\(\int \frac {1}{(d+e x) (a+b x+c x^2)^4} \, dx\) [497]

Optimal result

Integrand size = 20, antiderivative size = 773 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^4} \, dx=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^3}-\frac {6 a c \left (b^2-4 a c\right ) e^3-b \left (10 c^3 d^3+3 b^3 e^3-c^2 d e (15 b d-22 a e)+b c e^2 (2 b d-17 a e)\right )-c (2 c d-b e) \left (10 c^2 d^2-3 b^2 e^2-2 c e (5 b d-11 a e)\right ) x}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )^2}+\frac {b^5 c d e^4+2 b^6 e^5-64 a^3 c^3 e^5+b^4 c e^3 \left (c d^2-23 a e^2\right )-2 b^3 c^2 d e^2 \left (17 c d^2+5 a e^2\right )-4 b c^3 d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right )+2 b^2 c^2 e \left (25 c^2 d^4+48 a c d^2 e^2+43 a^2 e^4\right )-2 c (2 c d-b e) \left (10 c^4 d^4+b^4 e^4+b^2 c e^3 (3 b d-11 a e)-4 c^3 d^2 e (5 b d-8 a e)+c^2 e^2 \left (7 b^2 d^2-32 a b d e+38 a^2 e^2\right )\right ) x}{2 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 \left (a+b x+c x^2\right )}+\frac {\left (40 c^7 d^7+b^7 e^7-14 a b^5 c e^7+70 a^2 b^3 c^2 e^7-140 a^3 b c^3 e^7-28 c^6 d^5 e (5 b d-6 a e)+28 c^5 d^3 e^2 \left (6 b^2 d^2-15 a b d e+10 a^2 e^2\right )-70 c^4 d e^3 \left (b^3 d^3-4 a b^2 d^2 e+6 a^2 b d e^2-4 a^3 e^3\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2} \left (c d^2-b d e+a e^2\right )^4}+\frac {e^7 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {e^7 \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^4} \] Output:

-1/3*(b*c*d-b^2*e+2*a*c*e+c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^ 
2)/(c*x^2+b*x+a)^3-1/6*(6*a*c*(-4*a*c+b^2)*e^3-b*(10*c^3*d^3+3*b^3*e^3-c^2 
*d*e*(-22*a*e+15*b*d)+b*c*e^2*(-17*a*e+2*b*d))-c*(-b*e+2*c*d)*(10*c^2*d^2- 
3*b^2*e^2-2*c*e*(-11*a*e+5*b*d))*x)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^2/( 
c*x^2+b*x+a)^2+1/2*(b^5*c*d*e^4+2*b^6*e^5-64*a^3*c^3*e^5+b^4*c*e^3*(-23*a* 
e^2+c*d^2)-2*b^3*c^2*d*e^2*(5*a*e^2+17*c*d^2)-4*b*c^3*d*(19*a^2*e^4+16*a*c 
*d^2*e^2+5*c^2*d^4)+2*b^2*c^2*e*(43*a^2*e^4+48*a*c*d^2*e^2+25*c^2*d^4)-2*c 
*(-b*e+2*c*d)*(10*c^4*d^4+b^4*e^4+b^2*c*e^3*(-11*a*e+3*b*d)-4*c^3*d^2*e*(- 
8*a*e+5*b*d)+c^2*e^2*(38*a^2*e^2-32*a*b*d*e+7*b^2*d^2))*x)/(-4*a*c+b^2)^3/ 
(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)+(40*c^7*d^7+b^7*e^7-14*a*b^5*c*e^7+70* 
a^2*b^3*c^2*e^7-140*a^3*b*c^3*e^7-28*c^6*d^5*e*(-6*a*e+5*b*d)+28*c^5*d^3*e 
^2*(10*a^2*e^2-15*a*b*d*e+6*b^2*d^2)-70*c^4*d*e^3*(-4*a^3*e^3+6*a^2*b*d*e^ 
2-4*a*b^2*d^2*e+b^3*d^3))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^ 
2)^(7/2)/(a*e^2-b*d*e+c*d^2)^4+e^7*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^4-1/2*e^7 
*ln(c*x^2+b*x+a)/(a*e^2-b*d*e+c*d^2)^4
 

Mathematica [A] (verified)

Time = 2.00 (sec) , antiderivative size = 769, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^4} \, dx=\frac {1}{6} \left (\frac {-2 b^2 e+4 c (a e+c d x)+2 b c (d-e x)}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (a+x (b+c x))^3}+\frac {3 b^4 e^3+b^3 c e^2 (2 d+3 e x)+4 c^2 \left (6 a^2 e^3+5 c^2 d^3 x+11 a c d e^2 x\right )+2 b c^2 \left (5 c d^2 (d-3 e x)+11 a e^2 (d-e x)\right )+b^2 c e \left (-23 a e^2+c d (-15 d+4 e x)\right )}{\left (b^2-4 a c\right )^2 \left (c d^2+e (-b d+a e)\right )^2 (a+x (b+c x))^2}+\frac {3 \left (-2 b^6 e^5-b^5 c e^4 (d+2 e x)+8 c^3 \left (8 a^3 e^5+5 c^3 d^5 x+16 a c^2 d^3 e^2 x+19 a^2 c d e^4 x\right )+4 b c^3 \left (5 c^2 d^4 (d-5 e x)+16 a c d^2 e^2 (d-3 e x)+19 a^2 e^4 (d-e x)\right )-b^4 c e^3 \left (-23 a e^2+c d (d+2 e x)\right )+2 b^3 c^2 e^2 \left (c d^2 (17 d-e x)+a e^2 (5 d+11 e x)\right )+2 b^2 c^2 e \left (-43 a^2 e^4+2 a c d e^2 (-24 d+5 e x)+c^2 d^3 (-25 d+34 e x)\right )\right )}{\left (b^2-4 a c\right )^3 \left (-c d^2+e (b d-a e)\right )^3 (a+x (b+c x))}+\frac {6 \left (40 c^7 d^7+b^7 e^7-14 a b^5 c e^7+70 a^2 b^3 c^2 e^7-140 a^3 b c^3 e^7-28 c^6 d^5 e (5 b d-6 a e)+28 c^5 d^3 e^2 \left (6 b^2 d^2-15 a b d e+10 a^2 e^2\right )-70 c^4 d e^3 \left (b^3 d^3-4 a b^2 d^2 e+6 a^2 b d e^2-4 a^3 e^3\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{7/2} \left (c d^2+e (-b d+a e)\right )^4}+\frac {6 e^7 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {3 e^7 \log (a+x (b+c x))}{\left (c d^2+e (-b d+a e)\right )^4}\right ) \] Input:

Integrate[1/((d + e*x)*(a + b*x + c*x^2)^4),x]
 

Output:

((-2*b^2*e + 4*c*(a*e + c*d*x) + 2*b*c*(d - e*x))/((b^2 - 4*a*c)*(-(c*d^2) 
 + e*(b*d - a*e))*(a + x*(b + c*x))^3) + (3*b^4*e^3 + b^3*c*e^2*(2*d + 3*e 
*x) + 4*c^2*(6*a^2*e^3 + 5*c^2*d^3*x + 11*a*c*d*e^2*x) + 2*b*c^2*(5*c*d^2* 
(d - 3*e*x) + 11*a*e^2*(d - e*x)) + b^2*c*e*(-23*a*e^2 + c*d*(-15*d + 4*e* 
x)))/((b^2 - 4*a*c)^2*(c*d^2 + e*(-(b*d) + a*e))^2*(a + x*(b + c*x))^2) + 
(3*(-2*b^6*e^5 - b^5*c*e^4*(d + 2*e*x) + 8*c^3*(8*a^3*e^5 + 5*c^3*d^5*x + 
16*a*c^2*d^3*e^2*x + 19*a^2*c*d*e^4*x) + 4*b*c^3*(5*c^2*d^4*(d - 5*e*x) + 
16*a*c*d^2*e^2*(d - 3*e*x) + 19*a^2*e^4*(d - e*x)) - b^4*c*e^3*(-23*a*e^2 
+ c*d*(d + 2*e*x)) + 2*b^3*c^2*e^2*(c*d^2*(17*d - e*x) + a*e^2*(5*d + 11*e 
*x)) + 2*b^2*c^2*e*(-43*a^2*e^4 + 2*a*c*d*e^2*(-24*d + 5*e*x) + c^2*d^3*(- 
25*d + 34*e*x))))/((b^2 - 4*a*c)^3*(-(c*d^2) + e*(b*d - a*e))^3*(a + x*(b 
+ c*x))) + (6*(40*c^7*d^7 + b^7*e^7 - 14*a*b^5*c*e^7 + 70*a^2*b^3*c^2*e^7 
- 140*a^3*b*c^3*e^7 - 28*c^6*d^5*e*(5*b*d - 6*a*e) + 28*c^5*d^3*e^2*(6*b^2 
*d^2 - 15*a*b*d*e + 10*a^2*e^2) - 70*c^4*d*e^3*(b^3*d^3 - 4*a*b^2*d^2*e + 
6*a^2*b*d*e^2 - 4*a^3*e^3))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2 
 + 4*a*c)^(7/2)*(c*d^2 + e*(-(b*d) + a*e))^4) + (6*e^7*Log[d + e*x])/(c*d^ 
2 - b*d*e + a*e^2)^4 - (3*e^7*Log[a + x*(b + c*x)])/(c*d^2 + e*(-(b*d) + a 
*e))^4)/6
 

Rubi [A] (verified)

Time = 3.10 (sec) , antiderivative size = 887, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1165, 1235, 27, 1235, 27, 1200, 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.

\(\displaystyle \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^4} \, dx\)

\(\Big \downarrow \) 1165

\(\displaystyle -\frac {\int \frac {10 c^2 d^2-3 b^2 e^2-c e (5 b d-12 a e)+5 c e (2 c d-b e) x}{(d+e x) \left (c x^2+b x+a\right )^3}dx}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1235

\(\displaystyle -\frac {\frac {-c x (2 c d-b e) \left (-2 c e (5 b d-11 a e)-3 b^2 e^2+10 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (5 b d-12 a e)-3 b^2 e^2+10 c^2 d^2\right )+5 a c e (2 c d-b e)^2}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {3 \left (20 c^4 d^4-2 c^3 e (15 b d-22 a e) d^2+2 b^4 e^4+b^2 c e^3 (3 b d-16 a e)+2 c^2 e^2 \left (2 b^2 d^2-11 a b e d+16 a^2 e^2\right )+c e (2 c d-b e) \left (10 c^2 d^2-3 b^2 e^2-2 c e (5 b d-11 a e)\right ) x\right )}{(d+e x) \left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {-c x (2 c d-b e) \left (-2 c e (5 b d-11 a e)-3 b^2 e^2+10 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (5 b d-12 a e)-3 b^2 e^2+10 c^2 d^2\right )+5 a c e (2 c d-b e)^2}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}-\frac {3 \int \frac {20 c^4 d^4-2 c^3 e (15 b d-22 a e) d^2+2 b^4 e^4+b^2 c e^3 (3 b d-16 a e)+2 c^2 e^2 \left (2 b^2 d^2-11 a b e d+16 a^2 e^2\right )+c e (2 c d-b e) \left (10 c^2 d^2-3 b^2 e^2-2 c e (5 b d-11 a e)\right ) x}{(d+e x) \left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1235

\(\displaystyle -\frac {-e b^2+c d b+2 a c e+c (2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^3}-\frac {\frac {5 a c e (2 c d-b e)^2-c \left (10 c^2 d^2-3 b^2 e^2-2 c e (5 b d-11 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (10 c^2 d^2-3 b^2 e^2-c e (5 b d-12 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac {3 \left (\frac {2 e^5 b^6+c d e^4 b^5+c e^3 \left (c d^2-23 a e^2\right ) b^4-2 c^2 d e^2 \left (17 c d^2+5 a e^2\right ) b^3+2 c^2 e \left (25 c^2 d^4+48 a c e^2 d^2+43 a^2 e^4\right ) b^2-4 c^3 d \left (5 c^2 d^4+16 a c e^2 d^2+19 a^2 e^4\right ) b-64 a^3 c^3 e^5-2 c (2 c d-b e) \left (10 c^4 d^4-4 c^3 e (5 b d-8 a e) d^2+b^4 e^4+b^2 c e^3 (3 b d-11 a e)+c^2 e^2 \left (7 b^2 d^2-32 a b e d+38 a^2 e^2\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac {\int \frac {2 \left (20 c^6 d^6-2 c^5 e (25 b d-32 a e) d^4+2 c^4 e^2 \left (17 b^2 d^2-48 a b e d+38 a^2 e^2\right ) d^2-b^6 e^6-b^4 c e^5 (b d-12 a e)-b^2 c^2 e^4 \left (b^2 d^2-11 a b e d+48 a^2 e^2\right )-c^3 e^3 \left (b^3 d^3-10 a b^2 e d^2+38 a^2 b e^2 d-64 a^3 e^3\right )+c e (2 c d-b e) \left (10 c^4 d^4-4 c^3 e (5 b d-8 a e) d^2+b^4 e^4+b^2 c e^3 (3 b d-11 a e)+c^2 e^2 \left (7 b^2 d^2-32 a b e d+38 a^2 e^2\right )\right ) x\right )}{(d+e x) \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-e b^2+c d b+2 a c e+c (2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^3}-\frac {\frac {5 a c e (2 c d-b e)^2-c \left (10 c^2 d^2-3 b^2 e^2-2 c e (5 b d-11 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (10 c^2 d^2-3 b^2 e^2-c e (5 b d-12 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac {3 \left (\frac {2 e^5 b^6+c d e^4 b^5+c e^3 \left (c d^2-23 a e^2\right ) b^4-2 c^2 d e^2 \left (17 c d^2+5 a e^2\right ) b^3+2 c^2 e \left (25 c^2 d^4+48 a c e^2 d^2+43 a^2 e^4\right ) b^2-4 c^3 d \left (5 c^2 d^4+16 a c e^2 d^2+19 a^2 e^4\right ) b-64 a^3 c^3 e^5-2 c (2 c d-b e) \left (10 c^4 d^4-4 c^3 e (5 b d-8 a e) d^2+b^4 e^4+b^2 c e^3 (3 b d-11 a e)+c^2 e^2 \left (7 b^2 d^2-32 a b e d+38 a^2 e^2\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac {2 \int \frac {20 c^6 d^6-2 c^5 e (25 b d-32 a e) d^4+2 c^4 e^2 \left (17 b^2 d^2-48 a b e d+38 a^2 e^2\right ) d^2-b^6 e^6-b^4 c e^5 (b d-12 a e)-b^2 c^2 e^4 \left (b^2 d^2-11 a b e d+48 a^2 e^2\right )-c^3 e^3 \left (b^3 d^3-10 a b^2 e d^2+38 a^2 b e^2 d-64 a^3 e^3\right )+c e (2 c d-b e) \left (10 c^4 d^4-4 c^3 e (5 b d-8 a e) d^2+b^4 e^4+b^2 c e^3 (3 b d-11 a e)+c^2 e^2 \left (7 b^2 d^2-32 a b e d+38 a^2 e^2\right )\right ) x}{(d+e x) \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\)

\(\Big \downarrow \) 1200

\(\displaystyle -\frac {-e b^2+c d b+2 a c e+c (2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^3}-\frac {\frac {5 a c e (2 c d-b e)^2-c \left (10 c^2 d^2-3 b^2 e^2-2 c e (5 b d-11 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (10 c^2 d^2-3 b^2 e^2-c e (5 b d-12 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac {3 \left (\frac {2 e^5 b^6+c d e^4 b^5+c e^3 \left (c d^2-23 a e^2\right ) b^4-2 c^2 d e^2 \left (17 c d^2+5 a e^2\right ) b^3+2 c^2 e \left (25 c^2 d^4+48 a c e^2 d^2+43 a^2 e^4\right ) b^2-4 c^3 d \left (5 c^2 d^4+16 a c e^2 d^2+19 a^2 e^4\right ) b-64 a^3 c^3 e^5-2 c (2 c d-b e) \left (10 c^4 d^4-4 c^3 e (5 b d-8 a e) d^2+b^4 e^4+b^2 c e^3 (3 b d-11 a e)+c^2 e^2 \left (7 b^2 d^2-32 a b e d+38 a^2 e^2\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac {2 \int \left (\frac {20 c^7 d^7-14 c^6 e (5 b d-6 a e) d^5+14 c^5 e^2 \left (6 b^2 d^2-15 a b e d+10 a^2 e^2\right ) d^3-35 c^4 e^3 \left (b^3 d^3-4 a b^2 e d^2+6 a^2 b e^2 d-4 a^3 e^3\right ) d+b^7 e^7-102 a^3 b c^3 e^7+59 a^2 b^3 c^2 e^7-13 a b^5 c e^7+c \left (b^2-4 a c\right )^3 e^7 x}{\left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac {\left (b^2-4 a c\right )^3 e^8}{\left (c d^2-b e d+a e^2\right ) (d+e x)}\right )dx}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-e b^2+c d b+2 a c e+c (2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^3}-\frac {\frac {5 a c e (2 c d-b e)^2-c \left (10 c^2 d^2-3 b^2 e^2-2 c e (5 b d-11 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (10 c^2 d^2-3 b^2 e^2-c e (5 b d-12 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac {3 \left (\frac {2 e^5 b^6+c d e^4 b^5+c e^3 \left (c d^2-23 a e^2\right ) b^4-2 c^2 d e^2 \left (17 c d^2+5 a e^2\right ) b^3+2 c^2 e \left (25 c^2 d^4+48 a c e^2 d^2+43 a^2 e^4\right ) b^2-4 c^3 d \left (5 c^2 d^4+16 a c e^2 d^2+19 a^2 e^4\right ) b-64 a^3 c^3 e^5-2 c (2 c d-b e) \left (10 c^4 d^4-4 c^3 e (5 b d-8 a e) d^2+b^4 e^4+b^2 c e^3 (3 b d-11 a e)+c^2 e^2 \left (7 b^2 d^2-32 a b e d+38 a^2 e^2\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac {2 \left (-\frac {\left (b^2-4 a c\right )^3 \log (d+e x) e^7}{c d^2-b e d+a e^2}+\frac {\left (b^2-4 a c\right )^3 \log \left (c x^2+b x+a\right ) e^7}{2 \left (c d^2-b e d+a e^2\right )}-\frac {\left (40 c^7 d^7-28 c^6 e (5 b d-6 a e) d^5+28 c^5 e^2 \left (6 b^2 d^2-15 a b e d+10 a^2 e^2\right ) d^3-70 c^4 e^3 \left (b^3 d^3-4 a b^2 e d^2+6 a^2 b e^2 d-4 a^3 e^3\right ) d+b^7 e^7-140 a^3 b c^3 e^7+70 a^2 b^3 c^2 e^7-14 a b^5 c e^7\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right )}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\)

Input:

Int[1/((d + e*x)*(a + b*x + c*x^2)^4),x]
 

Output:

-1/3*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/((b^2 - 4*a*c)*(c*d^2 - 
 b*d*e + a*e^2)*(a + b*x + c*x^2)^3) - ((5*a*c*e*(2*c*d - b*e)^2 - (b*c*d 
- b^2*e + 2*a*c*e)*(10*c^2*d^2 - 3*b^2*e^2 - c*e*(5*b*d - 12*a*e)) - c*(2* 
c*d - b*e)*(10*c^2*d^2 - 3*b^2*e^2 - 2*c*e*(5*b*d - 11*a*e))*x)/(2*(b^2 - 
4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - (3*((b^5*c*d*e^4 + 2 
*b^6*e^5 - 64*a^3*c^3*e^5 + b^4*c*e^3*(c*d^2 - 23*a*e^2) - 2*b^3*c^2*d*e^2 
*(17*c*d^2 + 5*a*e^2) - 4*b*c^3*d*(5*c^2*d^4 + 16*a*c*d^2*e^2 + 19*a^2*e^4 
) + 2*b^2*c^2*e*(25*c^2*d^4 + 48*a*c*d^2*e^2 + 43*a^2*e^4) - 2*c*(2*c*d - 
b*e)*(10*c^4*d^4 + b^4*e^4 + b^2*c*e^3*(3*b*d - 11*a*e) - 4*c^3*d^2*e*(5*b 
*d - 8*a*e) + c^2*e^2*(7*b^2*d^2 - 32*a*b*d*e + 38*a^2*e^2))*x)/((b^2 - 4* 
a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)) - (2*(-(((40*c^7*d^7 + b^7 
*e^7 - 14*a*b^5*c*e^7 + 70*a^2*b^3*c^2*e^7 - 140*a^3*b*c^3*e^7 - 28*c^6*d^ 
5*e*(5*b*d - 6*a*e) + 28*c^5*d^3*e^2*(6*b^2*d^2 - 15*a*b*d*e + 10*a^2*e^2) 
 - 70*c^4*d*e^3*(b^3*d^3 - 4*a*b^2*d^2*e + 6*a^2*b*d*e^2 - 4*a^3*e^3))*Arc 
Tanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a 
*e^2))) - ((b^2 - 4*a*c)^3*e^7*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2) + ((b 
^2 - 4*a*c)^3*e^7*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2))))/((b^ 
2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))))/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a* 
e^2)))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1165

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) 
*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 
2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d 
+ e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p 
+ 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + 
 b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] 
 && IntQuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1200

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* 
x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In 
tegersQ[n]
 

rule 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2810\) vs. \(2(761)=1522\).

Time = 1.68 (sec) , antiderivative size = 2811, normalized size of antiderivative = 3.64

method	result	size
default	\(\text {Expression too large to display}\)	\(2811\)
risch	\(\text {Expression too large to display}\)	\(166452\)

Input:

int(1/(e*x+d)/(c*x^2+b*x+a)^4,x,method=_RETURNVERBOSE)
 

Output:

e^7*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^4-1/(a*e^2-b*d*e+c*d^2)^4*((c^3*(38*a^3* 
b*c^2*e^7-76*a^3*c^3*d*e^6-11*a^2*b^3*c*e^7-48*a^2*b^2*c^2*d*e^6+210*a^2*b 
*c^3*d^2*e^5-140*a^2*c^4*d^3*e^4+a*b^5*e^7+12*a*b^4*c*d*e^6-140*a*b^2*c^3* 
d^3*e^4+210*a*b*c^4*d^4*e^3-84*a*c^5*d^5*e^2-b^6*d*e^6+35*b^3*c^3*d^4*e^3- 
84*b^2*c^4*d^5*e^2+70*b*c^5*d^6*e-20*c^6*d^7)/(64*a^3*c^3-48*a^2*b^2*c^2+1 
2*a*b^4*c-b^6)*x^5-1/2*c^2*(64*a^4*c^3*e^7-238*a^3*b^2*c^2*e^7+316*a^3*b*c 
^3*d*e^6+64*a^3*c^4*d^2*e^5+67*a^2*b^4*c*e^7+288*a^2*b^3*c^2*d*e^6-1098*a^ 
2*b^2*c^3*d^2*e^5+700*a^2*b*c^4*d^3*e^4-6*a*b^6*e^7-72*a*b^5*c*d*e^6+12*a* 
b^4*c^2*d^2*e^5+700*a*b^3*c^3*d^3*e^4-1050*a*b^2*c^4*d^4*e^3+420*a*b*c^5*d 
^5*e^2+6*b^7*d*e^6-b^6*c*d^2*e^5-175*b^4*c^3*d^4*e^3+420*b^3*c^4*d^5*e^2-3 
50*b^2*c^5*d^6*e+100*b*c^6*d^7)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6) 
*x^4+1/6*c*(160*a^4*b*c^3*e^7-1088*a^4*c^4*d*e^6+578*a^3*b^3*c^2*e^7-1252* 
a^3*b^2*c^3*d*e^6+2784*a^3*b*c^4*d^2*e^5-2112*a^3*c^5*d^3*e^4-189*a^2*b^5* 
c*e^7-648*a^2*b^4*c^2*d*e^6+2742*a^2*b^3*c^3*d^2*e^5-3876*a^2*b^2*c^4*d^3* 
e^4+3360*a^2*b*c^5*d^4*e^3-1344*a^2*c^6*d^5*e^2+18*a*b^7*e^7+198*a*b^6*c*d 
*e^6-108*a*b^5*c^2*d^2*e^5-1516*a*b^4*c^3*d^3*e^4+2870*a*b^3*c^4*d^4*e^3-2 
268*a*b^2*c^5*d^5*e^2+1120*a*b*c^6*d^6*e-320*a*c^7*d^7-18*b^8*d*e^6+9*b^7* 
c*d^2*e^5-2*b^6*c^2*d^3*e^4+385*b^5*c^3*d^4*e^3-924*b^4*c^4*d^5*e^2+770*b^ 
3*c^5*d^6*e-220*b^2*c^6*d^7)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^ 
3-1/2*(160*a^5*c^4*e^7-328*a^4*b^2*c^3*e^7+352*a^4*b*c^4*d*e^6+192*a^4*...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^4} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x+d)/(c*x^2+b*x+a)^4,x, algorithm="fricas")
 

Output:

Timed out
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^4} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x+d)/(c*x**2+b*x+a)**4,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(e*x+d)/(c*x^2+b*x+a)^4,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3418 vs. \(2 (761) = 1522\).

Time = 0.41 (sec) , antiderivative size = 3418, normalized size of antiderivative = 4.42 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)/(c*x^2+b*x+a)^4,x, algorithm="giac")
 

Output:

e^8*log(abs(e*x + d))/(c^4*d^8*e - 4*b*c^3*d^7*e^2 + 6*b^2*c^2*d^6*e^3 + 4 
*a*c^3*d^6*e^3 - 4*b^3*c*d^5*e^4 - 12*a*b*c^2*d^5*e^4 + b^4*d^4*e^5 + 12*a 
*b^2*c*d^4*e^5 + 6*a^2*c^2*d^4*e^5 - 4*a*b^3*d^3*e^6 - 12*a^2*b*c*d^3*e^6 
+ 6*a^2*b^2*d^2*e^7 + 4*a^3*c*d^2*e^7 - 4*a^3*b*d*e^8 + a^4*e^9) - 1/2*e^7 
*log(c*x^2 + b*x + a)/(c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 + 4*a*c 
^3*d^6*e^2 - 4*b^3*c*d^5*e^3 - 12*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2 
*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b^3*d^3*e^5 - 12*a^2*b*c*d^3*e^5 + 6* 
a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8) - (40*c^7*d^7 
 - 140*b*c^6*d^6*e + 168*b^2*c^5*d^5*e^2 + 168*a*c^6*d^5*e^2 - 70*b^3*c^4* 
d^4*e^3 - 420*a*b*c^5*d^4*e^3 + 280*a*b^2*c^4*d^3*e^4 + 280*a^2*c^5*d^3*e^ 
4 - 420*a^2*b*c^4*d^2*e^5 + 280*a^3*c^4*d*e^6 + b^7*e^7 - 14*a*b^5*c*e^7 + 
 70*a^2*b^3*c^2*e^7 - 140*a^3*b*c^3*e^7)*arctan((2*c*x + b)/sqrt(-b^2 + 4* 
a*c))/((b^6*c^4*d^8 - 12*a*b^4*c^5*d^8 + 48*a^2*b^2*c^6*d^8 - 64*a^3*c^7*d 
^8 - 4*b^7*c^3*d^7*e + 48*a*b^5*c^4*d^7*e - 192*a^2*b^3*c^5*d^7*e + 256*a^ 
3*b*c^6*d^7*e + 6*b^8*c^2*d^6*e^2 - 68*a*b^6*c^3*d^6*e^2 + 240*a^2*b^4*c^4 
*d^6*e^2 - 192*a^3*b^2*c^5*d^6*e^2 - 256*a^4*c^6*d^6*e^2 - 4*b^9*c*d^5*e^3 
 + 36*a*b^7*c^2*d^5*e^3 - 48*a^2*b^5*c^3*d^5*e^3 - 320*a^3*b^3*c^4*d^5*e^3 
 + 768*a^4*b*c^5*d^5*e^3 + b^10*d^4*e^4 - 90*a^2*b^6*c^2*d^4*e^4 + 440*a^3 
*b^4*c^3*d^4*e^4 - 480*a^4*b^2*c^4*d^4*e^4 - 384*a^5*c^5*d^4*e^4 - 4*a*b^9 
*d^3*e^5 + 36*a^2*b^7*c*d^3*e^5 - 48*a^3*b^5*c^2*d^3*e^5 - 320*a^4*b^3*...
 

Mupad [B] (verification not implemented)

Time = 24.88 (sec) , antiderivative size = 13834, normalized size of antiderivative = 17.90 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \] Input:

int(1/((d + e*x)*(a + b*x + c*x^2)^4),x)
 

Output:

((352*a^5*c^3*e^5 - 11*a^2*b^6*e^5 + 2*b^5*c^3*d^5 - 2*b^8*d^2*e^3 - 26*a* 
b^3*c^4*d^5 + 132*a^2*b*c^5*d^5 + 124*a^3*b^4*c*e^5 + 64*a^3*c^5*d^4*e - 6 
*b^6*c^2*d^4*e + 6*b^7*c*d^3*e^2 - 438*a^4*b^2*c^2*e^5 + 224*a^4*c^4*d^2*e 
^3 + 7*a*b^7*d*e^4 + 266*a^2*b^3*c^3*d^3*e^2 + 69*a^2*b^4*c^2*d^2*e^3 - 68 
0*a^3*b^2*c^3*d^2*e^3 + 77*a*b^4*c^3*d^4*e + 11*a*b^6*c*d^2*e^3 - 78*a^2*b 
^5*c*d*e^4 + 124*a^4*b*c^3*d*e^4 - 69*a*b^5*c^2*d^3*e^2 - 378*a^2*b^2*c^4* 
d^4*e + 256*a^3*b*c^4*d^3*e^2 + 244*a^3*b^3*c^2*d*e^4)/(6*(64*a^3*c^6*d^6 
- a^3*b^6*e^6 + 64*a^6*c^3*e^6 - b^6*c^3*d^6 + b^9*d^3*e^3 + 12*a*b^4*c^4* 
d^6 + 12*a^4*b^4*c*e^6 - 3*a*b^8*d^2*e^4 + 3*a^2*b^7*d*e^5 + 3*b^7*c^2*d^5 
*e - 3*b^8*c*d^4*e^2 - 48*a^2*b^2*c^5*d^6 - 48*a^5*b^2*c^2*e^6 + 192*a^4*c 
^5*d^4*e^2 + 192*a^5*c^4*d^2*e^4 - 108*a^2*b^4*c^3*d^4*e^2 - 24*a^2*b^5*c^ 
2*d^3*e^3 + 48*a^3*b^2*c^4*d^4*e^2 + 224*a^3*b^3*c^3*d^3*e^3 - 108*a^3*b^4 
*c^2*d^2*e^4 + 48*a^4*b^2*c^3*d^2*e^4 - 36*a*b^5*c^3*d^5*e - 6*a*b^7*c*d^3 
*e^3 - 192*a^3*b*c^5*d^5*e - 36*a^3*b^5*c*d*e^5 - 192*a^5*b*c^3*d*e^5 + 33 
*a*b^6*c^2*d^4*e^2 + 144*a^2*b^3*c^4*d^5*e + 33*a^2*b^6*c*d^2*e^4 - 384*a^ 
4*b*c^4*d^3*e^3 + 144*a^4*b^3*c^2*d*e^5)) + (x^3*(320*a*c^7*d^5 - 18*b^7*c 
*e^5 + 220*b^2*c^6*d^5 + 189*a*b^5*c^2*e^5 - 160*a^3*b*c^4*e^5 + 1088*a^3* 
c^5*d*e^4 - 550*b^3*c^5*d^4*e - 9*b^6*c^2*d*e^4 - 578*a^2*b^3*c^3*e^5 + 10 
24*a^2*c^6*d^3*e^2 + 374*b^4*c^4*d^3*e^2 - 11*b^5*c^3*d^2*e^3 - 800*a*b*c^ 
6*d^4*e + 70*a*b^4*c^3*d*e^4 + 1248*a*b^2*c^5*d^3*e^2 - 1072*a*b^3*c^4*...
 

Reduce [B] (verification not implemented)

Time = 78.12 (sec) , antiderivative size = 17681, normalized size of antiderivative = 22.87 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^4} \, dx =\text {Too large to display} \] Input:

int(1/(e*x+d)/(c*x^2+b*x+a)^4,x)
 

Output:

( - 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**6*b**2* 
c**3*e**7 + 1680*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a 
**6*b*c**4*d*e**6 + 420*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b 
**2))*a**5*b**4*c**2*e**7 - 2520*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt( 
4*a*c - b**2))*a**5*b**3*c**3*e**7*x - 2520*sqrt(4*a*c - b**2)*atan((b + 2 
*c*x)/sqrt(4*a*c - b**2))*a**5*b**2*c**4*d**2*e**5 + 5040*sqrt(4*a*c - b** 
2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**5*b**2*c**4*d*e**6*x - 2520*sqr 
t(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**5*b**2*c**4*e**7*x 
**2 + 1680*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**5*b* 
c**5*d**3*e**4 + 5040*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b** 
2))*a**5*b*c**5*d*e**6*x**2 - 84*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt( 
4*a*c - b**2))*a**4*b**6*c*e**7 + 1260*sqrt(4*a*c - b**2)*atan((b + 2*c*x) 
/sqrt(4*a*c - b**2))*a**4*b**5*c**2*e**7*x - 1260*sqrt(4*a*c - b**2)*atan( 
(b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b**4*c**3*e**7*x**2 + 1680*sqrt(4*a*c 
 - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b**3*c**4*d**3*e**4 - 7 
560*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b**3*c**4 
*d**2*e**5*x + 5040*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2) 
)*a**4*b**3*c**4*d*e**6*x**2 - 5040*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sq 
rt(4*a*c - b**2))*a**4*b**3*c**4*e**7*x**3 - 2520*sqrt(4*a*c - b**2)*atan( 
(b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b**2*c**5*d**4*e**3 + 5040*sqrt(4*...