3.281 \(\int \frac{1}{x^2 (1+b x)} \, dx\)

Optimal. Leaf size=19 \[ -b \log (x)+b \log (b x+1)-\frac{1}{x} \]

[Out]

-x^(-1) - b*Log[x] + b*Log[1 + b*x]

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Rubi [A]  time = 0.0221505, antiderivative size = 19, 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 \[ -b \log (x)+b \log (b x+1)-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(1 + b*x)),x]

[Out]

-x^(-1) - b*Log[x] + b*Log[1 + b*x]

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Rubi in Sympy [A]  time = 3.66987, size = 15, normalized size = 0.79 \[ - b \log{\left (x \right )} + b \log{\left (b x + 1 \right )} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*log(x) + b*log(b*x + 1) - 1/x

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Mathematica [A]  time = 0.00548739, size = 19, normalized size = 1. \[ -b \log (x)+b \log (b x+1)-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(1 + b*x)),x]

[Out]

-x^(-1) - b*Log[x] + b*Log[1 + b*x]

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Maple [A]  time = 0.01, size = 20, normalized size = 1.1 \[ -{x}^{-1}-b\ln \left ( x \right ) +b\ln \left ( bx+1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x+1),x)

[Out]

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

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Maxima [A]  time = 1.33624, size = 26, normalized size = 1.37 \[ b \log \left (b x + 1\right ) - b \log \left (x\right ) - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*log(b*x + 1) - b*log(x) - 1/x

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Fricas [A]  time = 0.212271, size = 28, normalized size = 1.47 \[ \frac{b x \log \left (b x + 1\right ) - b x \log \left (x\right ) - 1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x*log(b*x + 1) - b*x*log(x) - 1)/x

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Sympy [A]  time = 1.12722, size = 14, normalized size = 0.74 \[ b \left (- \log{\left (x \right )} + \log{\left (x + \frac{1}{b} \right )}\right ) - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x+1),x)

[Out]

b*(-log(x) + log(x + 1/b)) - 1/x

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GIAC/XCAS [A]  time = 0.209177, size = 28, normalized size = 1.47 \[ b{\rm ln}\left ({\left | b x + 1 \right |}\right ) - b{\rm ln}\left ({\left | x \right |}\right ) - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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