\(\int \frac {1}{(b d+2 c d x)^2 (a+b x+c x^2)^2} \, dx\) [1175]

Optimal result

Integrand size = 24, antiderivative size = 98 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {12 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x)}-\frac {1}{\left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )}+\frac {12 c \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} d^2} \]

[Out]

-12*c/(-4*a*c+b^2)^2/d^2/(2*c*x+b)-1/(-4*a*c+b^2)/d^2/(2*c*x+b)/(c*x^2+b*x+a)+12*c*arctanh((2*c*x+b)/(-4*a*c+b
^2)^(1/2))/(-4*a*c+b^2)^(5/2)/d^2

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {701, 707, 632, 212} \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^2} \, dx=\frac {12 c \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{5/2}}-\frac {1}{d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )}-\frac {12 c}{d^2 \left (b^2-4 a c\right )^2 (b+2 c x)} \]

[In]

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

[Out]

(-12*c)/((b^2 - 4*a*c)^2*d^2*(b + 2*c*x)) - 1/((b^2 - 4*a*c)*d^2*(b + 2*c*x)*(a + b*x + c*x^2)) + (12*c*ArcTan
h[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*d^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 701

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*c*(d + e*x)^(m + 1
)*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Dist[2*c*e*((m + 2*p + 3)/(e*(p + 1)*(b^2 - 4*a*
c))), Int[(d + e*x)^m*(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]
&& EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && RationalQ[m] && IntegerQ[2*p]

Rule 707

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

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{\left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )}-\frac {(6 c) \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )} \, dx}{b^2-4 a c} \\ & = -\frac {12 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x)}-\frac {1}{\left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )}-\frac {(6 c) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2 d^2} \\ & = -\frac {12 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x)}-\frac {1}{\left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )}+\frac {(12 c) \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 )^2 d^2} \\ & = -\frac {12 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x)}-\frac {1}{\left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )}+\frac {12 c \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {\frac {8 c}{b+2 c x}+\frac {b+2 c x}{a+x (b+c x)}+\frac {12 c \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{\left (b^2-4 a c\right )^2 d^2} \]

[In]

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

[Out]

-(((8*c)/(b + 2*c*x) + (b + 2*c*x)/(a + x*(b + c*x)) + (12*c*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2
+ 4*a*c])/((b^2 - 4*a*c)^2*d^2))

Maple [A] (verified)

Time = 2.80 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00

method	result	size
default	\(\frac {-\frac {\frac {2 c x +b}{c \,x^{2}+b x +a}+\frac {12 c \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (4 a c -b^{2}\right )^{2}}-\frac {8 c}{\left (4 a c -b^{2}\right )^{2} \left (2 c x +b \right )}}{d^{2}}\)	\(98\)
risch	\(\frac {-\frac {12 c^{2} x^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {12 b c x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {8 a c +b^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}}{d^{2} \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}+6 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (1024 a^{5} c^{5} d^{4}-1280 a^{4} b^{2} c^{4} d^{4}+640 a^{3} b^{4} c^{3} d^{4}-160 a^{2} b^{6} c^{2} d^{4}+20 a \,b^{8} c \,d^{4}-b^{10} d^{4}\right ) \textit {\_Z}^{2}+c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (6144 a^{5} c^{6} d^{4}-7680 a^{4} b^{2} c^{5} d^{4}+3840 a^{3} b^{4} c^{4} d^{4}-960 a^{2} b^{6} c^{3} d^{4}+120 a \,b^{8} c^{2} d^{4}-6 b^{10} c \,d^{4}\right ) \textit {\_R}^{2}+4 c^{3}\right ) x +\left (3072 a^{5} b \,c^{5} d^{4}-3840 a^{4} b^{3} c^{4} d^{4}+1920 a^{3} b^{5} c^{3} d^{4}-480 a^{2} b^{7} c^{2} d^{4}+60 a \,b^{9} c \,d^{4}-3 b^{11} d^{4}\right ) \textit {\_R}^{2}+\left (64 a^{3} c^{4} d^{2}-48 a^{2} b^{2} d^{2} c^{3}+12 a \,b^{4} c^{2} d^{2}-b^{6} c \,d^{2}\right ) \textit {\_R} +2 b \,c^{2}\right )\right )\)	\(422\)

[In]

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

[Out]

1/d^2*(-1/(4*a*c-b^2)^2*((2*c*x+b)/(c*x^2+b*x+a)+12*c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2)))-8
/(4*a*c-b^2)^2*c/(2*c*x+b))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 312 vs. \(2 (94) = 188\).

Time = 0.36 (sec) , antiderivative size = 644, normalized size of antiderivative = 6.57 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^2} \, dx=\left [-\frac {b^{4} + 4 \, a b^{2} c - 32 \, a^{2} c^{2} + 12 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} - 6 \, {\left (2 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} + a b c + {\left (b^{2} c + 2 \, a c^{2}\right )} x\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 12 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x}{2 \, {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{2} x^{3} + 3 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} d^{2} x^{2} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} d^{2} x + {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} d^{2}}, -\frac {b^{4} + 4 \, a b^{2} c - 32 \, a^{2} c^{2} + 12 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} - 12 \, {\left (2 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} + a b c + {\left (b^{2} c + 2 \, a c^{2}\right )} x\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 12 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x}{2 \, {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{2} x^{3} + 3 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} d^{2} x^{2} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} d^{2} x + {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} d^{2}}\right ] \]

[In]

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

[Out]

[-(b^4 + 4*a*b^2*c - 32*a^2*c^2 + 12*(b^2*c^2 - 4*a*c^3)*x^2 - 6*(2*c^3*x^3 + 3*b*c^2*x^2 + a*b*c + (b^2*c + 2
*a*c^2)*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)) + 12*(b^3*c - 4*a*b*c^2)*x)/(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2*x^3 + 3*(b
^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^2*x^2 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2
*c^3 - 128*a^4*c^4)*d^2*x + (a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d^2), -(b^4 + 4*a*b^2*c - 3
2*a^2*c^2 + 12*(b^2*c^2 - 4*a*c^3)*x^2 - 12*(2*c^3*x^3 + 3*b*c^2*x^2 + a*b*c + (b^2*c + 2*a*c^2)*x)*sqrt(-b^2
+ 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 12*(b^3*c - 4*a*b*c^2)*x)/(2*(b^6*c^2 - 12*a*
b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2*x^3 + 3*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^2*
x^2 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*x + (a*b^7 - 12*a^2*b^5*c + 48*a^
3*b^3*c^2 - 64*a^4*b*c^3)*d^2)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (92) = 184\).

Time = 1.09 (sec) , antiderivative size = 459, normalized size of antiderivative = 4.68 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^2} \, dx=\frac {6 c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x + \frac {- 384 a^{3} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 288 a^{2} b^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 72 a b^{4} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b^{6} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b c}{12 c^{2}} \right )}}{d^{2}} - \frac {6 c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x + \frac {384 a^{3} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 288 a^{2} b^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 72 a b^{4} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 6 b^{6} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b c}{12 c^{2}} \right )}}{d^{2}} + \frac {- 8 a c - b^{2} - 12 b c x - 12 c^{2} x^{2}}{16 a^{3} b c^{2} d^{2} - 8 a^{2} b^{3} c d^{2} + a b^{5} d^{2} + x^{3} \cdot \left (32 a^{2} c^{4} d^{2} - 16 a b^{2} c^{3} d^{2} + 2 b^{4} c^{2} d^{2}\right ) + x^{2} \cdot \left (48 a^{2} b c^{3} d^{2} - 24 a b^{3} c^{2} d^{2} + 3 b^{5} c d^{2}\right ) + x \left (32 a^{3} c^{3} d^{2} - 6 a b^{4} c d^{2} + b^{6} d^{2}\right )} \]

[In]

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

[Out]

6*c*sqrt(-1/(4*a*c - b**2)**5)*log(x + (-384*a**3*c**4*sqrt(-1/(4*a*c - b**2)**5) + 288*a**2*b**2*c**3*sqrt(-1
/(4*a*c - b**2)**5) - 72*a*b**4*c**2*sqrt(-1/(4*a*c - b**2)**5) + 6*b**6*c*sqrt(-1/(4*a*c - b**2)**5) + 6*b*c)
/(12*c**2))/d**2 - 6*c*sqrt(-1/(4*a*c - b**2)**5)*log(x + (384*a**3*c**4*sqrt(-1/(4*a*c - b**2)**5) - 288*a**2
*b**2*c**3*sqrt(-1/(4*a*c - b**2)**5) + 72*a*b**4*c**2*sqrt(-1/(4*a*c - b**2)**5) - 6*b**6*c*sqrt(-1/(4*a*c -
b**2)**5) + 6*b*c)/(12*c**2))/d**2 + (-8*a*c - b**2 - 12*b*c*x - 12*c**2*x**2)/(16*a**3*b*c**2*d**2 - 8*a**2*b
**3*c*d**2 + a*b**5*d**2 + x**3*(32*a**2*c**4*d**2 - 16*a*b**2*c**3*d**2 + 2*b**4*c**2*d**2) + x**2*(48*a**2*b
*c**3*d**2 - 24*a*b**3*c**2*d**2 + 3*b**5*c*d**2) + x*(32*a**3*c**3*d**2 - 6*a*b**4*c*d**2 + b**6*d**2))

Maxima [F(-2)]

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

[In]

integrate(1/(2*c*d*x+b*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. 220 vs. \(2 (94) = 188\).

Time = 0.28 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.24 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {8 \, c^{5} d^{7}}{{\left (b^{4} c^{4} d^{8} - 8 \, a b^{2} c^{5} d^{8} + 16 \, a^{2} c^{6} d^{8}\right )} {\left (2 \, c d x + b d\right )}} - \frac {12 \, c \arctan \left (\frac {\frac {b^{2} d}{2 \, c d x + b d} - \frac {4 \, a c d}{2 \, c d x + b d}}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c} d^{2}} + \frac {4 \, c}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (2 \, c d x + b d\right )} {\left (\frac {b^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac {4 \, a c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - 1\right )} d} \]

[In]

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

[Out]

-8*c^5*d^7/((b^4*c^4*d^8 - 8*a*b^2*c^5*d^8 + 16*a^2*c^6*d^8)*(2*c*d*x + b*d)) - 12*c*arctan((b^2*d/(2*c*d*x +
b*d) - 4*a*c*d/(2*c*d*x + b*d))/sqrt(-b^2 + 4*a*c))/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(-b^2 + 4*a*c)*d^2) +
4*c/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*(2*c*d*x + b*d)*(b^2*d^2/(2*c*d*x + b*d)^2 - 4*a*c*d^2/(2*c*d*x + b*d)^2 -
1)*d)

Mupad [B] (verification not implemented)

Time = 9.72 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.37 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {\frac {b^2+8\,a\,c}{{\left (4\,a\,c-b^2\right )}^2}+\frac {12\,c^2\,x^2}{{\left (4\,a\,c-b^2\right )}^2}+\frac {12\,b\,c\,x}{{\left (4\,a\,c-b^2\right )}^2}}{x\,\left (b^2\,d^2+2\,a\,c\,d^2\right )+2\,c^2\,d^2\,x^3+a\,b\,d^2+3\,b\,c\,d^2\,x^2}-\frac {12\,c\,\mathrm {atan}\left (\frac {\frac {6\,c\,\left (16\,a^2\,b\,c^2\,d^2-8\,a\,b^3\,c\,d^2+b^5\,d^2\right )}{d^2\,{\left (4\,a\,c-b^2\right )}^{5/2}}+\frac {12\,c^2\,x\,\left (16\,a^2\,c^2\,d^2-8\,a\,b^2\,c\,d^2+b^4\,d^2\right )}{d^2\,{\left (4\,a\,c-b^2\right )}^{5/2}}}{6\,c}\right )}{d^2\,{\left (4\,a\,c-b^2\right )}^{5/2}} \]

[In]

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

[Out]

- ((8*a*c + b^2)/(4*a*c - b^2)^2 + (12*c^2*x^2)/(4*a*c - b^2)^2 + (12*b*c*x)/(4*a*c - b^2)^2)/(x*(b^2*d^2 + 2*
a*c*d^2) + 2*c^2*d^2*x^3 + a*b*d^2 + 3*b*c*d^2*x^2) - (12*c*atan(((6*c*(b^5*d^2 + 16*a^2*b*c^2*d^2 - 8*a*b^3*c
*d^2))/(d^2*(4*a*c - b^2)^(5/2)) + (12*c^2*x*(b^4*d^2 + 16*a^2*c^2*d^2 - 8*a*b^2*c*d^2))/(d^2*(4*a*c - b^2)^(5
/2)))/(6*c)))/(d^2*(4*a*c - b^2)^(5/2))