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

Optimal. Leaf size=28 \[ -\frac{b \log (x)}{a^2}+\frac{b \log (a+b x)}{a^2}-\frac{1}{a x} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0315487, antiderivative size = 28, 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 \[ -\frac{b \log (x)}{a^2}+\frac{b \log (a+b x)}{a^2}-\frac{1}{a x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.72571, size = 24, normalized size = 0.86 \[ - \frac{1}{a x} - \frac{b \log{\left (x \right )}}{a^{2}} + \frac{b \log{\left (a + b x \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00674428, size = 28, normalized size = 1. \[ -\frac{b \log (x)}{a^2}+\frac{b \log (a+b x)}{a^2}-\frac{1}{a x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a + b*x)),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 29, normalized size = 1. \[ -{\frac{1}{ax}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}}}+{\frac{b\ln \left ( bx+a \right ) }{{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x+a),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34077, size = 38, normalized size = 1.36 \[ \frac{b \log \left (b x + a\right )}{a^{2}} - \frac{b \log \left (x\right )}{a^{2}} - \frac{1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.199765, size = 35, normalized size = 1.25 \[ \frac{b x \log \left (b x + a\right ) - b x \log \left (x\right ) - a}{a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.27486, size = 19, normalized size = 0.68 \[ - \frac{1}{a x} + \frac{b \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x+a),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216976, size = 41, normalized size = 1.46 \[ \frac{b{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{2}} - \frac{b{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} - \frac{1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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