Integrand size = 22, antiderivative size = 234 \[ \int \frac {1}{(a+b x)^2 (c+d x)^2 (e+f x)^2} \, dx=-\frac {b^3}{(b c-a d)^2 (b e-a f)^2 (a+b x)}-\frac {d^3}{(b c-a d)^2 (d e-c f)^2 (c+d x)}-\frac {f^3}{(b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {2 b^3 (b d e+b c f-2 a d f) \log (a+b x)}{(b c-a d)^3 (b e-a f)^3}+\frac {2 d^3 (b d e-2 b c f+a d f) \log (c+d x)}{(b c-a d)^3 (d e-c f)^3}+\frac {2 f^3 (2 b d e-b c f-a d f) \log (e+f x)}{(b e-a f)^3 (d e-c f)^3} \] Output:
-b^3/(-a*d+b*c)^2/(-a*f+b*e)^2/(b*x+a)-d^3/(-a*d+b*c)^2/(-c*f+d*e)^2/(d*x+ c)-f^3/(-a*f+b*e)^2/(-c*f+d*e)^2/(f*x+e)-2*b^3*(-2*a*d*f+b*c*f+b*d*e)*ln(b *x+a)/(-a*d+b*c)^3/(-a*f+b*e)^3+2*d^3*(a*d*f-2*b*c*f+b*d*e)*ln(d*x+c)/(-a* d+b*c)^3/(-c*f+d*e)^3+2*f^3*(-a*d*f-b*c*f+2*b*d*e)*ln(f*x+e)/(-a*f+b*e)^3/ (-c*f+d*e)^3
Time = 0.29 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(a+b x)^2 (c+d x)^2 (e+f x)^2} \, dx=-\frac {b^3}{(b c-a d)^2 (b e-a f)^2 (a+b x)}-\frac {d^3}{(b c-a d)^2 (d e-c f)^2 (c+d x)}-\frac {f^3}{(b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {2 b^3 (b d e+b c f-2 a d f) \log (a+b x)}{(b c-a d)^3 (b e-a f)^3}-\frac {2 d^3 (b d e-2 b c f+a d f) \log (c+d x)}{(b c-a d)^3 (-d e+c f)^3}-\frac {2 f^3 (-2 b d e+b c f+a d f) \log (e+f x)}{(b e-a f)^3 (d e-c f)^3} \] Input:
Integrate[1/((a + b*x)^2*(c + d*x)^2*(e + f*x)^2),x]
Output:
-(b^3/((b*c - a*d)^2*(b*e - a*f)^2*(a + b*x))) - d^3/((b*c - a*d)^2*(d*e - c*f)^2*(c + d*x)) - f^3/((b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)) - (2*b^3* (b*d*e + b*c*f - 2*a*d*f)*Log[a + b*x])/((b*c - a*d)^3*(b*e - a*f)^3) - (2 *d^3*(b*d*e - 2*b*c*f + a*d*f)*Log[c + d*x])/((b*c - a*d)^3*(-(d*e) + c*f) ^3) - (2*f^3*(-2*b*d*e + b*c*f + a*d*f)*Log[e + f*x])/((b*e - a*f)^3*(d*e - c*f)^3)
Time = 0.67 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {99, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{(a+b x)^2 (c+d x)^2 (e+f x)^2} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {2 b^4 (-2 a d f+b c f+b d e)}{(a+b x) (b c-a d)^3 (b e-a f)^3}+\frac {b^4}{(a+b x)^2 (b c-a d)^2 (b e-a f)^2}-\frac {2 d^4 (a d f-2 b c f+b d e)}{(c+d x) (b c-a d)^3 (c f-d e)^3}+\frac {d^4}{(c+d x)^2 (b c-a d)^2 (c f-d e)^2}-\frac {2 f^4 (a d f+b c f-2 b d e)}{(e+f x) (b e-a f)^3 (d e-c f)^3}+\frac {f^4}{(e+f x)^2 (b e-a f)^2 (d e-c f)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b^3}{(a+b x) (b c-a d)^2 (b e-a f)^2}-\frac {2 b^3 \log (a+b x) (-2 a d f+b c f+b d e)}{(b c-a d)^3 (b e-a f)^3}-\frac {d^3}{(c+d x) (b c-a d)^2 (d e-c f)^2}+\frac {2 d^3 \log (c+d x) (a d f-2 b c f+b d e)}{(b c-a d)^3 (d e-c f)^3}-\frac {f^3}{(e+f x) (b e-a f)^2 (d e-c f)^2}+\frac {2 f^3 \log (e+f x) (-a d f-b c f+2 b d e)}{(b e-a f)^3 (d e-c f)^3}\) |
Input:
Int[1/((a + b*x)^2*(c + d*x)^2*(e + f*x)^2),x]
Output:
-(b^3/((b*c - a*d)^2*(b*e - a*f)^2*(a + b*x))) - d^3/((b*c - a*d)^2*(d*e - c*f)^2*(c + d*x)) - f^3/((b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)) - (2*b^3* (b*d*e + b*c*f - 2*a*d*f)*Log[a + b*x])/((b*c - a*d)^3*(b*e - a*f)^3) + (2 *d^3*(b*d*e - 2*b*c*f + a*d*f)*Log[c + d*x])/((b*c - a*d)^3*(d*e - c*f)^3) + (2*f^3*(2*b*d*e - b*c*f - a*d*f)*Log[e + f*x])/((b*e - a*f)^3*(d*e - c* f)^3)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 0.64 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {d^{3}}{\left (a d -b c \right )^{2} \left (c f -d e \right )^{2} \left (x d +c \right )}+\frac {2 d^{3} \left (a d f -2 b c f +b d e \right ) \ln \left (x d +c \right )}{\left (a d -b c \right )^{3} \left (c f -d e \right )^{3}}-\frac {b^{3}}{\left (a f -b e \right )^{2} \left (a d -b c \right )^{2} \left (b x +a \right )}+\frac {2 b^{3} \left (2 a d f -b c f -b d e \right ) \ln \left (b x +a \right )}{\left (a f -b e \right )^{3} \left (a d -b c \right )^{3}}-\frac {f^{3}}{\left (a f -b e \right )^{2} \left (c f -d e \right )^{2} \left (f x +e \right )}-\frac {2 f^{3} \left (a d f +b c f -2 b d e \right ) \ln \left (f x +e \right )}{\left (a f -b e \right )^{3} \left (c f -d e \right )^{3}}\) | \(235\) |
norman | \(\text {Expression too large to display}\) | \(1232\) |
risch | \(\text {Expression too large to display}\) | \(3320\) |
parallelrisch | \(\text {Expression too large to display}\) | \(4270\) |
Input:
int(1/(b*x+a)^2/(d*x+c)^2/(f*x+e)^2,x,method=_RETURNVERBOSE)
Output:
-d^3/(a*d-b*c)^2/(c*f-d*e)^2/(d*x+c)+2*d^3*(a*d*f-2*b*c*f+b*d*e)/(a*d-b*c) ^3/(c*f-d*e)^3*ln(d*x+c)-b^3/(a*f-b*e)^2/(a*d-b*c)^2/(b*x+a)+2*b^3*(2*a*d* f-b*c*f-b*d*e)/(a*f-b*e)^3/(a*d-b*c)^3*ln(b*x+a)-f^3/(a*f-b*e)^2/(c*f-d*e) ^2/(f*x+e)-2*f^3*(a*d*f+b*c*f-2*b*d*e)/(a*f-b*e)^3/(c*f-d*e)^3*ln(f*x+e)
Timed out. \[ \int \frac {1}{(a+b x)^2 (c+d x)^2 (e+f x)^2} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x+a)^2/(d*x+c)^2/(f*x+e)^2,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{(a+b x)^2 (c+d x)^2 (e+f x)^2} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x+a)**2/(d*x+c)**2/(f*x+e)**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 2096 vs. \(2 (234) = 468\).
Time = 0.11 (sec) , antiderivative size = 2096, normalized size of antiderivative = 8.96 \[ \int \frac {1}{(a+b x)^2 (c+d x)^2 (e+f x)^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(b*x+a)^2/(d*x+c)^2/(f*x+e)^2,x, algorithm="maxima")
Output:
-2*(b^4*d*e + (b^4*c - 2*a*b^3*d)*f)*log(b*x + a)/((b^6*c^3 - 3*a*b^5*c^2* d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*e^3 - 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d^3)*e^2*f + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3)*e*f^2 - (a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^ 5*b*c*d^2 - a^6*d^3)*f^3) + 2*(b*d^4*e - (2*b*c*d^3 - a*d^4)*f)*log(d*x + c)/((b^3*c^3*d^3 - 3*a*b^2*c^2*d^4 + 3*a^2*b*c*d^5 - a^3*d^6)*e^3 - 3*(b^3 *c^4*d^2 - 3*a*b^2*c^3*d^3 + 3*a^2*b*c^2*d^4 - a^3*c*d^5)*e^2*f + 3*(b^3*c ^5*d - 3*a*b^2*c^4*d^2 + 3*a^2*b*c^3*d^3 - a^3*c^2*d^4)*e*f^2 - (b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3)*f^3) + 2*(2*b*d*e*f^3 - (b *c + a*d)*f^4)*log(f*x + e)/(b^3*d^3*e^6 + a^3*c^3*f^6 - 3*(b^3*c*d^2 + a* b^2*d^3)*e^5*f + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^4*f^2 - (b^3* c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^3*f^3 + 3*(a*b^2*c^3 + 3* a^2*b*c^2*d + a^3*c*d^2)*e^2*f^4 - 3*(a^2*b*c^3 + a^3*c^2*d)*e*f^5) - ((b^ 3*c*d^2 + a*b^2*d^3)*e^3 - 2*(b^3*c^2*d + a^2*b*d^3)*e^2*f + (b^3*c^3 + a^ 3*d^3)*e*f^2 + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*f^3 + 2*(b^3*d^3*e^ 2*f - (b^3*c*d^2 + a*b^2*d^3)*e*f^2 + (b^3*c^2*d - a*b^2*c*d^2 + a^2*b*d^3 )*f^3)*x^2 + (2*b^3*d^3*e^3 - (b^3*c*d^2 + a*b^2*d^3)*e^2*f - (b^3*c^2*d + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + 2*a^3*d^3)*f^ 3)*x)/((a*b^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 + a^3*b^2*c*d^4)*e^5 - 2*(a*b^4* c^4*d - a^2*b^3*c^3*d^2 - a^3*b^2*c^2*d^3 + a^4*b*c*d^4)*e^4*f + (a*b^4...
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 1635, normalized size of antiderivative = 6.99 \[ \int \frac {1}{(a+b x)^2 (c+d x)^2 (e+f x)^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(b*x+a)^2/(d*x+c)^2/(f*x+e)^2,x, algorithm="giac")
Output:
-b^7/((b^8*c^2*e^2 - 2*a*b^7*c*d*e^2 + a^2*b^6*d^2*e^2 - 2*a*b^7*c^2*e*f + 4*a^2*b^6*c*d*e*f - 2*a^3*b^5*d^2*e*f + a^2*b^6*c^2*f^2 - 2*a^3*b^5*c*d*f ^2 + a^4*b^4*d^2*f^2)*(b*x + a)) + (b^4*d*e + b^4*c*f - 2*a*b^3*d*f)*log(b ^2*c*e/(b*x + a)^2 + b*d*e/(b*x + a) - a*b*d*e/(b*x + a)^2 + b*c*f/(b*x + a) - a*b*c*f/(b*x + a)^2 - 2*a*d*f/(b*x + a) + a^2*d*f/(b*x + a)^2 + d*f)/ (b^6*c^3*e^3 - 3*a*b^5*c^2*d*e^3 + 3*a^2*b^4*c*d^2*e^3 - a^3*b^3*d^3*e^3 - 3*a*b^5*c^3*e^2*f + 9*a^2*b^4*c^2*d*e^2*f - 9*a^3*b^3*c*d^2*e^2*f + 3*a^4 *b^2*d^3*e^2*f + 3*a^2*b^4*c^3*e*f^2 - 9*a^3*b^3*c^2*d*e*f^2 + 9*a^4*b^2*c *d^2*e*f^2 - 3*a^5*b*d^3*e*f^2 - a^3*b^3*c^3*f^3 + 3*a^4*b^2*c^2*d*f^3 - 3 *a^5*b*c*d^2*f^3 + a^6*d^3*f^3) - 2*I*(b^6*d^4*e^4 - 2*b^6*c*d^3*e^3*f - 2 *a*b^5*d^4*e^3*f + 6*a*b^5*c*d^3*e^2*f^2 - 2*b^6*c^3*d*e*f^3 + 6*a*b^5*c^2 *d^2*e*f^3 - 12*a^2*b^4*c*d^3*e*f^3 + 4*a^3*b^3*d^4*e*f^3 + b^6*c^4*f^4 - 2*a*b^5*c^3*d*f^4 + 4*a^3*b^3*c*d^3*f^4 - 2*a^4*b^2*d^4*f^4)*arctan((2*I*b ^3*c*e/(b*x + a) - 2*I*a*b^2*d*e/(b*x + a) + I*b^2*d*e - 2*I*a*b^2*c*f/(b* x + a) + I*b^2*c*f - 2*I*a*b*d*f + 2*I*a^2*b*d*f/(b*x + a))/abs(-b^2*d*e + b^2*c*f))/((b^6*c^3*d^2*e^5 - 3*a*b^5*c^2*d^3*e^5 + 3*a^2*b^4*c*d^4*e^5 - a^3*b^3*d^5*e^5 - 2*b^6*c^4*d*e^4*f + 3*a*b^5*c^3*d^2*e^4*f + 3*a^2*b^4*c ^2*d^3*e^4*f - 7*a^3*b^3*c*d^4*e^4*f + 3*a^4*b^2*d^5*e^4*f + b^6*c^5*e^3*f ^2 + 3*a*b^5*c^4*d*e^3*f^2 - 12*a^2*b^4*c^3*d^2*e^3*f^2 + 8*a^3*b^3*c^2*d^ 3*e^3*f^2 + 3*a^4*b^2*c*d^4*e^3*f^2 - 3*a^5*b*d^5*e^3*f^2 - 3*a*b^5*c^5...
Time = 4.82 (sec) , antiderivative size = 1940, normalized size of antiderivative = 8.29 \[ \int \frac {1}{(a+b x)^2 (c+d x)^2 (e+f x)^2} \, dx=\text {Too large to display} \] Input:
int(1/((e + f*x)^2*(a + b*x)^2*(c + d*x)^2),x)
Output:
- ((a*b^2*c^3*f^3 + a*b^2*d^3*e^3 + a^3*c*d^2*f^3 + b^3*c*d^2*e^3 + a^3*d^ 3*e*f^2 + b^3*c^3*e*f^2 - 2*a^2*b*c^2*d*f^3 - 2*a^2*b*d^3*e^2*f - 2*b^3*c^ 2*d*e^2*f)/(a^2*b^2*c^4*f^4 + a^2*b^2*d^4*e^4 + a^4*c^2*d^2*f^4 + b^4*c^2* d^2*e^4 + a^4*d^4*e^2*f^2 + b^4*c^4*e^2*f^2 - 2*a*b^3*c*d^3*e^4 - 2*a^3*b* c^3*d*f^4 - 2*a*b^3*c^4*e*f^3 - 2*a^3*b*d^4*e^3*f - 2*a^4*c*d^3*e*f^3 - 2* b^4*c^3*d*e^3*f + 2*a*b^3*c^2*d^2*e^3*f + 2*a*b^3*c^3*d*e^2*f^2 + 2*a^2*b^ 2*c*d^3*e^3*f + 2*a^2*b^2*c^3*d*e*f^3 + 2*a^3*b*c*d^3*e^2*f^2 + 2*a^3*b*c^ 2*d^2*e*f^3 - 6*a^2*b^2*c^2*d^2*e^2*f^2) + (2*x^2*(a^2*b*d^3*f^3 + b^3*c^2 *d*f^3 + b^3*d^3*e^2*f - a*b^2*c*d^2*f^3 - a*b^2*d^3*e*f^2 - b^3*c*d^2*e*f ^2))/(a^2*b^2*c^4*f^4 + a^2*b^2*d^4*e^4 + a^4*c^2*d^2*f^4 + b^4*c^2*d^2*e^ 4 + a^4*d^4*e^2*f^2 + b^4*c^4*e^2*f^2 - 2*a*b^3*c*d^3*e^4 - 2*a^3*b*c^3*d* f^4 - 2*a*b^3*c^4*e*f^3 - 2*a^3*b*d^4*e^3*f - 2*a^4*c*d^3*e*f^3 - 2*b^4*c^ 3*d*e^3*f + 2*a*b^3*c^2*d^2*e^3*f + 2*a*b^3*c^3*d*e^2*f^2 + 2*a^2*b^2*c*d^ 3*e^3*f + 2*a^2*b^2*c^3*d*e*f^3 + 2*a^3*b*c*d^3*e^2*f^2 + 2*a^3*b*c^2*d^2* e*f^3 - 6*a^2*b^2*c^2*d^2*e^2*f^2) - (x*(a*b^2*c^2*d*f^3 - 2*b^3*c^3*f^3 - 2*b^3*d^3*e^3 - 2*a^3*d^3*f^3 + a^2*b*c*d^2*f^3 + a*b^2*d^3*e^2*f + a^2*b *d^3*e*f^2 + b^3*c*d^2*e^2*f + b^3*c^2*d*e*f^2))/(a^2*b^2*c^4*f^4 + a^2*b^ 2*d^4*e^4 + a^4*c^2*d^2*f^4 + b^4*c^2*d^2*e^4 + a^4*d^4*e^2*f^2 + b^4*c^4* e^2*f^2 - 2*a*b^3*c*d^3*e^4 - 2*a^3*b*c^3*d*f^4 - 2*a*b^3*c^4*e*f^3 - 2*a^ 3*b*d^4*e^3*f - 2*a^4*c*d^3*e*f^3 - 2*b^4*c^3*d*e^3*f + 2*a*b^3*c^2*d^2...
Time = 0.18 (sec) , antiderivative size = 8516, normalized size of antiderivative = 36.39 \[ \int \frac {1}{(a+b x)^2 (c+d x)^2 (e+f x)^2} \, dx =\text {Too large to display} \] Input:
int(1/(b*x+a)^2/(d*x+c)^2/(f*x+e)^2,x)
Output:
(4*log(a + b*x)*a**3*b**3*c**4*d**2*e*f**5 + 4*log(a + b*x)*a**3*b**3*c**4 *d**2*f**6*x - 12*log(a + b*x)*a**3*b**3*c**3*d**3*e**2*f**4 - 8*log(a + b *x)*a**3*b**3*c**3*d**3*e*f**5*x + 4*log(a + b*x)*a**3*b**3*c**3*d**3*f**6 *x**2 + 12*log(a + b*x)*a**3*b**3*c**2*d**4*e**3*f**3 - 12*log(a + b*x)*a* *3*b**3*c**2*d**4*e*f**5*x**2 - 4*log(a + b*x)*a**3*b**3*c*d**5*e**4*f**2 + 8*log(a + b*x)*a**3*b**3*c*d**5*e**3*f**3*x + 12*log(a + b*x)*a**3*b**3* c*d**5*e**2*f**4*x**2 - 4*log(a + b*x)*a**3*b**3*d**6*e**4*f**2*x - 4*log( a + b*x)*a**3*b**3*d**6*e**3*f**3*x**2 + 2*log(a + b*x)*a**2*b**4*c**5*d*e *f**5 + 2*log(a + b*x)*a**2*b**4*c**5*d*f**6*x - 4*log(a + b*x)*a**2*b**4* c**4*d**2*e**2*f**4 + 2*log(a + b*x)*a**2*b**4*c**4*d**2*e*f**5*x + 6*log( a + b*x)*a**2*b**4*c**4*d**2*f**6*x**2 - 16*log(a + b*x)*a**2*b**4*c**3*d* *3*e**2*f**4*x - 12*log(a + b*x)*a**2*b**4*c**3*d**3*e*f**5*x**2 + 4*log(a + b*x)*a**2*b**4*c**3*d**3*f**6*x**3 + 4*log(a + b*x)*a**2*b**4*c**2*d**4 *e**4*f**2 + 16*log(a + b*x)*a**2*b**4*c**2*d**4*e**3*f**3*x - 12*log(a + b*x)*a**2*b**4*c**2*d**4*e*f**5*x**3 - 2*log(a + b*x)*a**2*b**4*c*d**5*e** 5*f - 2*log(a + b*x)*a**2*b**4*c*d**5*e**4*f**2*x + 12*log(a + b*x)*a**2*b **4*c*d**5*e**3*f**3*x**2 + 12*log(a + b*x)*a**2*b**4*c*d**5*e**2*f**4*x** 3 - 2*log(a + b*x)*a**2*b**4*d**6*e**5*f*x - 6*log(a + b*x)*a**2*b**4*d**6 *e**4*f**2*x**2 - 4*log(a + b*x)*a**2*b**4*d**6*e**3*f**3*x**3 - 2*log(a + b*x)*a*b**5*c**6*e*f**5 - 2*log(a + b*x)*a*b**5*c**6*f**6*x + 2*log(a ...