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

Optimal. Leaf size=177 \[ \frac{6 A b-a B}{a^7 x}+\frac{5 b (3 A b-a B)}{a^7 (a+b x)}+\frac{b (5 A b-2 a B)}{a^6 (a+b x)^2}+\frac{b (2 A b-a B)}{a^5 (a+b x)^3}+\frac{b (3 A b-2 a B)}{4 a^4 (a+b x)^4}+\frac{b (A b-a B)}{5 a^3 (a+b x)^5}+\frac{3 b \log (x) (7 A b-2 a B)}{a^8}-\frac{3 b (7 A b-2 a B) \log (a+b x)}{a^8}-\frac{A}{2 a^6 x^2} \]

[Out]

-A/(2*a^6*x^2) + (6*A*b - a*B)/(a^7*x) + (b*(A*b - a*B))/(5*a^3*(a + b*x)^5) + (b*(3*A*b - 2*a*B))/(4*a^4*(a +
b*x)^4) + (b*(2*A*b - a*B))/(a^5*(a + b*x)^3) + (b*(5*A*b - 2*a*B))/(a^6*(a + b*x)^2) + (5*b*(3*A*b - a*B))/(
a^7*(a + b*x)) + (3*b*(7*A*b - 2*a*B)*Log[x])/a^8 - (3*b*(7*A*b - 2*a*B)*Log[a + b*x])/a^8

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Rubi [A]  time = 0.202958, antiderivative size = 177, 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} \[ \frac{6 A b-a B}{a^7 x}+\frac{5 b (3 A b-a B)}{a^7 (a+b x)}+\frac{b (5 A b-2 a B)}{a^6 (a+b x)^2}+\frac{b (2 A b-a B)}{a^5 (a+b x)^3}+\frac{b (3 A b-2 a B)}{4 a^4 (a+b x)^4}+\frac{b (A b-a B)}{5 a^3 (a+b x)^5}+\frac{3 b \log (x) (7 A b-2 a B)}{a^8}-\frac{3 b (7 A b-2 a B) \log (a+b x)}{a^8}-\frac{A}{2 a^6 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-A/(2*a^6*x^2) + (6*A*b - a*B)/(a^7*x) + (b*(A*b - a*B))/(5*a^3*(a + b*x)^5) + (b*(3*A*b - 2*a*B))/(4*a^4*(a +
b*x)^4) + (b*(2*A*b - a*B))/(a^5*(a + b*x)^3) + (b*(5*A*b - 2*a*B))/(a^6*(a + b*x)^2) + (5*b*(3*A*b - a*B))/(
a^7*(a + b*x)) + (3*b*(7*A*b - 2*a*B)*Log[x])/a^8 - (3*b*(7*A*b - 2*a*B)*Log[a + b*x])/a^8

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 )^3} \, dx &=\int \frac{A+B x}{x^3 (a+b x)^6} \, dx\\ &=\int \left (\frac{A}{a^6 x^3}+\frac{-6 A b+a B}{a^7 x^2}-\frac{3 b (-7 A b+2 a B)}{a^8 x}+\frac{b^2 (-A b+a B)}{a^3 (a+b x)^6}+\frac{b^2 (-3 A b+2 a B)}{a^4 (a+b x)^5}+\frac{3 b^2 (-2 A b+a B)}{a^5 (a+b x)^4}+\frac{2 b^2 (-5 A b+2 a B)}{a^6 (a+b x)^3}+\frac{5 b^2 (-3 A b+a B)}{a^7 (a+b x)^2}+\frac{3 b^2 (-7 A b+2 a B)}{a^8 (a+b x)}\right ) \, dx\\ &=-\frac{A}{2 a^6 x^2}+\frac{6 A b-a B}{a^7 x}+\frac{b (A b-a B)}{5 a^3 (a+b x)^5}+\frac{b (3 A b-2 a B)}{4 a^4 (a+b x)^4}+\frac{b (2 A b-a B)}{a^5 (a+b x)^3}+\frac{b (5 A b-2 a B)}{a^6 (a+b x)^2}+\frac{5 b (3 A b-a B)}{a^7 (a+b x)}+\frac{3 b (7 A b-2 a B) \log (x)}{a^8}-\frac{3 b (7 A b-2 a B) \log (a+b x)}{a^8}\\ \end{align*}

Mathematica [A]  time = 0.183819, size = 162, normalized size = 0.92 \[ \frac{\frac{a \left (7 a^4 b^2 x^2 (137 A-110 B x)+5 a^3 b^3 x^3 (539 A-188 B x)+10 a^2 b^4 x^4 (329 A-54 B x)+2 a^5 b x (35 A-137 B x)-10 a^6 (A+2 B x)+30 a b^5 x^5 (63 A-4 B x)+420 A b^6 x^6\right )}{x^2 (a+b x)^5}+60 b \log (x) (7 A b-2 a B)+60 b (2 a B-7 A b) \log (a+b x)}{20 a^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(420*A*b^6*x^6 + 5*a^3*b^3*x^3*(539*A - 188*B*x) + 2*a^5*b*x*(35*A - 137*B*x) + 7*a^4*b^2*x^2*(137*A - 110
*B*x) + 10*a^2*b^4*x^4*(329*A - 54*B*x) + 30*a*b^5*x^5*(63*A - 4*B*x) - 10*a^6*(A + 2*B*x)))/(x^2*(a + b*x)^5)
+ 60*b*(7*A*b - 2*a*B)*Log[x] + 60*b*(-7*A*b + 2*a*B)*Log[a + b*x])/(20*a^8)

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Maple [A]  time = 0.016, size = 228, normalized size = 1.3 \begin{align*} -{\frac{A}{2\,{a}^{6}{x}^{2}}}+6\,{\frac{Ab}{{a}^{7}x}}-{\frac{B}{{a}^{6}x}}+21\,{\frac{A{b}^{2}\ln \left ( x \right ) }{{a}^{8}}}-6\,{\frac{b\ln \left ( x \right ) B}{{a}^{7}}}+2\,{\frac{A{b}^{2}}{{a}^{5} \left ( bx+a \right ) ^{3}}}-{\frac{bB}{{a}^{4} \left ( bx+a \right ) ^{3}}}+{\frac{3\,A{b}^{2}}{4\,{a}^{4} \left ( bx+a \right ) ^{4}}}-{\frac{bB}{2\,{a}^{3} \left ( bx+a \right ) ^{4}}}+15\,{\frac{A{b}^{2}}{{a}^{7} \left ( bx+a \right ) }}-5\,{\frac{bB}{{a}^{6} \left ( bx+a \right ) }}+5\,{\frac{A{b}^{2}}{{a}^{6} \left ( bx+a \right ) ^{2}}}-2\,{\frac{bB}{{a}^{5} \left ( bx+a \right ) ^{2}}}-21\,{\frac{{b}^{2}\ln \left ( bx+a \right ) A}{{a}^{8}}}+6\,{\frac{b\ln \left ( bx+a \right ) B}{{a}^{7}}}+{\frac{A{b}^{2}}{5\,{a}^{3} \left ( bx+a \right ) ^{5}}}-{\frac{bB}{5\,{a}^{2} \left ( bx+a \right ) ^{5}}} \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)^3,x)

[Out]

-1/2*A/a^6/x^2+6/a^7/x*A*b-1/a^6/x*B+21*b^2/a^8*ln(x)*A-6*b/a^7*ln(x)*B+2*b^2/a^5/(b*x+a)^3*A-b/a^4/(b*x+a)^3*
B+3/4*b^2/a^4/(b*x+a)^4*A-1/2*b/a^3/(b*x+a)^4*B+15*b^2/a^7/(b*x+a)*A-5*b/a^6/(b*x+a)*B+5*b^2/a^6/(b*x+a)^2*A-2
*b/a^5/(b*x+a)^2*B-21*b^2/a^8*ln(b*x+a)*A+6*b/a^7*ln(b*x+a)*B+1/5*b^2/a^3/(b*x+a)^5*A-1/5*b/a^2/(b*x+a)^5*B

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Maxima [A]  time = 1.07059, size = 327, normalized size = 1.85 \begin{align*} -\frac{10 \, A a^{6} + 60 \,{\left (2 \, B a b^{5} - 7 \, A b^{6}\right )} x^{6} + 270 \,{\left (2 \, B a^{2} b^{4} - 7 \, A a b^{5}\right )} x^{5} + 470 \,{\left (2 \, B a^{3} b^{3} - 7 \, A a^{2} b^{4}\right )} x^{4} + 385 \,{\left (2 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} x^{3} + 137 \,{\left (2 \, B a^{5} b - 7 \, A a^{4} b^{2}\right )} x^{2} + 10 \,{\left (2 \, B a^{6} - 7 \, A a^{5} b\right )} x}{20 \,{\left (a^{7} b^{5} x^{7} + 5 \, a^{8} b^{4} x^{6} + 10 \, a^{9} b^{3} x^{5} + 10 \, a^{10} b^{2} x^{4} + 5 \, a^{11} b x^{3} + a^{12} x^{2}\right )}} + \frac{3 \,{\left (2 \, B a b - 7 \, A b^{2}\right )} \log \left (b x + a\right )}{a^{8}} - \frac{3 \,{\left (2 \, B a b - 7 \, A b^{2}\right )} \log \left (x\right )}{a^{8}} \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)^3,x, algorithm="maxima")

[Out]

-1/20*(10*A*a^6 + 60*(2*B*a*b^5 - 7*A*b^6)*x^6 + 270*(2*B*a^2*b^4 - 7*A*a*b^5)*x^5 + 470*(2*B*a^3*b^3 - 7*A*a^
2*b^4)*x^4 + 385*(2*B*a^4*b^2 - 7*A*a^3*b^3)*x^3 + 137*(2*B*a^5*b - 7*A*a^4*b^2)*x^2 + 10*(2*B*a^6 - 7*A*a^5*b
)*x)/(a^7*b^5*x^7 + 5*a^8*b^4*x^6 + 10*a^9*b^3*x^5 + 10*a^10*b^2*x^4 + 5*a^11*b*x^3 + a^12*x^2) + 3*(2*B*a*b -
7*A*b^2)*log(b*x + a)/a^8 - 3*(2*B*a*b - 7*A*b^2)*log(x)/a^8

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Fricas [B]  time = 1.32573, size = 1029, normalized size = 5.81 \begin{align*} -\frac{10 \, A a^{7} + 60 \,{\left (2 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{6} + 270 \,{\left (2 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5}\right )} x^{5} + 470 \,{\left (2 \, B a^{4} b^{3} - 7 \, A a^{3} b^{4}\right )} x^{4} + 385 \,{\left (2 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3}\right )} x^{3} + 137 \,{\left (2 \, B a^{6} b - 7 \, A a^{5} b^{2}\right )} x^{2} + 10 \,{\left (2 \, B a^{7} - 7 \, A a^{6} b\right )} x - 60 \,{\left ({\left (2 \, B a b^{6} - 7 \, A b^{7}\right )} x^{7} + 5 \,{\left (2 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{6} + 10 \,{\left (2 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5}\right )} x^{5} + 10 \,{\left (2 \, B a^{4} b^{3} - 7 \, A a^{3} b^{4}\right )} x^{4} + 5 \,{\left (2 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3}\right )} x^{3} +{\left (2 \, B a^{6} b - 7 \, A a^{5} b^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) + 60 \,{\left ({\left (2 \, B a b^{6} - 7 \, A b^{7}\right )} x^{7} + 5 \,{\left (2 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{6} + 10 \,{\left (2 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5}\right )} x^{5} + 10 \,{\left (2 \, B a^{4} b^{3} - 7 \, A a^{3} b^{4}\right )} x^{4} + 5 \,{\left (2 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3}\right )} x^{3} +{\left (2 \, B a^{6} b - 7 \, A a^{5} b^{2}\right )} x^{2}\right )} \log \left (x\right )}{20 \,{\left (a^{8} b^{5} x^{7} + 5 \, a^{9} b^{4} x^{6} + 10 \, a^{10} b^{3} x^{5} + 10 \, a^{11} b^{2} x^{4} + 5 \, a^{12} b x^{3} + a^{13} 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)^3,x, algorithm="fricas")

[Out]

-1/20*(10*A*a^7 + 60*(2*B*a^2*b^5 - 7*A*a*b^6)*x^6 + 270*(2*B*a^3*b^4 - 7*A*a^2*b^5)*x^5 + 470*(2*B*a^4*b^3 -
7*A*a^3*b^4)*x^4 + 385*(2*B*a^5*b^2 - 7*A*a^4*b^3)*x^3 + 137*(2*B*a^6*b - 7*A*a^5*b^2)*x^2 + 10*(2*B*a^7 - 7*A
*a^6*b)*x - 60*((2*B*a*b^6 - 7*A*b^7)*x^7 + 5*(2*B*a^2*b^5 - 7*A*a*b^6)*x^6 + 10*(2*B*a^3*b^4 - 7*A*a^2*b^5)*x
^5 + 10*(2*B*a^4*b^3 - 7*A*a^3*b^4)*x^4 + 5*(2*B*a^5*b^2 - 7*A*a^4*b^3)*x^3 + (2*B*a^6*b - 7*A*a^5*b^2)*x^2)*l
og(b*x + a) + 60*((2*B*a*b^6 - 7*A*b^7)*x^7 + 5*(2*B*a^2*b^5 - 7*A*a*b^6)*x^6 + 10*(2*B*a^3*b^4 - 7*A*a^2*b^5)
*x^5 + 10*(2*B*a^4*b^3 - 7*A*a^3*b^4)*x^4 + 5*(2*B*a^5*b^2 - 7*A*a^4*b^3)*x^3 + (2*B*a^6*b - 7*A*a^5*b^2)*x^2)
*log(x))/(a^8*b^5*x^7 + 5*a^9*b^4*x^6 + 10*a^10*b^3*x^5 + 10*a^11*b^2*x^4 + 5*a^12*b*x^3 + a^13*x^2)

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Sympy [A]  time = 2.37338, size = 335, normalized size = 1.89 \begin{align*} - \frac{10 A a^{6} + x^{6} \left (- 420 A b^{6} + 120 B a b^{5}\right ) + x^{5} \left (- 1890 A a b^{5} + 540 B a^{2} b^{4}\right ) + x^{4} \left (- 3290 A a^{2} b^{4} + 940 B a^{3} b^{3}\right ) + x^{3} \left (- 2695 A a^{3} b^{3} + 770 B a^{4} b^{2}\right ) + x^{2} \left (- 959 A a^{4} b^{2} + 274 B a^{5} b\right ) + x \left (- 70 A a^{5} b + 20 B a^{6}\right )}{20 a^{12} x^{2} + 100 a^{11} b x^{3} + 200 a^{10} b^{2} x^{4} + 200 a^{9} b^{3} x^{5} + 100 a^{8} b^{4} x^{6} + 20 a^{7} b^{5} x^{7}} - \frac{3 b \left (- 7 A b + 2 B a\right ) \log{\left (x + \frac{- 21 A a b^{2} + 6 B a^{2} b - 3 a b \left (- 7 A b + 2 B a\right )}{- 42 A b^{3} + 12 B a b^{2}} \right )}}{a^{8}} + \frac{3 b \left (- 7 A b + 2 B a\right ) \log{\left (x + \frac{- 21 A a b^{2} + 6 B a^{2} b + 3 a b \left (- 7 A b + 2 B a\right )}{- 42 A b^{3} + 12 B a b^{2}} \right )}}{a^{8}} \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)**3,x)

[Out]

-(10*A*a**6 + x**6*(-420*A*b**6 + 120*B*a*b**5) + x**5*(-1890*A*a*b**5 + 540*B*a**2*b**4) + x**4*(-3290*A*a**2
*b**4 + 940*B*a**3*b**3) + x**3*(-2695*A*a**3*b**3 + 770*B*a**4*b**2) + x**2*(-959*A*a**4*b**2 + 274*B*a**5*b)
+ x*(-70*A*a**5*b + 20*B*a**6))/(20*a**12*x**2 + 100*a**11*b*x**3 + 200*a**10*b**2*x**4 + 200*a**9*b**3*x**5
+ 100*a**8*b**4*x**6 + 20*a**7*b**5*x**7) - 3*b*(-7*A*b + 2*B*a)*log(x + (-21*A*a*b**2 + 6*B*a**2*b - 3*a*b*(-
7*A*b + 2*B*a))/(-42*A*b**3 + 12*B*a*b**2))/a**8 + 3*b*(-7*A*b + 2*B*a)*log(x + (-21*A*a*b**2 + 6*B*a**2*b + 3
*a*b*(-7*A*b + 2*B*a))/(-42*A*b**3 + 12*B*a*b**2))/a**8

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Giac [A]  time = 1.25675, size = 277, normalized size = 1.56 \begin{align*} -\frac{3 \,{\left (2 \, B a b - 7 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{8}} + \frac{3 \,{\left (2 \, B a b^{2} - 7 \, A b^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{8} b} - \frac{10 \, A a^{7} + 60 \,{\left (2 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{6} + 270 \,{\left (2 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5}\right )} x^{5} + 470 \,{\left (2 \, B a^{4} b^{3} - 7 \, A a^{3} b^{4}\right )} x^{4} + 385 \,{\left (2 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3}\right )} x^{3} + 137 \,{\left (2 \, B a^{6} b - 7 \, A a^{5} b^{2}\right )} x^{2} + 10 \,{\left (2 \, B a^{7} - 7 \, A a^{6} b\right )} x}{20 \,{\left (b x + a\right )}^{5} a^{8} 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)^3,x, algorithm="giac")

[Out]

-3*(2*B*a*b - 7*A*b^2)*log(abs(x))/a^8 + 3*(2*B*a*b^2 - 7*A*b^3)*log(abs(b*x + a))/(a^8*b) - 1/20*(10*A*a^7 +
60*(2*B*a^2*b^5 - 7*A*a*b^6)*x^6 + 270*(2*B*a^3*b^4 - 7*A*a^2*b^5)*x^5 + 470*(2*B*a^4*b^3 - 7*A*a^3*b^4)*x^4 +
385*(2*B*a^5*b^2 - 7*A*a^4*b^3)*x^3 + 137*(2*B*a^6*b - 7*A*a^5*b^2)*x^2 + 10*(2*B*a^7 - 7*A*a^6*b)*x)/((b*x +
a)^5*a^8*x^2)