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

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

[Out]

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

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Rubi [A]  time = 0.0584225, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{b^4 \log (x)}{a^5}-\frac{b^4 \log (a+b x)}{a^5}+\frac{b^3}{a^4 x}-\frac{b^2}{2 a^3 x^2}+\frac{b}{3 a^2 x^3}-\frac{1}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a + b*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.0919, size = 61, normalized size = 0.9 \[ - \frac{1}{4 a x^{4}} + \frac{b}{3 a^{2} x^{3}} - \frac{b^{2}}{2 a^{3} x^{2}} + \frac{b^{3}}{a^{4} x} + \frac{b^{4} \log{\left (x \right )}}{a^{5}} - \frac{b^{4} \log{\left (a + b x \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00676188, size = 68, normalized size = 1. \[ \frac{b^4 \log (x)}{a^5}-\frac{b^4 \log (a+b x)}{a^5}+\frac{b^3}{a^4 x}-\frac{b^2}{2 a^3 x^2}+\frac{b}{3 a^2 x^3}-\frac{1}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a + b*x)),x]

[Out]

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

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Maple [A]  time = 0.013, size = 63, normalized size = 0.9 \[ -{\frac{1}{4\,a{x}^{4}}}+{\frac{b}{3\,{a}^{2}{x}^{3}}}-{\frac{{b}^{2}}{2\,{a}^{3}{x}^{2}}}+{\frac{{b}^{3}}{{a}^{4}x}}+{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{5}}}-{\frac{{b}^{4}\ln \left ( bx+a \right ) }{{a}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.34621, size = 84, normalized size = 1.24 \[ -\frac{b^{4} \log \left (b x + a\right )}{a^{5}} + \frac{b^{4} \log \left (x\right )}{a^{5}} + \frac{12 \, b^{3} x^{3} - 6 \, a b^{2} x^{2} + 4 \, a^{2} b x - 3 \, a^{3}}{12 \, a^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.200885, size = 88, normalized size = 1.29 \[ -\frac{12 \, b^{4} x^{4} \log \left (b x + a\right ) - 12 \, b^{4} x^{4} \log \left (x\right ) - 12 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 4 \, a^{3} b x + 3 \, a^{4}}{12 \, a^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.59872, size = 56, normalized size = 0.82 \[ \frac{- 3 a^{3} + 4 a^{2} b x - 6 a b^{2} x^{2} + 12 b^{3} x^{3}}{12 a^{4} x^{4}} + \frac{b^{4} \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(b*x+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.218773, size = 90, normalized size = 1.32 \[ -\frac{b^{4}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{5}} + \frac{b^{4}{\rm ln}\left ({\left | x \right |}\right )}{a^{5}} + \frac{12 \, a b^{3} x^{3} - 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x - 3 \, a^{4}}{12 \, a^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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