3.1617 \(\int \frac{x^5}{\left (a+\frac{b}{x}\right )^2} \, dx\)

Optimal. Leaf size=98 \[ \frac{b^7}{a^8 (a x+b)}+\frac{7 b^6 \log (a x+b)}{a^8}-\frac{6 b^5 x}{a^7}+\frac{5 b^4 x^2}{2 a^6}-\frac{4 b^3 x^3}{3 a^5}+\frac{3 b^2 x^4}{4 a^4}-\frac{2 b x^5}{5 a^3}+\frac{x^6}{6 a^2} \]

[Out]

(-6*b^5*x)/a^7 + (5*b^4*x^2)/(2*a^6) - (4*b^3*x^3)/(3*a^5) + (3*b^2*x^4)/(4*a^4)
- (2*b*x^5)/(5*a^3) + x^6/(6*a^2) + b^7/(a^8*(b + a*x)) + (7*b^6*Log[b + a*x])/
a^8

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Rubi [A]  time = 0.158466, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{b^7}{a^8 (a x+b)}+\frac{7 b^6 \log (a x+b)}{a^8}-\frac{6 b^5 x}{a^7}+\frac{5 b^4 x^2}{2 a^6}-\frac{4 b^3 x^3}{3 a^5}+\frac{3 b^2 x^4}{4 a^4}-\frac{2 b x^5}{5 a^3}+\frac{x^6}{6 a^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b/x)^2,x]

[Out]

(-6*b^5*x)/a^7 + (5*b^4*x^2)/(2*a^6) - (4*b^3*x^3)/(3*a^5) + (3*b^2*x^4)/(4*a^4)
- (2*b*x^5)/(5*a^3) + x^6/(6*a^2) + b^7/(a^8*(b + a*x)) + (7*b^6*Log[b + a*x])/
a^8

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{6}}{6 a^{2}} - \frac{2 b x^{5}}{5 a^{3}} + \frac{3 b^{2} x^{4}}{4 a^{4}} - \frac{4 b^{3} x^{3}}{3 a^{5}} + \frac{5 b^{4} \int x\, dx}{a^{6}} - \frac{6 b^{5} x}{a^{7}} + \frac{b^{7}}{a^{8} \left (a x + b\right )} + \frac{7 b^{6} \log{\left (a x + b \right )}}{a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(a+b/x)**2,x)

[Out]

x**6/(6*a**2) - 2*b*x**5/(5*a**3) + 3*b**2*x**4/(4*a**4) - 4*b**3*x**3/(3*a**5)
+ 5*b**4*Integral(x, x)/a**6 - 6*b**5*x/a**7 + b**7/(a**8*(a*x + b)) + 7*b**6*lo
g(a*x + b)/a**8

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Mathematica [A]  time = 0.0409777, size = 88, normalized size = 0.9 \[ \frac{10 a^6 x^6-24 a^5 b x^5+45 a^4 b^2 x^4-80 a^3 b^3 x^3+150 a^2 b^4 x^2+\frac{60 b^7}{a x+b}+420 b^6 \log (a x+b)-360 a b^5 x}{60 a^8} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b/x)^2,x]

[Out]

(-360*a*b^5*x + 150*a^2*b^4*x^2 - 80*a^3*b^3*x^3 + 45*a^4*b^2*x^4 - 24*a^5*b*x^5
+ 10*a^6*x^6 + (60*b^7)/(b + a*x) + 420*b^6*Log[b + a*x])/(60*a^8)

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Maple [A]  time = 0.012, size = 89, normalized size = 0.9 \[ -6\,{\frac{{b}^{5}x}{{a}^{7}}}+{\frac{5\,{b}^{4}{x}^{2}}{2\,{a}^{6}}}-{\frac{4\,{b}^{3}{x}^{3}}{3\,{a}^{5}}}+{\frac{3\,{b}^{2}{x}^{4}}{4\,{a}^{4}}}-{\frac{2\,b{x}^{5}}{5\,{a}^{3}}}+{\frac{{x}^{6}}{6\,{a}^{2}}}+{\frac{{b}^{7}}{{a}^{8} \left ( ax+b \right ) }}+7\,{\frac{{b}^{6}\ln \left ( ax+b \right ) }{{a}^{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(a+b/x)^2,x)

[Out]

-6*b^5*x/a^7+5/2*b^4*x^2/a^6-4/3*b^3*x^3/a^5+3/4*b^2*x^4/a^4-2/5*b*x^5/a^3+1/6*x
^6/a^2+b^7/a^8/(a*x+b)+7*b^6*ln(a*x+b)/a^8

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Maxima [A]  time = 1.44087, size = 124, normalized size = 1.27 \[ \frac{b^{7}}{a^{9} x + a^{8} b} + \frac{7 \, b^{6} \log \left (a x + b\right )}{a^{8}} + \frac{10 \, a^{5} x^{6} - 24 \, a^{4} b x^{5} + 45 \, a^{3} b^{2} x^{4} - 80 \, a^{2} b^{3} x^{3} + 150 \, a b^{4} x^{2} - 360 \, b^{5} x}{60 \, a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(a + b/x)^2,x, algorithm="maxima")

[Out]

b^7/(a^9*x + a^8*b) + 7*b^6*log(a*x + b)/a^8 + 1/60*(10*a^5*x^6 - 24*a^4*b*x^5 +
45*a^3*b^2*x^4 - 80*a^2*b^3*x^3 + 150*a*b^4*x^2 - 360*b^5*x)/a^7

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Fricas [A]  time = 0.219019, size = 144, normalized size = 1.47 \[ \frac{10 \, a^{7} x^{7} - 14 \, a^{6} b x^{6} + 21 \, a^{5} b^{2} x^{5} - 35 \, a^{4} b^{3} x^{4} + 70 \, a^{3} b^{4} x^{3} - 210 \, a^{2} b^{5} x^{2} - 360 \, a b^{6} x + 60 \, b^{7} + 420 \,{\left (a b^{6} x + b^{7}\right )} \log \left (a x + b\right )}{60 \,{\left (a^{9} x + a^{8} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(a + b/x)^2,x, algorithm="fricas")

[Out]

1/60*(10*a^7*x^7 - 14*a^6*b*x^6 + 21*a^5*b^2*x^5 - 35*a^4*b^3*x^4 + 70*a^3*b^4*x
^3 - 210*a^2*b^5*x^2 - 360*a*b^6*x + 60*b^7 + 420*(a*b^6*x + b^7)*log(a*x + b))/
(a^9*x + a^8*b)

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Sympy [A]  time = 1.5614, size = 99, normalized size = 1.01 \[ \frac{b^{7}}{a^{9} x + a^{8} b} + \frac{x^{6}}{6 a^{2}} - \frac{2 b x^{5}}{5 a^{3}} + \frac{3 b^{2} x^{4}}{4 a^{4}} - \frac{4 b^{3} x^{3}}{3 a^{5}} + \frac{5 b^{4} x^{2}}{2 a^{6}} - \frac{6 b^{5} x}{a^{7}} + \frac{7 b^{6} \log{\left (a x + b \right )}}{a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(a+b/x)**2,x)

[Out]

b**7/(a**9*x + a**8*b) + x**6/(6*a**2) - 2*b*x**5/(5*a**3) + 3*b**2*x**4/(4*a**4
) - 4*b**3*x**3/(3*a**5) + 5*b**4*x**2/(2*a**6) - 6*b**5*x/a**7 + 7*b**6*log(a*x
+ b)/a**8

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GIAC/XCAS [A]  time = 0.225835, size = 128, normalized size = 1.31 \[ \frac{7 \, b^{6}{\rm ln}\left ({\left | a x + b \right |}\right )}{a^{8}} + \frac{b^{7}}{{\left (a x + b\right )} a^{8}} + \frac{10 \, a^{10} x^{6} - 24 \, a^{9} b x^{5} + 45 \, a^{8} b^{2} x^{4} - 80 \, a^{7} b^{3} x^{3} + 150 \, a^{6} b^{4} x^{2} - 360 \, a^{5} b^{5} x}{60 \, a^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(a + b/x)^2,x, algorithm="giac")

[Out]

7*b^6*ln(abs(a*x + b))/a^8 + b^7/((a*x + b)*a^8) + 1/60*(10*a^10*x^6 - 24*a^9*b*
x^5 + 45*a^8*b^2*x^4 - 80*a^7*b^3*x^3 + 150*a^6*b^4*x^2 - 360*a^5*b^5*x)/a^12