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

Optimal. Leaf size=102 \[ \frac{A}{a^5 (a+b x)}+\frac{A}{2 a^4 (a+b x)^2}+\frac{A}{3 a^3 (a+b x)^3}+\frac{A}{4 a^2 (a+b x)^4}-\frac{A \log (a+b x)}{a^6}+\frac{A \log (x)}{a^6}+\frac{A b-a B}{5 a b (a+b x)^5} \]

[Out]

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

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Rubi [A]  time = 0.0742038, antiderivative size = 102, 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} \[ \frac{A}{a^5 (a+b x)}+\frac{A}{2 a^4 (a+b x)^2}+\frac{A}{3 a^3 (a+b x)^3}+\frac{A}{4 a^2 (a+b x)^4}-\frac{A \log (a+b x)}{a^6}+\frac{A \log (x)}{a^6}+\frac{A b-a B}{5 a b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0627001, size = 89, normalized size = 0.87 \[ \frac{\frac{a \left (470 a^2 A b^3 x^2+385 a^3 A b^2 x+137 a^4 A b-12 a^5 B+270 a A b^4 x^3+60 A b^5 x^4\right )}{b (a+b x)^5}-60 A \log (a+b x)+60 A \log (x)}{60 a^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(137*a^4*A*b - 12*a^5*B + 385*a^3*A*b^2*x + 470*a^2*A*b^3*x^2 + 270*a*A*b^4*x^3 + 60*A*b^5*x^4))/(b*(a + b
*x)^5) + 60*A*Log[x] - 60*A*Log[a + b*x])/(60*a^6)

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Maple [A]  time = 0.009, size = 98, normalized size = 1. \begin{align*}{\frac{A\ln \left ( x \right ) }{{a}^{6}}}+{\frac{A}{5\,a \left ( bx+a \right ) ^{5}}}-{\frac{B}{5\,b \left ( bx+a \right ) ^{5}}}-{\frac{A\ln \left ( bx+a \right ) }{{a}^{6}}}+{\frac{A}{{a}^{5} \left ( bx+a \right ) }}+{\frac{A}{2\,{a}^{4} \left ( bx+a \right ) ^{2}}}+{\frac{A}{3\,{a}^{3} \left ( bx+a \right ) ^{3}}}+{\frac{A}{4\,{a}^{2} \left ( bx+a \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08678, size = 185, normalized size = 1.81 \begin{align*} \frac{60 \, A b^{5} x^{4} + 270 \, A a b^{4} x^{3} + 470 \, A a^{2} b^{3} x^{2} + 385 \, A a^{3} b^{2} x - 12 \, B a^{5} + 137 \, A a^{4} b}{60 \,{\left (a^{5} b^{6} x^{5} + 5 \, a^{6} b^{5} x^{4} + 10 \, a^{7} b^{4} x^{3} + 10 \, a^{8} b^{3} x^{2} + 5 \, a^{9} b^{2} x + a^{10} b\right )}} - \frac{A \log \left (b x + a\right )}{a^{6}} + \frac{A \log \left (x\right )}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.29422, size = 551, normalized size = 5.4 \begin{align*} \frac{60 \, A a b^{5} x^{4} + 270 \, A a^{2} b^{4} x^{3} + 470 \, A a^{3} b^{3} x^{2} + 385 \, A a^{4} b^{2} x - 12 \, B a^{6} + 137 \, A a^{5} b - 60 \,{\left (A b^{6} x^{5} + 5 \, A a b^{5} x^{4} + 10 \, A a^{2} b^{4} x^{3} + 10 \, A a^{3} b^{3} x^{2} + 5 \, A a^{4} b^{2} x + A a^{5} b\right )} \log \left (b x + a\right ) + 60 \,{\left (A b^{6} x^{5} + 5 \, A a b^{5} x^{4} + 10 \, A a^{2} b^{4} x^{3} + 10 \, A a^{3} b^{3} x^{2} + 5 \, A a^{4} b^{2} x + A a^{5} b\right )} \log \left (x\right )}{60 \,{\left (a^{6} b^{6} x^{5} + 5 \, a^{7} b^{5} x^{4} + 10 \, a^{8} b^{4} x^{3} + 10 \, a^{9} b^{3} x^{2} + 5 \, a^{10} b^{2} x + a^{11} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.18633, size = 141, normalized size = 1.38 \begin{align*} \frac{A \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{6}} + \frac{137 A a^{4} b + 385 A a^{3} b^{2} x + 470 A a^{2} b^{3} x^{2} + 270 A a b^{4} x^{3} + 60 A b^{5} x^{4} - 12 B a^{5}}{60 a^{10} b + 300 a^{9} b^{2} x + 600 a^{8} b^{3} x^{2} + 600 a^{7} b^{4} x^{3} + 300 a^{6} b^{5} x^{4} + 60 a^{5} b^{6} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.22947, size = 128, normalized size = 1.25 \begin{align*} -\frac{A \log \left ({\left | b x + a \right |}\right )}{a^{6}} + \frac{A \log \left ({\left | x \right |}\right )}{a^{6}} + \frac{60 \, A a b^{5} x^{4} + 270 \, A a^{2} b^{4} x^{3} + 470 \, A a^{3} b^{3} x^{2} + 385 \, A a^{4} b^{2} x - 12 \, B a^{6} + 137 \, A a^{5} b}{60 \,{\left (b x + a\right )}^{5} a^{6} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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