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

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

[Out]

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

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Rubi [A]  time = 0.0230598, 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, Rules used = {44} \[ -\frac{b}{a^2 (a+b x)}-\frac{2 b \log (x)}{a^3}+\frac{2 b \log (a+b x)}{a^3}-\frac{1}{a^2 x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x)^2),x]

[Out]

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

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)^2} \, dx &=\int \left (\frac{1}{a^2 x^2}-\frac{2 b}{a^3 x}+\frac{b^2}{a^2 (a+b x)^2}+\frac{2 b^2}{a^3 (a+b x)}\right ) \, dx\\ &=-\frac{1}{a^2 x}-\frac{b}{a^2 (a+b x)}-\frac{2 b \log (x)}{a^3}+\frac{2 b \log (a+b x)}{a^3}\\ \end{align*}

Mathematica [A]  time = 0.0382501, size = 35, normalized size = 0.83 \[ -\frac{a \left (\frac{b}{a+b x}+\frac{1}{x}\right )-2 b \log (a+b x)+2 b \log (x)}{a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x)^2),x]

[Out]

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

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Maple [A]  time = 0.01, size = 43, normalized size = 1. \begin{align*} -{\frac{1}{{a}^{2}x}}-{\frac{b}{{a}^{2} \left ( bx+a \right ) }}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}}}+2\,{\frac{b\ln \left ( bx+a \right ) }{{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 0.931402, size = 61, normalized size = 1.45 \begin{align*} -\frac{2 \, b x + a}{a^{2} b x^{2} + a^{3} x} + \frac{2 \, b \log \left (b x + a\right )}{a^{3}} - \frac{2 \, b \log \left (x\right )}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.84507, size = 138, normalized size = 3.29 \begin{align*} -\frac{2 \, a b x + a^{2} - 2 \,{\left (b^{2} x^{2} + a b x\right )} \log \left (b x + a\right ) + 2 \,{\left (b^{2} x^{2} + a b x\right )} \log \left (x\right )}{a^{3} b x^{2} + a^{4} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a*b*x + a^2 - 2*(b^2*x^2 + a*b*x)*log(b*x + a) + 2*(b^2*x^2 + a*b*x)*log(x))/(a^3*b*x^2 + a^4*x)

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Sympy [A]  time = 0.404427, size = 36, normalized size = 0.86 \begin{align*} - \frac{a + 2 b x}{a^{3} x + a^{2} b x^{2}} + \frac{2 b \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b*x+a)**2,x)

[Out]

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

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Giac [A]  time = 1.06443, size = 70, normalized size = 1.67 \begin{align*} -\frac{2 \, b \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{3}} - \frac{b}{{\left (b x + a\right )} a^{2}} + \frac{b}{a^{3}{\left (\frac{a}{b x + a} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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