Integrand size = 46, antiderivative size = 234 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2} \, dx=-\frac {b^3}{(b c-a d)^2 (b e-a f)^2 (a+b x)}-\frac {d^3}{(b c-a d)^2 (d e-c f)^2 (c+d x)}-\frac {f^3}{(b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {2 b^3 (b d e+b c f-2 a d f) \log (a+b x)}{(b c-a d)^3 (b e-a f)^3}+\frac {2 d^3 (b d e-2 b c f+a d f) \log (c+d x)}{(b c-a d)^3 (d e-c f)^3}+\frac {2 f^3 (2 b d e-b c f-a d f) \log (e+f x)}{(b e-a f)^3 (d e-c f)^3} \]
-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 = 0.37 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2} \, dx=-\frac {b^3}{(b c-a d)^2 (b e-a f)^2 (a+b x)}-\frac {d^3}{(b c-a d)^2 (d e-c f)^2 (c+d x)}-\frac {f^3}{(b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {2 b^3 (b d e+b c f-2 a d f) \log (a+b x)}{(b c-a d)^3 (b e-a f)^3}-\frac {2 d^3 (b d e-2 b c f+a d f) \log (c+d x)}{(b c-a d)^3 (-d e+c f)^3}-\frac {2 f^3 (-2 b d e+b c f+a d f) \log (e+f x)}{(b e-a f)^3 (d e-c f)^3} \]
Integrate[(a*c*e + (b*c*e + a*d*e + a*c*f)*x + (b*d*e + b*c*f + a*d*f)*x^2 + b*d*f*x^3)^(-2),x]
-(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 = 0.63 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2462, 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.
\(\displaystyle \int \frac {1}{\left (x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3\right )^2} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (-\frac {2 b^4 (-2 a d f+b c f+b d e)}{(a+b x) (b c-a d)^3 (b e-a f)^3}+\frac {b^4}{(a+b x)^2 (b c-a d)^2 (b e-a f)^2}-\frac {2 d^4 (a d f-2 b c f+b d e)}{(c+d x) (b c-a d)^3 (c f-d e)^3}+\frac {d^4}{(c+d x)^2 (b c-a d)^2 (c f-d e)^2}-\frac {2 f^4 (a d f+b c f-2 b d e)}{(e+f x) (b e-a f)^3 (d e-c f)^3}+\frac {f^4}{(e+f x)^2 (b e-a f)^2 (d e-c f)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b^3}{(a+b x) (b c-a d)^2 (b e-a f)^2}-\frac {2 b^3 \log (a+b x) (-2 a d f+b c f+b d e)}{(b c-a d)^3 (b e-a f)^3}-\frac {d^3}{(c+d x) (b c-a d)^2 (d e-c f)^2}+\frac {2 d^3 \log (c+d x) (a d f-2 b c f+b d e)}{(b c-a d)^3 (d e-c f)^3}-\frac {f^3}{(e+f x) (b e-a f)^2 (d e-c f)^2}+\frac {2 f^3 \log (e+f x) (-a d f-b c f+2 b d e)}{(b e-a f)^3 (d e-c f)^3}\) |
-(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)
3.1.19.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.72 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {f^{3}}{\left (c f -e d \right )^{2} \left (a f -b e \right )^{2} \left (f x +e \right )}-\frac {2 f^{3} \left (a d f +f b c -2 b d e \right ) \ln \left (f x +e \right )}{\left (c f -e d \right )^{3} \left (a f -b e \right )^{3}}-\frac {d^{3}}{\left (c f -e d \right )^{2} \left (d a -b c \right )^{2} \left (d x +c \right )}+\frac {2 d^{3} \left (a d f -2 f b c +b d e \right ) \ln \left (d x +c \right )}{\left (c f -e d \right )^{3} \left (d a -b c \right )^{3}}-\frac {b^{3}}{\left (a f -b e \right )^{2} \left (d a -b c \right )^{2} \left (b x +a \right )}+\frac {2 b^{3} \left (2 a d f -f b c -b d e \right ) \ln \left (b x +a \right )}{\left (a f -b e \right )^{3} \left (d a -b c \right )^{3}}\) | \(235\) |
norman | \(\text {Expression too large to display}\) | \(1232\) |
risch | \(\text {Expression too large to display}\) | \(3349\) |
parallelrisch | \(\text {Expression too large to display}\) | \(4300\) |
int(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^2,x, method=_RETURNVERBOSE)
-f^3/(c*f-d*e)^2/(a*f-b*e)^2/(f*x+e)-2*f^3*(a*d*f+b*c*f-2*b*d*e)/(c*f-d*e) ^3/(a*f-b*e)^3*ln(f*x+e)-d^3/(c*f-d*e)^2/(a*d-b*c)^2/(d*x+c)+2*d^3*(a*d*f- 2*b*c*f+b*d*e)/(c*f-d*e)^3/(a*d-b*c)^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)
Timed out. \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2} \, dx=\text {Timed out} \]
integrate(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3 )^2,x, algorithm="fricas")
Timed out. \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2096 vs. \(2 (234) = 468\).
Time = 0.32 (sec) , antiderivative size = 2096, normalized size of antiderivative = 8.96 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2} \, dx=\text {Too large to display} \]
integrate(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3 )^2,x, algorithm="maxima")
-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...
Leaf count of result is larger than twice the leaf count of optimal. 1435 vs. \(2 (234) = 468\).
Time = 0.28 (sec) , antiderivative size = 1435, normalized size of antiderivative = 6.13 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2} \, dx=\text {Too large to display} \]
integrate(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3 )^2,x, algorithm="giac")
-2*(b^5*d*e + b^5*c*f - 2*a*b^4*d*f)*log(abs(b*x + a))/(b^7*c^3*e^3 - 3*a* b^6*c^2*d*e^3 + 3*a^2*b^5*c*d^2*e^3 - a^3*b^4*d^3*e^3 - 3*a*b^6*c^3*e^2*f + 9*a^2*b^5*c^2*d*e^2*f - 9*a^3*b^4*c*d^2*e^2*f + 3*a^4*b^3*d^3*e^2*f + 3* a^2*b^5*c^3*e*f^2 - 9*a^3*b^4*c^2*d*e*f^2 + 9*a^4*b^3*c*d^2*e*f^2 - 3*a^5* b^2*d^3*e*f^2 - a^3*b^4*c^3*f^3 + 3*a^4*b^3*c^2*d*f^3 - 3*a^5*b^2*c*d^2*f^ 3 + a^6*b*d^3*f^3) + 2*(b*d^5*e - 2*b*c*d^4*f + a*d^5*f)*log(abs(d*x + c)) /(b^3*c^3*d^4*e^3 - 3*a*b^2*c^2*d^5*e^3 + 3*a^2*b*c*d^6*e^3 - a^3*d^7*e^3 - 3*b^3*c^4*d^3*e^2*f + 9*a*b^2*c^3*d^4*e^2*f - 9*a^2*b*c^2*d^5*e^2*f + 3* a^3*c*d^6*e^2*f + 3*b^3*c^5*d^2*e*f^2 - 9*a*b^2*c^4*d^3*e*f^2 + 9*a^2*b*c^ 3*d^4*e*f^2 - 3*a^3*c^2*d^5*e*f^2 - b^3*c^6*d*f^3 + 3*a*b^2*c^5*d^2*f^3 - 3*a^2*b*c^4*d^3*f^3 + a^3*c^3*d^4*f^3) + 2*(2*b*d*e*f^4 - b*c*f^5 - a*d*f^ 5)*log(abs(f*x + e))/(b^3*d^3*e^6*f - 3*b^3*c*d^2*e^5*f^2 - 3*a*b^2*d^3*e^ 5*f^2 + 3*b^3*c^2*d*e^4*f^3 + 9*a*b^2*c*d^2*e^4*f^3 + 3*a^2*b*d^3*e^4*f^3 - b^3*c^3*e^3*f^4 - 9*a*b^2*c^2*d*e^3*f^4 - 9*a^2*b*c*d^2*e^3*f^4 - a^3*d^ 3*e^3*f^4 + 3*a*b^2*c^3*e^2*f^5 + 9*a^2*b*c^2*d*e^2*f^5 + 3*a^3*c*d^2*e^2* f^5 - 3*a^2*b*c^3*e*f^6 - 3*a^3*c^2*d*e*f^6 + a^3*c^3*f^7) - (2*b^3*d^3*e^ 2*f*x^2 - 2*b^3*c*d^2*e*f^2*x^2 - 2*a*b^2*d^3*e*f^2*x^2 + 2*b^3*c^2*d*f^3* x^2 - 2*a*b^2*c*d^2*f^3*x^2 + 2*a^2*b*d^3*f^3*x^2 + 2*b^3*d^3*e^3*x - b^3* c*d^2*e^2*f*x - a*b^2*d^3*e^2*f*x - b^3*c^2*d*e*f^2*x - a^2*b*d^3*e*f^2*x + 2*b^3*c^3*f^3*x - a*b^2*c^2*d*f^3*x - a^2*b*c*d^2*f^3*x + 2*a^3*d^3*f...
Time = 13.89 (sec) , antiderivative size = 1940, normalized size of antiderivative = 8.29 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2} \, dx=\text {Too large to display} \]
- ((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...