\(\int \frac {1}{(d+e x)^2 (a+b x+c x^2)^2} \, dx\) [2197]

Optimal result

Integrand size = 20, antiderivative size = 344 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {2 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )}+\frac {2 \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^3}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{\left (c d^2-b d e+a e^2\right )^3} \]

[Out]

-2*e*(c^2*d^2+b^2*e^2-c*e*(3*a*e+b*d))/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)+(-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)/(e*x+d)/(c*x^2+b*x+a)+2*(2*c^4*d^4-b^4*e^4-4*c^3*d^2*e*(-3*a*e+b*
d)-6*a*c^2*e^3*(a*e+2*b*d)+2*b^2*c*e^3*(3*a*e+b*d))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(3/2)/(
a*e^2-b*d*e+c*d^2)^3+2*e^3*(-b*e+2*c*d)*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^3-e^3*(-b*e+2*c*d)*ln(c*x^2+b*x+a)/(a*e^
2-b*d*e+c*d^2)^3

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 344, 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 = {754, 814, 648, 632, 212, 642} \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=\frac {2 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^3}-\frac {2 e \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]

[In]

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

[Out]

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

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )}-\frac {\int \frac {2 \left (c^2 d^2-b^2 e^2+3 a c e^2\right )+2 c e (2 c d-b e) x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx}{\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}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )}-\frac {\int \left (\frac {2 e^2 \left (-c^2 d^2-b^2 e^2+c e (b d+3 a e)\right )}{\left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {2 \left (b^2-4 a c\right ) e^4 (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {2 \left (c^4 d^4-b^4 e^4-2 c^3 d^2 e (b d-3 a e)-a c^2 e^3 (10 b d+3 a e)+b^2 c e^3 (2 b d+5 a e)+c \left (b^2-4 a c\right ) e^3 (2 c d-b e) x\right )}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}\right ) \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {2 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {2 \int \frac {c^4 d^4-b^4 e^4-2 c^3 d^2 e (b d-3 a e)-a c^2 e^3 (10 b d+3 a e)+b^2 c e^3 (2 b d+5 a e)+c \left (b^2-4 a c\right ) e^3 (2 c d-b e) x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {2 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {\left (e^3 (2 c d-b e)\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{\left (c d^2-b d e+a e^2\right )^3}-\frac {\left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {2 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{\left (c d^2-b d e+a e^2\right )^3}+\frac {\left (2 \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 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 ) \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {2 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )}+\frac {2 \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 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 )^{3/2} \left (c d^2-b d e+a e^2\right )^3}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{\left (c d^2-b d e+a e^2\right )^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {e^3}{\left (c d^2+e (-b d+a e)\right )^2 (d+e x)}+\frac {-b^3 e^2+b^2 c e (2 d-e x)+b c \left (3 a e^2-c d (d-2 e x)\right )-2 c^2 \left (c d^2 x+a e (2 d-e x)\right )}{\left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2 (a+x (b+c x))}-\frac {2 \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2} \left (-c d^2+e (b d-a e)\right )^3}-\frac {2 e^3 (-2 c d+b e) \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^3}+\frac {e^3 (-2 c d+b e) \log (a+x (b+c x))}{\left (c d^2+e (-b d+a e)\right )^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 25.60 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.54

method	result	size
default	\(-\frac {\frac {\frac {c \left (2 e^{4} a^{2} c -a \,b^{2} e^{4}+b^{3} d \,e^{3}-3 b^{2} c \,d^{2} e^{2}+4 d^{3} e b \,c^{2}-2 d^{4} c^{3}\right ) x}{4 a c -b^{2}}+\frac {3 a^{2} b c \,e^{4}-4 a^{2} c^{2} d \,e^{3}-a \,b^{3} e^{4}-a \,b^{2} c d \,e^{3}+6 a b \,c^{2} d^{2} e^{2}-4 a \,c^{3} d^{3} e +b^{4} d \,e^{3}-3 b^{3} c \,d^{2} e^{2}+3 b^{2} c^{2} d^{3} e -d^{4} c^{3} b}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (-4 a b \,c^{2} e^{4}+8 a \,c^{3} d \,e^{3}+b^{3} c \,e^{4}-2 b^{2} c^{2} d \,e^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{c}+\frac {4 \left (3 c^{2} a^{2} e^{4}-5 a \,b^{2} c \,e^{4}+10 a b \,c^{2} d \,e^{3}-6 c^{3} a \,d^{2} e^{2}+b^{4} e^{4}-2 b^{3} c d \,e^{3}+2 b \,c^{3} d^{3} e -c^{4} d^{4}-\frac {\left (-4 a b \,c^{2} e^{4}+8 a \,c^{3} d \,e^{3}+b^{3} c \,e^{4}-2 b^{2} c^{2} d \,e^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{\left (e^{2} a -b d e +c \,d^{2}\right )^{3}}-\frac {e^{3}}{\left (e^{2} a -b d e +c \,d^{2}\right )^{2} \left (e x +d \right )}-\frac {2 e^{3} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{3}}\)	\(529\)
risch	\(\text {Expression too large to display}\)	\(56540\)

[In]

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

[Out]

-1/(a*e^2-b*d*e+c*d^2)^3*((c*(2*a^2*c*e^4-a*b^2*e^4+b^3*d*e^3-3*b^2*c*d^2*e^2+4*b*c^2*d^3*e-2*c^3*d^4)/(4*a*c-
b^2)*x+(3*a^2*b*c*e^4-4*a^2*c^2*d*e^3-a*b^3*e^4-a*b^2*c*d*e^3+6*a*b*c^2*d^2*e^2-4*a*c^3*d^3*e+b^4*d*e^3-3*b^3*
c*d^2*e^2+3*b^2*c^2*d^3*e-b*c^3*d^4)/(4*a*c-b^2))/(c*x^2+b*x+a)+2/(4*a*c-b^2)*(1/2*(-4*a*b*c^2*e^4+8*a*c^3*d*e
^3+b^3*c*e^4-2*b^2*c^2*d*e^3)/c*ln(c*x^2+b*x+a)+2*(3*c^2*a^2*e^4-5*a*b^2*c*e^4+10*a*b*c^2*d*e^3-6*c^3*a*d^2*e^
2+b^4*e^4-2*b^3*c*d*e^3+2*b*c^3*d^3*e-c^4*d^4-1/2*(-4*a*b*c^2*e^4+8*a*c^3*d*e^3+b^3*c*e^4-2*b^2*c^2*d*e^3)*b/c
)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))))-e^3/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)-2*e^3/(a*e^2-b*d*e
+c*d^2)^3*(b*e-2*c*d)*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2739 vs. \(2 (340) = 680\).

Time = 53.34 (sec) , antiderivative size = 5498, normalized size of antiderivative = 15.98 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^2,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. 932 vs. \(2 (340) = 680\).

Time = 0.28 (sec) , antiderivative size = 932, normalized size of antiderivative = 2.71 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {e^{7}}{{\left (c^{2} d^{4} e^{4} - 2 \, b c d^{3} e^{5} + b^{2} d^{2} e^{6} + 2 \, a c d^{2} e^{6} - 2 \, a b d e^{7} + a^{2} e^{8}\right )} {\left (e x + d\right )}} - \frac {{\left (2 \, c d e^{3} - b e^{4}\right )} \log \left (c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {b e}{e x + d} - \frac {b d e}{{\left (e x + d\right )}^{2}} + \frac {a e^{2}}{{\left (e x + d\right )}^{2}}\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}} - \frac {2 \, {\left (2 \, c^{4} d^{4} e^{2} - 4 \, b c^{3} d^{3} e^{3} + 12 \, a c^{3} d^{2} e^{4} + 2 \, b^{3} c d e^{5} - 12 \, a b c^{2} d e^{5} - b^{4} e^{6} + 6 \, a b^{2} c e^{6} - 6 \, a^{2} c^{2} e^{6}\right )} \arctan \left (\frac {2 \, c d - \frac {2 \, c d^{2}}{e x + d} - b e + \frac {2 \, b d e}{e x + d} - \frac {2 \, a e^{2}}{e x + d}}{\sqrt {-b^{2} + 4 \, a c} e}\right )}{{\left (b^{2} c^{3} d^{6} - 4 \, a c^{4} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 12 \, a b c^{3} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - 9 \, a b^{2} c^{2} d^{4} e^{2} - 12 \, a^{2} c^{3} d^{4} e^{2} - b^{5} d^{3} e^{3} - 2 \, a b^{3} c d^{3} e^{3} + 24 \, a^{2} b c^{2} d^{3} e^{3} + 3 \, a b^{4} d^{2} e^{4} - 9 \, a^{2} b^{2} c d^{2} e^{4} - 12 \, a^{3} c^{2} d^{2} e^{4} - 3 \, a^{2} b^{3} d e^{5} + 12 \, a^{3} b c d e^{5} + a^{3} b^{2} e^{6} - 4 \, a^{4} c e^{6}\right )} \sqrt {-b^{2} + 4 \, a c} e^{2}} - \frac {\frac {2 \, c^{4} d^{3} e - 3 \, b c^{3} d^{2} e^{2} + 3 \, b^{2} c^{2} d e^{3} - 6 \, a c^{3} d e^{3} - b^{3} c e^{4} + 3 \, a b c^{2} e^{4}}{c d^{2} - b d e + a e^{2}} - \frac {2 \, c^{4} d^{4} e^{2} - 4 \, b c^{3} d^{3} e^{3} + 6 \, b^{2} c^{2} d^{2} e^{4} - 12 \, a c^{3} d^{2} e^{4} - 4 \, b^{3} c d e^{5} + 12 \, a b c^{2} d e^{5} + b^{4} e^{6} - 4 \, a b^{2} c e^{6} + 2 \, a^{2} c^{2} e^{6}}{{\left (c d^{2} - b d e + a e^{2}\right )} {\left (e x + d\right )} e}}{{\left (c d^{2} - b d e + a e^{2}\right )}^{2} {\left (b^{2} - 4 \, a c\right )} {\left (c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {b e}{e x + d} - \frac {b d e}{{\left (e x + d\right )}^{2}} + \frac {a e^{2}}{{\left (e x + d\right )}^{2}}\right )}} \]

[In]

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

[Out]

-e^7/((c^2*d^4*e^4 - 2*b*c*d^3*e^5 + b^2*d^2*e^6 + 2*a*c*d^2*e^6 - 2*a*b*d*e^7 + a^2*e^8)*(e*x + d)) - (2*c*d*
e^3 - b*e^4)*log(c - 2*c*d/(e*x + d) + c*d^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d)^2 + a*e^2/(e*x + d)
^2)/(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2
*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6) - 2*(2*c^4*d^4*e^2 - 4*b*c^3*d^3*e^3 + 12*a*c^3*d^2*e^4 + 2*
b^3*c*d*e^5 - 12*a*b*c^2*d*e^5 - b^4*e^6 + 6*a*b^2*c*e^6 - 6*a^2*c^2*e^6)*arctan((2*c*d - 2*c*d^2/(e*x + d) -
b*e + 2*b*d*e/(e*x + d) - 2*a*e^2/(e*x + d))/(sqrt(-b^2 + 4*a*c)*e))/((b^2*c^3*d^6 - 4*a*c^4*d^6 - 3*b^3*c^2*d
^5*e + 12*a*b*c^3*d^5*e + 3*b^4*c*d^4*e^2 - 9*a*b^2*c^2*d^4*e^2 - 12*a^2*c^3*d^4*e^2 - b^5*d^3*e^3 - 2*a*b^3*c
*d^3*e^3 + 24*a^2*b*c^2*d^3*e^3 + 3*a*b^4*d^2*e^4 - 9*a^2*b^2*c*d^2*e^4 - 12*a^3*c^2*d^2*e^4 - 3*a^2*b^3*d*e^5
+ 12*a^3*b*c*d*e^5 + a^3*b^2*e^6 - 4*a^4*c*e^6)*sqrt(-b^2 + 4*a*c)*e^2) - ((2*c^4*d^3*e - 3*b*c^3*d^2*e^2 + 3
*b^2*c^2*d*e^3 - 6*a*c^3*d*e^3 - b^3*c*e^4 + 3*a*b*c^2*e^4)/(c*d^2 - b*d*e + a*e^2) - (2*c^4*d^4*e^2 - 4*b*c^3
*d^3*e^3 + 6*b^2*c^2*d^2*e^4 - 12*a*c^3*d^2*e^4 - 4*b^3*c*d*e^5 + 12*a*b*c^2*d*e^5 + b^4*e^6 - 4*a*b^2*c*e^6 +
2*a^2*c^2*e^6)/((c*d^2 - b*d*e + a*e^2)*(e*x + d)*e))/((c*d^2 - b*d*e + a*e^2)^2*(b^2 - 4*a*c)*(c - 2*c*d/(e*
x + d) + c*d^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d)^2 + a*e^2/(e*x + d)^2))

Mupad [B] (verification not implemented)

Time = 14.00 (sec) , antiderivative size = 2239, normalized size of antiderivative = 6.51 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((a*b^2*e^3 + b*c^2*d^3 - 4*a^2*c*e^3 + b^3*d*e^2 + 4*a*c^2*d^2*e - 2*b^2*c*d^2*e - 3*a*b*c*d*e^2)/(4*a*c^3*d^
4 + 4*a^3*c*e^4 - a^2*b^2*e^4 - b^2*c^2*d^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d*e^3 + 2*b^3*c*d^3*e
- 8*a*b*c^2*d^3*e - 8*a^2*b*c*d*e^3 + 2*a*b^2*c*d^2*e^2) - (2*x^2*(3*a*c^2*e^3 - b^2*c*e^3 - c^3*d^2*e + b*c^2
*d*e^2))/(4*a*c^3*d^4 + 4*a^3*c*e^4 - a^2*b^2*e^4 - b^2*c^2*d^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d*
e^3 + 2*b^3*c*d^3*e - 8*a*b*c^2*d^3*e - 8*a^2*b*c*d*e^3 + 2*a*b^2*c*d^2*e^2) + (x*(2*b^3*e^3 + 2*c^3*d^3 - 7*a
*b*c*e^3 + 2*a*c^2*d*e^2 - b*c^2*d^2*e - b^2*c*d*e^2))/(4*a*c^3*d^4 + 4*a^3*c*e^4 - a^2*b^2*e^4 - b^2*c^2*d^4
- b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d*e^3 + 2*b^3*c*d^3*e - 8*a*b*c^2*d^3*e - 8*a^2*b*c*d*e^3 + 2*a*b^
2*c*d^2*e^2))/(a*d + x*(a*e + b*d) + x^2*(b*e + c*d) + c*e*x^3) - (log(b^3 + (-(4*a*c - b^2)^3)^(1/2) - 4*a*b*
c - 8*a*c^2*x + 2*b^2*c*x)*(b^7*e^4 - b^4*e^4*(-(4*a*c - b^2)^3)^(1/2) + 2*c^4*d^4*(-(4*a*c - b^2)^3)^(1/2) -
64*a^3*b*c^3*e^4 + 128*a^3*c^4*d*e^3 + 48*a^2*b^3*c^2*e^4 - 6*a^2*c^2*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^5*
c*e^4 - 2*b^6*c*d*e^3 + 6*a*b^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) + 24*a*b^4*c^2*d*e^3 - 4*b*c^3*d^3*e*(-(4*a*c -
b^2)^3)^(1/2) + 2*b^3*c*d*e^3*(-(4*a*c - b^2)^3)^(1/2) - 96*a^2*b^2*c^3*d*e^3 + 12*a*c^3*d^2*e^2*(-(4*a*c - b
^2)^3)^(1/2) - 12*a*b*c^2*d*e^3*(-(4*a*c - b^2)^3)^(1/2)))/(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) - (log(d + e*x)*(2*b*e^4 - 4*c*d*e^3))/(a^3*e^6 + c^3*d^6 - b^
3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^
5*e - 6*a*b*c*d^3*e^3) - (log((-(4*a*c - b^2)^3)^(1/2) - b^3 + 4*a*b*c + 8*a*c^2*x - 2*b^2*c*x)*(b^7*e^4 + b^4
*e^4*(-(4*a*c - b^2)^3)^(1/2) - 2*c^4*d^4*(-(4*a*c - b^2)^3)^(1/2) - 64*a^3*b*c^3*e^4 + 128*a^3*c^4*d*e^3 + 48
*a^2*b^3*c^2*e^4 + 6*a^2*c^2*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^5*c*e^4 - 2*b^6*c*d*e^3 - 6*a*b^2*c*e^4*(-(
4*a*c - b^2)^3)^(1/2) + 24*a*b^4*c^2*d*e^3 + 4*b*c^3*d^3*e*(-(4*a*c - b^2)^3)^(1/2) - 2*b^3*c*d*e^3*(-(4*a*c -
b^2)^3)^(1/2) - 96*a^2*b^2*c^3*d*e^3 - 12*a*c^3*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b*c^2*d*e^3*(-(4*a*c
- b^2)^3)^(1/2)))/(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)