3.1.7 \(\int \frac {b+a x}{(-p+x) (-q+x)} \, dx\) [7]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, 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 = {78} \begin {gather*} \frac {(a p+b) \log (p-x)}{p-q}-\frac {(a q+b) \log (q-x)}{p-q} \end {gather*}

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 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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 = 0.01, size = 34, normalized size = 0.85 \begin {gather*} \frac {(b+a p) \log (-p+x)-(b+a q) \log (-q+x)}{p-q} \end {gather*}

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)

________________________________________________________________________________________

Mathics [A]
time = 4.07, size = 34, normalized size = 0.85 \begin {gather*} \frac {-\text {Log}\left [-q+x\right ] \left (a q+b\right )+\text {Log}\left [-p+x\right ] \left (a p+b\right )}{p-q} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.04, size = 43, normalized size = 1.08

method	result	size
norman	\(\frac {\left (a p +b \right ) \ln \left (p -x \right )}{p -q}-\frac {\left (a q +b \right ) \ln \left (q -x \right )}{p -q}\)	\(41\)
default	\(\frac {\left (-a q -b \right ) \ln \left (q -x \right )}{p -q}+\frac {\left (a p +b \right ) \ln \left (p -x \right )}{p -q}\)	\(43\)
risch	\(-\frac {\ln \left (-q +x \right ) a q}{p -q}-\frac {\ln \left (-q +x \right ) b}{p -q}+\frac {\ln \left (p -x \right ) a p}{p -q}+\frac {\ln \left (p -x \right ) b}{p -q}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+b)/(-p+x)/(-q+x),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 40, normalized size = 1.00 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 34, normalized size = 0.85 \begin {gather*} \frac {{\left (a p + b\right )} \log \left (-p + x\right ) - {\left (a q + b\right )} \log \left (-q + x\right )}{p - q} \end {gather*}

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)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (26) = 52\)
time = 0.55, size = 144, normalized size = 3.60 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 35, normalized size = 0.88 \begin {gather*} \frac {\left (a q+b\right ) \ln \left |x-q\right |}{-p+q}+\frac {\left (a p+b\right ) \ln \left |x-p\right |}{p-q} \end {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 0.25, size = 40, normalized size = 1.00 \begin {gather*} \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} \end {gather*}

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)

________________________________________________________________________________________