\(\int \frac {(d+e x)^4}{(a+b x+c x^2)^4} \, dx\) [2214]

Optimal result

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

[Out]

-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)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {750, 818, 736, 632, 212} \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^4} \, dx=\frac {8 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac {2 (d+e x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {(b+2 c x) (d+e x)^4}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^3 \left (-2 a c e-2 b^2 e+5 c x (2 c d-b e)+5 b c d\right )}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2} \]

[In]

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

[Out]

-1/3*((b + 2*c*x)*(d + e*x)^4)/((b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + ((d + e*x)^3*(5*b*c*d - 2*b^2*e - 2*a*c*e
+ 5*c*(2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) - (2*(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)^3*(a + b*x + c*x^2)) + (8*(c*d^2 - b*d*e + a*e^2)*
(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(7/2)

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 736

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> 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] - Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && LtQ[p, -1]

Rule 750

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> 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] - Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && 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 818

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] - Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0] && LtQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\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 {\int \frac {(d+e x)^3 (-10 c d+4 b e-2 c e x)}{\left (a+b x+c x^2\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}+\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 {\left (2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx}{\left (b^2-4 a c\right )^2} \\ & = -\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 {\left (4 \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 )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^3} \\ & = -\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 {\left (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 )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^3} \\ & = -\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 ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}} \\ \end{align*}

Mathematica [B] (verified)

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

Time = 0.56 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.21 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^4} \, dx=\frac {1}{3} \left (\frac {6 \left (5 c^3 d^4+b^2 e^3 (-b d+a e)+2 c^2 d^2 e (-5 b d+3 a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right ) (b+2 c x)}{c \left (-b^2+4 a c\right )^3 (a+x (b+c x))}+\frac {b^4 e^4 x+b^3 e^3 (a e-4 c d x)+2 b^2 c e^2 \left (3 c d^2 x-2 a e (d+e x)\right )+b c \left (-3 a^2 e^4+c^2 d^3 (d-4 e x)+6 a c d e^2 (d+2 e x)\right )+2 c^2 \left (c^2 d^4 x+a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)\right )}{c^3 \left (-b^2+4 a c\right ) (a+x (b+c x))^3}+\frac {b^5 e^4-b^4 c e^3 (4 d+e x)+b c^2 \left (17 a^2 e^4+5 c^2 d^3 (d-4 e x)+6 a c d e^2 (d-2 e x)\right )+b^3 c e^2 \left (-7 a e^2+2 c d (3 d-e x)\right )+2 b^2 c^2 e \left (a e^2 (9 d+5 e x)+c d^2 (-5 d+6 e x)\right )+2 c^3 \left (5 c^2 d^4 x+6 a c d^2 e^2 x-a^2 e^3 (24 d+7 e x)\right )}{c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}+\frac {24 \left (5 c^3 d^4+b^2 e^3 (-b d+a e)+2 c^2 d^2 e (-5 b d+3 a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{7/2}}\right ) \]

[In]

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

[Out]

((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)/S
qrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2))/3

Maple [B] (verified)

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

Time = 17.28 (sec) , antiderivative size = 1205, normalized size of antiderivative = 4.65

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

[In]

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

[Out]

(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*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)/(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. 2305 vs. \(2 (251) = 502\).

Time = 0.35 (sec) , antiderivative size = 4631, normalized size of antiderivative = 17.88 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/3*(12*(5*(b^2*c^6 - 4*a*c^7)*d^4 - 10*(b^3*c^5 - 4*a*b*c^6)*d^3*e + 6*(b^4*c^4 - 3*a*b^2*c^5 - 4*a^2*c^6)*
d^2*e^2 - (b^5*c^3 + 2*a*b^3*c^4 - 24*a^2*b*c^5)*d*e^3 + (a*b^4*c^3 - 3*a^2*b^2*c^4 - 4*a^3*c^5)*e^4)*x^5 + (b
^7*c - 17*a*b^5*c^2 + 118*a^2*b^3*c^3 - 264*a^3*b*c^4)*d^4 + 2*(a*b^6*c - 22*a^2*b^4*c^2 + 8*a^3*b^2*c^3 + 256
*a^4*c^4)*d^3*e + 6*(a^2*b^5*c + 22*a^3*b^3*c^2 - 104*a^4*b*c^3)*d^2*e^2 - 4*(11*a^3*b^4*c - 28*a^4*b^2*c^2 -
64*a^5*c^3)*d*e^3 + (a^3*b^5 + 22*a^4*b^3*c - 104*a^5*b*c^2)*e^4 + 30*(5*(b^3*c^5 - 4*a*b*c^6)*d^4 - 10*(b^4*c
^4 - 4*a*b^2*c^5)*d^3*e + 6*(b^5*c^3 - 3*a*b^3*c^4 - 4*a^2*b*c^5)*d^2*e^2 - (b^6*c^2 + 2*a*b^4*c^3 - 24*a^2*b^
2*c^4)*d*e^3 + (a*b^5*c^2 - 3*a^2*b^3*c^3 - 4*a^3*b*c^4)*e^4)*x^4 + (10*(11*b^4*c^4 - 28*a*b^2*c^5 - 64*a^2*c^
6)*d^4 - 20*(11*b^5*c^3 - 28*a*b^3*c^4 - 64*a^2*b*c^5)*d^3*e + 12*(11*b^6*c^2 - 17*a*b^4*c^3 - 92*a^2*b^2*c^4
- 64*a^3*c^5)*d^2*e^2 - 2*(11*b^7*c + 38*a*b^5*c^2 - 232*a^2*b^3*c^3 - 384*a^3*b*c^4)*d*e^3 + (b^8 + 6*a*b^6*c
+ 62*a^2*b^4*c^2 - 440*a^3*b^2*c^3 + 128*a^4*c^4)*e^4)*x^3 + 3*(5*(b^5*c^3 + 12*a*b^3*c^4 - 64*a^2*b*c^5)*d^4
- 10*(b^6*c^2 + 12*a*b^4*c^3 - 64*a^2*b^2*c^4)*d^3*e + 6*(b^7*c + 13*a*b^5*c^2 - 52*a^2*b^3*c^3 - 64*a^3*b*c^
4)*d^2*e^2 - 2*(17*a*b^6*c - 44*a^2*b^4*c^2 - 64*a^3*b^2*c^3 - 128*a^4*c^4)*d*e^3 + (a*b^7 + 13*a^2*b^5*c - 52
*a^3*b^3*c^2 - 64*a^4*b*c^3)*e^4)*x^2 + 12*(5*a^3*c^4*d^4 - 10*a^3*b*c^3*d^3*e + (5*c^7*d^4 - 10*b*c^6*d^3*e +
6*(b^2*c^5 + a*c^6)*d^2*e^2 - (b^3*c^4 + 6*a*b*c^5)*d*e^3 + (a*b^2*c^4 + a^2*c^5)*e^4)*x^6 + 3*(5*b*c^6*d^4 -
10*b^2*c^5*d^3*e + 6*(b^3*c^4 + a*b*c^5)*d^2*e^2 - (b^4*c^3 + 6*a*b^2*c^4)*d*e^3 + (a*b^3*c^3 + a^2*b*c^4)*e^
4)*x^5 + 6*(a^3*b^2*c^2 + a^4*c^3)*d^2*e^2 - (a^3*b^3*c + 6*a^4*b*c^2)*d*e^3 + (a^4*b^2*c + a^5*c^2)*e^4 + 3*(
5*(b^2*c^5 + a*c^6)*d^4 - 10*(b^3*c^4 + a*b*c^5)*d^3*e + 6*(b^4*c^3 + 2*a*b^2*c^4 + a^2*c^5)*d^2*e^2 - (b^5*c^
2 + 7*a*b^3*c^3 + 6*a^2*b*c^4)*d*e^3 + (a*b^4*c^2 + 2*a^2*b^2*c^3 + a^3*c^4)*e^4)*x^4 + (5*(b^3*c^4 + 6*a*b*c^
5)*d^4 - 10*(b^4*c^3 + 6*a*b^2*c^4)*d^3*e + 6*(b^5*c^2 + 7*a*b^3*c^3 + 6*a^2*b*c^4)*d^2*e^2 - (b^6*c + 12*a*b^
4*c^2 + 36*a^2*b^2*c^3)*d*e^3 + (a*b^5*c + 7*a^2*b^3*c^2 + 6*a^3*b*c^3)*e^4)*x^3 + 3*(5*(a*b^2*c^4 + a^2*c^5)*
d^4 - 10*(a*b^3*c^3 + a^2*b*c^4)*d^3*e + 6*(a*b^4*c^2 + 2*a^2*b^2*c^3 + a^3*c^4)*d^2*e^2 - (a*b^5*c + 7*a^2*b^
3*c^2 + 6*a^3*b*c^3)*d*e^3 + (a^2*b^4*c + 2*a^3*b^2*c^2 + a^4*c^3)*e^4)*x^2 + 3*(5*a^2*b*c^4*d^4 - 10*a^2*b^2*
c^3*d^3*e + 6*(a^2*b^3*c^2 + a^3*b*c^3)*d^2*e^2 - (a^2*b^4*c + 6*a^3*b^2*c^2)*d*e^3 + (a^3*b^3*c + a^4*b*c^2)*
e^4)*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x
+ a)) - 3*((b^6*c^2 - 22*a*b^4*c^3 + 28*a^2*b^2*c^4 + 176*a^3*c^5)*d^4 - 2*(b^7*c - 22*a*b^5*c^2 + 28*a^2*b^3
*c^3 + 176*a^3*b*c^4)*d^3*e - 6*(a*b^6*c + 18*a^2*b^4*c^2 - 92*a^3*b^2*c^3 + 16*a^4*c^4)*d^2*e^2 + 40*(a^2*b^5
*c - 3*a^3*b^3*c^2 - 4*a^4*b*c^3)*d*e^3 - (a^2*b^6 + 18*a^3*b^4*c - 92*a^4*b^2*c^2 + 16*a^5*c^3)*e^4)*x)/(a^3*
b^8*c - 16*a^4*b^6*c^2 + 96*a^5*b^4*c^3 - 256*a^6*b^2*c^4 + 256*a^7*c^5 + (b^8*c^4 - 16*a*b^6*c^5 + 96*a^2*b^4
*c^6 - 256*a^3*b^2*c^7 + 256*a^4*c^8)*x^6 + 3*(b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 256*a^3*b^3*c^6 + 256
*a^4*b*c^7)*x^5 + 3*(b^10*c^2 - 15*a*b^8*c^3 + 80*a^2*b^6*c^4 - 160*a^3*b^4*c^5 + 256*a^5*c^7)*x^4 + (b^11*c -
10*a*b^9*c^2 + 320*a^3*b^5*c^4 - 1280*a^4*b^3*c^5 + 1536*a^5*b*c^6)*x^3 + 3*(a*b^10*c - 15*a^2*b^8*c^2 + 80*a
^3*b^6*c^3 - 160*a^4*b^4*c^4 + 256*a^6*c^6)*x^2 + 3*(a^2*b^9*c - 16*a^3*b^7*c^2 + 96*a^4*b^5*c^3 - 256*a^5*b^3
*c^4 + 256*a^6*b*c^5)*x), -1/3*(12*(5*(b^2*c^6 - 4*a*c^7)*d^4 - 10*(b^3*c^5 - 4*a*b*c^6)*d^3*e + 6*(b^4*c^4 -
3*a*b^2*c^5 - 4*a^2*c^6)*d^2*e^2 - (b^5*c^3 + 2*a*b^3*c^4 - 24*a^2*b*c^5)*d*e^3 + (a*b^4*c^3 - 3*a^2*b^2*c^4 -
4*a^3*c^5)*e^4)*x^5 + (b^7*c - 17*a*b^5*c^2 + 118*a^2*b^3*c^3 - 264*a^3*b*c^4)*d^4 + 2*(a*b^6*c - 22*a^2*b^4*
c^2 + 8*a^3*b^2*c^3 + 256*a^4*c^4)*d^3*e + 6*(a^2*b^5*c + 22*a^3*b^3*c^2 - 104*a^4*b*c^3)*d^2*e^2 - 4*(11*a^3*
b^4*c - 28*a^4*b^2*c^2 - 64*a^5*c^3)*d*e^3 + (a^3*b^5 + 22*a^4*b^3*c - 104*a^5*b*c^2)*e^4 + 30*(5*(b^3*c^5 - 4
*a*b*c^6)*d^4 - 10*(b^4*c^4 - 4*a*b^2*c^5)*d^3*e + 6*(b^5*c^3 - 3*a*b^3*c^4 - 4*a^2*b*c^5)*d^2*e^2 - (b^6*c^2
+ 2*a*b^4*c^3 - 24*a^2*b^2*c^4)*d*e^3 + (a*b^5*c^2 - 3*a^2*b^3*c^3 - 4*a^3*b*c^4)*e^4)*x^4 + (10*(11*b^4*c^4 -
28*a*b^2*c^5 - 64*a^2*c^6)*d^4 - 20*(11*b^5*c^3 - 28*a*b^3*c^4 - 64*a^2*b*c^5)*d^3*e + 12*(11*b^6*c^2 - 17*a*
b^4*c^3 - 92*a^2*b^2*c^4 - 64*a^3*c^5)*d^2*e^2 - 2*(11*b^7*c + 38*a*b^5*c^2 - 232*a^2*b^3*c^3 - 384*a^3*b*c^4)
*d*e^3 + (b^8 + 6*a*b^6*c + 62*a^2*b^4*c^2 - 440*a^3*b^2*c^3 + 128*a^4*c^4)*e^4)*x^3 + 3*(5*(b^5*c^3 + 12*a*b^
3*c^4 - 64*a^2*b*c^5)*d^4 - 10*(b^6*c^2 + 12*a*b^4*c^3 - 64*a^2*b^2*c^4)*d^3*e + 6*(b^7*c + 13*a*b^5*c^2 - 52*
a^2*b^3*c^3 - 64*a^3*b*c^4)*d^2*e^2 - 2*(17*a*b^6*c - 44*a^2*b^4*c^2 - 64*a^3*b^2*c^3 - 128*a^4*c^4)*d*e^3 + (
a*b^7 + 13*a^2*b^5*c - 52*a^3*b^3*c^2 - 64*a^4*b*c^3)*e^4)*x^2 - 24*(5*a^3*c^4*d^4 - 10*a^3*b*c^3*d^3*e + (5*c
^7*d^4 - 10*b*c^6*d^3*e + 6*(b^2*c^5 + a*c^6)*d^2*e^2 - (b^3*c^4 + 6*a*b*c^5)*d*e^3 + (a*b^2*c^4 + a^2*c^5)*e^
4)*x^6 + 3*(5*b*c^6*d^4 - 10*b^2*c^5*d^3*e + 6*(b^3*c^4 + a*b*c^5)*d^2*e^2 - (b^4*c^3 + 6*a*b^2*c^4)*d*e^3 + (
a*b^3*c^3 + a^2*b*c^4)*e^4)*x^5 + 6*(a^3*b^2*c^2 + a^4*c^3)*d^2*e^2 - (a^3*b^3*c + 6*a^4*b*c^2)*d*e^3 + (a^4*b
^2*c + a^5*c^2)*e^4 + 3*(5*(b^2*c^5 + a*c^6)*d^4 - 10*(b^3*c^4 + a*b*c^5)*d^3*e + 6*(b^4*c^3 + 2*a*b^2*c^4 + a
^2*c^5)*d^2*e^2 - (b^5*c^2 + 7*a*b^3*c^3 + 6*a^2*b*c^4)*d*e^3 + (a*b^4*c^2 + 2*a^2*b^2*c^3 + a^3*c^4)*e^4)*x^4
+ (5*(b^3*c^4 + 6*a*b*c^5)*d^4 - 10*(b^4*c^3 + 6*a*b^2*c^4)*d^3*e + 6*(b^5*c^2 + 7*a*b^3*c^3 + 6*a^2*b*c^4)*d
^2*e^2 - (b^6*c + 12*a*b^4*c^2 + 36*a^2*b^2*c^3)*d*e^3 + (a*b^5*c + 7*a^2*b^3*c^2 + 6*a^3*b*c^3)*e^4)*x^3 + 3*
(5*(a*b^2*c^4 + a^2*c^5)*d^4 - 10*(a*b^3*c^3 + a^2*b*c^4)*d^3*e + 6*(a*b^4*c^2 + 2*a^2*b^2*c^3 + a^3*c^4)*d^2*
e^2 - (a*b^5*c + 7*a^2*b^3*c^2 + 6*a^3*b*c^3)*d*e^3 + (a^2*b^4*c + 2*a^3*b^2*c^2 + a^4*c^3)*e^4)*x^2 + 3*(5*a^
2*b*c^4*d^4 - 10*a^2*b^2*c^3*d^3*e + 6*(a^2*b^3*c^2 + a^3*b*c^3)*d^2*e^2 - (a^2*b^4*c + 6*a^3*b^2*c^2)*d*e^3 +
(a^3*b^3*c + a^4*b*c^2)*e^4)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - 3*
((b^6*c^2 - 22*a*b^4*c^3 + 28*a^2*b^2*c^4 + 176*a^3*c^5)*d^4 - 2*(b^7*c - 22*a*b^5*c^2 + 28*a^2*b^3*c^3 + 176*
a^3*b*c^4)*d^3*e - 6*(a*b^6*c + 18*a^2*b^4*c^2 - 92*a^3*b^2*c^3 + 16*a^4*c^4)*d^2*e^2 + 40*(a^2*b^5*c - 3*a^3*
b^3*c^2 - 4*a^4*b*c^3)*d*e^3 - (a^2*b^6 + 18*a^3*b^4*c - 92*a^4*b^2*c^2 + 16*a^5*c^3)*e^4)*x)/(a^3*b^8*c - 16*
a^4*b^6*c^2 + 96*a^5*b^4*c^3 - 256*a^6*b^2*c^4 + 256*a^7*c^5 + (b^8*c^4 - 16*a*b^6*c^5 + 96*a^2*b^4*c^6 - 256*
a^3*b^2*c^7 + 256*a^4*c^8)*x^6 + 3*(b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 256*a^3*b^3*c^6 + 256*a^4*b*c^7)
*x^5 + 3*(b^10*c^2 - 15*a*b^8*c^3 + 80*a^2*b^6*c^4 - 160*a^3*b^4*c^5 + 256*a^5*c^7)*x^4 + (b^11*c - 10*a*b^9*c
^2 + 320*a^3*b^5*c^4 - 1280*a^4*b^3*c^5 + 1536*a^5*b*c^6)*x^3 + 3*(a*b^10*c - 15*a^2*b^8*c^2 + 80*a^3*b^6*c^3
- 160*a^4*b^4*c^4 + 256*a^6*c^6)*x^2 + 3*(a^2*b^9*c - 16*a^3*b^7*c^2 + 96*a^4*b^5*c^3 - 256*a^5*b^3*c^4 + 256*
a^6*b*c^5)*x)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or 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 = 0.27 (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=-\frac {8 \, {\left (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}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {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} + 12 \, 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} d e^{3} x + 3 \, a^{2} b^{4} e^{4} x + 66 \, a^{3} b^{2} c e^{4} x - 12 \, a^{4} c^{2} e^{4} x + b^{5} c d^{4} - 13 \, a b^{3} c^{2} d^{4} + 66 \, a^{2} b c^{3} d^{4} + 2 \, a b^{4} c d^{3} e - 36 \, a^{2} b^{2} c^{2} d^{3} e - 128 \, a^{3} c^{3} d^{3} e + 6 \, a^{2} b^{3} c d^{2} e^{2} + 156 \, a^{3} b c^{2} d^{2} e^{2} - 44 \, a^{3} b^{2} c d e^{3} - 64 \, a^{4} c^{2} d e^{3} + a^{3} b^{3} e^{4} + 26 \, a^{4} b c e^{4}}{3 \, {\left (b^{6} c - 12 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} - 64 \, a^{3} c^{4}\right )} {\left (c x^{2} + b x + a\right )}^{3}} \]

[In]

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

[Out]

-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 + 12*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*d*e^3*x + 3*a^2*b^4*e^4*x + 66*a^3*b^2*c*e^4*x - 12*a^4*c^2*e^4
*x + b^5*c*d^4 - 13*a*b^3*c^2*d^4 + 66*a^2*b*c^3*d^4 + 2*a*b^4*c*d^3*e - 36*a^2*b^2*c^2*d^3*e - 128*a^3*c^3*d^
3*e + 6*a^2*b^3*c*d^2*e^2 + 156*a^3*b*c^2*d^2*e^2 - 44*a^3*b^2*c*d*e^3 - 64*a^4*c^2*d*e^3 + a^3*b^3*e^4 + 26*a
^4*b*c*e^4)/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*(c*x^2 + b*x + a)^3)

Mupad [B] (verification not implemented)

Time = 10.89 (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=-\frac {\frac {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}{3\,c\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x^3\,\left (-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\right )}{3\,c\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {4\,c^2\,x^5\,\left (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\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {x^2\,\left (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\right )}{c\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x\,\left (-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\right )}{c\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {10\,b\,c\,x^4\,\left (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\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}}{x^2\,\left (3\,c\,a^2+3\,a\,b^2\right )+x^4\,\left (3\,b^2\,c+3\,a\,c^2\right )+a^3+x^3\,\left (b^3+6\,a\,c\,b\right )+c^3\,x^6+3\,b\,c^2\,x^5+3\,a^2\,b\,x}-\frac {8\,\mathrm {atan}\left (\frac {\left (\frac {4\,\left (c\,d^2-b\,d\,e+a\,e^2\right )\,\left (-64\,a^3\,b\,c^3+48\,a^2\,b^3\,c^2-12\,a\,b^5\,c+b^7\right )\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {8\,c\,x\,\left (c\,d^2-b\,d\,e+a\,e^2\right )\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}}\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{4\,a^2\,c\,e^4+4\,a\,b^2\,e^4-24\,a\,b\,c\,d\,e^3+24\,a\,c^2\,d^2\,e^2-4\,b^3\,d\,e^3+24\,b^2\,c\,d^2\,e^2-40\,b\,c^2\,d^3\,e+20\,c^3\,d^4}\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}} \]

[In]

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

[Out]

- ((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 - 192*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 + 4
8*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*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*(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) - (8*atan((((4*(a*e^2 + c*d^
2 - b*d*e)*(b^7 - 64*a^3*b*c^3 + 48*a^2*b^3*c^2 - 12*a*b^5*c)*(b^2*e^2 + 5*c^2*d^2 + a*c*e^2 - 5*b*c*d*e))/((4
*a*c - b^2)^(7/2)*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (8*c*x*(a*e^2 + c*d^2 - b*d*e)*(b^2*e^2
+ 5*c^2*d^2 + a*c*e^2 - 5*b*c*d*e))/(4*a*c - b^2)^(7/2))*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c))/(20
*c^3*d^4 + 4*a*b^2*e^4 + 4*a^2*c*e^4 - 4*b^3*d*e^3 + 24*a*c^2*d^2*e^2 + 24*b^2*c*d^2*e^2 - 40*b*c^2*d^3*e - 24
*a*b*c*d*e^3))*(a*e^2 + c*d^2 - b*d*e)*(b^2*e^2 + 5*c^2*d^2 + a*c*e^2 - 5*b*c*d*e))/(4*a*c - b^2)^(7/2)