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

Optimal. Leaf size=62 \[ \frac{b \log (x) (A b-a B)}{a^3}-\frac{b (A b-a B) \log (a+b x)}{a^3}+\frac{A b-a B}{a^2 x}-\frac{A}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.100591, antiderivative size = 62, 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 \[ \frac{b \log (x) (A b-a B)}{a^3}-\frac{b (A b-a B) \log (a+b x)}{a^3}+\frac{A b-a B}{a^2 x}-\frac{A}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.4616, size = 53, normalized size = 0.85 \[ - \frac{A}{2 a x^{2}} + \frac{A b - B a}{a^{2} x} + \frac{b \left (A b - B a\right ) \log{\left (x \right )}}{a^{3}} - \frac{b \left (A b - B a\right ) \log{\left (a + b x \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0518142, size = 58, normalized size = 0.94 \[ \frac{-\frac{a (a A+2 a B x-2 A b x)}{x^2}+2 b \log (x) (A b-a B)+2 b (a B-A b) \log (a+b x)}{2 a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 75, normalized size = 1.2 \[ -{\frac{A}{2\,a{x}^{2}}}+{\frac{Ab}{{a}^{2}x}}-{\frac{B}{ax}}+{\frac{A\ln \left ( x \right ){b}^{2}}{{a}^{3}}}-{\frac{bB\ln \left ( x \right ) }{{a}^{2}}}-{\frac{{b}^{2}\ln \left ( bx+a \right ) A}{{a}^{3}}}+{\frac{b\ln \left ( bx+a \right ) B}{{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^3/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.39457, size = 85, normalized size = 1.37 \[ \frac{{\left (B a b - A b^{2}\right )} \log \left (b x + a\right )}{a^{3}} - \frac{{\left (B a b - A b^{2}\right )} \log \left (x\right )}{a^{3}} - \frac{A a + 2 \,{\left (B a - A b\right )} x}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.213394, size = 93, normalized size = 1.5 \[ \frac{2 \,{\left (B a b - A b^{2}\right )} x^{2} \log \left (b x + a\right ) - 2 \,{\left (B a b - A b^{2}\right )} x^{2} \log \left (x\right ) - A a^{2} - 2 \,{\left (B a^{2} - A a b\right )} x}{2 \, a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.58721, size = 131, normalized size = 2.11 \[ - \frac{A a + x \left (- 2 A b + 2 B a\right )}{2 a^{2} x^{2}} - \frac{b \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{2} + B a^{2} b - a b \left (- A b + B a\right )}{- 2 A b^{3} + 2 B a b^{2}} \right )}}{a^{3}} + \frac{b \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{2} + B a^{2} b + a b \left (- A b + B a\right )}{- 2 A b^{3} + 2 B a b^{2}} \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.30958, size = 101, normalized size = 1.63 \[ -\frac{{\left (B a b - A b^{2}\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} + \frac{{\left (B a b^{2} - A b^{3}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{3} b} - \frac{A a^{2} + 2 \,{\left (B a^{2} - A a b\right )} x}{2 \, a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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