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

Optimal. Leaf size=111 \[ -\frac{\log (x) (4 A b-a B)}{a^5}+\frac{(4 A b-a B) \log (a+b x)}{a^5}-\frac{3 A b-a B}{a^4 (a+b x)}-\frac{A}{a^4 x}-\frac{2 A b-a B}{2 a^3 (a+b x)^2}-\frac{A b-a B}{3 a^2 (a+b x)^3} \]

[Out]

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

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Rubi [A]  time = 0.215459, antiderivative size = 111, 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 \[ -\frac{\log (x) (4 A b-a B)}{a^5}+\frac{(4 A b-a B) \log (a+b x)}{a^5}-\frac{3 A b-a B}{a^4 (a+b x)}-\frac{A}{a^4 x}-\frac{2 A b-a B}{2 a^3 (a+b x)^2}-\frac{A b-a B}{3 a^2 (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 48.6543, size = 95, normalized size = 0.86 \[ - \frac{A}{a^{4} x} - \frac{A b - B a}{3 a^{2} \left (a + b x\right )^{3}} - \frac{2 A b - B a}{2 a^{3} \left (a + b x\right )^{2}} - \frac{3 A b - B a}{a^{4} \left (a + b x\right )} - \frac{\left (4 A b - B a\right ) \log{\left (x \right )}}{a^{5}} + \frac{\left (4 A b - B a\right ) \log{\left (a + b x \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.116866, size = 102, normalized size = 0.92 \[ \frac{\frac{2 a^3 (a B-A b)}{(a+b x)^3}+\frac{3 a^2 (a B-2 A b)}{(a+b x)^2}+\frac{6 a (a B-3 A b)}{a+b x}+6 \log (x) (a B-4 A b)+6 (4 A b-a B) \log (a+b x)-\frac{6 a A}{x}}{6 a^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.018, size = 132, normalized size = 1.2 \[ -{\frac{A}{{a}^{4}x}}-4\,{\frac{Ab\ln \left ( x \right ) }{{a}^{5}}}+{\frac{\ln \left ( x \right ) B}{{a}^{4}}}-{\frac{Ab}{3\,{a}^{2} \left ( bx+a \right ) ^{3}}}+{\frac{B}{3\,a \left ( bx+a \right ) ^{3}}}-{\frac{Ab}{{a}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{B}{2\,{a}^{2} \left ( bx+a \right ) ^{2}}}-3\,{\frac{Ab}{{a}^{4} \left ( bx+a \right ) }}+{\frac{B}{{a}^{3} \left ( bx+a \right ) }}+4\,{\frac{\ln \left ( bx+a \right ) Ab}{{a}^{5}}}-{\frac{\ln \left ( bx+a \right ) B}{{a}^{4}}} \]

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)^2,x)

[Out]

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

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Maxima [A]  time = 0.702434, size = 181, normalized size = 1.63 \[ -\frac{6 \, A a^{3} - 6 \,{\left (B a b^{2} - 4 \, A b^{3}\right )} x^{3} - 15 \,{\left (B a^{2} b - 4 \, A a b^{2}\right )} x^{2} - 11 \,{\left (B a^{3} - 4 \, A a^{2} b\right )} x}{6 \,{\left (a^{4} b^{3} x^{4} + 3 \, a^{5} b^{2} x^{3} + 3 \, a^{6} b x^{2} + a^{7} x\right )}} - \frac{{\left (B a - 4 \, A b\right )} \log \left (b x + a\right )}{a^{5}} + \frac{{\left (B a - 4 \, A b\right )} \log \left (x\right )}{a^{5}} \]

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^2),x, algorithm="maxima")

[Out]

-1/6*(6*A*a^3 - 6*(B*a*b^2 - 4*A*b^3)*x^3 - 15*(B*a^2*b - 4*A*a*b^2)*x^2 - 11*(B
*a^3 - 4*A*a^2*b)*x)/(a^4*b^3*x^4 + 3*a^5*b^2*x^3 + 3*a^6*b*x^2 + a^7*x) - (B*a
- 4*A*b)*log(b*x + a)/a^5 + (B*a - 4*A*b)*log(x)/a^5

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Fricas [A]  time = 0.282765, size = 360, normalized size = 3.24 \[ -\frac{6 \, A a^{4} - 6 \,{\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 15 \,{\left (B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} - 11 \,{\left (B a^{4} - 4 \, A a^{3} b\right )} x + 6 \,{\left ({\left (B a b^{3} - 4 \, A b^{4}\right )} x^{4} + 3 \,{\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} + 3 \,{\left (B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} +{\left (B a^{4} - 4 \, A a^{3} b\right )} x\right )} \log \left (b x + a\right ) - 6 \,{\left ({\left (B a b^{3} - 4 \, A b^{4}\right )} x^{4} + 3 \,{\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} + 3 \,{\left (B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} +{\left (B a^{4} - 4 \, A a^{3} b\right )} x\right )} \log \left (x\right )}{6 \,{\left (a^{5} b^{3} x^{4} + 3 \, a^{6} b^{2} x^{3} + 3 \, a^{7} b x^{2} + a^{8} x\right )}} \]

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^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.64282, size = 204, normalized size = 1.84 \[ \frac{- 6 A a^{3} + x^{3} \left (- 24 A b^{3} + 6 B a b^{2}\right ) + x^{2} \left (- 60 A a b^{2} + 15 B a^{2} b\right ) + x \left (- 44 A a^{2} b + 11 B a^{3}\right )}{6 a^{7} x + 18 a^{6} b x^{2} + 18 a^{5} b^{2} x^{3} + 6 a^{4} b^{3} x^{4}} + \frac{\left (- 4 A b + B a\right ) \log{\left (x + \frac{- 4 A a b + B a^{2} - a \left (- 4 A b + B a\right )}{- 8 A b^{2} + 2 B a b} \right )}}{a^{5}} - \frac{\left (- 4 A b + B a\right ) \log{\left (x + \frac{- 4 A a b + B a^{2} + a \left (- 4 A b + B a\right )}{- 8 A b^{2} + 2 B a b} \right )}}{a^{5}} \]

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)**2,x)

[Out]

(-6*A*a**3 + x**3*(-24*A*b**3 + 6*B*a*b**2) + x**2*(-60*A*a*b**2 + 15*B*a**2*b)
+ x*(-44*A*a**2*b + 11*B*a**3))/(6*a**7*x + 18*a**6*b*x**2 + 18*a**5*b**2*x**3 +
6*a**4*b**3*x**4) + (-4*A*b + B*a)*log(x + (-4*A*a*b + B*a**2 - a*(-4*A*b + B*a
))/(-8*A*b**2 + 2*B*a*b))/a**5 - (-4*A*b + B*a)*log(x + (-4*A*a*b + B*a**2 + a*(
-4*A*b + B*a))/(-8*A*b**2 + 2*B*a*b))/a**5

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GIAC/XCAS [A]  time = 0.269359, size = 165, normalized size = 1.49 \[ \frac{{\left (B a - 4 \, A b\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{5}} - \frac{{\left (B a b - 4 \, A b^{2}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac{6 \, A a^{4} - 6 \,{\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 15 \,{\left (B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} - 11 \,{\left (B a^{4} - 4 \, A a^{3} b\right )} x}{6 \,{\left (b x + a\right )}^{3} a^{5} x} \]

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^2),x, algorithm="giac")

[Out]

(B*a - 4*A*b)*ln(abs(x))/a^5 - (B*a*b - 4*A*b^2)*ln(abs(b*x + a))/(a^5*b) - 1/6*
(6*A*a^4 - 6*(B*a^2*b^2 - 4*A*a*b^3)*x^3 - 15*(B*a^3*b - 4*A*a^2*b^2)*x^2 - 11*(
B*a^4 - 4*A*a^3*b)*x)/((b*x + a)^3*a^5*x)