3.630 \(\int \frac{A+B x}{x^3 (a^2+2 a b x+b^2 x^2)} \, dx\)

Optimal. Leaf size=85 \[ \frac{2 A b-a B}{a^3 x}+\frac{b (A b-a B)}{a^3 (a+b x)}+\frac{b \log (x) (3 A b-2 a B)}{a^4}-\frac{b (3 A b-2 a B) \log (a+b x)}{a^4}-\frac{A}{2 a^2 x^2} \]

[Out]

-A/(2*a^2*x^2) + (2*A*b - a*B)/(a^3*x) + (b*(A*b - a*B))/(a^3*(a + b*x)) + (b*(3*A*b - 2*a*B)*Log[x])/a^4 - (b
*(3*A*b - 2*a*B)*Log[a + b*x])/a^4

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Rubi [A]  time = 0.0714481, antiderivative size = 85, 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} \[ \frac{2 A b-a B}{a^3 x}+\frac{b (A b-a B)}{a^3 (a+b x)}+\frac{b \log (x) (3 A b-2 a B)}{a^4}-\frac{b (3 A b-2 a B) \log (a+b x)}{a^4}-\frac{A}{2 a^2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^3*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

-A/(2*a^2*x^2) + (2*A*b - a*B)/(a^3*x) + (b*(A*b - a*B))/(a^3*(a + b*x)) + (b*(3*A*b - 2*a*B)*Log[x])/a^4 - (b
*(3*A*b - 2*a*B)*Log[a + b*x])/a^4

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^3 \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac{A+B x}{x^3 (a+b x)^2} \, dx\\ &=\int \left (\frac{A}{a^2 x^3}+\frac{-2 A b+a B}{a^3 x^2}-\frac{b (-3 A b+2 a B)}{a^4 x}+\frac{b^2 (-A b+a B)}{a^3 (a+b x)^2}+\frac{b^2 (-3 A b+2 a B)}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac{A}{2 a^2 x^2}+\frac{2 A b-a B}{a^3 x}+\frac{b (A b-a B)}{a^3 (a+b x)}+\frac{b (3 A b-2 a B) \log (x)}{a^4}-\frac{b (3 A b-2 a B) \log (a+b x)}{a^4}\\ \end{align*}

Mathematica [A]  time = 0.069937, size = 85, normalized size = 1. \[ \frac{-\frac{a \left (a^2 (A+2 B x)+a b x (4 B x-3 A)-6 A b^2 x^2\right )}{x^2 (a+b x)}+2 b \log (x) (3 A b-2 a B)+2 b (2 a B-3 A b) \log (a+b x)}{2 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^3*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

(-((a*(-6*A*b^2*x^2 + a^2*(A + 2*B*x) + a*b*x*(-3*A + 4*B*x)))/(x^2*(a + b*x))) + 2*b*(3*A*b - 2*a*B)*Log[x] +
2*b*(-3*A*b + 2*a*B)*Log[a + b*x])/(2*a^4)

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Maple [A]  time = 0.011, size = 107, normalized size = 1.3 \begin{align*} -{\frac{A}{2\,{a}^{2}{x}^{2}}}+2\,{\frac{Ab}{{a}^{3}x}}-{\frac{B}{{a}^{2}x}}+3\,{\frac{A{b}^{2}\ln \left ( x \right ) }{{a}^{4}}}-2\,{\frac{b\ln \left ( x \right ) B}{{a}^{3}}}-3\,{\frac{{b}^{2}\ln \left ( bx+a \right ) A}{{a}^{4}}}+2\,{\frac{b\ln \left ( bx+a \right ) B}{{a}^{3}}}+{\frac{A{b}^{2}}{{a}^{3} \left ( bx+a \right ) }}-{\frac{bB}{{a}^{2} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^3/(b^2*x^2+2*a*b*x+a^2),x)

[Out]

-1/2*A/a^2/x^2+2/a^3/x*A*b-B/a^2/x+3*b^2/a^4*ln(x)*A-2*b/a^3*ln(x)*B-3*b^2/a^4*ln(b*x+a)*A+2*b/a^3*ln(b*x+a)*B
+b^2/a^3/(b*x+a)*A-b/a^2/(b*x+a)*B

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Maxima [A]  time = 1.0121, size = 134, normalized size = 1.58 \begin{align*} -\frac{A a^{2} + 2 \,{\left (2 \, B a b - 3 \, A b^{2}\right )} x^{2} +{\left (2 \, B a^{2} - 3 \, A a b\right )} x}{2 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} + \frac{{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (b x + a\right )}{a^{4}} - \frac{{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (x\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^3/(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

-1/2*(A*a^2 + 2*(2*B*a*b - 3*A*b^2)*x^2 + (2*B*a^2 - 3*A*a*b)*x)/(a^3*b*x^3 + a^4*x^2) + (2*B*a*b - 3*A*b^2)*l
og(b*x + a)/a^4 - (2*B*a*b - 3*A*b^2)*log(x)/a^4

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Fricas [A]  time = 1.28994, size = 321, normalized size = 3.78 \begin{align*} -\frac{A a^{3} + 2 \,{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2} +{\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} x - 2 \,{\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{3} +{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) + 2 \,{\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{3} +{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^3/(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

-1/2*(A*a^3 + 2*(2*B*a^2*b - 3*A*a*b^2)*x^2 + (2*B*a^3 - 3*A*a^2*b)*x - 2*((2*B*a*b^2 - 3*A*b^3)*x^3 + (2*B*a^
2*b - 3*A*a*b^2)*x^2)*log(b*x + a) + 2*((2*B*a*b^2 - 3*A*b^3)*x^3 + (2*B*a^2*b - 3*A*a*b^2)*x^2)*log(x))/(a^4*
b*x^3 + a^5*x^2)

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Sympy [B]  time = 0.779915, size = 184, normalized size = 2.16 \begin{align*} - \frac{A a^{2} + x^{2} \left (- 6 A b^{2} + 4 B a b\right ) + x \left (- 3 A a b + 2 B a^{2}\right )}{2 a^{4} x^{2} + 2 a^{3} b x^{3}} - \frac{b \left (- 3 A b + 2 B a\right ) \log{\left (x + \frac{- 3 A a b^{2} + 2 B a^{2} b - a b \left (- 3 A b + 2 B a\right )}{- 6 A b^{3} + 4 B a b^{2}} \right )}}{a^{4}} + \frac{b \left (- 3 A b + 2 B a\right ) \log{\left (x + \frac{- 3 A a b^{2} + 2 B a^{2} b + a b \left (- 3 A b + 2 B a\right )}{- 6 A b^{3} + 4 B a b^{2}} \right )}}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**3/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

-(A*a**2 + x**2*(-6*A*b**2 + 4*B*a*b) + x*(-3*A*a*b + 2*B*a**2))/(2*a**4*x**2 + 2*a**3*b*x**3) - b*(-3*A*b + 2
*B*a)*log(x + (-3*A*a*b**2 + 2*B*a**2*b - a*b*(-3*A*b + 2*B*a))/(-6*A*b**3 + 4*B*a*b**2))/a**4 + b*(-3*A*b + 2
*B*a)*log(x + (-3*A*a*b**2 + 2*B*a**2*b + a*b*(-3*A*b + 2*B*a))/(-6*A*b**3 + 4*B*a*b**2))/a**4

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Giac [A]  time = 1.1087, size = 143, normalized size = 1.68 \begin{align*} -\frac{{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac{{\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac{A a^{3} + 2 \,{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2} +{\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} x}{2 \,{\left (b x + a\right )} a^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^3/(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

-(2*B*a*b - 3*A*b^2)*log(abs(x))/a^4 + (2*B*a*b^2 - 3*A*b^3)*log(abs(b*x + a))/(a^4*b) - 1/2*(A*a^3 + 2*(2*B*a
^2*b - 3*A*a*b^2)*x^2 + (2*B*a^3 - 3*A*a^2*b)*x)/((b*x + a)*a^4*x^2)