3.642 \(\int \frac{A+B x}{x^4 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=166 \[ -\frac{10 b^2 \log (x) (2 A b-a B)}{a^7}+\frac{10 b^2 (2 A b-a B) \log (a+b x)}{a^7}-\frac{2 b^2 (5 A b-3 a B)}{a^6 (a+b x)}-\frac{2 b (5 A b-2 a B)}{a^6 x}-\frac{b^2 (4 A b-3 a B)}{2 a^5 (a+b x)^2}+\frac{4 A b-a B}{2 a^5 x^2}-\frac{b^2 (A b-a B)}{3 a^4 (a+b x)^3}-\frac{A}{3 a^4 x^3} \]

[Out]

-A/(3*a^4*x^3) + (4*A*b - a*B)/(2*a^5*x^2) - (2*b*(5*A*b - 2*a*B))/(a^6*x) - (b^
2*(A*b - a*B))/(3*a^4*(a + b*x)^3) - (b^2*(4*A*b - 3*a*B))/(2*a^5*(a + b*x)^2) -
(2*b^2*(5*A*b - 3*a*B))/(a^6*(a + b*x)) - (10*b^2*(2*A*b - a*B)*Log[x])/a^7 + (
10*b^2*(2*A*b - a*B)*Log[a + b*x])/a^7

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Rubi [A]  time = 0.362338, antiderivative size = 166, 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 \[ -\frac{10 b^2 \log (x) (2 A b-a B)}{a^7}+\frac{10 b^2 (2 A b-a B) \log (a+b x)}{a^7}-\frac{2 b^2 (5 A b-3 a B)}{a^6 (a+b x)}-\frac{2 b (5 A b-2 a B)}{a^6 x}-\frac{b^2 (4 A b-3 a B)}{2 a^5 (a+b x)^2}+\frac{4 A b-a B}{2 a^5 x^2}-\frac{b^2 (A b-a B)}{3 a^4 (a+b x)^3}-\frac{A}{3 a^4 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^4*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

-A/(3*a^4*x^3) + (4*A*b - a*B)/(2*a^5*x^2) - (2*b*(5*A*b - 2*a*B))/(a^6*x) - (b^
2*(A*b - a*B))/(3*a^4*(a + b*x)^3) - (b^2*(4*A*b - 3*a*B))/(2*a^5*(a + b*x)^2) -
(2*b^2*(5*A*b - 3*a*B))/(a^6*(a + b*x)) - (10*b^2*(2*A*b - a*B)*Log[x])/a^7 + (
10*b^2*(2*A*b - a*B)*Log[a + b*x])/a^7

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Rubi in Sympy [A]  time = 66.2014, size = 160, normalized size = 0.96 \[ - \frac{A}{3 a^{4} x^{3}} - \frac{b^{2} \left (A b - B a\right )}{3 a^{4} \left (a + b x\right )^{3}} - \frac{b^{2} \left (4 A b - 3 B a\right )}{2 a^{5} \left (a + b x\right )^{2}} + \frac{4 A b - B a}{2 a^{5} x^{2}} - \frac{2 b^{2} \left (5 A b - 3 B a\right )}{a^{6} \left (a + b x\right )} - \frac{2 b \left (5 A b - 2 B a\right )}{a^{6} x} - \frac{10 b^{2} \left (2 A b - B a\right ) \log{\left (x \right )}}{a^{7}} + \frac{10 b^{2} \left (2 A b - B a\right ) \log{\left (a + b x \right )}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**4/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

-A/(3*a**4*x**3) - b**2*(A*b - B*a)/(3*a**4*(a + b*x)**3) - b**2*(4*A*b - 3*B*a)
/(2*a**5*(a + b*x)**2) + (4*A*b - B*a)/(2*a**5*x**2) - 2*b**2*(5*A*b - 3*B*a)/(a
**6*(a + b*x)) - 2*b*(5*A*b - 2*B*a)/(a**6*x) - 10*b**2*(2*A*b - B*a)*log(x)/a**
7 + 10*b**2*(2*A*b - B*a)*log(a + b*x)/a**7

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Mathematica [A]  time = 0.466425, size = 148, normalized size = 0.89 \[ \frac{\frac{a \left (a^5 (-(2 A+3 B x))+3 a^4 b x (2 A+5 B x)+10 a^3 b^2 x^2 (11 B x-3 A)+10 a^2 b^3 x^3 (15 B x-22 A)+60 a b^4 x^4 (B x-5 A)-120 A b^5 x^5\right )}{x^3 (a+b x)^3}-60 b^2 \log (x) (2 A b-a B)+60 b^2 (2 A b-a B) \log (a+b x)}{6 a^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^4*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

((a*(-120*A*b^5*x^5 + 60*a*b^4*x^4*(-5*A + B*x) - a^5*(2*A + 3*B*x) + 3*a^4*b*x*
(2*A + 5*B*x) + 10*a^3*b^2*x^2*(-3*A + 11*B*x) + 10*a^2*b^3*x^3*(-22*A + 15*B*x)
))/(x^3*(a + b*x)^3) - 60*b^2*(2*A*b - a*B)*Log[x] + 60*b^2*(2*A*b - a*B)*Log[a
+ b*x])/(6*a^7)

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Maple [A]  time = 0.02, size = 200, normalized size = 1.2 \[ -{\frac{A}{3\,{a}^{4}{x}^{3}}}+2\,{\frac{Ab}{{a}^{5}{x}^{2}}}-{\frac{B}{2\,{a}^{4}{x}^{2}}}-10\,{\frac{{b}^{2}A}{{a}^{6}x}}+4\,{\frac{Bb}{{a}^{5}x}}-20\,{\frac{A{b}^{3}\ln \left ( x \right ) }{{a}^{7}}}+10\,{\frac{{b}^{2}B\ln \left ( x \right ) }{{a}^{6}}}-{\frac{A{b}^{3}}{3\,{a}^{4} \left ( bx+a \right ) ^{3}}}+{\frac{{b}^{2}B}{3\,{a}^{3} \left ( bx+a \right ) ^{3}}}-2\,{\frac{A{b}^{3}}{{a}^{5} \left ( bx+a \right ) ^{2}}}+{\frac{3\,{b}^{2}B}{2\,{a}^{4} \left ( bx+a \right ) ^{2}}}+20\,{\frac{{b}^{3}\ln \left ( bx+a \right ) A}{{a}^{7}}}-10\,{\frac{{b}^{2}\ln \left ( bx+a \right ) B}{{a}^{6}}}-10\,{\frac{A{b}^{3}}{{a}^{6} \left ( bx+a \right ) }}+6\,{\frac{{b}^{2}B}{{a}^{5} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^4/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

-1/3*A/a^4/x^3+2/a^5/x^2*A*b-1/2/a^4/x^2*B-10/a^6*b^2/x*A+4/a^5*b/x*B-20*b^3/a^7
*ln(x)*A+10*b^2/a^6*ln(x)*B-1/3/a^4*b^3/(b*x+a)^3*A+1/3/a^3*b^2/(b*x+a)^3*B-2/a^
5*b^3/(b*x+a)^2*A+3/2/a^4*b^2/(b*x+a)^2*B+20*b^3/a^7*ln(b*x+a)*A-10*b^2/a^6*ln(b
*x+a)*B-10/a^6*b^3/(b*x+a)*A+6/a^5*b^2/(b*x+a)*B

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Maxima [A]  time = 0.694815, size = 261, normalized size = 1.57 \[ -\frac{2 \, A a^{5} - 60 \,{\left (B a b^{4} - 2 \, A b^{5}\right )} x^{5} - 150 \,{\left (B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{4} - 110 \,{\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{3} - 15 \,{\left (B a^{4} b - 2 \, A a^{3} b^{2}\right )} x^{2} + 3 \,{\left (B a^{5} - 2 \, A a^{4} b\right )} x}{6 \,{\left (a^{6} b^{3} x^{6} + 3 \, a^{7} b^{2} x^{5} + 3 \, a^{8} b x^{4} + a^{9} x^{3}\right )}} - \frac{10 \,{\left (B a b^{2} - 2 \, A b^{3}\right )} \log \left (b x + a\right )}{a^{7}} + \frac{10 \,{\left (B a b^{2} - 2 \, A b^{3}\right )} \log \left (x\right )}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^2*x^4),x, algorithm="maxima")

[Out]

-1/6*(2*A*a^5 - 60*(B*a*b^4 - 2*A*b^5)*x^5 - 150*(B*a^2*b^3 - 2*A*a*b^4)*x^4 - 1
10*(B*a^3*b^2 - 2*A*a^2*b^3)*x^3 - 15*(B*a^4*b - 2*A*a^3*b^2)*x^2 + 3*(B*a^5 - 2
*A*a^4*b)*x)/(a^6*b^3*x^6 + 3*a^7*b^2*x^5 + 3*a^8*b*x^4 + a^9*x^3) - 10*(B*a*b^2
- 2*A*b^3)*log(b*x + a)/a^7 + 10*(B*a*b^2 - 2*A*b^3)*log(x)/a^7

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Fricas [A]  time = 0.275815, size = 450, normalized size = 2.71 \[ -\frac{2 \, A a^{6} - 60 \,{\left (B a^{2} b^{4} - 2 \, A a b^{5}\right )} x^{5} - 150 \,{\left (B a^{3} b^{3} - 2 \, A a^{2} b^{4}\right )} x^{4} - 110 \,{\left (B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} x^{3} - 15 \,{\left (B a^{5} b - 2 \, A a^{4} b^{2}\right )} x^{2} + 3 \,{\left (B a^{6} - 2 \, A a^{5} b\right )} x + 60 \,{\left ({\left (B a b^{5} - 2 \, A b^{6}\right )} x^{6} + 3 \,{\left (B a^{2} b^{4} - 2 \, A a b^{5}\right )} x^{5} + 3 \,{\left (B a^{3} b^{3} - 2 \, A a^{2} b^{4}\right )} x^{4} +{\left (B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) - 60 \,{\left ({\left (B a b^{5} - 2 \, A b^{6}\right )} x^{6} + 3 \,{\left (B a^{2} b^{4} - 2 \, A a b^{5}\right )} x^{5} + 3 \,{\left (B a^{3} b^{3} - 2 \, A a^{2} b^{4}\right )} x^{4} +{\left (B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{7} b^{3} x^{6} + 3 \, a^{8} b^{2} x^{5} + 3 \, a^{9} b x^{4} + a^{10} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^2*x^4),x, algorithm="fricas")

[Out]

-1/6*(2*A*a^6 - 60*(B*a^2*b^4 - 2*A*a*b^5)*x^5 - 150*(B*a^3*b^3 - 2*A*a^2*b^4)*x
^4 - 110*(B*a^4*b^2 - 2*A*a^3*b^3)*x^3 - 15*(B*a^5*b - 2*A*a^4*b^2)*x^2 + 3*(B*a
^6 - 2*A*a^5*b)*x + 60*((B*a*b^5 - 2*A*b^6)*x^6 + 3*(B*a^2*b^4 - 2*A*a*b^5)*x^5
+ 3*(B*a^3*b^3 - 2*A*a^2*b^4)*x^4 + (B*a^4*b^2 - 2*A*a^3*b^3)*x^3)*log(b*x + a)
- 60*((B*a*b^5 - 2*A*b^6)*x^6 + 3*(B*a^2*b^4 - 2*A*a*b^5)*x^5 + 3*(B*a^3*b^3 - 2
*A*a^2*b^4)*x^4 + (B*a^4*b^2 - 2*A*a^3*b^3)*x^3)*log(x))/(a^7*b^3*x^6 + 3*a^8*b^
2*x^5 + 3*a^9*b*x^4 + a^10*x^3)

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Sympy [A]  time = 4.796, size = 291, normalized size = 1.75 \[ \frac{- 2 A a^{5} + x^{5} \left (- 120 A b^{5} + 60 B a b^{4}\right ) + x^{4} \left (- 300 A a b^{4} + 150 B a^{2} b^{3}\right ) + x^{3} \left (- 220 A a^{2} b^{3} + 110 B a^{3} b^{2}\right ) + x^{2} \left (- 30 A a^{3} b^{2} + 15 B a^{4} b\right ) + x \left (6 A a^{4} b - 3 B a^{5}\right )}{6 a^{9} x^{3} + 18 a^{8} b x^{4} + 18 a^{7} b^{2} x^{5} + 6 a^{6} b^{3} x^{6}} + \frac{10 b^{2} \left (- 2 A b + B a\right ) \log{\left (x + \frac{- 20 A a b^{3} + 10 B a^{2} b^{2} - 10 a b^{2} \left (- 2 A b + B a\right )}{- 40 A b^{4} + 20 B a b^{3}} \right )}}{a^{7}} - \frac{10 b^{2} \left (- 2 A b + B a\right ) \log{\left (x + \frac{- 20 A a b^{3} + 10 B a^{2} b^{2} + 10 a b^{2} \left (- 2 A b + B a\right )}{- 40 A b^{4} + 20 B a b^{3}} \right )}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**4/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

(-2*A*a**5 + x**5*(-120*A*b**5 + 60*B*a*b**4) + x**4*(-300*A*a*b**4 + 150*B*a**2
*b**3) + x**3*(-220*A*a**2*b**3 + 110*B*a**3*b**2) + x**2*(-30*A*a**3*b**2 + 15*
B*a**4*b) + x*(6*A*a**4*b - 3*B*a**5))/(6*a**9*x**3 + 18*a**8*b*x**4 + 18*a**7*b
**2*x**5 + 6*a**6*b**3*x**6) + 10*b**2*(-2*A*b + B*a)*log(x + (-20*A*a*b**3 + 10
*B*a**2*b**2 - 10*a*b**2*(-2*A*b + B*a))/(-40*A*b**4 + 20*B*a*b**3))/a**7 - 10*b
**2*(-2*A*b + B*a)*log(x + (-20*A*a*b**3 + 10*B*a**2*b**2 + 10*a*b**2*(-2*A*b +
B*a))/(-40*A*b**4 + 20*B*a*b**3))/a**7

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GIAC/XCAS [A]  time = 0.268686, size = 236, normalized size = 1.42 \[ \frac{10 \,{\left (B a b^{2} - 2 \, A b^{3}\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{7}} - \frac{10 \,{\left (B a b^{3} - 2 \, A b^{4}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{7} b} + \frac{60 \, B a b^{4} x^{5} - 120 \, A b^{5} x^{5} + 150 \, B a^{2} b^{3} x^{4} - 300 \, A a b^{4} x^{4} + 110 \, B a^{3} b^{2} x^{3} - 220 \, A a^{2} b^{3} x^{3} + 15 \, B a^{4} b x^{2} - 30 \, A a^{3} b^{2} x^{2} - 3 \, B a^{5} x + 6 \, A a^{4} b x - 2 \, A a^{5}}{6 \,{\left (b x^{2} + a x\right )}^{3} a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^2*x^4),x, algorithm="giac")

[Out]

10*(B*a*b^2 - 2*A*b^3)*ln(abs(x))/a^7 - 10*(B*a*b^3 - 2*A*b^4)*ln(abs(b*x + a))/
(a^7*b) + 1/6*(60*B*a*b^4*x^5 - 120*A*b^5*x^5 + 150*B*a^2*b^3*x^4 - 300*A*a*b^4*
x^4 + 110*B*a^3*b^2*x^3 - 220*A*a^2*b^3*x^3 + 15*B*a^4*b*x^2 - 30*A*a^3*b^2*x^2
- 3*B*a^5*x + 6*A*a^4*b*x - 2*A*a^5)/((b*x^2 + a*x)^3*a^6)