3.283 \(\int \left (\frac{b}{x}+\frac{1}{x^2 (1+b x)}\right ) \, dx\)

Optimal. Leaf size=14 \[ b \log (b x+1)-\frac{1}{x} \]

[Out]

-x^(-1) + b*Log[1 + b*x]

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Rubi [A]  time = 0.0236048, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ b \log (b x+1)-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]  Int[b/x + 1/(x^2*(1 + b*x)),x]

[Out]

-x^(-1) + b*Log[1 + b*x]

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Rubi in Sympy [A]  time = 3.75086, size = 10, normalized size = 0.71 \[ b \log{\left (b x + 1 \right )} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(b/x+1/x**2/(b*x+1),x)

[Out]

b*log(b*x + 1) - 1/x

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Mathematica [A]  time = 0.00696411, size = 14, normalized size = 1. \[ b \log (b x+1)-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[b/x + 1/(x^2*(1 + b*x)),x]

[Out]

-x^(-1) + b*Log[1 + b*x]

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Maple [A]  time = 0.003, size = 15, normalized size = 1.1 \[ -{x}^{-1}+b\ln \left ( bx+1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(b/x+1/x^2/(b*x+1),x)

[Out]

-1/x+b*ln(b*x+1)

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Maxima [A]  time = 1.34128, size = 19, normalized size = 1.36 \[ b \log \left (b x + 1\right ) - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(b/x + 1/((b*x + 1)*x^2),x, algorithm="maxima")

[Out]

b*log(b*x + 1) - 1/x

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Fricas [A]  time = 0.208525, size = 20, normalized size = 1.43 \[ \frac{b x \log \left (b x + 1\right ) - 1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(b/x + 1/((b*x + 1)*x^2),x, algorithm="fricas")

[Out]

(b*x*log(b*x + 1) - 1)/x

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Sympy [A]  time = 1.11623, size = 10, normalized size = 0.71 \[ b \log{\left (b x + 1 \right )} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(b/x+1/x**2/(b*x+1),x)

[Out]

b*log(b*x + 1) - 1/x

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GIAC/XCAS [A]  time = 0.208019, size = 20, normalized size = 1.43 \[ b{\rm ln}\left ({\left | b x + 1 \right |}\right ) - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(b/x + 1/((b*x + 1)*x^2),x, algorithm="giac")

[Out]

b*ln(abs(b*x + 1)) - 1/x