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

Optimal. Leaf size=157 \[ -\frac{5 A b-a B}{a^6 (a+b x)}-\frac{4 A b-a B}{2 a^5 (a+b x)^2}-\frac{3 A b-a B}{3 a^4 (a+b x)^3}-\frac{2 A b-a B}{4 a^3 (a+b x)^4}-\frac{A b-a B}{5 a^2 (a+b x)^5}-\frac{\log (x) (6 A b-a B)}{a^7}+\frac{(6 A b-a B) \log (a+b x)}{a^7}-\frac{A}{a^6 x} \]

[Out]

-(A/(a^6*x)) - (A*b - a*B)/(5*a^2*(a + b*x)^5) - (2*A*b - a*B)/(4*a^3*(a + b*x)^4) - (3*A*b - a*B)/(3*a^4*(a +
b*x)^3) - (4*A*b - a*B)/(2*a^5*(a + b*x)^2) - (5*A*b - a*B)/(a^6*(a + b*x)) - ((6*A*b - a*B)*Log[x])/a^7 + ((
6*A*b - a*B)*Log[a + b*x])/a^7

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Rubi [A]  time = 0.164546, antiderivative size = 157, 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} \[ -\frac{5 A b-a B}{a^6 (a+b x)}-\frac{4 A b-a B}{2 a^5 (a+b x)^2}-\frac{3 A b-a B}{3 a^4 (a+b x)^3}-\frac{2 A b-a B}{4 a^3 (a+b x)^4}-\frac{A b-a B}{5 a^2 (a+b x)^5}-\frac{\log (x) (6 A b-a B)}{a^7}+\frac{(6 A b-a B) \log (a+b x)}{a^7}-\frac{A}{a^6 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(A/(a^6*x)) - (A*b - a*B)/(5*a^2*(a + b*x)^5) - (2*A*b - a*B)/(4*a^3*(a + b*x)^4) - (3*A*b - a*B)/(3*a^4*(a +
b*x)^3) - (4*A*b - a*B)/(2*a^5*(a + b*x)^2) - (5*A*b - a*B)/(a^6*(a + b*x)) - ((6*A*b - a*B)*Log[x])/a^7 + ((
6*A*b - a*B)*Log[a + b*x])/a^7

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 )^3} \, dx &=\int \frac{A+B x}{x^2 (a+b x)^6} \, dx\\ &=\int \left (\frac{A}{a^6 x^2}+\frac{-6 A b+a B}{a^7 x}-\frac{b (-A b+a B)}{a^2 (a+b x)^6}-\frac{b (-2 A b+a B)}{a^3 (a+b x)^5}-\frac{b (-3 A b+a B)}{a^4 (a+b x)^4}-\frac{b (-4 A b+a B)}{a^5 (a+b x)^3}-\frac{b (-5 A b+a B)}{a^6 (a+b x)^2}-\frac{b (-6 A b+a B)}{a^7 (a+b x)}\right ) \, dx\\ &=-\frac{A}{a^6 x}-\frac{A b-a B}{5 a^2 (a+b x)^5}-\frac{2 A b-a B}{4 a^3 (a+b x)^4}-\frac{3 A b-a B}{3 a^4 (a+b x)^3}-\frac{4 A b-a B}{2 a^5 (a+b x)^2}-\frac{5 A b-a B}{a^6 (a+b x)}-\frac{(6 A b-a B) \log (x)}{a^7}+\frac{(6 A b-a B) \log (a+b x)}{a^7}\\ \end{align*}

Mathematica [A]  time = 0.0945075, size = 142, normalized size = 0.9 \[ \frac{\frac{12 a^5 (a B-A b)}{(a+b x)^5}+\frac{15 a^4 (a B-2 A b)}{(a+b x)^4}+\frac{20 a^3 (a B-3 A b)}{(a+b x)^3}+\frac{30 a^2 (a B-4 A b)}{(a+b x)^2}+\frac{60 a (a B-5 A b)}{a+b x}+60 \log (x) (a B-6 A b)+60 (6 A b-a B) \log (a+b x)-\frac{60 a A}{x}}{60 a^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-60*a*A)/x + (12*a^5*(-(A*b) + a*B))/(a + b*x)^5 + (15*a^4*(-2*A*b + a*B))/(a + b*x)^4 + (20*a^3*(-3*A*b + a
*B))/(a + b*x)^3 + (30*a^2*(-4*A*b + a*B))/(a + b*x)^2 + (60*a*(-5*A*b + a*B))/(a + b*x) + 60*(-6*A*b + a*B)*L
og[x] + 60*(6*A*b - a*B)*Log[a + b*x])/(60*a^7)

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Maple [A]  time = 0.013, size = 186, normalized size = 1.2 \begin{align*} -{\frac{A}{{a}^{6}x}}-6\,{\frac{Ab\ln \left ( x \right ) }{{a}^{7}}}+{\frac{\ln \left ( x \right ) B}{{a}^{6}}}-{\frac{Ab}{2\,{a}^{3} \left ( bx+a \right ) ^{4}}}+{\frac{B}{4\,{a}^{2} \left ( bx+a \right ) ^{4}}}-{\frac{Ab}{{a}^{4} \left ( bx+a \right ) ^{3}}}+{\frac{B}{3\,{a}^{3} \left ( bx+a \right ) ^{3}}}-2\,{\frac{Ab}{{a}^{5} \left ( bx+a \right ) ^{2}}}+{\frac{B}{2\,{a}^{4} \left ( bx+a \right ) ^{2}}}-5\,{\frac{Ab}{{a}^{6} \left ( bx+a \right ) }}+{\frac{B}{{a}^{5} \left ( bx+a \right ) }}+6\,{\frac{\ln \left ( bx+a \right ) Ab}{{a}^{7}}}-{\frac{\ln \left ( bx+a \right ) B}{{a}^{6}}}-{\frac{Ab}{5\,{a}^{2} \left ( bx+a \right ) ^{5}}}+{\frac{B}{5\,a \left ( bx+a \right ) ^{5}}} \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)^3,x)

[Out]

-A/a^6/x-6/a^7*ln(x)*A*b+1/a^6*ln(x)*B-1/2/a^3/(b*x+a)^4*A*b+1/4/a^2/(b*x+a)^4*B-1/a^4/(b*x+a)^3*A*b+1/3/a^3/(
b*x+a)^3*B-2/a^5/(b*x+a)^2*A*b+1/2/a^4/(b*x+a)^2*B-5/a^6/(b*x+a)*A*b+1/a^5/(b*x+a)*B+6/a^7*ln(b*x+a)*A*b-1/a^6
*ln(b*x+a)*B-1/5/a^2/(b*x+a)^5*A*b+1/5/a/(b*x+a)^5*B

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Maxima [A]  time = 1.06907, size = 273, normalized size = 1.74 \begin{align*} -\frac{60 \, A a^{5} - 60 \,{\left (B a b^{4} - 6 \, A b^{5}\right )} x^{5} - 270 \,{\left (B a^{2} b^{3} - 6 \, A a b^{4}\right )} x^{4} - 470 \,{\left (B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} x^{3} - 385 \,{\left (B a^{4} b - 6 \, A a^{3} b^{2}\right )} x^{2} - 137 \,{\left (B a^{5} - 6 \, A a^{4} b\right )} x}{60 \,{\left (a^{6} b^{5} x^{6} + 5 \, a^{7} b^{4} x^{5} + 10 \, a^{8} b^{3} x^{4} + 10 \, a^{9} b^{2} x^{3} + 5 \, a^{10} b x^{2} + a^{11} x\right )}} - \frac{{\left (B a - 6 \, A b\right )} \log \left (b x + a\right )}{a^{7}} + \frac{{\left (B a - 6 \, A b\right )} \log \left (x\right )}{a^{7}} \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)^3,x, algorithm="maxima")

[Out]

-1/60*(60*A*a^5 - 60*(B*a*b^4 - 6*A*b^5)*x^5 - 270*(B*a^2*b^3 - 6*A*a*b^4)*x^4 - 470*(B*a^3*b^2 - 6*A*a^2*b^3)
*x^3 - 385*(B*a^4*b - 6*A*a^3*b^2)*x^2 - 137*(B*a^5 - 6*A*a^4*b)*x)/(a^6*b^5*x^6 + 5*a^7*b^4*x^5 + 10*a^8*b^3*
x^4 + 10*a^9*b^2*x^3 + 5*a^10*b*x^2 + a^11*x) - (B*a - 6*A*b)*log(b*x + a)/a^7 + (B*a - 6*A*b)*log(x)/a^7

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Fricas [B]  time = 1.42771, size = 907, normalized size = 5.78 \begin{align*} -\frac{60 \, A a^{6} - 60 \,{\left (B a^{2} b^{4} - 6 \, A a b^{5}\right )} x^{5} - 270 \,{\left (B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} x^{4} - 470 \,{\left (B a^{4} b^{2} - 6 \, A a^{3} b^{3}\right )} x^{3} - 385 \,{\left (B a^{5} b - 6 \, A a^{4} b^{2}\right )} x^{2} - 137 \,{\left (B a^{6} - 6 \, A a^{5} b\right )} x + 60 \,{\left ({\left (B a b^{5} - 6 \, A b^{6}\right )} x^{6} + 5 \,{\left (B a^{2} b^{4} - 6 \, A a b^{5}\right )} x^{5} + 10 \,{\left (B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} x^{4} + 10 \,{\left (B a^{4} b^{2} - 6 \, A a^{3} b^{3}\right )} x^{3} + 5 \,{\left (B a^{5} b - 6 \, A a^{4} b^{2}\right )} x^{2} +{\left (B a^{6} - 6 \, A a^{5} b\right )} x\right )} \log \left (b x + a\right ) - 60 \,{\left ({\left (B a b^{5} - 6 \, A b^{6}\right )} x^{6} + 5 \,{\left (B a^{2} b^{4} - 6 \, A a b^{5}\right )} x^{5} + 10 \,{\left (B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} x^{4} + 10 \,{\left (B a^{4} b^{2} - 6 \, A a^{3} b^{3}\right )} x^{3} + 5 \,{\left (B a^{5} b - 6 \, A a^{4} b^{2}\right )} x^{2} +{\left (B a^{6} - 6 \, A a^{5} b\right )} x\right )} \log \left (x\right )}{60 \,{\left (a^{7} b^{5} x^{6} + 5 \, a^{8} b^{4} x^{5} + 10 \, a^{9} b^{3} x^{4} + 10 \, a^{10} b^{2} x^{3} + 5 \, a^{11} b x^{2} + a^{12} x\right )}} \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)^3,x, algorithm="fricas")

[Out]

-1/60*(60*A*a^6 - 60*(B*a^2*b^4 - 6*A*a*b^5)*x^5 - 270*(B*a^3*b^3 - 6*A*a^2*b^4)*x^4 - 470*(B*a^4*b^2 - 6*A*a^
3*b^3)*x^3 - 385*(B*a^5*b - 6*A*a^4*b^2)*x^2 - 137*(B*a^6 - 6*A*a^5*b)*x + 60*((B*a*b^5 - 6*A*b^6)*x^6 + 5*(B*
a^2*b^4 - 6*A*a*b^5)*x^5 + 10*(B*a^3*b^3 - 6*A*a^2*b^4)*x^4 + 10*(B*a^4*b^2 - 6*A*a^3*b^3)*x^3 + 5*(B*a^5*b -
6*A*a^4*b^2)*x^2 + (B*a^6 - 6*A*a^5*b)*x)*log(b*x + a) - 60*((B*a*b^5 - 6*A*b^6)*x^6 + 5*(B*a^2*b^4 - 6*A*a*b^
5)*x^5 + 10*(B*a^3*b^3 - 6*A*a^2*b^4)*x^4 + 10*(B*a^4*b^2 - 6*A*a^3*b^3)*x^3 + 5*(B*a^5*b - 6*A*a^4*b^2)*x^2 +
(B*a^6 - 6*A*a^5*b)*x)*log(x))/(a^7*b^5*x^6 + 5*a^8*b^4*x^5 + 10*a^9*b^3*x^4 + 10*a^10*b^2*x^3 + 5*a^11*b*x^2
+ a^12*x)

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Sympy [B]  time = 1.64411, size = 275, normalized size = 1.75 \begin{align*} \frac{- 60 A a^{5} + x^{5} \left (- 360 A b^{5} + 60 B a b^{4}\right ) + x^{4} \left (- 1620 A a b^{4} + 270 B a^{2} b^{3}\right ) + x^{3} \left (- 2820 A a^{2} b^{3} + 470 B a^{3} b^{2}\right ) + x^{2} \left (- 2310 A a^{3} b^{2} + 385 B a^{4} b\right ) + x \left (- 822 A a^{4} b + 137 B a^{5}\right )}{60 a^{11} x + 300 a^{10} b x^{2} + 600 a^{9} b^{2} x^{3} + 600 a^{8} b^{3} x^{4} + 300 a^{7} b^{4} x^{5} + 60 a^{6} b^{5} x^{6}} + \frac{\left (- 6 A b + B a\right ) \log{\left (x + \frac{- 6 A a b + B a^{2} - a \left (- 6 A b + B a\right )}{- 12 A b^{2} + 2 B a b} \right )}}{a^{7}} - \frac{\left (- 6 A b + B a\right ) \log{\left (x + \frac{- 6 A a b + B a^{2} + a \left (- 6 A b + B a\right )}{- 12 A b^{2} + 2 B a b} \right )}}{a^{7}} \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)**3,x)

[Out]

(-60*A*a**5 + x**5*(-360*A*b**5 + 60*B*a*b**4) + x**4*(-1620*A*a*b**4 + 270*B*a**2*b**3) + x**3*(-2820*A*a**2*
b**3 + 470*B*a**3*b**2) + x**2*(-2310*A*a**3*b**2 + 385*B*a**4*b) + x*(-822*A*a**4*b + 137*B*a**5))/(60*a**11*
x + 300*a**10*b*x**2 + 600*a**9*b**2*x**3 + 600*a**8*b**3*x**4 + 300*a**7*b**4*x**5 + 60*a**6*b**5*x**6) + (-6
*A*b + B*a)*log(x + (-6*A*a*b + B*a**2 - a*(-6*A*b + B*a))/(-12*A*b**2 + 2*B*a*b))/a**7 - (-6*A*b + B*a)*log(x
+ (-6*A*a*b + B*a**2 + a*(-6*A*b + B*a))/(-12*A*b**2 + 2*B*a*b))/a**7

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Giac [A]  time = 1.16387, size = 227, normalized size = 1.45 \begin{align*} \frac{{\left (B a - 6 \, A b\right )} \log \left ({\left | x \right |}\right )}{a^{7}} - \frac{{\left (B a b - 6 \, A b^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{7} b} - \frac{60 \, A a^{6} - 60 \,{\left (B a^{2} b^{4} - 6 \, A a b^{5}\right )} x^{5} - 270 \,{\left (B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} x^{4} - 470 \,{\left (B a^{4} b^{2} - 6 \, A a^{3} b^{3}\right )} x^{3} - 385 \,{\left (B a^{5} b - 6 \, A a^{4} b^{2}\right )} x^{2} - 137 \,{\left (B a^{6} - 6 \, A a^{5} b\right )} x}{60 \,{\left (b x + a\right )}^{5} a^{7} 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)^3,x, algorithm="giac")

[Out]

(B*a - 6*A*b)*log(abs(x))/a^7 - (B*a*b - 6*A*b^2)*log(abs(b*x + a))/(a^7*b) - 1/60*(60*A*a^6 - 60*(B*a^2*b^4 -
6*A*a*b^5)*x^5 - 270*(B*a^3*b^3 - 6*A*a^2*b^4)*x^4 - 470*(B*a^4*b^2 - 6*A*a^3*b^3)*x^3 - 385*(B*a^5*b - 6*A*a
^4*b^2)*x^2 - 137*(B*a^6 - 6*A*a^5*b)*x)/((b*x + a)^5*a^7*x)