\(\int \frac {(d+e x)^3}{(a+b x+c x^2)^4} \, dx\) [493]

Optimal result

Integrand size = 20, antiderivative size = 282 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^2 \left (10 b c d-3 b^2 e-8 a c e+10 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {(d+e x) \left (25 b^2 c d e+20 a c^2 d e-3 b^3 e^2-6 b c \left (5 c d^2+3 a e^2\right )-c \left (60 c^2 d^2+11 b^2 e^2-4 c e (15 b d-4 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}} \] Output:

-1/3*(2*c*x+b)*(e*x+d)^3/(-4*a*c+b^2)/(c*x^2+b*x+a)^3+1/6*(e*x+d)^2*(10*b* 
c*d-3*b^2*e-8*a*c*e+10*c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^2+1/ 
3*(e*x+d)*(25*b^2*c*d*e+20*a*c^2*d*e-3*b^3*e^2-6*b*c*(3*a*e^2+5*c*d^2)-c*( 
60*c^2*d^2+11*b^2*e^2-4*c*e*(-4*a*e+15*b*d))*x)/(-4*a*c+b^2)^3/(c*x^2+b*x+ 
a)+2*(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))*arctanh((2*c*x 
+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(7/2)
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.42 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^4} \, dx=\frac {1}{6} \left (\frac {3 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2+2 c e (-5 b d+3 a e)\right ) (b+2 c x)}{c \left (-b^2+4 a c\right )^3 (a+x (b+c x))}+\frac {2 \left (-b^3 e^3 x+b^2 e^2 (-a e+3 c d x)+2 c \left (a^2 e^3+c^2 d^3 x-3 a c d e (d+e x)\right )+b c \left (c d^2 (d-3 e x)+3 a e^2 (d+e x)\right )\right )}{c^2 \left (-b^2+4 a c\right ) (a+x (b+c x))^3}+\frac {-2 b^4 e^3+b^3 c e^2 (6 d-e x)+4 c^2 \left (-6 a^2 e^3+5 c^2 d^3 x+3 a c d e^2 x\right )+2 b c^2 \left (5 c d^2 (d-3 e x)+3 a e^2 (d-e x)\right )+3 b^2 c e \left (3 a e^2+c d (-5 d+4 e x)\right )}{c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-\frac {12 (-2 c d+b e) \left (10 c^2 d^2+b^2 e^2+2 c e (-5 b d+3 a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{7/2}}\right ) \] Input:

Integrate[(d + e*x)^3/(a + b*x + c*x^2)^4,x]
 

Output:

((3*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 + 2*c*e*(-5*b*d + 3*a*e))*(b + 2*c 
*x))/(c*(-b^2 + 4*a*c)^3*(a + x*(b + c*x))) + (2*(-(b^3*e^3*x) + b^2*e^2*( 
-(a*e) + 3*c*d*x) + 2*c*(a^2*e^3 + c^2*d^3*x - 3*a*c*d*e*(d + e*x)) + b*c* 
(c*d^2*(d - 3*e*x) + 3*a*e^2*(d + e*x))))/(c^2*(-b^2 + 4*a*c)*(a + x*(b + 
c*x))^3) + (-2*b^4*e^3 + b^3*c*e^2*(6*d - e*x) + 4*c^2*(-6*a^2*e^3 + 5*c^2 
*d^3*x + 3*a*c*d*e^2*x) + 2*b*c^2*(5*c*d^2*(d - 3*e*x) + 3*a*e^2*(d - e*x) 
) + 3*b^2*c*e*(3*a*e^2 + c*d*(-5*d + 4*e*x)))/(c^2*(b^2 - 4*a*c)^2*(a + x* 
(b + c*x))^2) - (12*(-2*c*d + b*e)*(10*c^2*d^2 + b^2*e^2 + 2*c*e*(-5*b*d + 
 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2))/6
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1163, 25, 1234, 27, 1224, 1083, 219}

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.

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

\(\Big \downarrow \) 1163

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1234

\(\displaystyle -\frac {-\frac {\int \frac {2 (d+e x) \left (30 c^2 d^2+3 b^2 e^2-c e (25 b d-8 a e)+5 c e (2 c d-b e) x\right )}{\left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^2 \left (-8 a c e-3 b^2 e+10 c x (2 c d-b e)+10 b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}}{3 \left (b^2-4 a c\right )}-\frac {(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {(d+e x) \left (30 c^2 d^2+3 b^2 e^2-c e (25 b d-8 a e)+5 c e (2 c d-b e) x\right )}{\left (c x^2+b x+a\right )^2}dx}{b^2-4 a c}-\frac {(d+e x)^2 \left (-8 a c e-3 b^2 e+10 c x (2 c d-b e)+10 b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}}{3 \left (b^2-4 a c\right )}-\frac {(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

\(\Big \downarrow \) 1224

\(\displaystyle -\frac {-\frac {-\frac {3 (2 c d-b e) \left (6 a c e^2+b^2 e^2-10 b c d e+10 c^2 d^2\right ) \int \frac {1}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {2 x (2 c d-b e) \left (-c e (a e+15 b d)+4 b^2 e^2+15 c^2 d^2\right )-b^2 \left (11 a e^3+25 c d^2 e\right )+6 b c d \left (13 a e^2+5 c d^2\right )-16 a c e \left (a e^2+5 c d^2\right )+3 b^3 d e^2}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{b^2-4 a c}-\frac {(d+e x)^2 \left (-8 a c e-3 b^2 e+10 c x (2 c d-b e)+10 b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}}{3 \left (b^2-4 a c\right )}-\frac {(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

\(\Big \downarrow \) 1083

\(\displaystyle -\frac {-\frac {\frac {6 (2 c d-b e) \left (6 a c e^2+b^2 e^2-10 b c d e+10 c^2 d^2\right ) \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{b^2-4 a c}-\frac {2 x (2 c d-b e) \left (-c e (a e+15 b d)+4 b^2 e^2+15 c^2 d^2\right )-b^2 \left (11 a e^3+25 c d^2 e\right )+6 b c d \left (13 a e^2+5 c d^2\right )-16 a c e \left (a e^2+5 c d^2\right )+3 b^3 d e^2}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{b^2-4 a c}-\frac {(d+e x)^2 \left (-8 a c e-3 b^2 e+10 c x (2 c d-b e)+10 b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}}{3 \left (b^2-4 a c\right )}-\frac {(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {\frac {6 (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (6 a c e^2+b^2 e^2-10 b c d e+10 c^2 d^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {2 x (2 c d-b e) \left (-c e (a e+15 b d)+4 b^2 e^2+15 c^2 d^2\right )-b^2 \left (11 a e^3+25 c d^2 e\right )+6 b c d \left (13 a e^2+5 c d^2\right )-16 a c e \left (a e^2+5 c d^2\right )+3 b^3 d e^2}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{b^2-4 a c}-\frac {(d+e x)^2 \left (-8 a c e-3 b^2 e+10 c x (2 c d-b e)+10 b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}}{3 \left (b^2-4 a c\right )}-\frac {(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

Input:

Int[(d + e*x)^3/(a + b*x + c*x^2)^4,x]
 

Output:

-1/3*((b + 2*c*x)*(d + e*x)^3)/((b^2 - 4*a*c)*(a + b*x + c*x^2)^3) - (-1/2 
*((d + e*x)^2*(10*b*c*d - 3*b^2*e - 8*a*c*e + 10*c*(2*c*d - b*e)*x))/((b^2 
 - 4*a*c)*(a + b*x + c*x^2)^2) - (-((3*b^3*d*e^2 - 16*a*c*e*(5*c*d^2 + a*e 
^2) + 6*b*c*d*(5*c*d^2 + 13*a*e^2) - b^2*(25*c*d^2*e + 11*a*e^3) + 2*(2*c* 
d - b*e)*(15*c^2*d^2 + 4*b^2*e^2 - c*e*(15*b*d + a*e))*x)/((b^2 - 4*a*c)*( 
a + b*x + c*x^2))) + (6*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2 + 
 6*a*c*e^2)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2))/( 
b^2 - 4*a*c))/(3*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1163

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^m*(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)* 
(b^2 - 4*a*c))), x] - Simp[1/((p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 1 
)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 
1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[ 
m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, 
e, m, p, x]
 

rule 1224

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - ( 
b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x))*((a + b*x + c*x^2)^(p 
+ 1)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c 
*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(c*(p + 1)*(b^2 - 4*a*c))   Int[(a + 
b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, - 
1] &&  !(IntegerQ[p] && NeQ[a, 0] && NiceSqrtQ[b^2 - 4*a*c])
 

rule 1234

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( 
(f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 
 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g 
*(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)* 
(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1 
] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(884\) vs. \(2(272)=544\).

Time = 0.98 (sec) , antiderivative size = 885, normalized size of antiderivative = 3.14

method	result	size
default	\(\frac {-\frac {\left (6 a b c \,e^{3}-12 d \,e^{2} a \,c^{2}+b^{3} e^{3}-12 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-20 d^{3} c^{3}\right ) c^{2} x^{5}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {5 \left (6 a b c \,e^{3}-12 d \,e^{2} a \,c^{2}+b^{3} e^{3}-12 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-20 d^{3} c^{3}\right ) b c \,x^{4}}{2 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {\left (16 a c +11 b^{2}\right ) \left (6 a b c \,e^{3}-12 d \,e^{2} a \,c^{2}+b^{3} e^{3}-12 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-20 d^{3} c^{3}\right ) x^{3}}{6 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {\left (32 a^{3} c^{2} e^{3}+24 a^{2} b^{2} c \,e^{3}-96 a^{2} b \,c^{2} d \,e^{2}+17 a \,b^{4} e^{3}-102 a \,b^{3} c d \,e^{2}+240 a \,b^{2} c^{2} d^{2} e -160 a b \,c^{3} d^{3}-6 b^{5} d \,e^{2}+15 b^{4} c \,d^{2} e -10 b^{3} c^{2} d^{3}\right ) x^{2}}{2 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {\left (20 a^{3} b c \,e^{3}+24 a^{3} c^{2} d \,e^{2}+20 a^{2} b^{3} e^{3}-132 a^{2} b^{2} c d \,e^{2}+132 a^{2} b \,c^{2} d^{2} e -88 a^{2} c^{3} d^{3}-6 a \,b^{4} d \,e^{2}+54 a \,b^{3} c \,d^{2} e -36 d^{3} a \,c^{2} b^{2}-3 b^{5} d^{2} e +2 b^{4} c \,d^{3}\right ) x}{2 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {32 a^{4} c \,e^{3}+22 a^{3} b^{2} e^{3}-156 a^{3} b c d \,e^{2}+192 a^{3} c^{2} d^{2} e -6 a^{2} b^{3} d \,e^{2}+54 a^{2} b^{2} c \,d^{2} e -132 a^{2} b \,c^{2} d^{3}-3 a \,b^{4} d^{2} e +26 a \,b^{3} c \,d^{3}-2 b^{5} d^{3}}{6 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}}{\left (c \,x^{2}+b x +a \right )^{3}}-\frac {2 \left (6 a b c \,e^{3}-12 d \,e^{2} a \,c^{2}+b^{3} e^{3}-12 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-20 d^{3} c^{3}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \sqrt {4 a c -b^{2}}}\)	\(885\)
risch	\(\text {Expression too large to display}\)	\(1974\)

Input:

int((e*x+d)^3/(c*x^2+b*x+a)^4,x,method=_RETURNVERBOSE)
 

Output:

(-(6*a*b*c*e^3-12*a*c^2*d*e^2+b^3*e^3-12*b^2*c*d*e^2+30*b*c^2*d^2*e-20*c^3 
*d^3)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*c^2*x^5-5/2*(6*a*b*c*e^3- 
12*a*c^2*d*e^2+b^3*e^3-12*b^2*c*d*e^2+30*b*c^2*d^2*e-20*c^3*d^3)/(64*a^3*c 
^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*b*c*x^4-1/6*(16*a*c+11*b^2)*(6*a*b*c*e^3 
-12*a*c^2*d*e^2+b^3*e^3-12*b^2*c*d*e^2+30*b*c^2*d^2*e-20*c^3*d^3)/(64*a^3* 
c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3-1/2*(32*a^3*c^2*e^3+24*a^2*b^2*c*e^ 
3-96*a^2*b*c^2*d*e^2+17*a*b^4*e^3-102*a*b^3*c*d*e^2+240*a*b^2*c^2*d^2*e-16 
0*a*b*c^3*d^3-6*b^5*d*e^2+15*b^4*c*d^2*e-10*b^3*c^2*d^3)/(64*a^3*c^3-48*a^ 
2*b^2*c^2+12*a*b^4*c-b^6)*x^2-1/2*(20*a^3*b*c*e^3+24*a^3*c^2*d*e^2+20*a^2* 
b^3*e^3-132*a^2*b^2*c*d*e^2+132*a^2*b*c^2*d^2*e-88*a^2*c^3*d^3-6*a*b^4*d*e 
^2+54*a*b^3*c*d^2*e-36*a*b^2*c^2*d^3-3*b^5*d^2*e+2*b^4*c*d^3)/(64*a^3*c^3- 
48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x-1/6*(32*a^4*c*e^3+22*a^3*b^2*e^3-156*a^3* 
b*c*d*e^2+192*a^3*c^2*d^2*e-6*a^2*b^3*d*e^2+54*a^2*b^2*c*d^2*e-132*a^2*b*c 
^2*d^3-3*a*b^4*d^2*e+26*a*b^3*c*d^3-2*b^5*d^3)/(64*a^3*c^3-48*a^2*b^2*c^2+ 
12*a*b^4*c-b^6))/(c*x^2+b*x+a)^3-2*(6*a*b*c*e^3-12*a*c^2*d*e^2+b^3*e^3-12* 
b^2*c*d*e^2+30*b*c^2*d^2*e-20*c^3*d^3)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4 
*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1794 vs. \(2 (272) = 544\).

Time = 0.12 (sec) , antiderivative size = 3608, normalized size of antiderivative = 12.79 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^3/(c*x^2+b*x+a)^4,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2057 vs. \(2 (287) = 574\).

Time = 10.62 (sec) , antiderivative size = 2057, normalized size of antiderivative = 7.29 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)**3/(c*x**2+b*x+a)**4,x)
 

Output:

sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c* 
d*e + 10*c**2*d**2)*log(x + (-256*a**4*c**4*sqrt(-1/(4*a*c - b**2)**7)*(b* 
e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) + 256*a**3 
*b**2*c**3*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e** 
2 - 10*b*c*d*e + 10*c**2*d**2) - 96*a**2*b**4*c**2*sqrt(-1/(4*a*c - b**2)* 
*7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) + 1 
6*a*b**6*c*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e** 
2 - 10*b*c*d*e + 10*c**2*d**2) + 6*a*b**2*c*e**3 - 12*a*b*c**2*d*e**2 - b* 
*8*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b 
*c*d*e + 10*c**2*d**2) + b**4*e**3 - 12*b**3*c*d*e**2 + 30*b**2*c**2*d**2* 
e - 20*b*c**3*d**3)/(12*a*b*c**2*e**3 - 24*a*c**3*d*e**2 + 2*b**3*c*e**3 - 
 24*b**2*c**2*d*e**2 + 60*b*c**3*d**2*e - 40*c**4*d**3)) - sqrt(-1/(4*a*c 
- b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d 
**2)*log(x + (256*a**4*c**4*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a* 
c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) - 256*a**3*b**2*c**3*sqrt( 
-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 
 10*c**2*d**2) + 96*a**2*b**4*c**2*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d 
)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) - 16*a*b**6*c*sqrt( 
-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 
 10*c**2*d**2) + 6*a*b**2*c*e**3 - 12*a*b*c**2*d*e**2 + b**8*sqrt(-1/(4...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^3/(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. 848 vs. \(2 (272) = 544\).

Time = 0.33 (sec) , antiderivative size = 848, normalized size of antiderivative = 3.01 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^4} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^3/(c*x^2+b*x+a)^4,x, algorithm="giac")
 

Output:

-2*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 + 12*a*c^2*d*e^2 - b^3*e^ 
3 - 6*a*b*c*e^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^6 - 12*a*b^4*c 
 + 48*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(-b^2 + 4*a*c)) - 1/6*(120*c^5*d^3*x^5 
 - 180*b*c^4*d^2*e*x^5 + 72*b^2*c^3*d*e^2*x^5 + 72*a*c^4*d*e^2*x^5 - 6*b^3 
*c^2*e^3*x^5 - 36*a*b*c^3*e^3*x^5 + 300*b*c^4*d^3*x^4 - 450*b^2*c^3*d^2*e* 
x^4 + 180*b^3*c^2*d*e^2*x^4 + 180*a*b*c^3*d*e^2*x^4 - 15*b^4*c*e^3*x^4 - 9 
0*a*b^2*c^2*e^3*x^4 + 220*b^2*c^3*d^3*x^3 + 320*a*c^4*d^3*x^3 - 330*b^3*c^ 
2*d^2*e*x^3 - 480*a*b*c^3*d^2*e*x^3 + 132*b^4*c*d*e^2*x^3 + 324*a*b^2*c^2* 
d*e^2*x^3 + 192*a^2*c^3*d*e^2*x^3 - 11*b^5*e^3*x^3 - 82*a*b^3*c*e^3*x^3 - 
96*a^2*b*c^2*e^3*x^3 + 30*b^3*c^2*d^3*x^2 + 480*a*b*c^3*d^3*x^2 - 45*b^4*c 
*d^2*e*x^2 - 720*a*b^2*c^2*d^2*e*x^2 + 18*b^5*d*e^2*x^2 + 306*a*b^3*c*d*e^ 
2*x^2 + 288*a^2*b*c^2*d*e^2*x^2 - 51*a*b^4*e^3*x^2 - 72*a^2*b^2*c*e^3*x^2 
- 96*a^3*c^2*e^3*x^2 - 6*b^4*c*d^3*x + 108*a*b^2*c^2*d^3*x + 264*a^2*c^3*d 
^3*x + 9*b^5*d^2*e*x - 162*a*b^3*c*d^2*e*x - 396*a^2*b*c^2*d^2*e*x + 18*a* 
b^4*d*e^2*x + 396*a^2*b^2*c*d*e^2*x - 72*a^3*c^2*d*e^2*x - 60*a^2*b^3*e^3* 
x - 60*a^3*b*c*e^3*x + 2*b^5*d^3 - 26*a*b^3*c*d^3 + 132*a^2*b*c^2*d^3 + 3* 
a*b^4*d^2*e - 54*a^2*b^2*c*d^2*e - 192*a^3*c^2*d^2*e + 6*a^2*b^3*d*e^2 + 1 
56*a^3*b*c*d*e^2 - 22*a^3*b^2*e^3 - 32*a^4*c*e^3)/((b^6 - 12*a*b^4*c + 48* 
a^2*b^2*c^2 - 64*a^3*c^3)*(c*x^2 + b*x + a)^3)
 

Mupad [B] (verification not implemented)

Time = 5.96 (sec) , antiderivative size = 1126, normalized size of antiderivative = 3.99 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^4} \, dx =\text {Too large to display} \] Input:

int((d + e*x)^3/(a + b*x + c*x^2)^4,x)
 

Output:

((x*(2*b^4*c*d^3 - 3*b^5*d^2*e + 20*a^2*b^3*e^3 - 88*a^2*c^3*d^3 - 36*a*b^ 
2*c^2*d^3 + 24*a^3*c^2*d*e^2 + 20*a^3*b*c*e^3 - 6*a*b^4*d*e^2 + 54*a*b^3*c 
*d^2*e + 132*a^2*b*c^2*d^2*e - 132*a^2*b^2*c*d*e^2))/(2*(b^6 - 64*a^3*c^3 
+ 48*a^2*b^2*c^2 - 12*a*b^4*c)) - (2*b^5*d^3 - 32*a^4*c*e^3 - 22*a^3*b^2*e 
^3 + 132*a^2*b*c^2*d^3 + 6*a^2*b^3*d*e^2 - 192*a^3*c^2*d^2*e - 26*a*b^3*c* 
d^3 + 3*a*b^4*d^2*e + 156*a^3*b*c*d*e^2 - 54*a^2*b^2*c*d^2*e)/(6*(b^6 - 64 
*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (x^2*(17*a*b^4*e^3 - 6*b^5*d*e^ 
2 + 32*a^3*c^2*e^3 - 10*b^3*c^2*d^3 + 24*a^2*b^2*c*e^3 - 160*a*b*c^3*d^3 + 
 15*b^4*c*d^2*e - 102*a*b^3*c*d*e^2 + 240*a*b^2*c^2*d^2*e - 96*a^2*b*c^2*d 
*e^2))/(2*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (x^3*(16*a*c 
 + 11*b^2)*(b^3*e^3 - 20*c^3*d^3 + 6*a*b*c*e^3 - 12*a*c^2*d*e^2 + 30*b*c^2 
*d^2*e - 12*b^2*c*d*e^2))/(6*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4 
*c)) + (c^2*x^5*(b^3*e^3 - 20*c^3*d^3 + 6*a*b*c*e^3 - 12*a*c^2*d*e^2 + 30* 
b*c^2*d^2*e - 12*b^2*c*d*e^2))/(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b 
^4*c) + (5*b*c*x^4*(b^3*e^3 - 20*c^3*d^3 + 6*a*b*c*e^3 - 12*a*c^2*d*e^2 + 
30*b*c^2*d^2*e - 12*b^2*c*d*e^2))/(2*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 
12*a*b^4*c)))/(x^2*(3*a*b^2 + 3*a^2*c) + x^4*(3*a*c^2 + 3*b^2*c) + a^3 + x 
^3*(b^3 + 6*a*b*c) + c^3*x^6 + 3*b*c^2*x^5 + 3*a^2*b*x) + (2*atan(((((b*e 
- 2*c*d)*(b^7 - 64*a^3*b*c^3 + 48*a^2*b^3*c^2 - 12*a*b^5*c)*(b^2*e^2 + 10* 
c^2*d^2 + 6*a*c*e^2 - 10*b*c*d*e))/((4*a*c - b^2)^(7/2)*(b^6 - 64*a^3*c...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 4152, normalized size of antiderivative = 14.72 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^4} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^3/(c*x^2+b*x+a)^4,x)
 

Output:

( - 72*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b**2*c 
*e**3 + 144*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b 
*c**2*d*e**2 - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))* 
a**3*b**4*e**3 + 144*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2 
))*a**3*b**3*c*d*e**2 - 216*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c 
 - b**2))*a**3*b**3*c*e**3*x - 360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqr 
t(4*a*c - b**2))*a**3*b**2*c**2*d**2*e + 432*sqrt(4*a*c - b**2)*atan((b + 
2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c**2*d*e**2*x - 216*sqrt(4*a*c - b**2 
)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c**2*e**3*x**2 + 240*sqrt 
(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c**3*d**3 + 432 
*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c**3*d*e** 
2*x**2 - 36*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b 
**5*e**3*x + 432*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a 
**2*b**4*c*d*e**2*x - 252*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - 
 b**2))*a**2*b**4*c*e**3*x**2 - 1080*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/s 
qrt(4*a*c - b**2))*a**2*b**3*c**2*d**2*e*x + 864*sqrt(4*a*c - b**2)*atan(( 
b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**3*c**2*d*e**2*x**2 - 432*sqrt(4*a*c 
 - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**3*c**2*e**3*x**3 + 7 
20*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c**3* 
d**3*x - 1080*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a...