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 = 0.0268245, 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, Rules used = {72} \[ \frac{(a p+b) \log (p-x)}{p-q}-\frac{(a q+b) \log (q-x)}{p-q} \]

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)

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{b+a x}{(-p+x) (-q+x)} \, dx &=\int \left (\frac{-b-a p}{(p-q) (p-x)}+\frac{b+a q}{(p-q) (q-x)}\right ) \, dx\\ &=\frac{(b+a p) \log (p-x)}{p-q}-\frac{(b+a q) \log (q-x)}{p-q}\\ \end{align*}

Mathematica [A]  time = 0.0171396, 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 = 0.007, size = 66, normalized size = 1.7 \begin{align*} -{\frac{\ln \left ( -q+x \right ) aq}{p-q}}-{\frac{\ln \left ( -q+x \right ) b}{p-q}}+{\frac{\ln \left ( -p+x \right ) ap}{p-q}}+{\frac{\ln \left ( -p+x \right ) b}{p-q}} \end{align*}

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

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Maxima [A]  time = 0.936923, size = 54, normalized size = 1.35 \begin{align*} \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} \end{align*}

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 = 1.94076, size = 77, normalized size = 1.92 \begin{align*} \frac{{\left (a p + b\right )} \log \left (-p + x\right ) -{\left (a q + b\right )} \log \left (-q + x\right )}{p - q} \end{align*}

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 [B]  time = 0.655328, size = 144, normalized size = 3.6 \begin{align*} \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} \end{align*}

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 [A]  time = 1.06803, size = 57, normalized size = 1.42 \begin{align*} \frac{{\left (a p + b\right )} \log \left ({\left | -p + x \right |}\right )}{p - q} - \frac{{\left (a q + b\right )} \log \left ({\left | -q + x \right |}\right )}{p - q} \end{align*}

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