3.1642 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^3 x^7} \, dx\)

Optimal. Leaf size=89 \[ -\frac{10 a^3 \log (x)}{b^6}+\frac{10 a^3 \log (a x+b)}{b^6}-\frac{4 a^3}{b^5 (a x+b)}-\frac{a^3}{2 b^4 (a x+b)^2}-\frac{6 a^2}{b^5 x}+\frac{3 a}{2 b^4 x^2}-\frac{1}{3 b^3 x^3} \]

[Out]

-1/(3*b^3*x^3) + (3*a)/(2*b^4*x^2) - (6*a^2)/(b^5*x) - a^3/(2*b^4*(b + a*x)^2) -
(4*a^3)/(b^5*(b + a*x)) - (10*a^3*Log[x])/b^6 + (10*a^3*Log[b + a*x])/b^6

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Rubi [A]  time = 0.128025, antiderivative size = 89, 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 \[ -\frac{10 a^3 \log (x)}{b^6}+\frac{10 a^3 \log (a x+b)}{b^6}-\frac{4 a^3}{b^5 (a x+b)}-\frac{a^3}{2 b^4 (a x+b)^2}-\frac{6 a^2}{b^5 x}+\frac{3 a}{2 b^4 x^2}-\frac{1}{3 b^3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^3*x^7),x]

[Out]

-1/(3*b^3*x^3) + (3*a)/(2*b^4*x^2) - (6*a^2)/(b^5*x) - a^3/(2*b^4*(b + a*x)^2) -
(4*a^3)/(b^5*(b + a*x)) - (10*a^3*Log[x])/b^6 + (10*a^3*Log[b + a*x])/b^6

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Rubi in Sympy [A]  time = 18.8407, size = 87, normalized size = 0.98 \[ - \frac{a^{3}}{2 b^{4} \left (a x + b\right )^{2}} - \frac{4 a^{3}}{b^{5} \left (a x + b\right )} - \frac{10 a^{3} \log{\left (x \right )}}{b^{6}} + \frac{10 a^{3} \log{\left (a x + b \right )}}{b^{6}} - \frac{6 a^{2}}{b^{5} x} + \frac{3 a}{2 b^{4} x^{2}} - \frac{1}{3 b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**3/x**7,x)

[Out]

-a**3/(2*b**4*(a*x + b)**2) - 4*a**3/(b**5*(a*x + b)) - 10*a**3*log(x)/b**6 + 10
*a**3*log(a*x + b)/b**6 - 6*a**2/(b**5*x) + 3*a/(2*b**4*x**2) - 1/(3*b**3*x**3)

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Mathematica [A]  time = 0.127842, size = 79, normalized size = 0.89 \[ -\frac{-60 a^3 \log (a x+b)+60 a^3 \log (x)+\frac{b \left (60 a^4 x^4+90 a^3 b x^3+20 a^2 b^2 x^2-5 a b^3 x+2 b^4\right )}{x^3 (a x+b)^2}}{6 b^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^3*x^7),x]

[Out]

-((b*(2*b^4 - 5*a*b^3*x + 20*a^2*b^2*x^2 + 90*a^3*b*x^3 + 60*a^4*x^4))/(x^3*(b +
a*x)^2) + 60*a^3*Log[x] - 60*a^3*Log[b + a*x])/(6*b^6)

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Maple [A]  time = 0.017, size = 84, normalized size = 0.9 \[ -{\frac{1}{3\,{b}^{3}{x}^{3}}}+{\frac{3\,a}{2\,{b}^{4}{x}^{2}}}-6\,{\frac{{a}^{2}}{{b}^{5}x}}-{\frac{{a}^{3}}{2\,{b}^{4} \left ( ax+b \right ) ^{2}}}-4\,{\frac{{a}^{3}}{{b}^{5} \left ( ax+b \right ) }}-10\,{\frac{{a}^{3}\ln \left ( x \right ) }{{b}^{6}}}+10\,{\frac{{a}^{3}\ln \left ( ax+b \right ) }{{b}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^3/x^7,x)

[Out]

-1/3/b^3/x^3+3/2*a/b^4/x^2-6*a^2/b^5/x-1/2*a^3/b^4/(a*x+b)^2-4*a^3/b^5/(a*x+b)-1
0*a^3*ln(x)/b^6+10*a^3*ln(a*x+b)/b^6

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Maxima [A]  time = 1.44994, size = 131, normalized size = 1.47 \[ -\frac{60 \, a^{4} x^{4} + 90 \, a^{3} b x^{3} + 20 \, a^{2} b^{2} x^{2} - 5 \, a b^{3} x + 2 \, b^{4}}{6 \,{\left (a^{2} b^{5} x^{5} + 2 \, a b^{6} x^{4} + b^{7} x^{3}\right )}} + \frac{10 \, a^{3} \log \left (a x + b\right )}{b^{6}} - \frac{10 \, a^{3} \log \left (x\right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^7),x, algorithm="maxima")

[Out]

-1/6*(60*a^4*x^4 + 90*a^3*b*x^3 + 20*a^2*b^2*x^2 - 5*a*b^3*x + 2*b^4)/(a^2*b^5*x
^5 + 2*a*b^6*x^4 + b^7*x^3) + 10*a^3*log(a*x + b)/b^6 - 10*a^3*log(x)/b^6

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Fricas [A]  time = 0.231806, size = 190, normalized size = 2.13 \[ -\frac{60 \, a^{4} b x^{4} + 90 \, a^{3} b^{2} x^{3} + 20 \, a^{2} b^{3} x^{2} - 5 \, a b^{4} x + 2 \, b^{5} - 60 \,{\left (a^{5} x^{5} + 2 \, a^{4} b x^{4} + a^{3} b^{2} x^{3}\right )} \log \left (a x + b\right ) + 60 \,{\left (a^{5} x^{5} + 2 \, a^{4} b x^{4} + a^{3} b^{2} x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{2} b^{6} x^{5} + 2 \, a b^{7} x^{4} + b^{8} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^7),x, algorithm="fricas")

[Out]

-1/6*(60*a^4*b*x^4 + 90*a^3*b^2*x^3 + 20*a^2*b^3*x^2 - 5*a*b^4*x + 2*b^5 - 60*(a
^5*x^5 + 2*a^4*b*x^4 + a^3*b^2*x^3)*log(a*x + b) + 60*(a^5*x^5 + 2*a^4*b*x^4 + a
^3*b^2*x^3)*log(x))/(a^2*b^6*x^5 + 2*a*b^7*x^4 + b^8*x^3)

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Sympy [A]  time = 2.29455, size = 92, normalized size = 1.03 \[ \frac{10 a^{3} \left (- \log{\left (x \right )} + \log{\left (x + \frac{b}{a} \right )}\right )}{b^{6}} - \frac{60 a^{4} x^{4} + 90 a^{3} b x^{3} + 20 a^{2} b^{2} x^{2} - 5 a b^{3} x + 2 b^{4}}{6 a^{2} b^{5} x^{5} + 12 a b^{6} x^{4} + 6 b^{7} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**3/x**7,x)

[Out]

10*a**3*(-log(x) + log(x + b/a))/b**6 - (60*a**4*x**4 + 90*a**3*b*x**3 + 20*a**2
*b**2*x**2 - 5*a*b**3*x + 2*b**4)/(6*a**2*b**5*x**5 + 12*a*b**6*x**4 + 6*b**7*x*
*3)

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GIAC/XCAS [A]  time = 0.228485, size = 116, normalized size = 1.3 \[ \frac{10 \, a^{3}{\rm ln}\left ({\left | a x + b \right |}\right )}{b^{6}} - \frac{10 \, a^{3}{\rm ln}\left ({\left | x \right |}\right )}{b^{6}} - \frac{60 \, a^{4} b x^{4} + 90 \, a^{3} b^{2} x^{3} + 20 \, a^{2} b^{3} x^{2} - 5 \, a b^{4} x + 2 \, b^{5}}{6 \,{\left (a x + b\right )}^{2} b^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^7),x, algorithm="giac")

[Out]

10*a^3*ln(abs(a*x + b))/b^6 - 10*a^3*ln(abs(x))/b^6 - 1/6*(60*a^4*b*x^4 + 90*a^3
*b^2*x^3 + 20*a^2*b^3*x^2 - 5*a*b^4*x + 2*b^5)/((a*x + b)^2*b^6*x^3)