3.12 \(\int \frac{x}{(-a+x) (-b+x) (-c+x)} \, dx\)
Optimal. Leaf size=68 \[ \frac{a \log (a-x)}{(a-b) (a-c)}-\frac{b \log (b-x)}{(a-b) (b-c)}+\frac{c \log (c-x)}{(a-c) (b-c)} \]
[Out]
(a*Log[a - x])/((a - b)*(a - c)) - (b*Log[b - x])/((a - b)*(b - c)) + (c*Log[c - x])/((a - c)*(b - c))
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Rubi [A] time = 0.0515693, antiderivative size = 68, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used =
{148} \[ \frac{a \log (a-x)}{(a-b) (a-c)}-\frac{b \log (b-x)}{(a-b) (b-c)}+\frac{c \log (c-x)}{(a-c) (b-c)} \]
Antiderivative was successfully verified.
[In]
Int[x/((-a + x)*(-b + x)*(-c + x)),x]
[Out]
(a*Log[a - x])/((a - b)*(a - c)) - (b*Log[b - x])/((a - b)*(b - c)) + (c*Log[c - x])/((a - c)*(b - c))
Rule 148
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g
, h, m}, x] && (IntegersQ[m, n, p] || (IGtQ[n, 0] && IGtQ[p, 0]))
Rubi steps
\begin{align*} \int \frac{x}{(-a+x) (-b+x) (-c+x)} \, dx &=\int \left (-\frac{a}{(a-b) (a-c) (a-x)}+\frac{b}{(a-b) (b-c) (b-x)}+\frac{c}{(a-c) (-b+c) (c-x)}\right ) \, dx\\ &=\frac{a \log (a-x)}{(a-b) (a-c)}-\frac{b \log (b-x)}{(a-b) (b-c)}+\frac{c \log (c-x)}{(a-c) (b-c)}\\ \end{align*}
Mathematica [A] time = 0.0300064, size = 62, normalized size = 0.91 \[ \frac{a (b-c) \log (x-a)+b (c-a) \log (x-b)+c (a-b) \log (x-c)}{(a-b) (a-c) (b-c)} \]
Antiderivative was successfully verified.
[In]
Integrate[x/((-a + x)*(-b + x)*(-c + x)),x]
[Out]
(a*(b - c)*Log[-a + x] + b*(-a + c)*Log[-b + x] + (a - b)*c*Log[-c + x])/((a - b)*(a - c)*(b - c))
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Maple [A] time = 0.008, size = 69, normalized size = 1. \begin{align*} -{\frac{b\ln \left ( -b+x \right ) }{ \left ( a-b \right ) \left ( b-c \right ) }}+{\frac{c\ln \left ( -c+x \right ) }{ \left ( b-c \right ) \left ( a-c \right ) }}+{\frac{a\ln \left ( -a+x \right ) }{ \left ( a-b \right ) \left ( a-c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x/(-a+x)/(-b+x)/(-c+x),x)
[Out]
-b/(a-b)/(b-c)*ln(-b+x)+c/(b-c)/(a-c)*ln(-c+x)+1/(a-b)/(a-c)*a*ln(-a+x)
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Maxima [A] time = 0.961421, size = 105, normalized size = 1.54 \begin{align*} \frac{a \log \left (-a + x\right )}{a^{2} - a b -{\left (a - b\right )} c} - \frac{b \log \left (-b + x\right )}{a b - b^{2} -{\left (a - b\right )} c} + \frac{c \log \left (-c + x\right )}{a b -{\left (a + b\right )} c + c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(-a+x)/(-b+x)/(-c+x),x, algorithm="maxima")
[Out]
a*log(-a + x)/(a^2 - a*b - (a - b)*c) - b*log(-b + x)/(a*b - b^2 - (a - b)*c) + c*log(-c + x)/(a*b - (a + b)*c
+ c^2)
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Fricas [A] time = 2.76227, size = 166, normalized size = 2.44 \begin{align*} \frac{{\left (a - b\right )} c \log \left (-c + x\right ) +{\left (a b - a c\right )} \log \left (-a + x\right ) -{\left (a b - b c\right )} \log \left (-b + x\right )}{a^{2} b - a b^{2} +{\left (a - b\right )} c^{2} -{\left (a^{2} - b^{2}\right )} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(-a+x)/(-b+x)/(-c+x),x, algorithm="fricas")
[Out]
((a - b)*c*log(-c + x) + (a*b - a*c)*log(-a + x) - (a*b - b*c)*log(-b + x))/(a^2*b - a*b^2 + (a - b)*c^2 - (a^
2 - b^2)*c)
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Sympy [B] time = 85.2064, size = 4216, normalized size = 62. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(-a+x)/(-b+x)/(-c+x),x)
[Out]
a*log(x + (-a**8*b**3/((a - b)**2*(a - c)**2) + a**8*b**2*c/((a - b)**2*(a - c)**2) + a**8*b*c**2/((a - b)**2*
(a - c)**2) - a**8*c**3/((a - b)**2*(a - c)**2) + a**7*b**4/((a - b)**2*(a - c)**2) + 8*a**7*b**3*c/((a - b)**
2*(a - c)**2) - 18*a**7*b**2*c**2/((a - b)**2*(a - c)**2) + 8*a**7*b*c**3/((a - b)**2*(a - c)**2) + a**7*c**4/
((a - b)**2*(a - c)**2) + a**6*b**5/((a - b)**2*(a - c)**2) - 18*a**6*b**4*c/((a - b)**2*(a - c)**2) + 17*a**6
*b**3*c**2/((a - b)**2*(a - c)**2) - a**6*b**3/((a - b)*(a - c)) + 17*a**6*b**2*c**3/((a - b)**2*(a - c)**2) +
a**6*b**2*c/((a - b)*(a - c)) - 18*a**6*b*c**4/((a - b)**2*(a - c)**2) + a**6*b*c**2/((a - b)*(a - c)) + a**6
*c**5/((a - b)**2*(a - c)**2) - a**6*c**3/((a - b)*(a - c)) - a**5*b**6/((a - b)**2*(a - c)**2) + 8*a**5*b**5*
c/((a - b)**2*(a - c)**2) + 17*a**5*b**4*c**2/((a - b)**2*(a - c)**2) + 2*a**5*b**4/((a - b)*(a - c)) - 48*a**
5*b**3*c**3/((a - b)**2*(a - c)**2) - a**5*b**3*c/((a - b)*(a - c)) + 17*a**5*b**2*c**4/((a - b)**2*(a - c)**2
) - 2*a**5*b**2*c**2/((a - b)*(a - c)) + 8*a**5*b*c**5/((a - b)**2*(a - c)**2) - a**5*b*c**3/((a - b)*(a - c))
- a**5*c**6/((a - b)**2*(a - c)**2) + 2*a**5*c**4/((a - b)*(a - c)) + a**4*b**6*c/((a - b)**2*(a - c)**2) - 1
8*a**4*b**5*c**2/((a - b)**2*(a - c)**2) - a**4*b**5/((a - b)*(a - c)) + 17*a**4*b**4*c**3/((a - b)**2*(a - c)
**2) - a**4*b**4*c/((a - b)*(a - c)) + 17*a**4*b**3*c**4/((a - b)**2*(a - c)**2) + 2*a**4*b**3*c**2/((a - b)*(
a - c)) - 18*a**4*b**2*c**5/((a - b)**2*(a - c)**2) + 2*a**4*b**2*c**3/((a - b)*(a - c)) + a**4*b**2*c + a**4*
b*c**6/((a - b)**2*(a - c)**2) - a**4*b*c**4/((a - b)*(a - c)) + a**4*b*c**2 - a**4*c**5/((a - b)*(a - c)) + a
**3*b**6*c**2/((a - b)**2*(a - c)**2) + 8*a**3*b**5*c**3/((a - b)**2*(a - c)**2) + a**3*b**5*c/((a - b)*(a - c
)) - 18*a**3*b**4*c**4/((a - b)**2*(a - c)**2) - 2*a**3*b**4*c**2/((a - b)*(a - c)) + 8*a**3*b**3*c**5/((a - b
)**2*(a - c)**2) + 2*a**3*b**3*c**3/((a - b)*(a - c)) - 4*a**3*b**3*c + a**3*b**2*c**6/((a - b)**2*(a - c)**2)
- 2*a**3*b**2*c**4/((a - b)*(a - c)) + 2*a**3*b**2*c**2 + a**3*b*c**5/((a - b)*(a - c)) - 4*a**3*b*c**3 - a**
2*b**6*c**3/((a - b)**2*(a - c)**2) + a**2*b**5*c**4/((a - b)**2*(a - c)**2) + a**2*b**5*c**2/((a - b)*(a - c)
) + a**2*b**4*c**5/((a - b)**2*(a - c)**2) - a**2*b**4*c**3/((a - b)*(a - c)) + a**2*b**4*c - a**2*b**3*c**6/(
(a - b)**2*(a - c)**2) - a**2*b**3*c**4/((a - b)*(a - c)) + 2*a**2*b**3*c**2 + a**2*b**2*c**5/((a - b)*(a - c)
) + 2*a**2*b**2*c**3 + a**2*b*c**4 - a*b**5*c**3/((a - b)*(a - c)) + 2*a*b**4*c**4/((a - b)*(a - c)) + a*b**4*
c**2 - a*b**3*c**5/((a - b)*(a - c)) - 4*a*b**3*c**3 + a*b**2*c**4)/(2*a**3*b**3 - 3*a**3*b**2*c - 3*a**3*b*c*
*2 + 2*a**3*c**3 - 3*a**2*b**3*c + 12*a**2*b**2*c**2 - 3*a**2*b*c**3 - 3*a*b**3*c**2 - 3*a*b**2*c**3 + 2*b**3*
c**3))/((a - b)*(a - c)) - b*log(x + (-a**6*b**5/((a - b)**2*(b - c)**2) + a**6*b**4*c/((a - b)**2*(b - c)**2)
+ a**6*b**3*c**2/((a - b)**2*(b - c)**2) - a**6*b**2*c**3/((a - b)**2*(b - c)**2) + a**5*b**6/((a - b)**2*(b
- c)**2) + 8*a**5*b**5*c/((a - b)**2*(b - c)**2) - 18*a**5*b**4*c**2/((a - b)**2*(b - c)**2) + a**5*b**4/((a -
b)*(b - c)) + 8*a**5*b**3*c**3/((a - b)**2*(b - c)**2) - a**5*b**3*c/((a - b)*(b - c)) + a**5*b**2*c**4/((a -
b)**2*(b - c)**2) - a**5*b**2*c**2/((a - b)*(b - c)) + a**5*b*c**3/((a - b)*(b - c)) + a**4*b**7/((a - b)**2*
(b - c)**2) - 18*a**4*b**6*c/((a - b)**2*(b - c)**2) + 17*a**4*b**5*c**2/((a - b)**2*(b - c)**2) - 2*a**4*b**5
/((a - b)*(b - c)) + 17*a**4*b**4*c**3/((a - b)**2*(b - c)**2) + a**4*b**4*c/((a - b)*(b - c)) - 18*a**4*b**3*
c**4/((a - b)**2*(b - c)**2) + 2*a**4*b**3*c**2/((a - b)*(b - c)) + a**4*b**2*c**5/((a - b)**2*(b - c)**2) + a
**4*b**2*c**3/((a - b)*(b - c)) + a**4*b**2*c - 2*a**4*b*c**4/((a - b)*(b - c)) + a**4*b*c**2 - a**3*b**8/((a
- b)**2*(b - c)**2) + 8*a**3*b**7*c/((a - b)**2*(b - c)**2) + 17*a**3*b**6*c**2/((a - b)**2*(b - c)**2) + a**3
*b**6/((a - b)*(b - c)) - 48*a**3*b**5*c**3/((a - b)**2*(b - c)**2) + a**3*b**5*c/((a - b)*(b - c)) + 17*a**3*
b**4*c**4/((a - b)**2*(b - c)**2) - 2*a**3*b**4*c**2/((a - b)*(b - c)) + 8*a**3*b**3*c**5/((a - b)**2*(b - c)*
*2) - 2*a**3*b**3*c**3/((a - b)*(b - c)) - 4*a**3*b**3*c - a**3*b**2*c**6/((a - b)**2*(b - c)**2) + a**3*b**2*
c**4/((a - b)*(b - c)) + 2*a**3*b**2*c**2 + a**3*b*c**5/((a - b)*(b - c)) - 4*a**3*b*c**3 + a**2*b**8*c/((a -
b)**2*(b - c)**2) - 18*a**2*b**7*c**2/((a - b)**2*(b - c)**2) + 17*a**2*b**6*c**3/((a - b)**2*(b - c)**2) - a*
*2*b**6*c/((a - b)*(b - c)) + 17*a**2*b**5*c**4/((a - b)**2*(b - c)**2) + 2*a**2*b**5*c**2/((a - b)*(b - c)) -
18*a**2*b**4*c**5/((a - b)**2*(b - c)**2) - 2*a**2*b**4*c**3/((a - b)*(b - c)) + a**2*b**4*c + a**2*b**3*c**6
/((a - b)**2*(b - c)**2) + 2*a**2*b**3*c**4/((a - b)*(b - c)) + 2*a**2*b**3*c**2 - a**2*b**2*c**5/((a - b)*(b
- c)) + 2*a**2*b**2*c**3 + a**2*b*c**4 + a*b**8*c**2/((a - b)**2*(b - c)**2) + 8*a*b**7*c**3/((a - b)**2*(b -
c)**2) - 18*a*b**6*c**4/((a - b)**2*(b - c)**2) - a*b**6*c**2/((a - b)*(b - c)) + 8*a*b**5*c**5/((a - b)**2*(b
- c)**2) + a*b**5*c**3/((a - b)*(b - c)) + a*b**4*c**6/((a - b)**2*(b - c)**2) + a*b**4*c**4/((a - b)*(b - c)
) + a*b**4*c**2 - a*b**3*c**5/((a - b)*(b - c)) - 4*a*b**3*c**3 + a*b**2*c**4 - b**8*c**3/((a - b)**2*(b - c)*
*2) + b**7*c**4/((a - b)**2*(b - c)**2) + b**6*c**5/((a - b)**2*(b - c)**2) + b**6*c**3/((a - b)*(b - c)) - b*
*5*c**6/((a - b)**2*(b - c)**2) - 2*b**5*c**4/((a - b)*(b - c)) + b**4*c**5/((a - b)*(b - c)))/(2*a**3*b**3 -
3*a**3*b**2*c - 3*a**3*b*c**2 + 2*a**3*c**3 - 3*a**2*b**3*c + 12*a**2*b**2*c**2 - 3*a**2*b*c**3 - 3*a*b**3*c**
2 - 3*a*b**2*c**3 + 2*b**3*c**3))/((a - b)*(b - c)) + c*log(x + (-a**6*b**3*c**2/((a - c)**2*(b - c)**2) + a**
6*b**2*c**3/((a - c)**2*(b - c)**2) + a**6*b*c**4/((a - c)**2*(b - c)**2) - a**6*c**5/((a - c)**2*(b - c)**2)
+ a**5*b**4*c**2/((a - c)**2*(b - c)**2) + 8*a**5*b**3*c**3/((a - c)**2*(b - c)**2) - a**5*b**3*c/((a - c)*(b
- c)) - 18*a**5*b**2*c**4/((a - c)**2*(b - c)**2) + a**5*b**2*c**2/((a - c)*(b - c)) + 8*a**5*b*c**5/((a - c)*
*2*(b - c)**2) + a**5*b*c**3/((a - c)*(b - c)) + a**5*c**6/((a - c)**2*(b - c)**2) - a**5*c**4/((a - c)*(b - c
)) + a**4*b**5*c**2/((a - c)**2*(b - c)**2) - 18*a**4*b**4*c**3/((a - c)**2*(b - c)**2) + 2*a**4*b**4*c/((a -
c)*(b - c)) + 17*a**4*b**3*c**4/((a - c)**2*(b - c)**2) - a**4*b**3*c**2/((a - c)*(b - c)) + 17*a**4*b**2*c**5
/((a - c)**2*(b - c)**2) - 2*a**4*b**2*c**3/((a - c)*(b - c)) + a**4*b**2*c - 18*a**4*b*c**6/((a - c)**2*(b -
c)**2) - a**4*b*c**4/((a - c)*(b - c)) + a**4*b*c**2 + a**4*c**7/((a - c)**2*(b - c)**2) + 2*a**4*c**5/((a - c
)*(b - c)) - a**3*b**6*c**2/((a - c)**2*(b - c)**2) + 8*a**3*b**5*c**3/((a - c)**2*(b - c)**2) - a**3*b**5*c/(
(a - c)*(b - c)) + 17*a**3*b**4*c**4/((a - c)**2*(b - c)**2) - a**3*b**4*c**2/((a - c)*(b - c)) - 48*a**3*b**3
*c**5/((a - c)**2*(b - c)**2) + 2*a**3*b**3*c**3/((a - c)*(b - c)) - 4*a**3*b**3*c + 17*a**3*b**2*c**6/((a - c
)**2*(b - c)**2) + 2*a**3*b**2*c**4/((a - c)*(b - c)) + 2*a**3*b**2*c**2 + 8*a**3*b*c**7/((a - c)**2*(b - c)**
2) - a**3*b*c**5/((a - c)*(b - c)) - 4*a**3*b*c**3 - a**3*c**8/((a - c)**2*(b - c)**2) - a**3*c**6/((a - c)*(b
- c)) + a**2*b**6*c**3/((a - c)**2*(b - c)**2) - 18*a**2*b**5*c**4/((a - c)**2*(b - c)**2) + a**2*b**5*c**2/(
(a - c)*(b - c)) + 17*a**2*b**4*c**5/((a - c)**2*(b - c)**2) - 2*a**2*b**4*c**3/((a - c)*(b - c)) + a**2*b**4*
c + 17*a**2*b**3*c**6/((a - c)**2*(b - c)**2) + 2*a**2*b**3*c**4/((a - c)*(b - c)) + 2*a**2*b**3*c**2 - 18*a**
2*b**2*c**7/((a - c)**2*(b - c)**2) - 2*a**2*b**2*c**5/((a - c)*(b - c)) + 2*a**2*b**2*c**3 + a**2*b*c**8/((a
- c)**2*(b - c)**2) + a**2*b*c**6/((a - c)*(b - c)) + a**2*b*c**4 + a*b**6*c**4/((a - c)**2*(b - c)**2) + 8*a*
b**5*c**5/((a - c)**2*(b - c)**2) + a*b**5*c**3/((a - c)*(b - c)) - 18*a*b**4*c**6/((a - c)**2*(b - c)**2) - a
*b**4*c**4/((a - c)*(b - c)) + a*b**4*c**2 + 8*a*b**3*c**7/((a - c)**2*(b - c)**2) - a*b**3*c**5/((a - c)*(b -
c)) - 4*a*b**3*c**3 + a*b**2*c**8/((a - c)**2*(b - c)**2) + a*b**2*c**6/((a - c)*(b - c)) + a*b**2*c**4 - b**
6*c**5/((a - c)**2*(b - c)**2) + b**5*c**6/((a - c)**2*(b - c)**2) - b**5*c**4/((a - c)*(b - c)) + b**4*c**7/(
(a - c)**2*(b - c)**2) + 2*b**4*c**5/((a - c)*(b - c)) - b**3*c**8/((a - c)**2*(b - c)**2) - b**3*c**6/((a - c
)*(b - c)))/(2*a**3*b**3 - 3*a**3*b**2*c - 3*a**3*b*c**2 + 2*a**3*c**3 - 3*a**2*b**3*c + 12*a**2*b**2*c**2 - 3
*a**2*b*c**3 - 3*a*b**3*c**2 - 3*a*b**2*c**3 + 2*b**3*c**3))/((a - c)*(b - c))
________________________________________________________________________________________
Giac [A] time = 1.06321, size = 109, normalized size = 1.6 \begin{align*} \frac{a \log \left ({\left | -a + x \right |}\right )}{a^{2} - a b - a c + b c} - \frac{b \log \left ({\left | -b + x \right |}\right )}{a b - b^{2} - a c + b c} + \frac{c \log \left ({\left | -c + x \right |}\right )}{a b - a c - b c + c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(-a+x)/(-b+x)/(-c+x),x, algorithm="giac")
[Out]
a*log(abs(-a + x))/(a^2 - a*b - a*c + b*c) - b*log(abs(-b + x))/(a*b - b^2 - a*c + b*c) + c*log(abs(-c + x))/(
a*b - a*c - b*c + c^2)