435\,{a}^{7}{b}^{8}{x}^{2}}{2}}+2002\,{a}^{6}{b}^{9}{x}^{5/2}+1001\,{a}^{5}{b}^{10}{x}^{3}+390\,{a}^{4}{b}^{11}{x}^{7/2}+{\frac{455\,{a}^{3}{b}^{12}{x}^{4}}{4}}+{\frac{70\,{a}^{2}{b}^{13}}{3}{x}^{{\frac{9}{2}}}}+3\,a{b}^{14}{x}^{5}+{\frac{2\,{b}^{15}}{11}{x}^{{\frac{11}{2}}}}+1365\,{a}^{11}{b}^{4}\ln \left ( x \right ) -910\,{\frac{{a}^{12}{b}^{3}}{\sqrt{x}}}+6006\,{a}^{10}{b}^{5}\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^15/x^3,x)

[Out]

-1/2*a^15/x^2-10*a^14*b/x^(3/2)-105*a^13*b^2/x+5005*a^9*b^6*x+4290*a^8*b^7*x^(3/
2)+6435/2*a^7*b^8*x^2+2002*a^6*b^9*x^(5/2)+1001*a^5*b^10*x^3+390*a^4*b^11*x^(7/2
)+455/4*a^3*b^12*x^4+70/3*a^2*b^13*x^(9/2)+3*a*b^14*x^5+2/11*b^15*x^(11/2)+1365*
a^11*b^4*ln(x)-910*a^12*b^3/x^(1/2)+6006*a^10*b^5*x^(1/2)

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Maxima [A]  time = 1.42875, size = 220, normalized size = 1.16 \[ \frac{2}{11} \, b^{15} x^{\frac{11}{2}} + 3 \, a b^{14} x^{5} + \frac{70}{3} \, a^{2} b^{13} x^{\frac{9}{2}} + \frac{455}{4} \, a^{3} b^{12} x^{4} + 390 \, a^{4} b^{11} x^{\frac{7}{2}} + 1001 \, a^{5} b^{10} x^{3} + 2002 \, a^{6} b^{9} x^{\frac{5}{2}} + \frac{6435}{2} \, a^{7} b^{8} x^{2} + 4290 \, a^{8} b^{7} x^{\frac{3}{2}} + 5005 \, a^{9} b^{6} x + 1365 \, a^{11} b^{4} \log \left (x\right ) + 6006 \, a^{10} b^{5} \sqrt{x} - \frac{1820 \, a^{12} b^{3} x^{\frac{3}{2}} + 210 \, a^{13} b^{2} x + 20 \, a^{14} b \sqrt{x} + a^{15}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^3,x, algorithm="maxima")

[Out]

2/11*b^15*x^(11/2) + 3*a*b^14*x^5 + 70/3*a^2*b^13*x^(9/2) + 455/4*a^3*b^12*x^4 +
390*a^4*b^11*x^(7/2) + 1001*a^5*b^10*x^3 + 2002*a^6*b^9*x^(5/2) + 6435/2*a^7*b^
8*x^2 + 4290*a^8*b^7*x^(3/2) + 5005*a^9*b^6*x + 1365*a^11*b^4*log(x) + 6006*a^10
*b^5*sqrt(x) - 1/2*(1820*a^12*b^3*x^(3/2) + 210*a^13*b^2*x + 20*a^14*b*sqrt(x) +
a^15)/x^2

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Fricas [A]  time = 0.234782, size = 232, normalized size = 1.22 \[ \frac{396 \, a b^{14} x^{7} + 15015 \, a^{3} b^{12} x^{6} + 132132 \, a^{5} b^{10} x^{5} + 424710 \, a^{7} b^{8} x^{4} + 660660 \, a^{9} b^{6} x^{3} + 360360 \, a^{11} b^{4} x^{2} \log \left (\sqrt{x}\right ) - 13860 \, a^{13} b^{2} x - 66 \, a^{15} + 8 \,{\left (3 \, b^{15} x^{7} + 385 \, a^{2} b^{13} x^{6} + 6435 \, a^{4} b^{11} x^{5} + 33033 \, a^{6} b^{9} x^{4} + 70785 \, a^{8} b^{7} x^{3} + 99099 \, a^{10} b^{5} x^{2} - 15015 \, a^{12} b^{3} x - 165 \, a^{14} b\right )} \sqrt{x}}{132 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^3,x, algorithm="fricas")

[Out]

1/132*(396*a*b^14*x^7 + 15015*a^3*b^12*x^6 + 132132*a^5*b^10*x^5 + 424710*a^7*b^
8*x^4 + 660660*a^9*b^6*x^3 + 360360*a^11*b^4*x^2*log(sqrt(x)) - 13860*a^13*b^2*x
- 66*a^15 + 8*(3*b^15*x^7 + 385*a^2*b^13*x^6 + 6435*a^4*b^11*x^5 + 33033*a^6*b^
9*x^4 + 70785*a^8*b^7*x^3 + 99099*a^10*b^5*x^2 - 15015*a^12*b^3*x - 165*a^14*b)*
sqrt(x))/x^2

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Sympy [A]  time = 18.0269, size = 196, normalized size = 1.03 \[ - \frac{a^{15}}{2 x^{2}} - \frac{10 a^{14} b}{x^{\frac{3}{2}}} - \frac{105 a^{13} b^{2}}{x} - \frac{910 a^{12} b^{3}}{\sqrt{x}} + 1365 a^{11} b^{4} \log{\left (x \right )} + 6006 a^{10} b^{5} \sqrt{x} + 5005 a^{9} b^{6} x + 4290 a^{8} b^{7} x^{\frac{3}{2}} + \frac{6435 a^{7} b^{8} x^{2}}{2} + 2002 a^{6} b^{9} x^{\frac{5}{2}} + 1001 a^{5} b^{10} x^{3} + 390 a^{4} b^{11} x^{\frac{7}{2}} + \frac{455 a^{3} b^{12} x^{4}}{4} + \frac{70 a^{2} b^{13} x^{\frac{9}{2}}}{3} + 3 a b^{14} x^{5} + \frac{2 b^{15} x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**15/x**3,x)

[Out]

-a**15/(2*x**2) - 10*a**14*b/x**(3/2) - 105*a**13*b**2/x - 910*a**12*b**3/sqrt(x
) + 1365*a**11*b**4*log(x) + 6006*a**10*b**5*sqrt(x) + 5005*a**9*b**6*x + 4290*a
**8*b**7*x**(3/2) + 6435*a**7*b**8*x**2/2 + 2002*a**6*b**9*x**(5/2) + 1001*a**5*
b**10*x**3 + 390*a**4*b**11*x**(7/2) + 455*a**3*b**12*x**4/4 + 70*a**2*b**13*x**
(9/2)/3 + 3*a*b**14*x**5 + 2*b**15*x**(11/2)/11

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GIAC/XCAS [A]  time = 0.2245, size = 221, normalized size = 1.16 \[ \frac{2}{11} \, b^{15} x^{\frac{11}{2}} + 3 \, a b^{14} x^{5} + \frac{70}{3} \, a^{2} b^{13} x^{\frac{9}{2}} + \frac{455}{4} \, a^{3} b^{12} x^{4} + 390 \, a^{4} b^{11} x^{\frac{7}{2}} + 1001 \, a^{5} b^{10} x^{3} + 2002 \, a^{6} b^{9} x^{\frac{5}{2}} + \frac{6435}{2} \, a^{7} b^{8} x^{2} + 4290 \, a^{8} b^{7} x^{\frac{3}{2}} + 5005 \, a^{9} b^{6} x + 1365 \, a^{11} b^{4}{\rm ln}\left ({\left | x \right |}\right ) + 6006 \, a^{10} b^{5} \sqrt{x} - \frac{1820 \, a^{12} b^{3} x^{\frac{3}{2}} + 210 \, a^{13} b^{2} x + 20 \, a^{14} b \sqrt{x} + a^{15}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^3,x, algorithm="giac")

[Out]

2/11*b^15*x^(11/2) + 3*a*b^14*x^5 + 70/3*a^2*b^13*x^(9/2) + 455/4*a^3*b^12*x^4 +
390*a^4*b^11*x^(7/2) + 1001*a^5*b^10*x^3 + 2002*a^6*b^9*x^(5/2) + 6435/2*a^7*b^
8*x^2 + 4290*a^8*b^7*x^(3/2) + 5005*a^9*b^6*x + 1365*a^11*b^4*ln(abs(x)) + 6006*
a^10*b^5*sqrt(x) - 1/2*(1820*a^12*b^3*x^(3/2) + 210*a^13*b^2*x + 20*a^14*b*sqrt(
x) + a^15)/x^2