3.165 \(\int \frac{1}{x^3 (a+b x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0402526, antiderivative size = 42, 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 \[ \frac{b^2 \log (x)}{a^3}-\frac{b^2 \log (a+b x)}{a^3}+\frac{b}{a^2 x}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.59428, size = 37, normalized size = 0.88 \[ - \frac{1}{2 a x^{2}} + \frac{b}{a^{2} x} + \frac{b^{2} \log{\left (x \right )}}{a^{3}} - \frac{b^{2} \log{\left (a + b x \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00651453, size = 42, normalized size = 1. \[ \frac{b^2 \log (x)}{a^3}-\frac{b^2 \log (a+b x)}{a^3}+\frac{b}{a^2 x}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x)),x]

[Out]

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

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Maple [A]  time = 0.011, size = 41, normalized size = 1. \[ -{\frac{1}{2\,a{x}^{2}}}+{\frac{b}{{a}^{2}x}}+{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{3}}}-{\frac{{b}^{2}\ln \left ( bx+a \right ) }{{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.34059, size = 54, normalized size = 1.29 \[ -\frac{b^{2} \log \left (b x + a\right )}{a^{3}} + \frac{b^{2} \log \left (x\right )}{a^{3}} + \frac{2 \, b x - a}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.199177, size = 55, normalized size = 1.31 \[ -\frac{2 \, b^{2} x^{2} \log \left (b x + a\right ) - 2 \, b^{2} x^{2} \log \left (x\right ) - 2 \, a b x + a^{2}}{2 \, a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.40797, size = 31, normalized size = 0.74 \[ \frac{- a + 2 b x}{2 a^{2} x^{2}} + \frac{b^{2} \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b*x+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.207017, size = 61, normalized size = 1.45 \[ -\frac{b^{2}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{3}} + \frac{b^{2}{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} + \frac{2 \, a b x - a^{2}}{2 \, a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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