3.1619 \(\int \frac{x^3}{\left (a+\frac{b}{x}\right )^2} \, dx\)

Optimal. Leaf size=72 \[ \frac{b^5}{a^6 (a x+b)}+\frac{5 b^4 \log (a x+b)}{a^6}-\frac{4 b^3 x}{a^5}+\frac{3 b^2 x^2}{2 a^4}-\frac{2 b x^3}{3 a^3}+\frac{x^4}{4 a^2} \]

[Out]

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

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Rubi [A]  time = 0.112584, antiderivative size = 72, 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 \[ \frac{b^5}{a^6 (a x+b)}+\frac{5 b^4 \log (a x+b)}{a^6}-\frac{4 b^3 x}{a^5}+\frac{3 b^2 x^2}{2 a^4}-\frac{2 b x^3}{3 a^3}+\frac{x^4}{4 a^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b/x)^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4/(4*a**2) - 2*b*x**3/(3*a**3) + 3*b**2*Integral(x, x)/a**4 - 4*b**3*x/a**5 +
b**5/(a**6*(a*x + b)) + 5*b**4*log(a*x + b)/a**6

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Mathematica [A]  time = 0.0327599, size = 66, normalized size = 0.92 \[ \frac{3 a^4 x^4-8 a^3 b x^3+18 a^2 b^2 x^2+\frac{12 b^5}{a x+b}+60 b^4 \log (a x+b)-48 a b^3 x}{12 a^6} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b/x)^2,x]

[Out]

(-48*a*b^3*x + 18*a^2*b^2*x^2 - 8*a^3*b*x^3 + 3*a^4*x^4 + (12*b^5)/(b + a*x) + 6
0*b^4*Log[b + a*x])/(12*a^6)

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Maple [A]  time = 0.01, size = 67, normalized size = 0.9 \[ -4\,{\frac{{b}^{3}x}{{a}^{5}}}+{\frac{3\,{b}^{2}{x}^{2}}{2\,{a}^{4}}}-{\frac{2\,b{x}^{3}}{3\,{a}^{3}}}+{\frac{{x}^{4}}{4\,{a}^{2}}}+{\frac{{b}^{5}}{{a}^{6} \left ( ax+b \right ) }}+5\,{\frac{{b}^{4}\ln \left ( ax+b \right ) }{{a}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b/x)^2,x)

[Out]

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

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Maxima [A]  time = 1.43685, size = 95, normalized size = 1.32 \[ \frac{b^{5}}{a^{7} x + a^{6} b} + \frac{5 \, b^{4} \log \left (a x + b\right )}{a^{6}} + \frac{3 \, a^{3} x^{4} - 8 \, a^{2} b x^{3} + 18 \, a b^{2} x^{2} - 48 \, b^{3} x}{12 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a + b/x)^2,x, algorithm="maxima")

[Out]

b^5/(a^7*x + a^6*b) + 5*b^4*log(a*x + b)/a^6 + 1/12*(3*a^3*x^4 - 8*a^2*b*x^3 + 1
8*a*b^2*x^2 - 48*b^3*x)/a^5

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Fricas [A]  time = 0.233988, size = 115, normalized size = 1.6 \[ \frac{3 \, a^{5} x^{5} - 5 \, a^{4} b x^{4} + 10 \, a^{3} b^{2} x^{3} - 30 \, a^{2} b^{3} x^{2} - 48 \, a b^{4} x + 12 \, b^{5} + 60 \,{\left (a b^{4} x + b^{5}\right )} \log \left (a x + b\right )}{12 \,{\left (a^{7} x + a^{6} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a + b/x)^2,x, algorithm="fricas")

[Out]

1/12*(3*a^5*x^5 - 5*a^4*b*x^4 + 10*a^3*b^2*x^3 - 30*a^2*b^3*x^2 - 48*a*b^4*x + 1
2*b^5 + 60*(a*b^4*x + b^5)*log(a*x + b))/(a^7*x + a^6*b)

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Sympy [A]  time = 1.44939, size = 71, normalized size = 0.99 \[ \frac{b^{5}}{a^{7} x + a^{6} b} + \frac{x^{4}}{4 a^{2}} - \frac{2 b x^{3}}{3 a^{3}} + \frac{3 b^{2} x^{2}}{2 a^{4}} - \frac{4 b^{3} x}{a^{5}} + \frac{5 b^{4} \log{\left (a x + b \right )}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(a+b/x)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.232923, size = 99, normalized size = 1.38 \[ \frac{5 \, b^{4}{\rm ln}\left ({\left | a x + b \right |}\right )}{a^{6}} + \frac{b^{5}}{{\left (a x + b\right )} a^{6}} + \frac{3 \, a^{6} x^{4} - 8 \, a^{5} b x^{3} + 18 \, a^{4} b^{2} x^{2} - 48 \, a^{3} b^{3} x}{12 \, a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a + b/x)^2,x, algorithm="giac")

[Out]

5*b^4*ln(abs(a*x + b))/a^6 + b^5/((a*x + b)*a^6) + 1/12*(3*a^6*x^4 - 8*a^5*b*x^3
+ 18*a^4*b^2*x^2 - 48*a^3*b^3*x)/a^8