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

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

[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 = 0.150553, 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 \[ -\frac{A \log (a+b x)}{a^6}+\frac{A \log (x)}{a^6}+\frac{A}{a^5 (a+b x)}+\frac{A}{2 a^4 (a+b x)^2}+\frac{A}{3 a^3 (a+b x)^3}+\frac{A}{4 a^2 (a+b x)^4}+\frac{A b-a B}{5 a b (a+b x)^5} \]

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

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Rubi in Sympy [A]  time = 48.9083, size = 88, normalized size = 0.86 \[ \frac{A}{4 a^{2} \left (a + b x\right )^{4}} + \frac{A}{3 a^{3} \left (a + b x\right )^{3}} + \frac{A}{2 a^{4} \left (a + b x\right )^{2}} + \frac{A}{a^{5} \left (a + b x\right )} + \frac{A \log{\left (x \right )}}{a^{6}} - \frac{A \log{\left (a + b x \right )}}{a^{6}} + \frac{A b - B a}{5 a b \left (a + b x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

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 + (A*b - B*a)/(5*a*b*(a +
b*x)**5)

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Mathematica [A]  time = 0.11433, size = 89, normalized size = 0.87 \[ \frac{\frac{a \left (-12 a^5 B+137 a^4 A b+385 a^3 A b^2 x+470 a^2 A b^3 x^2+270 a A b^4 x^3+60 A b^5 x^4\right )}{b (a+b x)^5}-60 A \log (a+b x)+60 A \log (x)}{60 a^6} \]

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 = 0.016, size = 98, normalized size = 1. \[{\frac{A\ln \left ( x \right ) }{{a}^{6}}}+{\frac{A}{5\,a \left ( bx+a \right ) ^{5}}}-{\frac{B}{5\, \left ( bx+a \right ) ^{5}b}}+{\frac{A}{4\,{a}^{2} \left ( bx+a \right ) ^{4}}}-{\frac{A\ln \left ( bx+a \right ) }{{a}^{6}}}+{\frac{A}{{a}^{5} \left ( bx+a \right ) }}+{\frac{A}{2\,{a}^{4} \left ( bx+a \right ) ^{2}}}+{\frac{A}{3\,{a}^{3} \left ( bx+a \right ) ^{3}}} \]

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+1/4*A/a^2/(b*x+a)^4-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

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Maxima [A]  time = 0.693347, size = 185, normalized size = 1.81 \[ \frac{60 \, A b^{5} x^{4} + 270 \, A a b^{4} x^{3} + 470 \, A a^{2} b^{3} x^{2} + 385 \, A a^{3} b^{2} x - 12 \, B a^{5} + 137 \, A a^{4} b}{60 \,{\left (a^{5} b^{6} x^{5} + 5 \, a^{6} b^{5} x^{4} + 10 \, a^{7} b^{4} x^{3} + 10 \, a^{8} b^{3} x^{2} + 5 \, a^{9} b^{2} x + a^{10} b\right )}} - \frac{A \log \left (b x + a\right )}{a^{6}} + \frac{A \log \left (x\right )}{a^{6}} \]

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)^3*x),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 [A]  time = 0.30097, size = 338, normalized size = 3.31 \[ \frac{60 \, A a b^{5} x^{4} + 270 \, A a^{2} b^{4} x^{3} + 470 \, A a^{3} b^{3} x^{2} + 385 \, A a^{4} b^{2} x - 12 \, B a^{6} + 137 \, A a^{5} b - 60 \,{\left (A b^{6} x^{5} + 5 \, A a b^{5} x^{4} + 10 \, A a^{2} b^{4} x^{3} + 10 \, A a^{3} b^{3} x^{2} + 5 \, A a^{4} b^{2} x + A a^{5} b\right )} \log \left (b x + a\right ) + 60 \,{\left (A b^{6} x^{5} + 5 \, A a b^{5} x^{4} + 10 \, A a^{2} b^{4} x^{3} + 10 \, A a^{3} b^{3} x^{2} + 5 \, A a^{4} b^{2} x + A a^{5} b\right )} \log \left (x\right )}{60 \,{\left (a^{6} b^{6} x^{5} + 5 \, a^{7} b^{5} x^{4} + 10 \, a^{8} b^{4} x^{3} + 10 \, a^{9} b^{3} x^{2} + 5 \, a^{10} b^{2} x + a^{11} b\right )}} \]

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)^3*x),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 = 3.78258, size = 141, normalized size = 1.38 \[ \frac{A \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{6}} + \frac{137 A a^{4} b + 385 A a^{3} b^{2} x + 470 A a^{2} b^{3} x^{2} + 270 A a b^{4} x^{3} + 60 A b^{5} x^{4} - 12 B a^{5}}{60 a^{10} b + 300 a^{9} b^{2} x + 600 a^{8} b^{3} x^{2} + 600 a^{7} b^{4} x^{3} + 300 a^{6} b^{5} x^{4} + 60 a^{5} b^{6} x^{5}} \]

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/XCAS [A]  time = 0.26999, size = 128, normalized size = 1.25 \[ -\frac{A{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{6}} + \frac{A{\rm ln}\left ({\left | x \right |}\right )}{a^{6}} + \frac{60 \, A a b^{5} x^{4} + 270 \, A a^{2} b^{4} x^{3} + 470 \, A a^{3} b^{3} x^{2} + 385 \, A a^{4} b^{2} x - 12 \, B a^{6} + 137 \, A a^{5} b}{60 \,{\left (b x + a\right )}^{5} a^{6} b} \]

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)^3*x),x, algorithm="giac")

[Out]

-A*ln(abs(b*x + a))/a^6 + A*ln(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)