3.1605 \(\int \frac{x^4}{a+\frac{b}{x}} \, dx\)

Optimal. Leaf size=70 \[ -\frac{b^5 \log (a x+b)}{a^6}+\frac{b^4 x}{a^5}-\frac{b^3 x^2}{2 a^4}+\frac{b^2 x^3}{3 a^3}-\frac{b x^4}{4 a^2}+\frac{x^5}{5 a} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0998011, antiderivative size = 70, 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 \[ -\frac{b^5 \log (a x+b)}{a^6}+\frac{b^4 x}{a^5}-\frac{b^3 x^2}{2 a^4}+\frac{b^2 x^3}{3 a^3}-\frac{b x^4}{4 a^2}+\frac{x^5}{5 a} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b/x),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ b^{4} \int \frac{1}{a^{5}}\, dx + \frac{x^{5}}{5 a} - \frac{b x^{4}}{4 a^{2}} + \frac{b^{2} x^{3}}{3 a^{3}} - \frac{b^{3} \int x\, dx}{a^{4}} - \frac{b^{5} \log{\left (a x + b \right )}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(a+b/x),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00745688, size = 70, normalized size = 1. \[ -\frac{b^5 \log (a x+b)}{a^6}+\frac{b^4 x}{a^5}-\frac{b^3 x^2}{2 a^4}+\frac{b^2 x^3}{3 a^3}-\frac{b x^4}{4 a^2}+\frac{x^5}{5 a} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b/x),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 63, normalized size = 0.9 \[{\frac{{b}^{4}x}{{a}^{5}}}-{\frac{{b}^{3}{x}^{2}}{2\,{a}^{4}}}+{\frac{{b}^{2}{x}^{3}}{3\,{a}^{3}}}-{\frac{b{x}^{4}}{4\,{a}^{2}}}+{\frac{{x}^{5}}{5\,a}}-{\frac{{b}^{5}\ln \left ( ax+b \right ) }{{a}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(a+b/x),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.44359, size = 86, normalized size = 1.23 \[ -\frac{b^{5} \log \left (a x + b\right )}{a^{6}} + \frac{12 \, a^{4} x^{5} - 15 \, a^{3} b x^{4} + 20 \, a^{2} b^{2} x^{3} - 30 \, a b^{3} x^{2} + 60 \, b^{4} x}{60 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(a + b/x),x, algorithm="maxima")

[Out]

-b^5*log(a*x + b)/a^6 + 1/60*(12*a^4*x^5 - 15*a^3*b*x^4 + 20*a^2*b^2*x^3 - 30*a*
b^3*x^2 + 60*b^4*x)/a^5

_______________________________________________________________________________________

Fricas [A]  time = 0.220855, size = 85, normalized size = 1.21 \[ \frac{12 \, a^{5} x^{5} - 15 \, a^{4} b x^{4} + 20 \, a^{3} b^{2} x^{3} - 30 \, a^{2} b^{3} x^{2} + 60 \, a b^{4} x - 60 \, b^{5} \log \left (a x + b\right )}{60 \, a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(a + b/x),x, algorithm="fricas")

[Out]

1/60*(12*a^5*x^5 - 15*a^4*b*x^4 + 20*a^3*b^2*x^3 - 30*a^2*b^3*x^2 + 60*a*b^4*x -
60*b^5*log(a*x + b))/a^6

_______________________________________________________________________________________

Sympy [A]  time = 1.16754, size = 61, normalized size = 0.87 \[ \frac{x^{5}}{5 a} - \frac{b x^{4}}{4 a^{2}} + \frac{b^{2} x^{3}}{3 a^{3}} - \frac{b^{3} x^{2}}{2 a^{4}} + \frac{b^{4} x}{a^{5}} - \frac{b^{5} \log{\left (a x + b \right )}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(a+b/x),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.229322, size = 88, normalized size = 1.26 \[ -\frac{b^{5}{\rm ln}\left ({\left | a x + b \right |}\right )}{a^{6}} + \frac{12 \, a^{4} x^{5} - 15 \, a^{3} b x^{4} + 20 \, a^{2} b^{2} x^{3} - 30 \, a b^{3} x^{2} + 60 \, b^{4} x}{60 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(a + b/x),x, algorithm="giac")

[Out]

-b^5*ln(abs(a*x + b))/a^6 + 1/60*(12*a^4*x^5 - 15*a^3*b*x^4 + 20*a^2*b^2*x^3 - 3
0*a*b^3*x^2 + 60*b^4*x)/a^5