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

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

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Rubi [A]  time = 0.012942, 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, Rules used = {44} \[ -\frac{b \log (x)}{a^2}+\frac{b \log (a+b x)}{a^2}-\frac{1}{a x} \]

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^2 (a+b x)} \, dx &=\int \left (\frac{1}{a x^2}-\frac{b}{a^2 x}+\frac{b^2}{a^2 (a+b x)}\right ) \, dx\\ &=-\frac{1}{a x}-\frac{b \log (x)}{a^2}+\frac{b \log (a+b x)}{a^2}\\ \end{align*}

Mathematica [A]  time = 0.0041583, 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

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Maple [A]  time = 0.009, size = 29, normalized size = 1. \begin{align*} -{\frac{1}{ax}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}}}+{\frac{b\ln \left ( bx+a \right ) }{{a}^{2}}} \end{align*}

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

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Maxima [A]  time = 0.958913, size = 38, normalized size = 1.36 \begin{align*} \frac{b \log \left (b x + a\right )}{a^{2}} - \frac{b \log \left (x\right )}{a^{2}} - \frac{1}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.99516, size = 61, normalized size = 2.18 \begin{align*} \frac{b x \log \left (b x + a\right ) - b x \log \left (x\right ) - a}{a^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.325009, size = 19, normalized size = 0.68 \begin{align*} - \frac{1}{a x} + \frac{b \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{2}} \end{align*}

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

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Giac [A]  time = 1.07897, size = 41, normalized size = 1.46 \begin{align*} \frac{b \log \left ({\left | b x + a \right |}\right )}{a^{2}} - \frac{b \log \left ({\left | x \right |}\right )}{a^{2}} - \frac{1}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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