\(\int \frac {(d+e x)^4}{(a+b x+c x^2)^4} \, dx\) [492]

Optimal result

Integrand size = 20, antiderivative size = 259 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {(b+2 c x) (d+e x)^4}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^3 \left (5 b c d-2 b^2 e-2 a c e+5 c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {8 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}} \] Output:

-1/3*(2*c*x+b)*(e*x+d)^4/(-4*a*c+b^2)/(c*x^2+b*x+a)^3+1/3*(e*x+d)^3*(5*b*c 
*d-2*b^2*e-2*a*c*e+5*c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^2-2*(5 
*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4* 
a*c+b^2)^3/(c*x^2+b*x+a)+8*(a*e^2-b*d*e+c*d^2)*(5*c^2*d^2+b^2*e^2-c*e*(-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 [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(572\) vs. \(2(259)=518\).

Time = 0.64 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.21 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^4} \, dx=\frac {1}{3} \left (\frac {6 \left (5 c^3 d^4+b^2 e^3 (-b d+a e)+2 c^2 d^2 e (-5 b d+3 a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right ) (b+2 c x)}{c \left (-b^2+4 a c\right )^3 (a+x (b+c x))}+\frac {b^4 e^4 x+b^3 e^3 (a e-4 c d x)+2 b^2 c e^2 \left (3 c d^2 x-2 a e (d+e x)\right )+b c \left (-3 a^2 e^4+c^2 d^3 (d-4 e x)+6 a c d e^2 (d+2 e x)\right )+2 c^2 \left (c^2 d^4 x+a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)\right )}{c^3 \left (-b^2+4 a c\right ) (a+x (b+c x))^3}+\frac {b^5 e^4-b^4 c e^3 (4 d+e x)+b c^2 \left (17 a^2 e^4+5 c^2 d^3 (d-4 e x)+6 a c d e^2 (d-2 e x)\right )+b^3 c e^2 \left (-7 a e^2+2 c d (3 d-e x)\right )+2 b^2 c^2 e \left (a e^2 (9 d+5 e x)+c d^2 (-5 d+6 e x)\right )+2 c^3 \left (5 c^2 d^4 x+6 a c d^2 e^2 x-a^2 e^3 (24 d+7 e x)\right )}{c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}+\frac {24 \left (5 c^3 d^4+b^2 e^3 (-b d+a e)+2 c^2 d^2 e (-5 b d+3 a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{7/2}}\right ) \] Input:

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

Output:

((6*(5*c^3*d^4 + b^2*e^3*(-(b*d) + a*e) + 2*c^2*d^2*e*(-5*b*d + 3*a*e) + c 
*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2))*(b + 2*c*x))/(c*(-b^2 + 4*a*c)^3*( 
a + x*(b + c*x))) + (b^4*e^4*x + b^3*e^3*(a*e - 4*c*d*x) + 2*b^2*c*e^2*(3* 
c*d^2*x - 2*a*e*(d + e*x)) + b*c*(-3*a^2*e^4 + c^2*d^3*(d - 4*e*x) + 6*a*c 
*d*e^2*(d + 2*e*x)) + 2*c^2*(c^2*d^4*x + a^2*e^3*(4*d + e*x) - 2*a*c*d^2*e 
*(2*d + 3*e*x)))/(c^3*(-b^2 + 4*a*c)*(a + x*(b + c*x))^3) + (b^5*e^4 - b^4 
*c*e^3*(4*d + e*x) + b*c^2*(17*a^2*e^4 + 5*c^2*d^3*(d - 4*e*x) + 6*a*c*d*e 
^2*(d - 2*e*x)) + b^3*c*e^2*(-7*a*e^2 + 2*c*d*(3*d - e*x)) + 2*b^2*c^2*e*( 
a*e^2*(9*d + 5*e*x) + c*d^2*(-5*d + 6*e*x)) + 2*c^3*(5*c^2*d^4*x + 6*a*c*d 
^2*e^2*x - a^2*e^3*(24*d + 7*e*x)))/(c^3*(b^2 - 4*a*c)^2*(a + x*(b + c*x)) 
^2) + (24*(5*c^3*d^4 + b^2*e^3*(-(b*d) + a*e) + 2*c^2*d^2*e*(-5*b*d + 3*a* 
e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 
 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2))/3
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1163, 27, 1227, 1153, 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.

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

\(\Big \downarrow \) 1163

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1227

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

\(\Big \downarrow \) 1153

\(\displaystyle -\frac {2 \left (-\frac {3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \left (-\frac {2 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{b^2-4 a c}-\frac {(d+e x)^3 \left (-2 a c e-2 b^2 e+5 c x (2 c d-b e)+5 b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\right )}{3 \left (b^2-4 a c\right )}-\frac {(b+2 c x) (d+e x)^4}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

\(\Big \downarrow \) 1083

\(\displaystyle -\frac {2 \left (-\frac {3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \left (\frac {4 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{b^2-4 a c}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{b^2-4 a c}-\frac {(d+e x)^3 \left (-2 a c e-2 b^2 e+5 c x (2 c d-b e)+5 b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\right )}{3 \left (b^2-4 a c\right )}-\frac {(b+2 c x) (d+e x)^4}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {2 \left (-\frac {3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \left (\frac {4 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (a e^2-b d e+c d^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{b^2-4 a c}-\frac {(d+e x)^3 \left (-2 a c e-2 b^2 e+5 c x (2 c d-b e)+5 b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\right )}{3 \left (b^2-4 a c\right )}-\frac {(b+2 c x) (d+e x)^4}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

Input:

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

Output:

-1/3*((b + 2*c*x)*(d + e*x)^4)/((b^2 - 4*a*c)*(a + b*x + c*x^2)^3) - (2*(- 
1/2*((d + e*x)^3*(5*b*c*d - 2*b^2*e - 2*a*c*e + 5*c*(2*c*d - b*e)*x))/((b^ 
2 - 4*a*c)*(a + b*x + c*x^2)^2) - (3*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a 
*e))*(-(((d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b* 
x + c*x^2))) + (4*(c*d^2 - b*d*e + a*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 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 1153

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

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 1227

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)*( 
(b*f - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[m*((b*( 
e*f + d*g) - 2*(c*d*f + a*e*g))/((p + 1)*(b^2 - 4*a*c)))   Int[(d + e*x)^(m 
 - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] 
&& EqQ[Simplify[m + 2*p + 3], 0] && LtQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1204\) vs. \(2(251)=502\).

Time = 1.02 (sec) , antiderivative size = 1205, normalized size of antiderivative = 4.65

method	result	size
default	\(\text {Expression too large to display}\)	\(1205\)
risch	\(\text {Expression too large to display}\)	\(2656\)

Input:

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

Output:

(4*(a^2*c*e^4+a*b^2*e^4-6*a*b*c*d*e^3+6*a*c^2*d^2*e^2-b^3*d*e^3+6*b^2*c*d^ 
2*e^2-10*b*c^2*d^3*e+5*c^3*d^4)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6) 
*c^2*x^5+10*(a^2*c*e^4+a*b^2*e^4-6*a*b*c*d*e^3+6*a*c^2*d^2*e^2-b^3*d*e^3+6 
*b^2*c*d^2*e^2-10*b*c^2*d^3*e+5*c^3*d^4)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b 
^4*c-b^6)*b*c*x^4-1/3*(32*a^3*c^3*e^4-102*a^2*b^2*c^2*e^4+192*a^2*b*c^3*d* 
e^3-192*a^2*c^4*d^2*e^2-10*a*b^4*c*e^4+164*a*b^3*c^2*d*e^3-324*a*b^2*c^3*d 
^2*e^2+320*a*b*c^4*d^3*e-160*a*c^5*d^4-b^6*e^4+22*b^5*c*d*e^3-132*b^4*c^2* 
d^2*e^2+220*b^3*c^3*d^3*e-110*b^2*c^4*d^4)/c/(64*a^3*c^3-48*a^2*b^2*c^2+12 
*a*b^4*c-b^6)*x^3+(16*a^3*b*c^2*e^4-64*a^3*c^3*d*e^3+17*a^2*b^3*c*e^4-48*a 
^2*b^2*c^2*d*e^3+96*a^2*b*c^3*d^2*e^2+a*b^5*e^4-34*a*b^4*c*d*e^3+102*a*b^3 
*c^2*d^2*e^2-160*a*b^2*c^3*d^3*e+80*a*b*c^4*d^4+6*b^5*c*d^2*e^2-10*b^4*c^2 
*d^3*e+5*b^3*c^3*d^4)/c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2-(4* 
a^4*c^2*e^4-22*a^3*b^2*c*e^4+40*a^3*b*c^2*d*e^3+24*a^3*c^3*d^2*e^2-a^2*b^4 
*e^4+40*a^2*b^3*c*d*e^3-132*a^2*b^2*c^2*d^2*e^2+88*a^2*b*c^3*d^3*e-44*a^2* 
c^4*d^4-6*a*b^4*c*d^2*e^2+36*a*b^3*c^2*d^3*e-18*a*b^2*c^3*d^4-2*b^5*c*d^3* 
e+b^4*c^2*d^4)/c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x+1/3*(26*a^4* 
b*c*e^4-64*a^4*c^2*d*e^3+a^3*b^3*e^4-44*a^3*b^2*c*d*e^3+156*a^3*b*c^2*d^2* 
e^2-128*a^3*c^3*d^3*e+6*a^2*b^3*c*d^2*e^2-36*a^2*b^2*c^2*d^3*e+66*a^2*b*c^ 
3*d^4+2*a*b^4*c*d^3*e-13*a*b^3*c^2*d^4+b^5*c*d^4)/c/(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+8*(a^2*c*e^4+a*b^2*e^4-6*a*b*c*d*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2305 vs. \(2 (251) = 502\).

Time = 0.15 (sec) , antiderivative size = 4631, normalized size of antiderivative = 17.88 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^4/(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. 2547 vs. \(2 (257) = 514\).

Time = 58.29 (sec) , antiderivative size = 2547, normalized size of antiderivative = 9.83 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \] Input:

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

Output:

-4*sqrt(-1/(4*a*c - b**2)**7)*(a*e**2 - b*d*e + c*d**2)*(a*c*e**2 + b**2*e 
**2 - 5*b*c*d*e + 5*c**2*d**2)*log(x + (-1024*a**4*c**4*sqrt(-1/(4*a*c - b 
**2)**7)*(a*e**2 - b*d*e + c*d**2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c 
**2*d**2) + 1024*a**3*b**2*c**3*sqrt(-1/(4*a*c - b**2)**7)*(a*e**2 - b*d*e 
 + c*d**2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) - 384*a**2*b** 
4*c**2*sqrt(-1/(4*a*c - b**2)**7)*(a*e**2 - b*d*e + c*d**2)*(a*c*e**2 + b* 
*2*e**2 - 5*b*c*d*e + 5*c**2*d**2) + 4*a**2*b*c*e**4 + 64*a*b**6*c*sqrt(-1 
/(4*a*c - b**2)**7)*(a*e**2 - b*d*e + c*d**2)*(a*c*e**2 + b**2*e**2 - 5*b* 
c*d*e + 5*c**2*d**2) + 4*a*b**3*e**4 - 24*a*b**2*c*d*e**3 + 24*a*b*c**2*d* 
*2*e**2 - 4*b**8*sqrt(-1/(4*a*c - b**2)**7)*(a*e**2 - b*d*e + c*d**2)*(a*c 
*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) - 4*b**4*d*e**3 + 24*b**3*c*d 
**2*e**2 - 40*b**2*c**2*d**3*e + 20*b*c**3*d**4)/(8*a**2*c**2*e**4 + 8*a*b 
**2*c*e**4 - 48*a*b*c**2*d*e**3 + 48*a*c**3*d**2*e**2 - 8*b**3*c*d*e**3 + 
48*b**2*c**2*d**2*e**2 - 80*b*c**3*d**3*e + 40*c**4*d**4)) + 4*sqrt(-1/(4* 
a*c - b**2)**7)*(a*e**2 - b*d*e + c*d**2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d* 
e + 5*c**2*d**2)*log(x + (1024*a**4*c**4*sqrt(-1/(4*a*c - b**2)**7)*(a*e** 
2 - b*d*e + c*d**2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) - 102 
4*a**3*b**2*c**3*sqrt(-1/(4*a*c - b**2)**7)*(a*e**2 - b*d*e + c*d**2)*(a*c 
*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) + 384*a**2*b**4*c**2*sqrt(-1/ 
(4*a*c - b**2)**7)*(a*e**2 - b*d*e + c*d**2)*(a*c*e**2 + b**2*e**2 - 5*...
 

Maxima [F(-2)]

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

integrate((e*x+d)^4/(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. 1167 vs. \(2 (251) = 502\).

Time = 0.42 (sec) , antiderivative size = 1167, normalized size of antiderivative = 4.51 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^4} \, dx =\text {Too large to display} \] Input:

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

Output:

-8*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 + 6*a*c^2*d^2*e^2 - b^3*d 
*e^3 - 6*a*b*c*d*e^3 + a*b^2*e^4 + a^2*c*e^4)*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)) - 1/3*(60*c^6*d^4*x^5 - 120*b*c^5*d^3*e*x^5 + 72*b^2*c^4*d^2*e^2*x^ 
5 + 72*a*c^5*d^2*e^2*x^5 - 12*b^3*c^3*d*e^3*x^5 - 72*a*b*c^4*d*e^3*x^5 + 1 
2*a*b^2*c^3*e^4*x^5 + 12*a^2*c^4*e^4*x^5 + 150*b*c^5*d^4*x^4 - 300*b^2*c^4 
*d^3*e*x^4 + 180*b^3*c^3*d^2*e^2*x^4 + 180*a*b*c^4*d^2*e^2*x^4 - 30*b^4*c^ 
2*d*e^3*x^4 - 180*a*b^2*c^3*d*e^3*x^4 + 30*a*b^3*c^2*e^4*x^4 + 30*a^2*b*c^ 
3*e^4*x^4 + 110*b^2*c^4*d^4*x^3 + 160*a*c^5*d^4*x^3 - 220*b^3*c^3*d^3*e*x^ 
3 - 320*a*b*c^4*d^3*e*x^3 + 132*b^4*c^2*d^2*e^2*x^3 + 324*a*b^2*c^3*d^2*e^ 
2*x^3 + 192*a^2*c^4*d^2*e^2*x^3 - 22*b^5*c*d*e^3*x^3 - 164*a*b^3*c^2*d*e^3 
*x^3 - 192*a^2*b*c^3*d*e^3*x^3 + b^6*e^4*x^3 + 10*a*b^4*c*e^4*x^3 + 102*a^ 
2*b^2*c^2*e^4*x^3 - 32*a^3*c^3*e^4*x^3 + 15*b^3*c^3*d^4*x^2 + 240*a*b*c^4* 
d^4*x^2 - 30*b^4*c^2*d^3*e*x^2 - 480*a*b^2*c^3*d^3*e*x^2 + 18*b^5*c*d^2*e^ 
2*x^2 + 306*a*b^3*c^2*d^2*e^2*x^2 + 288*a^2*b*c^3*d^2*e^2*x^2 - 102*a*b^4* 
c*d*e^3*x^2 - 144*a^2*b^2*c^2*d*e^3*x^2 - 192*a^3*c^3*d*e^3*x^2 + 3*a*b^5* 
e^4*x^2 + 51*a^2*b^3*c*e^4*x^2 + 48*a^3*b*c^2*e^4*x^2 - 3*b^4*c^2*d^4*x + 
54*a*b^2*c^3*d^4*x + 132*a^2*c^4*d^4*x + 6*b^5*c*d^3*e*x - 108*a*b^3*c^2*d 
^3*e*x - 264*a^2*b*c^3*d^3*e*x + 18*a*b^4*c*d^2*e^2*x + 396*a^2*b^2*c^2*d^ 
2*e^2*x - 72*a^3*c^3*d^2*e^2*x - 120*a^2*b^3*c*d*e^3*x - 120*a^3*b*c^2*...
 

Mupad [B] (verification not implemented)

Time = 6.15 (sec) , antiderivative size = 1463, normalized size of antiderivative = 5.65 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^4} \, dx =\text {Too large to display} \] Input:

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

Output:

- ((b^5*c*d^4 + a^3*b^3*e^4 - 13*a*b^3*c^2*d^4 + 66*a^2*b*c^3*d^4 - 128*a^ 
3*c^3*d^3*e - 64*a^4*c^2*d*e^3 + 26*a^4*b*c*e^4 + 2*a*b^4*c*d^3*e - 44*a^3 
*b^2*c*d*e^3 - 36*a^2*b^2*c^2*d^3*e + 6*a^2*b^3*c*d^2*e^2 + 156*a^3*b*c^2* 
d^2*e^2)/(3*c*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (x^3*(b^ 
6*e^4 + 160*a*c^5*d^4 - 32*a^3*c^3*e^4 + 110*b^2*c^4*d^4 - 220*b^3*c^3*d^3 
*e + 102*a^2*b^2*c^2*e^4 + 192*a^2*c^4*d^2*e^2 + 132*b^4*c^2*d^2*e^2 + 10* 
a*b^4*c*e^4 - 22*b^5*c*d*e^3 - 320*a*b*c^4*d^3*e - 164*a*b^3*c^2*d*e^3 - 1 
92*a^2*b*c^3*d*e^3 + 324*a*b^2*c^3*d^2*e^2))/(3*c*(b^6 - 64*a^3*c^3 + 48*a 
^2*b^2*c^2 - 12*a*b^4*c)) + (4*c^2*x^5*(5*c^3*d^4 + a*b^2*e^4 + a^2*c*e^4 
- b^3*d*e^3 + 6*a*c^2*d^2*e^2 + 6*b^2*c*d^2*e^2 - 10*b*c^2*d^3*e - 6*a*b*c 
*d*e^3))/(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c) + (x^2*(a*b^5*e^ 
4 + 5*b^3*c^3*d^4 + 17*a^2*b^3*c*e^4 + 16*a^3*b*c^2*e^4 - 64*a^3*c^3*d*e^3 
 - 10*b^4*c^2*d^3*e + 6*b^5*c*d^2*e^2 + 80*a*b*c^4*d^4 - 34*a*b^4*c*d*e^3 
- 160*a*b^2*c^3*d^3*e + 102*a*b^3*c^2*d^2*e^2 + 96*a^2*b*c^3*d^2*e^2 - 48* 
a^2*b^2*c^2*d*e^3))/(c*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + 
 (x*(a^2*b^4*e^4 + 44*a^2*c^4*d^4 - 4*a^4*c^2*e^4 - b^4*c^2*d^4 + 18*a*b^2 
*c^3*d^4 + 22*a^3*b^2*c*e^4 - 24*a^3*c^3*d^2*e^2 + 2*b^5*c*d^3*e + 132*a^2 
*b^2*c^2*d^2*e^2 - 36*a*b^3*c^2*d^3*e + 6*a*b^4*c*d^2*e^2 - 88*a^2*b*c^3*d 
^3*e - 40*a^2*b^3*c*d*e^3 - 40*a^3*b*c^2*d*e^3))/(c*(b^6 - 64*a^3*c^3 + 48 
*a^2*b^2*c^2 - 12*a*b^4*c)) + (10*b*c*x^4*(5*c^3*d^4 + a*b^2*e^4 + a^2*...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 5492, normalized size of antiderivative = 21.20 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^4} \, dx =\text {Too large to display} \] Input:

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

Output:

(24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**5*b*c**2*e* 
*4 + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b**3* 
c*e**4 - 144*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4* 
b**2*c**2*d*e**3 + 72*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b** 
2))*a**4*b**2*c**2*e**4*x + 144*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4 
*a*c - b**2))*a**4*b*c**3*d**2*e**2 + 72*sqrt(4*a*c - b**2)*atan((b + 2*c* 
x)/sqrt(4*a*c - b**2))*a**4*b*c**3*e**4*x**2 - 24*sqrt(4*a*c - b**2)*atan( 
(b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**4*c*d*e**3 + 72*sqrt(4*a*c - b**2) 
*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**4*c*e**4*x + 144*sqrt(4*a*c 
- b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**3*c**2*d**2*e**2 - 43 
2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**3*c**2*d 
*e**3*x + 144*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3 
*b**3*c**2*e**4*x**2 - 240*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c 
- b**2))*a**3*b**2*c**3*d**3*e + 432*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/s 
qrt(4*a*c - b**2))*a**3*b**2*c**3*d**2*e**2*x - 432*sqrt(4*a*c - b**2)*ata 
n((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c**3*d*e**3*x**2 + 144*sqrt(4* 
a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c**3*e**4*x**3 
+ 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c**4* 
d**4 + 432*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b* 
c**4*d**2*e**2*x**2 + 72*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c...