3.1613 \(\int \frac{1}{\left (a+\frac{b}{x}\right ) x^4} \, dx\)

Optimal. Leaf size=42 \[ \frac{a^2 \log (x)}{b^3}-\frac{a^2 \log (a x+b)}{b^3}+\frac{a}{b^2 x}-\frac{1}{2 b x^2} \]

[Out]

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

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Rubi [A]  time = 0.0552284, antiderivative size = 42, 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 \[ \frac{a^2 \log (x)}{b^3}-\frac{a^2 \log (a x+b)}{b^3}+\frac{a}{b^2 x}-\frac{1}{2 b x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.39716, size = 37, normalized size = 0.88 \[ \frac{a^{2} \log{\left (x \right )}}{b^{3}} - \frac{a^{2} \log{\left (a x + b \right )}}{b^{3}} + \frac{a}{b^{2} x} - \frac{1}{2 b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.00818869, size = 42, normalized size = 1. \[ \frac{a^2 \log (x)}{b^3}-\frac{a^2 \log (a x+b)}{b^3}+\frac{a}{b^2 x}-\frac{1}{2 b x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)*x^4),x]

[Out]

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

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Maple [A]  time = 0.013, size = 41, normalized size = 1. \[ -{\frac{1}{2\,b{x}^{2}}}+{\frac{a}{{b}^{2}x}}+{\frac{{a}^{2}\ln \left ( x \right ) }{{b}^{3}}}-{\frac{{a}^{2}\ln \left ( ax+b \right ) }{{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)/x^4,x)

[Out]

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

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Maxima [A]  time = 1.43762, size = 54, normalized size = 1.29 \[ -\frac{a^{2} \log \left (a x + b\right )}{b^{3}} + \frac{a^{2} \log \left (x\right )}{b^{3}} + \frac{2 \, a x - b}{2 \, b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)*x^4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.226613, size = 55, normalized size = 1.31 \[ -\frac{2 \, a^{2} x^{2} \log \left (a x + b\right ) - 2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, a b x + b^{2}}{2 \, b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.39915, size = 31, normalized size = 0.74 \[ \frac{a^{2} \left (\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}\right )}{b^{3}} + \frac{2 a x - b}{2 b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)/x**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.223123, size = 61, normalized size = 1.45 \[ -\frac{a^{2}{\rm ln}\left ({\left | a x + b \right |}\right )}{b^{3}} + \frac{a^{2}{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} + \frac{2 \, a b x - b^{2}}{2 \, b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)*x^4),x, algorithm="giac")

[Out]

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