3.23.19 \(\int \frac {1}{(d+e x) (a+b x+c x^2)^4} \, dx\) [2219]

Optimal. Leaf size=771 \[ -\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^3}-\frac {5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (10 c^2 d^2-3 b^2 e^2-c e (5 b d-12 a e)\right )-c (2 c d-b e) \left (10 c^2 d^2-3 b^2 e^2-2 c e (5 b d-11 a e)\right ) x}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )^2}+\frac {b^5 c d e^4+2 b^6 e^5-64 a^3 c^3 e^5+b^4 c e^3 \left (c d^2-23 a e^2\right )-2 b^3 c^2 d e^2 \left (17 c d^2+5 a e^2\right )-4 b c^3 d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right )+2 b^2 c^2 e \left (25 c^2 d^4+48 a c d^2 e^2+43 a^2 e^4\right )-2 c (2 c d-b e) \left (10 c^4 d^4+b^4 e^4+b^2 c e^3 (3 b d-11 a e)-4 c^3 d^2 e (5 b d-8 a e)+c^2 e^2 \left (7 b^2 d^2-32 a b d e+38 a^2 e^2\right )\right ) x}{2 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 \left (a+b x+c x^2\right )}+\frac {\left (40 c^7 d^7+b^7 e^7-14 a b^5 c e^7+70 a^2 b^3 c^2 e^7-140 a^3 b c^3 e^7-28 c^6 d^5 e (5 b d-6 a e)+28 c^5 d^3 e^2 \left (6 b^2 d^2-15 a b d e+10 a^2 e^2\right )-70 c^4 d e^3 \left (b^3 d^3-4 a b^2 d^2 e+6 a^2 b d e^2-4 a^3 e^3\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2} \left (c d^2-b d e+a e^2\right )^4}+\frac {e^7 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {e^7 \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^4} \]

[Out]

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*(-5*a*c*e*(-b
*e+2*c*d)^2+(2*a*c*e-b^2*e+b*c*d)*(10*c^2*d^2-3*b^2*e^2-c*e*(-12*a*e+5*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*l
n(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

________________________________________________________________________________________

Rubi [A]
time = 5.41, antiderivative size = 771, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {754, 836, 814, 648, 632, 212, 642} \begin {gather*} \frac {\left (-140 a^3 b c^3 e^7+70 a^2 b^3 c^2 e^7+28 c^5 d^3 e^2 \left (10 a^2 e^2-15 a b d e+6 b^2 d^2\right )-70 c^4 d e^3 \left (-4 a^3 e^3+6 a^2 b d e^2-4 a b^2 d^2 e+b^3 d^3\right )-14 a b^5 c e^7-28 c^6 d^5 e (5 b d-6 a e)+b^7 e^7+40 c^7 d^7\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} \left (a e^2-b d e+c d^2\right )^4}+\frac {-64 a^3 c^3 e^5+2 b^2 c^2 e \left (43 a^2 e^4+48 a c d^2 e^2+25 c^2 d^4\right )-2 c x (2 c d-b e) \left (c^2 e^2 \left (38 a^2 e^2-32 a b d e+7 b^2 d^2\right )+b^2 c e^3 (3 b d-11 a e)-4 c^3 d^2 e (5 b d-8 a e)+b^4 e^4+10 c^4 d^4\right )-4 b c^3 d \left (19 a^2 e^4+16 a c d^2 e^2+5 c^2 d^4\right )+b^4 c e^3 \left (c d^2-23 a e^2\right )-2 b^3 c^2 d e^2 \left (5 a e^2+17 c d^2\right )+2 b^6 e^5+b^5 c d e^4}{2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^3}-\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}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )^2}-\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 )}-\frac {e^7 \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^4}+\frac {e^7 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-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)/(6*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*
(a + b*x + c*x^2)^2) + (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)/(2*(b^2 - 4*a*c)^3*(c*d^2 - b*d*e + a*e^2
)^3*(a + b*x + c*x^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))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(7/2)*(c*d^2
- b*d*e + a*e^2)^4) + (e^7*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^4 - (e^7*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*
d*e + a*e^2)^4)

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 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

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

Rule 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> 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] + Dist[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] && 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] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), 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] && IntegerQ[m]

Rule 836

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] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps

\begin {align*} \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 {\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 (a+b x+c x^2\right )^3} \, dx}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\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 {5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+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 )-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 {\int \frac {3 \left (20 c^4 d^4+2 b^4 e^4-2 c^3 d^2 e (15 b d-22 a e)+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 d e+16 a^2 e^2\right )\right )+3 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 (a+b x+c x^2\right )^2} \, dx}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\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 {5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+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 )-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 {\int \frac {6 \left (20 c^6 d^6-b^6 e^6-2 c^5 d^4 e (25 b d-32 a e)-b^4 c e^5 (b d-12 a e)+2 c^4 d^2 e^2 \left (17 b^2 d^2-48 a b d e+38 a^2 e^2\right )-b^2 c^2 e^4 \left (b^2 d^2-11 a b d e+48 a^2 e^2\right )-c^3 e^3 \left (b^3 d^3-10 a b^2 d^2 e+38 a^2 b d e^2-64 a^3 e^3\right )\right )+6 c e (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}{(d+e x) \left (a+b x+c x^2\right )} \, dx}{6 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\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 {5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+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 )-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 {\int \left (-\frac {6 \left (b^2-4 a c\right )^3 e^8}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {6 \left (20 c^7 d^7+b^7 e^7-13 a b^5 c e^7+59 a^2 b^3 c^2 e^7-102 a^3 b c^3 e^7-14 c^6 d^5 e (5 b d-6 a e)+14 c^5 d^3 e^2 \left (6 b^2 d^2-15 a b d e+10 a^2 e^2\right )-35 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 )+c \left (b^2-4 a c\right )^3 e^7 x\right )}{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx}{6 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\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 {5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+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 )-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 {e^7 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {\int \frac {20 c^7 d^7+b^7 e^7-13 a b^5 c e^7+59 a^2 b^3 c^2 e^7-102 a^3 b c^3 e^7-14 c^6 d^5 e (5 b d-6 a e)+14 c^5 d^3 e^2 \left (6 b^2 d^2-15 a b d e+10 a^2 e^2\right )-35 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 )+c \left (b^2-4 a c\right )^3 e^7 x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\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 {5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+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 )-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 {e^7 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {e^7 \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^4}-\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 ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\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 {5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+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 )-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 {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}+\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 {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 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\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 {5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+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 )-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 ) \tanh ^{-1}\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}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 2.05, size = 769, normalized size = 1.00 \begin {gather*} \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 ) \tan ^{-1}\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2810\) vs. \(2(758)=1516\).
time = 1.17, size = 2811, normalized size = 3.65

method	result	size
default	\(\text {Expression too large to display}\)	\(2811\)
risch	\(\text {Expression too large to display}\)	\(13500\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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+12*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^2-350*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^3-2268*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*c^5*d^2*e^5+27*a^3*b^4*c^2*e^7+518*a^3*b^3*c^3*d*e^6-1632*a^3*b^2*c^4*d^2*e^5+992*a^3*b*c^5*d^3*e^
4+32*a^3*c^6*d^4*e^3+13*a^2*b^6*c*e^7-48*a^2*b^5*c^2*d*e^6-213*a^2*b^4*c^3*d^2*e^5+1286*a^2*b^3*c^4*d^3*e^4-17
04*a^2*b^2*c^5*d^4*e^3+672*a^2*b*c^6*d^5*e^2-2*a*b^8*e^7-12*a*b^7*c*d*e^6+33*a*b^6*c^2*d^2*e^5+46*a*b^5*c^3*d^
3*e^4-379*a*b^4*c^4*d^4*e^3+714*a*b^3*c^5*d^5*e^2-560*a*b^2*c^6*d^6*e+160*a*b*c^7*d^7+2*b^9*d*e^6-3*b^8*c*d^2*
e^5+2*b^7*c^2*d^3*e^4-18*b^6*c^3*d^4*e^3+42*b^5*c^4*d^5*e^2-35*b^4*c^5*d^6*e+10*b^3*c^6*d^7)/(64*a^3*c^3-48*a^
2*b^2*c^2+12*a*b^4*c-b^6)*x^2-1/2*(44*a^5*b*c^3*e^7+232*a^5*c^4*d*e^6-172*a^4*b^3*c^2*e^7+156*a^4*b^2*c^3*d*e^
6-540*a^4*b*c^4*d^2*e^5+488*a^4*c^5*d^3*e^4+54*a^3*b^5*c*e^7+202*a^3*b^4*c^2*d*e^6-602*a^3*b^3*c^3*d^2*e^5+836
*a^3*b^2*c^4*d^3*e^4-828*a^3*b*c^5*d^4*e^3+344*a^3*c^6*d^5*e^2-5*a^2*b^7*e^7-66*a^2*b^6*c*d*e^6+45*a^2*b^5*c^2
*d^2*e^5+310*a^2*b^4*c^3*d^3*e^4-604*a^2*b^3*c^4*d^4*e^3+540*a^2*b^2*c^5*d^5*e^2-308*a^2*b*c^6*d^6*e+88*a^2*c^
7*d^7+6*a*b^8*d*e^6+6*a*b^7*c*d^2*e^5-36*a*b^6*c^2*d^3*e^4-24*a*b^5*c^3*d^4*e^3+138*a*b^4*c^4*d^5*e^2-126*a*b^
3*c^5*d^6*e+36*a*b^2*c^6*d^7-b^9*d^2*e^5+2*b^8*c*d^3*e^4+2*b^7*c^2*d^4*e^3-8*b^6*c^3*d^5*e^2+7*b^5*c^4*d^6*e-2
*b^4*c^5*d^7)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x-1/6*(352*a^6*c^3*e^7-438*a^5*b^2*c^2*e^7-228*a^5*b*
c^3*d*e^6+576*a^5*c^4*d^2*e^5+124*a^4*b^4*c*e^7+682*a^4*b^3*c^2*d*e^6-1242*a^4*b^2*c^3*d^2*e^5+156*a^4*b*c^4*d
^3*e^4+288*a^4*c^5*d^4*e^3-11*a^3*b^6*e^7-202*a^3*b^5*c*d*e^6-51*a^3*b^4*c^2*d^2*e^5+1190*a^3*b^3*c^3*d^3*e^4-
1314*a^3*b^2*c^4*d^4*e^3+324*a^3*b*c^5*d^5*e^2+64*a^3*c^6*d^6*e+18*a^2*b^7*d*e^6+78*a^2*b^6*c*d^2*e^5-216*a^2*
b^5*c^2*d^3*e^4-120*a^2*b^4*c^3*d^4*e^3+618*a^2*b^3*c^4*d^5*e^2-510*a^2*b^2*c^5*d^6*e+132*a^2*b*c^6*d^7-9*a*b^
8*d^2*e^5+2*a*b^7*c*d^3*e^4+74*a*b^6*c^2*d^4*e^3-144*a*b^5*c^3*d^5*e^2+103*a*b^4*c^4*d^6*e-26*a*b^3*c^5*d^7+2*
b^9*d^3*e^4-8*b^8*c*d^4*e^3+12*b^7*c^2*d^5*e^2-8*b^6*c^3*d^6*e+2*b^5*c^4*d^7)/(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+1/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*(1/2*(64*a^3*c^4*e^7-48*a^2*b^2*c^3*e
^7+12*a*b^4*c^2*e^7-b^6*c*e^7)/c*ln(c*x^2+b*x+a)+2*(102*a^3*b*c^3*e^7-140*a^3*c^4*d*e^6-59*a^2*b^3*c^2*e^7+210
*a^2*b*c^4*d^2*e^5-140*a^2*c^5*d^3*e^4+13*a*b^5*c*e^7-140*a*b^2*c^4*d^3*e^4+210*a*b*c^5*d^4*e^3-84*a*c^6*d^5*e
^2-b^7*e^7+35*b^3*c^4*d^4*e^3-84*b^2*c^5*d^5*e^2+70*b*c^6*d^6*e-20*c^7*d^7-1/2*(64*a^3*c^4*e^7-48*a^2*b^2*c^3*
e^7+12*a*b^4*c^2*e^7-b^6*c*e^7)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))))+e^7*ln(e*x+d)/(a*
e^2-b*d*e+c*d^2)^4

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(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

________________________________________________________________________________________

Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3230 vs. \(2 (754) = 1508\).
time = 1.27, size = 3230, normalized size = 4.19 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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) + e^8*log(abs(x*e + 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) - (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 + 2
80*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*c^3*d^3*e^5 + 768*a^5*b*c^4*d^3*e^5 + 6*a^2*b^8*d^2*e^6 - 68*a
^3*b^6*c*d^2*e^6 + 240*a^4*b^4*c^2*d^2*e^6 - 192*a^5*b^2*c^3*d^2*e^6 - 256*a^6*c^4*d^2*e^6 - 4*a^3*b^7*d*e^7 +
48*a^4*b^5*c*d*e^7 - 192*a^5*b^3*c^2*d*e^7 + 256*a^6*b*c^3*d*e^7 + a^4*b^6*e^8 - 12*a^5*b^4*c*e^8 + 48*a^6*b^
2*c^2*e^8 - 64*a^7*c^3*e^8)*sqrt(-b^2 + 4*a*c)) - 1/6*(2*b^5*c^4*d^7 - 26*a*b^3*c^5*d^7 + 132*a^2*b*c^6*d^7 -
8*b^6*c^3*d^6*e + 103*a*b^4*c^4*d^6*e - 510*a^2*b^2*c^5*d^6*e + 64*a^3*c^6*d^6*e + 12*b^7*c^2*d^5*e^2 - 144*a*
b^5*c^3*d^5*e^2 + 618*a^2*b^3*c^4*d^5*e^2 + 324*a^3*b*c^5*d^5*e^2 - 8*b^8*c*d^4*e^3 + 74*a*b^6*c^2*d^4*e^3 - 1
20*a^2*b^4*c^3*d^4*e^3 - 1314*a^3*b^2*c^4*d^4*e^3 + 288*a^4*c^5*d^4*e^3 + 2*b^9*d^3*e^4 + 2*a*b^7*c*d^3*e^4 -
216*a^2*b^5*c^2*d^3*e^4 + 1190*a^3*b^3*c^3*d^3*e^4 + 156*a^4*b*c^4*d^3*e^4 - 9*a*b^8*d^2*e^5 + 78*a^2*b^6*c*d^
2*e^5 - 51*a^3*b^4*c^2*d^2*e^5 - 1242*a^4*b^2*c^3*d^2*e^5 + 576*a^5*c^4*d^2*e^5 + 18*a^2*b^7*d*e^6 - 202*a^3*b
^5*c*d*e^6 + 682*a^4*b^3*c^2*d*e^6 - 228*a^5*b*c^3*d*e^6 - 11*a^3*b^6*e^7 + 124*a^4*b^4*c*e^7 - 438*a^5*b^2*c^
2*e^7 + 352*a^6*c^3*e^7 + 6*(20*c^9*d^7 - 70*b*c^8*d^6*e + 84*b^2*c^7*d^5*e^2 + 84*a*c^8*d^5*e^2 - 35*b^3*c^6*
d^4*e^3 - 210*a*b*c^7*d^4*e^3 + 140*a*b^2*c^6*d^3*e^4 + 140*a^2*c^7*d^3*e^4 - 210*a^2*b*c^6*d^2*e^5 + b^6*c^3*
d*e^6 - 12*a*b^4*c^4*d*e^6 + 48*a^2*b^2*c^5*d*e^6 + 76*a^3*c^6*d*e^6 - a*b^5*c^3*e^7 + 11*a^2*b^3*c^4*e^7 - 38
*a^3*b*c^5*e^7)*x^5 + 3*(100*b*c^8*d^7 - 350*b^2*c^7*d^6*e + 420*b^3*c^6*d^5*e^2 + 420*a*b*c^7*d^5*e^2 - 175*b
^4*c^5*d^4*e^3 - 1050*a*b^2*c^6*d^4*e^3 + 700*a*b^3*c^5*d^3*e^4 + 700*a^2*b*c^6*d^3*e^4 - b^6*c^3*d^2*e^5 + 12
*a*b^4*c^4*d^2*e^5 - 1098*a^2*b^2*c^5*d^2*e^5 + 64*a^3*c^6*d^2*e^5 + 6*b^7*c^2*d*e^6 - 72*a*b^5*c^3*d*e^6 + 28
8*a^2*b^3*c^4*d*e^6 + 316*a^3*b*c^5*d*e^6 - 6*a*b^6*c^2*e^7 + 67*a^2*b^4*c^3*e^7 - 238*a^3*b^2*c^4*e^7 + 64*a^
4*c^5*e^7)*x^4 + (220*b^2*c^7*d^7 + 320*a*c^8*d^7 - 770*b^3*c^6*d^6*e - 1120*a*b*c^7*d^6*e + 924*b^4*c^5*d^5*e
^2 + 2268*a*b^2*c^6*d^5*e^2 + 1344*a^2*c^7*d^5*e^2 - 385*b^5*c^4*d^4*e^3 - 2870*a*b^3*c^5*d^4*e^3 - 3360*a^2*b
*c^6*d^4*e^3 + 2*b^6*c^3*d^3*e^4 + 1516*a*b^4*c^4*d^3*e^4 + 3876*a^2*b^2*c^5*d^3*e^4 + 2112*a^3*c^6*d^3*e^4 -
9*b^7*c^2*d^2*e^5 + 108*a*b^5*c^3*d^2*e^5 - 2742*a^2*b^3*c^4*d^2*e^5 - 2784*a^3*b*c^5*d^2*e^5 + 18*b^8*c*d*e^6
- 198*a*b^6*c^2*d*e^6 + 648*a^2*b^4*c^3*d*e^6 + 1252*a^3*b^2*c^4*d*e^6 + 1088*a^4*c^5*d*e^6 - 18*a*b^7*c*e^7
+ 189*a^2*b^5*c^2*e^7 - 578*a^3*b^3*c^3*e^7 - 160*a^4*b*c^4*e^7)*x^3 + 3*(10*b^3*c^6*d^7 + 160*a*b*c^7*d^7 - 3
5*b^4*c^5*d^6*e - 560*a*b^2*c^6*d^6*e + 42*b^5*c^4*d^5*e^2 + 714*a*b^3*c^5*d^5*e^2 + 672*a^2*b*c^6*d^5*e^2 - 1
8*b^6*c^3*d^4*e^3 - 379*a*b^4*c^4*d^4*e^3 - 1704*a^2*b^2*c^5*d^4*e^3 + 32*a^3*c^6*d^4*e^3 + 2*b^7*c^2*d^3*e^4
+ 46*a*b^5*c^3*d^3*e^4 + 1286*a^2*b^3*c^4*d^3*e^4 + 992*a^3*b*c^5*d^3*e^4 - 3*b^8*c*d^2*e^5 + 33*a*b^6*c^2*d^2
*e^5 - 213*a^2*b^4*c^3*d^2*e^5 - 1632*a^3*b^2*c^4*d^2*e^5 + 192*a^4*c^5*d^2*e^5 + 2*b^9*d*e^6 - 12*a*b^7*c*d*e
^6 - 48*a^2*b^5*c^2*d*e^6 + 518*a^3*b^3*c^3*d*e^6 + 352*a^4*b*c^4*d*e^6 - 2*a*b^8*e^7 + 13*a^2*b^6*c*e^7 + 27*
a^3*b^4*c^2*e^7 - 328*a^4*b^2*c^3*e^7 + 160*a^5*c^4*e^7)*x^2 - 3*(2*b^4*c^5*d^7 - 36*a*b^2*c^6*d^7 - 88*a^2*c^
7*d^7 - 7*b^5*c^4*d^6*e + 126*a*b^3*c^5*d^6*e + 308*a^2*b*c^6*d^6*e + 8*b^6*c^3*d^5*e^2 - 138*a*b^4*c^4*d^5*e^
2 - 540*a^2*b^2*c^5*d^5*e^2 - 344*a^3*c^6*d^5*e^2 - 2*b^7*c^2*d^4*e^3 + 24*a*b^5*c^3*d^4*e^3 + 604*a^2*b^3*c^4
*d^4*e^3 + 828*a^3*b*c^5*d^4*e^3 - 2*b^8*c*d^3*e^4 + 36*a*b^6*c^2*d^3*e^4 - 310*a^2*b^4*c^3*d^3*e^4 - 836*a^3*
b^2*c^4*d^3*e^4 - 488*a^4*c^5*d^3*e^4 + b^9*d^2...

________________________________________________________________________________________

Mupad [B]
time = 33.20, size = 2500, normalized size = 3.24 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((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 + 12
4*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 - 680*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 + 1
89*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 + 1024*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*d^2*e^3 - 1536*a^2*b*c^5*d^2*e^3 + 1092*a^2*b^2*c^4*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^2*(16
0*a^4*c^4*e^5 - 2*b^8*e^5 + 10*b^3*c^5*d^5 - 25*b^4*c^4*d^4*e + 27*a^2*b^4*c^2*e^5 - 328*a^3*b^2*c^3*e^5 + 32*
a^3*c^5*d^2*e^3 + 17*b^5*c^3*d^3*e^2 - b^6*c^2*d^2*e^3 + 160*a*b*c^6*d^5 + 13*a*b^6*c*e^5 + b^7*c*d*e^4 - 792*
a^2*b^2*c^4*d^2*e^3 - 400*a*b^2*c^5*d^4*e - 21*a*b^5*c^2*d*e^4 + 512*a^3*b*c^4*d*e^4 + 304*a*b^3*c^4*d^3*e^2 -
50*a*b^4*c^3*d^2*e^3 + 512*a^2*b*c^5*d^3*e^2 + 190*a^2*b^3*c^3*d*e^4))/(2*(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^4*(100*b*c^7*d^5 + 64*a^3*c^5*e^5 - 6*b^
6*c^2*e^5 + 67*a*b^4*c^3*e^5 - 250*b^2*c^6*d^4*e - 5*b^5*c^3*d*e^4 - 238*a^2*b^2*c^4*e^5 + 170*b^3*c^5*d^3*e^2
- 5*b^4*c^4*d^2*e^3 + 320*a*b*c^6*d^3*e^2 + 50*a*b^3*c^4*d*e^4 + 380*a^2*b*c^5*d*e^4 - 480*a*b^2*c^5*d^2*e^3)
)/(2*(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 + 4
8*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^
5*(20*c^8*d^5 - b^5*c^3*e^5 + 11*a*b^3*c^4*e^5 - 38*a^2*b*c^5*e^5 + 64*a*c^7*d^3*e^2 + 76*a^2*c^6*d*e^4 - b^4*
c^4*d*e^4 + 34*b^2*c^6*d^3*e^2 - b^3*c^5*d^2*e^3 - 50*b*c^7*d^4*e - 96*a*b*c^6*d^2*e^3 + 10*a*b^2*c^5*d*e^4))/
(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...

________________________________________________________________________________________