3.7 \(\int \frac{b+a x}{(-p+x) (-q+x)} \, dx\)

Optimal. Leaf size=40 \[ \frac{(a p+b) \log (p-x)}{p-q}-\frac{(a q+b) \log (q-x)}{p-q} \]

[Out]

((b + a*p)*Log[p - x])/(p - q) - ((b + a*q)*Log[q - x])/(p - q)

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Rubi [A]  time = 0.0637384, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{(a p+b) \log (p-x)}{p-q}-\frac{(a q+b) \log (q-x)}{p-q} \]

Antiderivative was successfully verified.

[In]  Int[(b + a*x)/((-p + x)*(-q + x)),x]

[Out]

((b + a*p)*Log[p - x])/(p - q) - ((b + a*q)*Log[q - x])/(p - q)

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Rubi in Sympy [A]  time = 4.94215, size = 26, normalized size = 0.65 \[ \frac{\left (a p + b\right ) \log{\left (p - x \right )}}{p - q} - \frac{\left (a q + b\right ) \log{\left (q - x \right )}}{p - q} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x+b)/(-p+x)/(-q+x),x)

[Out]

(a*p + b)*log(p - x)/(p - q) - (a*q + b)*log(q - x)/(p - q)

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Mathematica [A]  time = 0.0225985, size = 34, normalized size = 0.85 \[ \frac{(a p+b) \log (x-p)-(a q+b) \log (x-q)}{p-q} \]

Antiderivative was successfully verified.

[In]  Integrate[(b + a*x)/((-p + x)*(-q + x)),x]

[Out]

((b + a*p)*Log[-p + x] - (b + a*q)*Log[-q + x])/(p - q)

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Maple [A]  time = 0.005, size = 66, normalized size = 1.7 \[{\frac{\ln \left ( -p+x \right ) ap}{p-q}}+{\frac{\ln \left ( -p+x \right ) b}{p-q}}-{\frac{\ln \left ( -q+x \right ) aq}{p-q}}-{\frac{\ln \left ( -q+x \right ) b}{p-q}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x+b)/(-p+x)/(-q+x),x)

[Out]

1/(p-q)*ln(-p+x)*a*p+1/(p-q)*ln(-p+x)*b-1/(p-q)*ln(-q+x)*a*q-1/(p-q)*ln(-q+x)*b

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Maxima [A]  time = 1.37392, size = 54, normalized size = 1.35 \[ \frac{{\left (a p + b\right )} \log \left (-p + x\right )}{p - q} - \frac{{\left (a q + b\right )} \log \left (-q + x\right )}{p - q} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + b)/((p - x)*(q - x)),x, algorithm="maxima")

[Out]

(a*p + b)*log(-p + x)/(p - q) - (a*q + b)*log(-q + x)/(p - q)

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Fricas [A]  time = 0.201657, size = 46, normalized size = 1.15 \[ \frac{{\left (a p + b\right )} \log \left (-p + x\right ) -{\left (a q + b\right )} \log \left (-q + x\right )}{p - q} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + b)/((p - x)*(q - x)),x, algorithm="fricas")

[Out]

((a*p + b)*log(-p + x) - (a*q + b)*log(-q + x))/(p - q)

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Sympy [A]  time = 1.03251, size = 144, normalized size = 3.6 \[ \frac{\left (a p + b\right ) \log{\left (x + \frac{- 2 a p q - b p - b q - \frac{p^{2} \left (a p + b\right )}{p - q} + \frac{2 p q \left (a p + b\right )}{p - q} - \frac{q^{2} \left (a p + b\right )}{p - q}}{a p + a q + 2 b} \right )}}{p - q} - \frac{\left (a q + b\right ) \log{\left (x + \frac{- 2 a p q - b p - b q + \frac{p^{2} \left (a q + b\right )}{p - q} - \frac{2 p q \left (a q + b\right )}{p - q} + \frac{q^{2} \left (a q + b\right )}{p - q}}{a p + a q + 2 b} \right )}}{p - q} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x+b)/(-p+x)/(-q+x),x)

[Out]

(a*p + b)*log(x + (-2*a*p*q - b*p - b*q - p**2*(a*p + b)/(p - q) + 2*p*q*(a*p +
b)/(p - q) - q**2*(a*p + b)/(p - q))/(a*p + a*q + 2*b))/(p - q) - (a*q + b)*log(
x + (-2*a*p*q - b*p - b*q + p**2*(a*q + b)/(p - q) - 2*p*q*(a*q + b)/(p - q) + q
**2*(a*q + b)/(p - q))/(a*p + a*q + 2*b))/(p - q)

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GIAC/XCAS [A]  time = 0.209679, size = 57, normalized size = 1.42 \[ \frac{{\left (a p + b\right )}{\rm ln}\left ({\left | -p + x \right |}\right )}{p - q} - \frac{{\left (a q + b\right )}{\rm ln}\left ({\left | -q + x \right |}\right )}{p - q} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + b)/((p - x)*(q - x)),x, algorithm="giac")

[Out]

(a*p + b)*ln(abs(-p + x))/(p - q) - (a*q + b)*ln(abs(-q + x))/(p - q)