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]

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

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Rubi [A]  time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.41, size = 34, normalized size = 0.85 \[ \frac {{\left (a p + b\right )} \log \left (-p + x\right ) - {\left (a q + b\right )} \log \left (-q + x\right )}{p - q} \]

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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giac [A]  time = 1.18, size = 42, normalized size = 1.05 \[ \frac {{\left (a p + b\right )} \log \left ({\left | -p + x \right |}\right )}{p - q} - \frac {{\left (a q + b\right )} \log \left ({\left | -q + x \right |}\right )}{p - q} \]

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)

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maple [A]  time = 0.01, size = 66, normalized size = 1.65 \[ \frac {a p \ln \left (-p +x \right )}{p -q}-\frac {a q \ln \left (-q +x \right )}{p -q}+\frac {b \ln \left (-p +x \right )}{p -q}-\frac {b \ln \left (-q +x \right )}{p -q} \]

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(-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 = 0.45, size = 40, normalized size = 1.00 \[ \frac {{\left (a p + b\right )} \log \left (-p + x\right )}{p - q} - \frac {{\left (a q + b\right )} \log \left (-q + x\right )}{p - q} \]

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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mupad [B]  time = 0.25, size = 40, normalized size = 1.00 \[ \frac {\ln \left (x-p\right )\,\left (b+a\,p\right )}{p-q}-\frac {\ln \left (x-q\right )\,\left (b+a\,q\right )}{p-q} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + a*x)/((p - x)*(q - x)),x)

[Out]

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

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sympy [B]  time = 0.88, size = 144, normalized size = 3.60 \[ \frac {\left (a p + b\right ) \log {\left (x + \frac {- 2 a p q - b p - b q - \frac {p^{2} \left (a p + b\right )}{p - q} + \frac {2 p q \left (a p + b\right )}{p - q} - \frac {q^{2} \left (a p + b\right )}{p - q}}{a p + a q + 2 b} \right )}}{p - q} - \frac {\left (a q + b\right ) \log {\left (x + \frac {- 2 a p q - b p - b q + \frac {p^{2} \left (a q + b\right )}{p - q} - \frac {2 p q \left (a q + b\right )}{p - q} + \frac {q^{2} \left (a q + b\right )}{p - q}}{a p + a q + 2 b} \right )}}{p - q} \]

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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