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

Optimal. Leaf size=135 \[ \frac{4 A b-a B}{a^5 x}+\frac{3 b (2 A b-a B)}{a^5 (a+b x)}+\frac{b (3 A b-2 a B)}{2 a^4 (a+b x)^2}+\frac{b (A b-a B)}{3 a^3 (a+b x)^3}+\frac{2 b \log (x) (5 A b-2 a B)}{a^6}-\frac{2 b (5 A b-2 a B) \log (a+b x)}{a^6}-\frac{A}{2 a^4 x^2} \]

[Out]

-A/(2*a^4*x^2) + (4*A*b - a*B)/(a^5*x) + (b*(A*b - a*B))/(3*a^3*(a + b*x)^3) + (b*(3*A*b - 2*a*B))/(2*a^4*(a +
b*x)^2) + (3*b*(2*A*b - a*B))/(a^5*(a + b*x)) + (2*b*(5*A*b - 2*a*B)*Log[x])/a^6 - (2*b*(5*A*b - 2*a*B)*Log[a
+ b*x])/a^6

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Rubi [A]  time = 0.121375, antiderivative size = 135, 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} \[ \frac{4 A b-a B}{a^5 x}+\frac{3 b (2 A b-a B)}{a^5 (a+b x)}+\frac{b (3 A b-2 a B)}{2 a^4 (a+b x)^2}+\frac{b (A b-a B)}{3 a^3 (a+b x)^3}+\frac{2 b \log (x) (5 A b-2 a B)}{a^6}-\frac{2 b (5 A b-2 a B) \log (a+b x)}{a^6}-\frac{A}{2 a^4 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-A/(2*a^4*x^2) + (4*A*b - a*B)/(a^5*x) + (b*(A*b - a*B))/(3*a^3*(a + b*x)^3) + (b*(3*A*b - 2*a*B))/(2*a^4*(a +
b*x)^2) + (3*b*(2*A*b - a*B))/(a^5*(a + b*x)) + (2*b*(5*A*b - 2*a*B)*Log[x])/a^6 - (2*b*(5*A*b - 2*a*B)*Log[a
+ b*x])/a^6

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 )^2} \, dx &=\int \frac{A+B x}{x^3 (a+b x)^4} \, dx\\ &=\int \left (\frac{A}{a^4 x^3}+\frac{-4 A b+a B}{a^5 x^2}-\frac{2 b (-5 A b+2 a B)}{a^6 x}+\frac{b^2 (-A b+a B)}{a^3 (a+b x)^4}+\frac{b^2 (-3 A b+2 a B)}{a^4 (a+b x)^3}+\frac{3 b^2 (-2 A b+a B)}{a^5 (a+b x)^2}+\frac{2 b^2 (-5 A b+2 a B)}{a^6 (a+b x)}\right ) \, dx\\ &=-\frac{A}{2 a^4 x^2}+\frac{4 A b-a B}{a^5 x}+\frac{b (A b-a B)}{3 a^3 (a+b x)^3}+\frac{b (3 A b-2 a B)}{2 a^4 (a+b x)^2}+\frac{3 b (2 A b-a B)}{a^5 (a+b x)}+\frac{2 b (5 A b-2 a B) \log (x)}{a^6}-\frac{2 b (5 A b-2 a B) \log (a+b x)}{a^6}\\ \end{align*}

Mathematica [A]  time = 0.121143, size = 123, normalized size = 0.91 \[ \frac{\frac{a \left (10 a^2 b^2 x^2 (11 A-6 B x)+a^3 b x (15 A-44 B x)-3 a^4 (A+2 B x)+6 a b^3 x^3 (25 A-4 B x)+60 A b^4 x^4\right )}{x^2 (a+b x)^3}+12 b \log (x) (5 A b-2 a B)+12 b (2 a B-5 A b) \log (a+b x)}{6 a^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(60*A*b^4*x^4 + a^3*b*x*(15*A - 44*B*x) + 10*a^2*b^2*x^2*(11*A - 6*B*x) + 6*a*b^3*x^3*(25*A - 4*B*x) - 3*a
^4*(A + 2*B*x)))/(x^2*(a + b*x)^3) + 12*b*(5*A*b - 2*a*B)*Log[x] + 12*b*(-5*A*b + 2*a*B)*Log[a + b*x])/(6*a^6)

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Maple [A]  time = 0.013, size = 168, normalized size = 1.2 \begin{align*} -{\frac{A}{2\,{a}^{4}{x}^{2}}}+4\,{\frac{Ab}{{a}^{5}x}}-{\frac{B}{{a}^{4}x}}+10\,{\frac{A{b}^{2}\ln \left ( x \right ) }{{a}^{6}}}-4\,{\frac{b\ln \left ( x \right ) B}{{a}^{5}}}+6\,{\frac{A{b}^{2}}{{a}^{5} \left ( bx+a \right ) }}-3\,{\frac{bB}{{a}^{4} \left ( bx+a \right ) }}+{\frac{3\,A{b}^{2}}{2\,{a}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{bB}{{a}^{3} \left ( bx+a \right ) ^{2}}}-10\,{\frac{{b}^{2}\ln \left ( bx+a \right ) A}{{a}^{6}}}+4\,{\frac{b\ln \left ( bx+a \right ) B}{{a}^{5}}}+{\frac{A{b}^{2}}{3\,{a}^{3} \left ( bx+a \right ) ^{3}}}-{\frac{bB}{3\,{a}^{2} \left ( bx+a \right ) ^{3}}} \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)^2,x)

[Out]

-1/2*A/a^4/x^2+4/a^5/x*A*b-1/a^4/x*B+10*b^2/a^6*ln(x)*A-4*b/a^5*ln(x)*B+6*b^2/a^5/(b*x+a)*A-3*b/a^4/(b*x+a)*B+
3/2*b^2/a^4/(b*x+a)^2*A-b/a^3/(b*x+a)^2*B-10*b^2/a^6*ln(b*x+a)*A+4*b/a^5*ln(b*x+a)*B+1/3*b^2/a^3/(b*x+a)^3*A-1
/3*b/a^2/(b*x+a)^3*B

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Maxima [A]  time = 0.993066, size = 232, normalized size = 1.72 \begin{align*} -\frac{3 \, A a^{4} + 12 \,{\left (2 \, B a b^{3} - 5 \, A b^{4}\right )} x^{4} + 30 \,{\left (2 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{3} + 22 \,{\left (2 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{2} + 3 \,{\left (2 \, B a^{4} - 5 \, A a^{3} b\right )} x}{6 \,{\left (a^{5} b^{3} x^{5} + 3 \, a^{6} b^{2} x^{4} + 3 \, a^{7} b x^{3} + a^{8} x^{2}\right )}} + \frac{2 \,{\left (2 \, B a b - 5 \, A b^{2}\right )} \log \left (b x + a\right )}{a^{6}} - \frac{2 \,{\left (2 \, B a b - 5 \, A b^{2}\right )} \log \left (x\right )}{a^{6}} \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)^2,x, algorithm="maxima")

[Out]

-1/6*(3*A*a^4 + 12*(2*B*a*b^3 - 5*A*b^4)*x^4 + 30*(2*B*a^2*b^2 - 5*A*a*b^3)*x^3 + 22*(2*B*a^3*b - 5*A*a^2*b^2)
*x^2 + 3*(2*B*a^4 - 5*A*a^3*b)*x)/(a^5*b^3*x^5 + 3*a^6*b^2*x^4 + 3*a^7*b*x^3 + a^8*x^2) + 2*(2*B*a*b - 5*A*b^2
)*log(b*x + a)/a^6 - 2*(2*B*a*b - 5*A*b^2)*log(x)/a^6

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Fricas [B]  time = 1.28574, size = 668, normalized size = 4.95 \begin{align*} -\frac{3 \, A a^{5} + 12 \,{\left (2 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + 30 \,{\left (2 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} + 22 \,{\left (2 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} + 3 \,{\left (2 \, B a^{5} - 5 \, A a^{4} b\right )} x - 12 \,{\left ({\left (2 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 3 \,{\left (2 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + 3 \,{\left (2 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} +{\left (2 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) + 12 \,{\left ({\left (2 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 3 \,{\left (2 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + 3 \,{\left (2 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} +{\left (2 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2}\right )} \log \left (x\right )}{6 \,{\left (a^{6} b^{3} x^{5} + 3 \, a^{7} b^{2} x^{4} + 3 \, a^{8} b x^{3} + a^{9} 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)^2,x, algorithm="fricas")

[Out]

-1/6*(3*A*a^5 + 12*(2*B*a^2*b^3 - 5*A*a*b^4)*x^4 + 30*(2*B*a^3*b^2 - 5*A*a^2*b^3)*x^3 + 22*(2*B*a^4*b - 5*A*a^
3*b^2)*x^2 + 3*(2*B*a^5 - 5*A*a^4*b)*x - 12*((2*B*a*b^4 - 5*A*b^5)*x^5 + 3*(2*B*a^2*b^3 - 5*A*a*b^4)*x^4 + 3*(
2*B*a^3*b^2 - 5*A*a^2*b^3)*x^3 + (2*B*a^4*b - 5*A*a^3*b^2)*x^2)*log(b*x + a) + 12*((2*B*a*b^4 - 5*A*b^5)*x^5 +
3*(2*B*a^2*b^3 - 5*A*a*b^4)*x^4 + 3*(2*B*a^3*b^2 - 5*A*a^2*b^3)*x^3 + (2*B*a^4*b - 5*A*a^3*b^2)*x^2)*log(x))/
(a^6*b^3*x^5 + 3*a^7*b^2*x^4 + 3*a^8*b*x^3 + a^9*x^2)

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Sympy [B]  time = 1.30901, size = 264, normalized size = 1.96 \begin{align*} - \frac{3 A a^{4} + x^{4} \left (- 60 A b^{4} + 24 B a b^{3}\right ) + x^{3} \left (- 150 A a b^{3} + 60 B a^{2} b^{2}\right ) + x^{2} \left (- 110 A a^{2} b^{2} + 44 B a^{3} b\right ) + x \left (- 15 A a^{3} b + 6 B a^{4}\right )}{6 a^{8} x^{2} + 18 a^{7} b x^{3} + 18 a^{6} b^{2} x^{4} + 6 a^{5} b^{3} x^{5}} - \frac{2 b \left (- 5 A b + 2 B a\right ) \log{\left (x + \frac{- 10 A a b^{2} + 4 B a^{2} b - 2 a b \left (- 5 A b + 2 B a\right )}{- 20 A b^{3} + 8 B a b^{2}} \right )}}{a^{6}} + \frac{2 b \left (- 5 A b + 2 B a\right ) \log{\left (x + \frac{- 10 A a b^{2} + 4 B a^{2} b + 2 a b \left (- 5 A b + 2 B a\right )}{- 20 A b^{3} + 8 B a b^{2}} \right )}}{a^{6}} \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)**2,x)

[Out]

-(3*A*a**4 + x**4*(-60*A*b**4 + 24*B*a*b**3) + x**3*(-150*A*a*b**3 + 60*B*a**2*b**2) + x**2*(-110*A*a**2*b**2
+ 44*B*a**3*b) + x*(-15*A*a**3*b + 6*B*a**4))/(6*a**8*x**2 + 18*a**7*b*x**3 + 18*a**6*b**2*x**4 + 6*a**5*b**3*
x**5) - 2*b*(-5*A*b + 2*B*a)*log(x + (-10*A*a*b**2 + 4*B*a**2*b - 2*a*b*(-5*A*b + 2*B*a))/(-20*A*b**3 + 8*B*a*
b**2))/a**6 + 2*b*(-5*A*b + 2*B*a)*log(x + (-10*A*a*b**2 + 4*B*a**2*b + 2*a*b*(-5*A*b + 2*B*a))/(-20*A*b**3 +
8*B*a*b**2))/a**6

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Giac [A]  time = 1.20709, size = 212, normalized size = 1.57 \begin{align*} -\frac{2 \,{\left (2 \, B a b - 5 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac{2 \,{\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{6} b} - \frac{3 \, A a^{5} + 12 \,{\left (2 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + 30 \,{\left (2 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} + 22 \,{\left (2 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} + 3 \,{\left (2 \, B a^{5} - 5 \, A a^{4} b\right )} x}{6 \,{\left (b x + a\right )}^{3} a^{6} 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)^2,x, algorithm="giac")

[Out]

-2*(2*B*a*b - 5*A*b^2)*log(abs(x))/a^6 + 2*(2*B*a*b^2 - 5*A*b^3)*log(abs(b*x + a))/(a^6*b) - 1/6*(3*A*a^5 + 12
*(2*B*a^2*b^3 - 5*A*a*b^4)*x^4 + 30*(2*B*a^3*b^2 - 5*A*a^2*b^3)*x^3 + 22*(2*B*a^4*b - 5*A*a^3*b^2)*x^2 + 3*(2*
B*a^5 - 5*A*a^4*b)*x)/((b*x + a)^3*a^6*x^2)