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

Optimal. Leaf size=344 \[ -\frac {2 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )}+\frac {2 \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^3}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{\left (c d^2-b d e+a e^2\right )^3} \]

[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

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Rubi [A]
time = 0.44, antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {754, 814, 648, 632, 212, 642} \begin {gather*} -\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 {2 \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 ) \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 (a e^2-b d e+c d^2\right )^3}-\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx &=-\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*}

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Mathematica [A]
time = 0.41, size = 339, normalized size = 0.99 \begin {gather*} -\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 ) \tan ^{-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+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} \end {gather*}

Antiderivative was successfully verified.

[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

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Maple [A]
time = 0.99, size = 529, normalized size = 1.54

method	result	size
default	\(-\frac {\frac {\frac {c \left (2 e^{4} a^{2} c -a \,b^{2} e^{4}+b^{3} d \,e^{3}-3 b^{2} c \,d^{2} e^{2}+4 d^{3} e b \,c^{2}-2 d^{4} c^{3}\right ) x}{4 a c -b^{2}}+\frac {3 a^{2} b c \,e^{4}-4 a^{2} c^{2} d \,e^{3}-a \,b^{3} e^{4}-a \,b^{2} c d \,e^{3}+6 a b \,c^{2} d^{2} e^{2}-4 a \,c^{3} d^{3} e +b^{4} d \,e^{3}-3 b^{3} c \,d^{2} e^{2}+3 b^{2} c^{2} d^{3} e -d^{4} b \,c^{3}}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (-4 a b \,c^{2} e^{4}+8 d \,e^{3} c^{3} a +b^{3} c \,e^{4}-2 b^{2} c^{2} d \,e^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{c}+\frac {4 \left (3 e^{4} a^{2} c^{2}-5 a \,b^{2} c \,e^{4}+10 a b \,c^{2} d \,e^{3}-6 d^{2} e^{2} c^{3} a +b^{4} e^{4}-2 b^{3} c d \,e^{3}+2 b \,c^{3} d^{3} e -c^{4} d^{4}-\frac {\left (-4 a b \,c^{2} e^{4}+8 d \,e^{3} c^{3} a +b^{3} c \,e^{4}-2 b^{2} c^{2} d \,e^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{\left (e^{2} a -b d e +c \,d^{2}\right )^{3}}-\frac {e^{3}}{\left (e^{2} a -b d e +c \,d^{2}\right )^{2} \left (e x +d \right )}-\frac {2 e^{3} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{3}}\)	\(529\)
risch	\(\text {Expression too large to display}\)	\(3757\)

Verification of antiderivative is not currently implemented for this CAS.

[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*e^4*a^2*c^2-5*a*b^2*c*e^4+10*a*b*c^2*d*e^3-6*d^2*e^2*c^3*
a+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)

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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)^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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2689 vs. \(2 (349) = 698\).
time = 142.31, size = 5398, normalized size = 15.69 \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)^2/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

[-(2*(b^2*c^4 - 4*a*c^5)*d^5*x + (b^3*c^3 - 4*a*b*c^4)*d^5 - (2*c^5*d^5*x^2 + 2*b*c^4*d^5*x + 2*a*c^4*d^5 - ((
b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*x^3 + (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*x^2 + (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)
*x)*e^5 + (2*(b^3*c^2 - 6*a*b*c^3)*d*x^3 + (b^4*c - 6*a*b^2*c^2 - 6*a^2*c^3)*d*x^2 - (b^5 - 8*a*b^3*c + 18*a^2
*b*c^2)*d*x - (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*d)*e^4 + 2*(6*a*c^4*d^2*x^3 + b^3*c^2*d^2*x^2 + (b^4*c - 6*a*b
^2*c^2 + 6*a^2*c^3)*d^2*x + (a*b^3*c - 6*a^2*b*c^2)*d^2)*e^3 - 4*(b*c^4*d^3*x^3 - 2*a*b*c^3*d^3*x - 3*a^2*c^3*
d^3 + (b^2*c^3 - 3*a*c^4)*d^3*x^2)*e^2 + 2*(c^5*d^4*x^3 - b*c^4*d^4*x^2 - 2*a*b*c^3*d^4 - (2*b^2*c^3 - a*c^4)*
d^4*x)*e)*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)) + (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + 2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*x^2 + (2*a*b^5 - 15*
a^2*b^3*c + 28*a^3*b*c^2)*x)*e^5 - (2*(b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d*x^2 + 2*(b^6 - 7*a*b^4*c + 11*a^2*
b^2*c^2 + 4*a^3*c^3)*d*x - (a^2*b^3*c - 4*a^3*b*c^2)*d)*e^4 + (4*(b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d^2*x^2 +
(3*b^5*c - 22*a*b^3*c^2 + 40*a^2*b*c^3)*d^2*x - (b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*d^2)*e^3 - (4*(b^3*c^3 - 4*
a*b*c^4)*d^3*x^2 - 4*(a*b^2*c^3 - 4*a^2*c^4)*d^3*x - 3*(b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d^3)*e^2 + (2*(b^2*
c^4 - 4*a*c^5)*d^4*x^2 - 3*(b^3*c^3 - 4*a*b*c^4)*d^4*x - (3*b^4*c^2 - 16*a*b^2*c^3 + 16*a^2*c^4)*d^4)*e - (((b
^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*x^2 + (a*b^5 - 8*a^2*b^3*c + 16*a^
3*b*c^2)*x)*e^5 - (2*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*x^3 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*x^2 -
(b^6 - 10*a*b^4*c + 32*a^2*b^2*c^2 - 32*a^3*c^3)*d*x - (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d)*e^4 - 2*((b^4*
c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2*x^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2*x + (a*b^4*c - 8*a^2*b^2*c^
2 + 16*a^3*c^3)*d^2)*e^3)*log(c*x^2 + b*x + a) + 2*(((b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 8*a*b^4
*c + 16*a^2*b^2*c^2)*x^2 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*e^5 - (2*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^
4)*d*x^3 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*x^2 - (b^6 - 10*a*b^4*c + 32*a^2*b^2*c^2 - 32*a^3*c^3)*d*x -
(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d)*e^4 - 2*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2*x^2 + (b^5*c - 8*a*
b^3*c^2 + 16*a^2*b*c^3)*d^2*x + (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*d^2)*e^3)*log(x*e + d))/((b^4*c^4 - 8*a
*b^2*c^5 + 16*a^2*c^6)*d^7*x^2 + (b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d^7*x + (a*b^4*c^3 - 8*a^2*b^2*c^4 + 1
6*a^3*c^5)*d^7 + ((a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*x^3 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x^2 +
(a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*x)*e^7 - (3*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d*x^3 + (3*a^2*b^6
- 25*a^3*b^4*c + 56*a^4*b^2*c^2 - 16*a^5*c^3)*d*x^2 + 2*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d*x - (a^4*b^4
- 8*a^5*b^2*c + 16*a^6*c^2)*d)*e^6 + 3*((a*b^6*c - 7*a^2*b^4*c^2 + 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^2*x^3 + (a*b
^7 - 8*a^2*b^5*c + 16*a^3*b^3*c^2)*d^2*x^2 + (a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*d^2*x - (a^3*b^5 - 8*a^4
*b^3*c + 16*a^5*b*c^2)*d^2)*e^5 - ((b^7*c - 2*a*b^5*c^2 - 32*a^2*b^3*c^3 + 96*a^3*b*c^4)*d^3*x^3 + (b^8 - 5*a*
b^6*c - 11*a^2*b^4*c^2 + 72*a^3*b^2*c^3 - 48*a^4*c^4)*d^3*x^2 - (2*a*b^7 - 19*a^2*b^5*c + 56*a^3*b^3*c^2 - 48*
a^4*b*c^3)*d^3*x - 3*(a^2*b^6 - 7*a^3*b^4*c + 8*a^4*b^2*c^2 + 16*a^5*c^3)*d^3)*e^4 + (3*(b^6*c^2 - 7*a*b^4*c^3
+ 8*a^2*b^2*c^4 + 16*a^3*c^5)*d^4*x^3 + (2*b^7*c - 19*a*b^5*c^2 + 56*a^2*b^3*c^3 - 48*a^3*b*c^4)*d^4*x^2 - (b
^8 - 5*a*b^6*c - 11*a^2*b^4*c^2 + 72*a^3*b^2*c^3 - 48*a^4*c^4)*d^4*x - (a*b^7 - 2*a^2*b^5*c - 32*a^3*b^3*c^2 +
96*a^4*b*c^3)*d^4)*e^3 - 3*((b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d^5*x^3 - (a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*
a^3*c^5)*d^5*x^2 - (b^7*c - 8*a*b^5*c^2 + 16*a^2*b^3*c^3)*d^5*x - (a*b^6*c - 7*a^2*b^4*c^2 + 8*a^3*b^2*c^3 + 1
6*a^4*c^4)*d^5)*e^2 + ((b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^6*x^3 - 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)
*d^6*x^2 - (3*b^6*c^2 - 25*a*b^4*c^3 + 56*a^2*b^2*c^4 - 16*a^3*c^5)*d^6*x - 3*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*
a^3*b*c^4)*d^6)*e), -(2*(b^2*c^4 - 4*a*c^5)*d^5*x + (b^3*c^3 - 4*a*b*c^4)*d^5 - 2*(2*c^5*d^5*x^2 + 2*b*c^4*d^5
*x + 2*a*c^4*d^5 - ((b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*x^3 + (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*x^2 + (a*b^4 - 6*a
^2*b^2*c + 6*a^3*c^2)*x)*e^5 + (2*(b^3*c^2 - 6*a*b*c^3)*d*x^3 + (b^4*c - 6*a*b^2*c^2 - 6*a^2*c^3)*d*x^2 - (b^5
- 8*a*b^3*c + 18*a^2*b*c^2)*d*x - (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*d)*e^4 + 2*(6*a*c^4*d^2*x^3 + b^3*c^2*d^2
*x^2 + (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d^2*x + (a*b^3*c - 6*a^2*b*c^2)*d^2)*e^3 - 4*(b*c^4*d^3*x^3 - 2*a*b*c
^3*d^3*x - 3*a^2*c^3*d^3 + (b^2*c^3 - 3*a*c^4)*d^3*x^2)*e^2 + 2*(c^5*d^4*x^3 - b*c^4*d^4*x^2 - 2*a*b*c^3*d^4 -
(2*b^2*c^3 - a*c^4)*d^4*x)*e)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (a^2
*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + 2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*x^2 + (2*a*b^5 - 15*a^2*b^3*c + 28*
a^3*b*c^2)*x)*e^5 - (2*(b^5*c - 6*a*b^3*c^2 + 8...

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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)**2/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 899 vs. \(2 (349) = 698\).
time = 0.63, size = 899, normalized size = 2.61 \begin {gather*} -\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 {{\left (2 \, c d - \frac {2 \, c d^{2}}{x e + d} - b e + \frac {2 \, b d e}{x e + d} - \frac {2 \, a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt {-b^{2} + 4 \, a c}}\right ) e^{\left (-2\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}} - \frac {{\left (2 \, c d e^{3} - b e^{4}\right )} \log \left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + 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 {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 (x e + d\right )}} - \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 {{\left (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}\right )} e^{\left (-1\right )}}{{\left (c d^{2} - b d e + a e^{2}\right )} {\left (x e + d\right )}}}{{\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}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 4.48, size = 2239, normalized size = 6.51 \begin {gather*} \frac {\frac {-4\,a^2\,c\,e^3+a\,b^2\,e^3-3\,a\,b\,c\,d\,e^2+4\,a\,c^2\,d^2\,e+b^3\,d\,e^2-2\,b^2\,c\,d^2\,e+b\,c^2\,d^3}{4\,a^3\,c\,e^4-a^2\,b^2\,e^4-8\,a^2\,b\,c\,d\,e^3+8\,a^2\,c^2\,d^2\,e^2+2\,a\,b^3\,d\,e^3+2\,a\,b^2\,c\,d^2\,e^2-8\,a\,b\,c^2\,d^3\,e+4\,a\,c^3\,d^4-b^4\,d^2\,e^2+2\,b^3\,c\,d^3\,e-b^2\,c^2\,d^4}-\frac {2\,x^2\,\left (-b^2\,c\,e^3+b\,c^2\,d\,e^2-c^3\,d^2\,e+3\,a\,c^2\,e^3\right )}{4\,a^3\,c\,e^4-a^2\,b^2\,e^4-8\,a^2\,b\,c\,d\,e^3+8\,a^2\,c^2\,d^2\,e^2+2\,a\,b^3\,d\,e^3+2\,a\,b^2\,c\,d^2\,e^2-8\,a\,b\,c^2\,d^3\,e+4\,a\,c^3\,d^4-b^4\,d^2\,e^2+2\,b^3\,c\,d^3\,e-b^2\,c^2\,d^4}+\frac {x\,\left (2\,b^3\,e^3-b^2\,c\,d\,e^2-b\,c^2\,d^2\,e-7\,a\,b\,c\,e^3+2\,c^3\,d^3+2\,a\,c^2\,d\,e^2\right )}{4\,a^3\,c\,e^4-a^2\,b^2\,e^4-8\,a^2\,b\,c\,d\,e^3+8\,a^2\,c^2\,d^2\,e^2+2\,a\,b^3\,d\,e^3+2\,a\,b^2\,c\,d^2\,e^2-8\,a\,b\,c^2\,d^3\,e+4\,a\,c^3\,d^4-b^4\,d^2\,e^2+2\,b^3\,c\,d^3\,e-b^2\,c^2\,d^4}}{c\,e\,x^3+\left (b\,e+c\,d\right )\,x^2+\left (a\,e+b\,d\right )\,x+a\,d}-\frac {\ln \left (b^3+\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-4\,a\,b\,c-8\,a\,c^2\,x+2\,b^2\,c\,x\right )\,\left (b^7\,e^4-b^4\,e^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+2\,c^4\,d^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-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\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-12\,a\,b^5\,c\,e^4-2\,b^6\,c\,d\,e^3+6\,a\,b^2\,c\,e^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+24\,a\,b^4\,c^2\,d\,e^3-4\,b\,c^3\,d^3\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+2\,b^3\,c\,d\,e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-96\,a^2\,b^2\,c^3\,d\,e^3+12\,a\,c^3\,d^2\,e^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-12\,a\,b\,c^2\,d\,e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )}{64\,a^6\,c^3\,e^6-48\,a^5\,b^2\,c^2\,e^6-192\,a^5\,b\,c^3\,d\,e^5+192\,a^5\,c^4\,d^2\,e^4+12\,a^4\,b^4\,c\,e^6+144\,a^4\,b^3\,c^2\,d\,e^5+48\,a^4\,b^2\,c^3\,d^2\,e^4-384\,a^4\,b\,c^4\,d^3\,e^3+192\,a^4\,c^5\,d^4\,e^2-a^3\,b^6\,e^6-36\,a^3\,b^5\,c\,d\,e^5-108\,a^3\,b^4\,c^2\,d^2\,e^4+224\,a^3\,b^3\,c^3\,d^3\,e^3+48\,a^3\,b^2\,c^4\,d^4\,e^2-192\,a^3\,b\,c^5\,d^5\,e+64\,a^3\,c^6\,d^6+3\,a^2\,b^7\,d\,e^5+33\,a^2\,b^6\,c\,d^2\,e^4-24\,a^2\,b^5\,c^2\,d^3\,e^3-108\,a^2\,b^4\,c^3\,d^4\,e^2+144\,a^2\,b^3\,c^4\,d^5\,e-48\,a^2\,b^2\,c^5\,d^6-3\,a\,b^8\,d^2\,e^4-6\,a\,b^7\,c\,d^3\,e^3+33\,a\,b^6\,c^2\,d^4\,e^2-36\,a\,b^5\,c^3\,d^5\,e+12\,a\,b^4\,c^4\,d^6+b^9\,d^3\,e^3-3\,b^8\,c\,d^4\,e^2+3\,b^7\,c^2\,d^5\,e-b^6\,c^3\,d^6}-\frac {\ln \left (d+e\,x\right )\,\left (2\,b\,e^4-4\,c\,d\,e^3\right )}{a^3\,e^6-3\,a^2\,b\,d\,e^5+3\,a^2\,c\,d^2\,e^4+3\,a\,b^2\,d^2\,e^4-6\,a\,b\,c\,d^3\,e^3+3\,a\,c^2\,d^4\,e^2-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6}-\frac {\ln \left (\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3+4\,a\,b\,c+8\,a\,c^2\,x-2\,b^2\,c\,x\right )\,\left (b^7\,e^4+b^4\,e^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-2\,c^4\,d^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-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\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-12\,a\,b^5\,c\,e^4-2\,b^6\,c\,d\,e^3-6\,a\,b^2\,c\,e^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+24\,a\,b^4\,c^2\,d\,e^3+4\,b\,c^3\,d^3\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-2\,b^3\,c\,d\,e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-96\,a^2\,b^2\,c^3\,d\,e^3-12\,a\,c^3\,d^2\,e^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+12\,a\,b\,c^2\,d\,e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )}{64\,a^6\,c^3\,e^6-48\,a^5\,b^2\,c^2\,e^6-192\,a^5\,b\,c^3\,d\,e^5+192\,a^5\,c^4\,d^2\,e^4+12\,a^4\,b^4\,c\,e^6+144\,a^4\,b^3\,c^2\,d\,e^5+48\,a^4\,b^2\,c^3\,d^2\,e^4-384\,a^4\,b\,c^4\,d^3\,e^3+192\,a^4\,c^5\,d^4\,e^2-a^3\,b^6\,e^6-36\,a^3\,b^5\,c\,d\,e^5-108\,a^3\,b^4\,c^2\,d^2\,e^4+224\,a^3\,b^3\,c^3\,d^3\,e^3+48\,a^3\,b^2\,c^4\,d^4\,e^2-192\,a^3\,b\,c^5\,d^5\,e+64\,a^3\,c^6\,d^6+3\,a^2\,b^7\,d\,e^5+33\,a^2\,b^6\,c\,d^2\,e^4-24\,a^2\,b^5\,c^2\,d^3\,e^3-108\,a^2\,b^4\,c^3\,d^4\,e^2+144\,a^2\,b^3\,c^4\,d^5\,e-48\,a^2\,b^2\,c^5\,d^6-3\,a\,b^8\,d^2\,e^4-6\,a\,b^7\,c\,d^3\,e^3+33\,a\,b^6\,c^2\,d^4\,e^2-36\,a\,b^5\,c^3\,d^5\,e+12\,a\,b^4\,c^4\,d^6+b^9\,d^3\,e^3-3\,b^8\,c\,d^4\,e^2+3\,b^7\,c^2\,d^5\,e-b^6\,c^3\,d^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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