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

Optimal. Leaf size=43 \[ -\frac{\log (x) (A b-a B)}{a^2}+\frac{(A b-a B) \log (a+b x)}{a^2}-\frac{A}{a x} \]

[Out]

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

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Rubi [A]  time = 0.0750495, antiderivative size = 43, 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 \[ -\frac{\log (x) (A b-a B)}{a^2}+\frac{(A b-a B) \log (a+b x)}{a^2}-\frac{A}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.7459, size = 34, normalized size = 0.79 \[ - \frac{A}{a x} - \frac{\left (A b - B a\right ) \log{\left (x \right )}}{a^{2}} + \frac{\left (A b - B a\right ) \log{\left (a + b x \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0272837, size = 42, normalized size = 0.98 \[ \frac{\log (x) (a B-A b)}{a^2}+\frac{(A b-a B) \log (a+b x)}{a^2}-\frac{A}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 51, normalized size = 1.2 \[ -{\frac{A}{ax}}-{\frac{A\ln \left ( x \right ) b}{{a}^{2}}}+{\frac{\ln \left ( x \right ) B}{a}}+{\frac{\ln \left ( bx+a \right ) Ab}{{a}^{2}}}-{\frac{\ln \left ( bx+a \right ) B}{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^2/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.37365, size = 58, normalized size = 1.35 \[ -\frac{{\left (B a - A b\right )} \log \left (b x + a\right )}{a^{2}} + \frac{{\left (B a - A b\right )} \log \left (x\right )}{a^{2}} - \frac{A}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.206465, size = 55, normalized size = 1.28 \[ -\frac{{\left (B a - A b\right )} x \log \left (b x + a\right ) -{\left (B a - A b\right )} x \log \left (x\right ) + A a}{a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.00614, size = 95, normalized size = 2.21 \[ - \frac{A}{a x} + \frac{\left (- A b + B a\right ) \log{\left (x + \frac{- A a b + B a^{2} - a \left (- A b + B a\right )}{- 2 A b^{2} + 2 B a b} \right )}}{a^{2}} - \frac{\left (- A b + B a\right ) \log{\left (x + \frac{- A a b + B a^{2} + a \left (- A b + B a\right )}{- 2 A b^{2} + 2 B a b} \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.411661, size = 69, normalized size = 1.6 \[ \frac{{\left (B a - A b\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} - \frac{A}{a x} - \frac{{\left (B a b - A b^{2}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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