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

Optimal. Leaf size=88 \[ -\frac{\log (x) (3 A b-a B)}{a^4}+\frac{(3 A b-a B) \log (a+b x)}{a^4}-\frac{2 A b-a B}{a^3 (a+b x)}-\frac{A}{a^3 x}-\frac{A b-a B}{2 a^2 (a+b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.164836, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{\log (x) (3 A b-a B)}{a^4}+\frac{(3 A b-a B) \log (a+b x)}{a^4}-\frac{2 A b-a B}{a^3 (a+b x)}-\frac{A}{a^3 x}-\frac{A b-a B}{2 a^2 (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 29.3914, size = 75, normalized size = 0.85 \[ - \frac{A}{a^{3} x} - \frac{A b - B a}{2 a^{2} \left (a + b x\right )^{2}} - \frac{2 A b - B a}{a^{3} \left (a + b x\right )} - \frac{\left (3 A b - B a\right ) \log{\left (x \right )}}{a^{4}} + \frac{\left (3 A b - B a\right ) \log{\left (a + b x \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0796838, size = 81, normalized size = 0.92 \[ \frac{\frac{a^2 (a B-A b)}{(a+b x)^2}+\frac{2 a (a B-2 A b)}{a+b x}+2 \log (x) (a B-3 A b)+2 (3 A b-a B) \log (a+b x)-\frac{2 a A}{x}}{2 a^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 105, normalized size = 1.2 \[ -{\frac{A}{{a}^{3}x}}-3\,{\frac{A\ln \left ( x \right ) b}{{a}^{4}}}+{\frac{\ln \left ( x \right ) B}{{a}^{3}}}-2\,{\frac{Ab}{{a}^{3} \left ( bx+a \right ) }}+{\frac{B}{{a}^{2} \left ( bx+a \right ) }}+3\,{\frac{\ln \left ( bx+a \right ) Ab}{{a}^{4}}}-{\frac{\ln \left ( bx+a \right ) B}{{a}^{3}}}-{\frac{Ab}{2\,{a}^{2} \left ( bx+a \right ) ^{2}}}+{\frac{B}{2\,a \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^2/(b*x+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.34988, size = 135, normalized size = 1.53 \[ -\frac{2 \, A a^{2} - 2 \,{\left (B a b - 3 \, A b^{2}\right )} x^{2} - 3 \,{\left (B a^{2} - 3 \, A a b\right )} x}{2 \,{\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} - \frac{{\left (B a - 3 \, A b\right )} \log \left (b x + a\right )}{a^{4}} + \frac{{\left (B a - 3 \, A b\right )} \log \left (x\right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.21139, size = 252, normalized size = 2.86 \[ -\frac{2 \, A a^{3} - 2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} - 3 \,{\left (B a^{3} - 3 \, A a^{2} b\right )} x + 2 \,{\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{3} + 2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} +{\left (B a^{3} - 3 \, A a^{2} b\right )} x\right )} \log \left (b x + a\right ) - 2 \,{\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{3} + 2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} +{\left (B a^{3} - 3 \, A a^{2} b\right )} x\right )} \log \left (x\right )}{2 \,{\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.83799, size = 168, normalized size = 1.91 \[ \frac{- 2 A a^{2} + x^{2} \left (- 6 A b^{2} + 2 B a b\right ) + x \left (- 9 A a b + 3 B a^{2}\right )}{2 a^{5} x + 4 a^{4} b x^{2} + 2 a^{3} b^{2} x^{3}} + \frac{\left (- 3 A b + B a\right ) \log{\left (x + \frac{- 3 A a b + B a^{2} - a \left (- 3 A b + B a\right )}{- 6 A b^{2} + 2 B a b} \right )}}{a^{4}} - \frac{\left (- 3 A b + B a\right ) \log{\left (x + \frac{- 3 A a b + B a^{2} + a \left (- 3 A b + B a\right )}{- 6 A b^{2} + 2 B a b} \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*A*a**2 + x**2*(-6*A*b**2 + 2*B*a*b) + x*(-9*A*a*b + 3*B*a**2))/(2*a**5*x + 4
*a**4*b*x**2 + 2*a**3*b**2*x**3) + (-3*A*b + B*a)*log(x + (-3*A*a*b + B*a**2 - a
*(-3*A*b + B*a))/(-6*A*b**2 + 2*B*a*b))/a**4 - (-3*A*b + B*a)*log(x + (-3*A*a*b
+ B*a**2 + a*(-3*A*b + B*a))/(-6*A*b**2 + 2*B*a*b))/a**4

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GIAC/XCAS [A]  time = 0.256167, size = 134, normalized size = 1.52 \[ \frac{{\left (B a - 3 \, A b\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{{\left (B a b - 3 \, A b^{2}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac{2 \, A a^{3} - 2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} - 3 \,{\left (B a^{3} - 3 \, A a^{2} b\right )} x}{2 \,{\left (b x + a\right )}^{2} a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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