3.1614 \(\int \frac{1}{\left (a+\frac{b}{x}\right ) x^5} \, dx\)

Optimal. Leaf size=56 \[ -\frac{a^3 \log (x)}{b^4}+\frac{a^3 \log (a x+b)}{b^4}-\frac{a^2}{b^3 x}+\frac{a}{2 b^2 x^2}-\frac{1}{3 b x^3} \]

[Out]

-1/(3*b*x^3) + a/(2*b^2*x^2) - a^2/(b^3*x) - (a^3*Log[x])/b^4 + (a^3*Log[b + a*x
])/b^4

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Rubi [A]  time = 0.0712666, antiderivative size = 56, 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 \[ -\frac{a^3 \log (x)}{b^4}+\frac{a^3 \log (a x+b)}{b^4}-\frac{a^2}{b^3 x}+\frac{a}{2 b^2 x^2}-\frac{1}{3 b x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)*x^5),x]

[Out]

-1/(3*b*x^3) + a/(2*b^2*x^2) - a^2/(b^3*x) - (a^3*Log[x])/b^4 + (a^3*Log[b + a*x
])/b^4

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Rubi in Sympy [A]  time = 11.086, size = 49, normalized size = 0.88 \[ - \frac{a^{3} \log{\left (x \right )}}{b^{4}} + \frac{a^{3} \log{\left (a x + b \right )}}{b^{4}} - \frac{a^{2}}{b^{3} x} + \frac{a}{2 b^{2} x^{2}} - \frac{1}{3 b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)/x**5,x)

[Out]

-a**3*log(x)/b**4 + a**3*log(a*x + b)/b**4 - a**2/(b**3*x) + a/(2*b**2*x**2) - 1
/(3*b*x**3)

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Mathematica [A]  time = 0.00972204, size = 56, normalized size = 1. \[ -\frac{a^3 \log (x)}{b^4}+\frac{a^3 \log (a x+b)}{b^4}-\frac{a^2}{b^3 x}+\frac{a}{2 b^2 x^2}-\frac{1}{3 b x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)*x^5),x]

[Out]

-1/(3*b*x^3) + a/(2*b^2*x^2) - a^2/(b^3*x) - (a^3*Log[x])/b^4 + (a^3*Log[b + a*x
])/b^4

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Maple [A]  time = 0.011, size = 53, normalized size = 1. \[ -{\frac{1}{3\,b{x}^{3}}}+{\frac{a}{2\,{b}^{2}{x}^{2}}}-{\frac{{a}^{2}}{{b}^{3}x}}-{\frac{{a}^{3}\ln \left ( x \right ) }{{b}^{4}}}+{\frac{{a}^{3}\ln \left ( ax+b \right ) }{{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)/x^5,x)

[Out]

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

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Maxima [A]  time = 1.41971, size = 69, normalized size = 1.23 \[ \frac{a^{3} \log \left (a x + b\right )}{b^{4}} - \frac{a^{3} \log \left (x\right )}{b^{4}} - \frac{6 \, a^{2} x^{2} - 3 \, a b x + 2 \, b^{2}}{6 \, b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)*x^5),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.227135, size = 73, normalized size = 1.3 \[ \frac{6 \, a^{3} x^{3} \log \left (a x + b\right ) - 6 \, a^{3} x^{3} \log \left (x\right ) - 6 \, a^{2} b x^{2} + 3 \, a b^{2} x - 2 \, b^{3}}{6 \, b^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)*x^5),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.54706, size = 44, normalized size = 0.79 \[ \frac{a^{3} \left (- \log{\left (x \right )} + \log{\left (x + \frac{b}{a} \right )}\right )}{b^{4}} - \frac{6 a^{2} x^{2} - 3 a b x + 2 b^{2}}{6 b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)/x**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.231316, size = 76, normalized size = 1.36 \[ \frac{a^{3}{\rm ln}\left ({\left | a x + b \right |}\right )}{b^{4}} - \frac{a^{3}{\rm ln}\left ({\left | x \right |}\right )}{b^{4}} - \frac{6 \, a^{2} b x^{2} - 3 \, a b^{2} x + 2 \, b^{3}}{6 \, b^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)*x^5),x, algorithm="giac")

[Out]

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