3.1615 \(\int \frac{1}{\left (a+\frac{b}{x}\right ) x^6} \, dx\)

Optimal. Leaf size=68 \[ \frac{a^4 \log (x)}{b^5}-\frac{a^4 \log (a x+b)}{b^5}+\frac{a^3}{b^4 x}-\frac{a^2}{2 b^3 x^2}+\frac{a}{3 b^2 x^3}-\frac{1}{4 b x^4} \]

[Out]

-1/(4*b*x^4) + a/(3*b^2*x^3) - a^2/(2*b^3*x^2) + a^3/(b^4*x) + (a^4*Log[x])/b^5
- (a^4*Log[b + a*x])/b^5

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Rubi [A]  time = 0.0845574, antiderivative size = 68, 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 \[ \frac{a^4 \log (x)}{b^5}-\frac{a^4 \log (a x+b)}{b^5}+\frac{a^3}{b^4 x}-\frac{a^2}{2 b^3 x^2}+\frac{a}{3 b^2 x^3}-\frac{1}{4 b x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)*x^6),x]

[Out]

-1/(4*b*x^4) + a/(3*b^2*x^3) - a^2/(2*b^3*x^2) + a^3/(b^4*x) + (a^4*Log[x])/b^5
- (a^4*Log[b + a*x])/b^5

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Rubi in Sympy [A]  time = 12.9768, size = 61, normalized size = 0.9 \[ \frac{a^{4} \log{\left (x \right )}}{b^{5}} - \frac{a^{4} \log{\left (a x + b \right )}}{b^{5}} + \frac{a^{3}}{b^{4} x} - \frac{a^{2}}{2 b^{3} x^{2}} + \frac{a}{3 b^{2} x^{3}} - \frac{1}{4 b x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)/x**6,x)

[Out]

a**4*log(x)/b**5 - a**4*log(a*x + b)/b**5 + a**3/(b**4*x) - a**2/(2*b**3*x**2) +
a/(3*b**2*x**3) - 1/(4*b*x**4)

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Mathematica [A]  time = 0.00818325, size = 68, normalized size = 1. \[ \frac{a^4 \log (x)}{b^5}-\frac{a^4 \log (a x+b)}{b^5}+\frac{a^3}{b^4 x}-\frac{a^2}{2 b^3 x^2}+\frac{a}{3 b^2 x^3}-\frac{1}{4 b x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)*x^6),x]

[Out]

-1/(4*b*x^4) + a/(3*b^2*x^3) - a^2/(2*b^3*x^2) + a^3/(b^4*x) + (a^4*Log[x])/b^5
- (a^4*Log[b + a*x])/b^5

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Maple [A]  time = 0.012, size = 63, normalized size = 0.9 \[ -{\frac{1}{4\,b{x}^{4}}}+{\frac{a}{3\,{b}^{2}{x}^{3}}}-{\frac{{a}^{2}}{2\,{b}^{3}{x}^{2}}}+{\frac{{a}^{3}}{{b}^{4}x}}+{\frac{{a}^{4}\ln \left ( x \right ) }{{b}^{5}}}-{\frac{{a}^{4}\ln \left ( ax+b \right ) }{{b}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)/x^6,x)

[Out]

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

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Maxima [A]  time = 1.43607, size = 84, normalized size = 1.24 \[ -\frac{a^{4} \log \left (a x + b\right )}{b^{5}} + \frac{a^{4} \log \left (x\right )}{b^{5}} + \frac{12 \, a^{3} x^{3} - 6 \, a^{2} b x^{2} + 4 \, a b^{2} x - 3 \, b^{3}}{12 \, b^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)*x^6),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.224339, size = 88, normalized size = 1.29 \[ -\frac{12 \, a^{4} x^{4} \log \left (a x + b\right ) - 12 \, a^{4} x^{4} \log \left (x\right ) - 12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 4 \, a b^{3} x + 3 \, b^{4}}{12 \, b^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)*x^6),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.67637, size = 56, normalized size = 0.82 \[ \frac{a^{4} \left (\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}\right )}{b^{5}} + \frac{12 a^{3} x^{3} - 6 a^{2} b x^{2} + 4 a b^{2} x - 3 b^{3}}{12 b^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)/x**6,x)

[Out]

a**4*(log(x) - log(x + b/a))/b**5 + (12*a**3*x**3 - 6*a**2*b*x**2 + 4*a*b**2*x -
3*b**3)/(12*b**4*x**4)

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GIAC/XCAS [A]  time = 0.222521, size = 90, normalized size = 1.32 \[ -\frac{a^{4}{\rm ln}\left ({\left | a x + b \right |}\right )}{b^{5}} + \frac{a^{4}{\rm ln}\left ({\left | x \right |}\right )}{b^{5}} + \frac{12 \, a^{3} b x^{3} - 6 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x - 3 \, b^{4}}{12 \, b^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)*x^6),x, algorithm="giac")

[Out]

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