3.629 \(\int \frac{A+B x}{x^2 (a^2+2 a b x+b^2 x^2)} \, dx\)

Optimal. Leaf size=65 \[ -\frac{A b-a B}{a^2 (a+b x)}-\frac{\log (x) (2 A b-a B)}{a^3}+\frac{(2 A b-a B) \log (a+b x)}{a^3}-\frac{A}{a^2 x} \]

[Out]

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

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Rubi [A]  time = 0.0505052, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 77} \[ -\frac{A b-a B}{a^2 (a+b x)}-\frac{\log (x) (2 A b-a B)}{a^3}+\frac{(2 A b-a B) \log (a+b x)}{a^3}-\frac{A}{a^2 x} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^2*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0388051, size = 56, normalized size = 0.86 \[ \frac{\frac{a (a B-A b)}{a+b x}+\log (x) (a B-2 A b)+(2 A b-a B) \log (a+b x)-\frac{a A}{x}}{a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^2*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

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

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Maple [A]  time = 0.012, size = 78, normalized size = 1.2 \begin{align*} -{\frac{A}{{a}^{2}x}}-2\,{\frac{Ab\ln \left ( x \right ) }{{a}^{3}}}+{\frac{\ln \left ( x \right ) B}{{a}^{2}}}+2\,{\frac{\ln \left ( bx+a \right ) Ab}{{a}^{3}}}-{\frac{\ln \left ( bx+a \right ) B}{{a}^{2}}}-{\frac{Ab}{{a}^{2} \left ( bx+a \right ) }}+{\frac{B}{a \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06314, size = 90, normalized size = 1.38 \begin{align*} -\frac{A a -{\left (B a - 2 \, A b\right )} x}{a^{2} b x^{2} + a^{3} x} - \frac{{\left (B a - 2 \, A b\right )} \log \left (b x + a\right )}{a^{3}} + \frac{{\left (B a - 2 \, A b\right )} \log \left (x\right )}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.34274, size = 227, normalized size = 3.49 \begin{align*} -\frac{A a^{2} -{\left (B a^{2} - 2 \, A a b\right )} x +{\left ({\left (B a b - 2 \, A b^{2}\right )} x^{2} +{\left (B a^{2} - 2 \, A a b\right )} x\right )} \log \left (b x + a\right ) -{\left ({\left (B a b - 2 \, A b^{2}\right )} x^{2} +{\left (B a^{2} - 2 \, A a b\right )} 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((B*x+A)/x^2/(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.655717, size = 128, normalized size = 1.97 \begin{align*} \frac{- A a + x \left (- 2 A b + B a\right )}{a^{3} x + a^{2} b x^{2}} + \frac{\left (- 2 A b + B a\right ) \log{\left (x + \frac{- 2 A a b + B a^{2} - a \left (- 2 A b + B a\right )}{- 4 A b^{2} + 2 B a b} \right )}}{a^{3}} - \frac{\left (- 2 A b + B a\right ) \log{\left (x + \frac{- 2 A a b + B a^{2} + a \left (- 2 A b + B a\right )}{- 4 A b^{2} + 2 B a b} \right )}}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**2/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

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

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Giac [A]  time = 1.09354, size = 96, normalized size = 1.48 \begin{align*} \frac{{\left (B a - 2 \, A b\right )} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac{B a x - 2 \, A b x - A a}{{\left (b x^{2} + a x\right )} a^{2}} - \frac{{\left (B a b - 2 \, A b^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{3} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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