\(\int \frac {1}{(b d+2 c d x)^4 (a+b x+c x^2)} \, dx\) [57]

Optimal result

Integrand size = 24, antiderivative size = 86 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )} \, dx=\frac {2}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3}+\frac {2}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)}-\frac {2 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} d^4} \] Output:

2/3/(-4*a*c+b^2)/d^4/(2*c*x+b)^3+2/(-4*a*c+b^2)^2/d^4/(2*c*x+b)-2*arctanh( 
(2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(5/2)/d^4
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )} \, dx=\frac {2 \left (\frac {b^2-4 a c}{(b+2 c x)^3}+\frac {3}{b+2 c x}+\frac {3 \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}\right )}{3 \left (b^2-4 a c\right )^2 d^4} \] Input:

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

Output:

(2*((b^2 - 4*a*c)/(b + 2*c*x)^3 + 3/(b + 2*c*x) + (3*ArcTan[(b + 2*c*x)/Sq 
rt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c]))/(3*(b^2 - 4*a*c)^2*d^4)
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1117, 27, 1117, 1083, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{\left (a+b x+c x^2\right ) (b d+2 c d x)^4} \, dx\)

\(\Big \downarrow \) 1117

\(\displaystyle \frac {\int \frac {1}{d^2 (b+2 c x)^2 \left (c x^2+b x+a\right )}dx}{d^2 \left (b^2-4 a c\right )}+\frac {2}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {1}{(b+2 c x)^2 \left (c x^2+b x+a\right )}dx}{d^4 \left (b^2-4 a c\right )}+\frac {2}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3}\)

\(\Big \downarrow \) 1117

\(\displaystyle \frac {\frac {\int \frac {1}{c x^2+b x+a}dx}{b^2-4 a c}+\frac {2}{\left (b^2-4 a c\right ) (b+2 c x)}}{d^4 \left (b^2-4 a c\right )}+\frac {2}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\frac {2}{\left (b^2-4 a c\right ) (b+2 c x)}-\frac {2 \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{b^2-4 a c}}{d^4 \left (b^2-4 a c\right )}+\frac {2}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {2}{\left (b^2-4 a c\right ) (b+2 c x)}-\frac {2 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}}{d^4 \left (b^2-4 a c\right )}+\frac {2}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1083

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

rule 1117

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> 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] + Simp[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] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] & 
& (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || IntegerQ[(m + 2*p + 3) 
/2])
 

Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.98

method	result	size
default	\(\frac {\frac {2}{\left (4 a c -b^{2}\right )^{2} \left (2 c x +b \right )}-\frac {2}{3 \left (4 a c -b^{2}\right ) \left (2 c x +b \right )^{3}}+\frac {2 \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {5}{2}}}}{d^{4}}\)	\(84\)
risch	\(\frac {\frac {8 c^{2} x^{2}}{16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}}+\frac {8 b c x}{16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}}-\frac {8 \left (a c -b^{2}\right )}{3 \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}}{d^{4} \left (2 c x +b \right )^{3}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (1024 a^{5} d^{8} c^{5}-1280 a^{4} b^{2} c^{4} d^{8}+640 a^{3} b^{4} d^{8} c^{3}-160 a^{2} b^{6} d^{8} c^{2}+20 a \,b^{8} d^{8} c -b^{10} d^{8}\right ) \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (6144 a^{5} c^{6} d^{8}-7680 a^{4} b^{2} c^{5} d^{8}+3840 a^{3} b^{4} c^{4} d^{8}-960 a^{2} b^{6} c^{3} d^{8}+120 a \,b^{8} c^{2} d^{8}-6 b^{10} c \,d^{8}\right ) \textit {\_R}^{2}+4 c \right ) x +\left (3072 a^{5} b \,c^{5} d^{8}-3840 a^{4} b^{3} c^{4} d^{8}+1920 a^{3} b^{5} c^{3} d^{8}-480 a^{2} b^{7} c^{2} d^{8}+60 a \,b^{9} c \,d^{8}-3 b^{11} d^{8}\right ) \textit {\_R}^{2}+\left (-64 a^{3} d^{4} c^{3}+48 a^{2} b^{2} c^{2} d^{4}-12 b^{4} c \,d^{4} a +b^{6} d^{4}\right ) \textit {\_R} +2 b \right )\right )\)	\(398\)

Input:

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

Output:

1/d^4*(2/(4*a*c-b^2)^2/(2*c*x+b)-2/3/(4*a*c-b^2)/(2*c*x+b)^3+2/(4*a*c-b^2) 
^(5/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (80) = 160\).

Time = 0.10 (sec) , antiderivative size = 627, normalized size of antiderivative = 7.29 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )} \, dx=\left [\frac {8 \, b^{4} - 40 \, a b^{2} c + 32 \, a^{2} c^{2} + 24 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 3 \, {\left (8 \, c^{3} x^{3} + 12 \, b c^{2} x^{2} + 6 \, b^{2} c x + b^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 24 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x}{3 \, {\left (8 \, {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} d^{4} x^{3} + 12 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} d^{4} x^{2} + 6 \, {\left (b^{8} c - 12 \, a b^{6} c^{2} + 48 \, a^{2} b^{4} c^{3} - 64 \, a^{3} b^{2} c^{4}\right )} d^{4} x + {\left (b^{9} - 12 \, a b^{7} c + 48 \, a^{2} b^{5} c^{2} - 64 \, a^{3} b^{3} c^{3}\right )} d^{4}\right )}}, \frac {2 \, {\left (4 \, b^{4} - 20 \, a b^{2} c + 16 \, a^{2} c^{2} + 12 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} - 3 \, {\left (8 \, c^{3} x^{3} + 12 \, b c^{2} x^{2} + 6 \, b^{2} c x + b^{3}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 12 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )}}{3 \, {\left (8 \, {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} d^{4} x^{3} + 12 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} d^{4} x^{2} + 6 \, {\left (b^{8} c - 12 \, a b^{6} c^{2} + 48 \, a^{2} b^{4} c^{3} - 64 \, a^{3} b^{2} c^{4}\right )} d^{4} x + {\left (b^{9} - 12 \, a b^{7} c + 48 \, a^{2} b^{5} c^{2} - 64 \, a^{3} b^{3} c^{3}\right )} d^{4}\right )}}\right ] \] Input:

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

Output:

[1/3*(8*b^4 - 40*a*b^2*c + 32*a^2*c^2 + 24*(b^2*c^2 - 4*a*c^3)*x^2 + 3*(8* 
c^3*x^3 + 12*b*c^2*x^2 + 6*b^2*c*x + b^3)*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 
)) + 24*(b^3*c - 4*a*b*c^2)*x)/(8*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 
 - 64*a^3*c^6)*d^4*x^3 + 12*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64* 
a^3*b*c^5)*d^4*x^2 + 6*(b^8*c - 12*a*b^6*c^2 + 48*a^2*b^4*c^3 - 64*a^3*b^2 
*c^4)*d^4*x + (b^9 - 12*a*b^7*c + 48*a^2*b^5*c^2 - 64*a^3*b^3*c^3)*d^4), 2 
/3*(4*b^4 - 20*a*b^2*c + 16*a^2*c^2 + 12*(b^2*c^2 - 4*a*c^3)*x^2 - 3*(8*c^ 
3*x^3 + 12*b*c^2*x^2 + 6*b^2*c*x + b^3)*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)/(8*(b^6 
*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^4*x^3 + 12*(b^7*c^2 - 
 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^4*x^2 + 6*(b^8*c - 12*a*b 
^6*c^2 + 48*a^2*b^4*c^3 - 64*a^3*b^2*c^4)*d^4*x + (b^9 - 12*a*b^7*c + 48*a 
^2*b^5*c^2 - 64*a^3*b^3*c^3)*d^4)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (82) = 164\).

Time = 0.88 (sec) , antiderivative size = 442, normalized size of antiderivative = 5.14 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )} \, dx=\frac {- 8 a c + 8 b^{2} + 24 b c x + 24 c^{2} x^{2}}{48 a^{2} b^{3} c^{2} d^{4} - 24 a b^{5} c d^{4} + 3 b^{7} d^{4} + x^{3} \cdot \left (384 a^{2} c^{5} d^{4} - 192 a b^{2} c^{4} d^{4} + 24 b^{4} c^{3} d^{4}\right ) + x^{2} \cdot \left (576 a^{2} b c^{4} d^{4} - 288 a b^{3} c^{3} d^{4} + 36 b^{5} c^{2} d^{4}\right ) + x \left (288 a^{2} b^{2} c^{3} d^{4} - 144 a b^{4} c^{2} d^{4} + 18 b^{6} c d^{4}\right )} - \frac {\sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x + \frac {- 64 a^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 48 a^{2} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 12 a b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + b}{2 c} \right )}}{d^{4}} + \frac {\sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x + \frac {64 a^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 48 a^{2} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 12 a b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + b}{2 c} \right )}}{d^{4}} \] Input:

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

Output:

(-8*a*c + 8*b**2 + 24*b*c*x + 24*c**2*x**2)/(48*a**2*b**3*c**2*d**4 - 24*a 
*b**5*c*d**4 + 3*b**7*d**4 + x**3*(384*a**2*c**5*d**4 - 192*a*b**2*c**4*d* 
*4 + 24*b**4*c**3*d**4) + x**2*(576*a**2*b*c**4*d**4 - 288*a*b**3*c**3*d** 
4 + 36*b**5*c**2*d**4) + x*(288*a**2*b**2*c**3*d**4 - 144*a*b**4*c**2*d**4 
 + 18*b**6*c*d**4)) - sqrt(-1/(4*a*c - b**2)**5)*log(x + (-64*a**3*c**3*sq 
rt(-1/(4*a*c - b**2)**5) + 48*a**2*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5) - 
12*a*b**4*c*sqrt(-1/(4*a*c - b**2)**5) + b**6*sqrt(-1/(4*a*c - b**2)**5) + 
 b)/(2*c))/d**4 + sqrt(-1/(4*a*c - b**2)**5)*log(x + (64*a**3*c**3*sqrt(-1 
/(4*a*c - b**2)**5) - 48*a**2*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5) + 12*a* 
b**4*c*sqrt(-1/(4*a*c - b**2)**5) - b**6*sqrt(-1/(4*a*c - b**2)**5) + b)/( 
2*c))/d**4
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.49 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )} \, dx=\frac {2 \, \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} d^{4} - 8 \, a b^{2} c d^{4} + 16 \, a^{2} c^{2} d^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {8 \, {\left (3 \, c^{2} x^{2} + 3 \, b c x + b^{2} - a c\right )}}{3 \, {\left (b^{4} d^{4} - 8 \, a b^{2} c d^{4} + 16 \, a^{2} c^{2} d^{4}\right )} {\left (2 \, c x + b\right )}^{3}} \] Input:

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

Output:

2*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4*d^4 - 8*a*b^2*c*d^4 + 16*a^ 
2*c^2*d^4)*sqrt(-b^2 + 4*a*c)) + 8/3*(3*c^2*x^2 + 3*b*c*x + b^2 - a*c)/((b 
^4*d^4 - 8*a*b^2*c*d^4 + 16*a^2*c^2*d^4)*(2*c*x + b)^3)
 

Mupad [B] (verification not implemented)

Time = 5.79 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.55 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )} \, dx=\frac {\frac {8\,c^2\,x^2}{{\left (4\,a\,c-b^2\right )}^2}-\frac {8\,\left (a\,c-b^2\right )}{3\,{\left (4\,a\,c-b^2\right )}^2}+\frac {8\,b\,c\,x}{{\left (4\,a\,c-b^2\right )}^2}}{b^3\,d^4+6\,b^2\,c\,d^4\,x+12\,b\,c^2\,d^4\,x^2+8\,c^3\,d^4\,x^3}+\frac {2\,\mathrm {atan}\left (\frac {16\,a^2\,b\,c^2\,d^4-8\,a\,b^3\,c\,d^4+b^5\,d^4}{d^4\,{\left (4\,a\,c-b^2\right )}^{5/2}}+\frac {2\,c\,x\,\left (16\,a^2\,c^2\,d^4-8\,a\,b^2\,c\,d^4+b^4\,d^4\right )}{d^4\,{\left (4\,a\,c-b^2\right )}^{5/2}}\right )}{d^4\,{\left (4\,a\,c-b^2\right )}^{5/2}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 402, normalized size of antiderivative = 4.67 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )} \, dx=\frac {2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{4}+12 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{3} c x +24 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c^{2} x^{2}+16 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b \,c^{3} x^{3}-\frac {32 a^{2} b \,c^{2}}{3}+\frac {32 a \,b^{3} c}{3}+16 a \,b^{2} c^{2} x -\frac {64 a \,c^{4} x^{3}}{3}-2 b^{5}-4 b^{4} c x +\frac {16 b^{2} c^{3} x^{3}}{3}}{b \,d^{4} \left (512 a^{3} c^{6} x^{3}-384 a^{2} b^{2} c^{5} x^{3}+96 a \,b^{4} c^{4} x^{3}-8 b^{6} c^{3} x^{3}+768 a^{3} b \,c^{5} x^{2}-576 a^{2} b^{3} c^{4} x^{2}+144 a \,b^{5} c^{3} x^{2}-12 b^{7} c^{2} x^{2}+384 a^{3} b^{2} c^{4} x -288 a^{2} b^{4} c^{3} x +72 a \,b^{6} c^{2} x -6 b^{8} c x +64 a^{3} b^{3} c^{3}-48 a^{2} b^{5} c^{2}+12 a \,b^{7} c -b^{9}\right )} \] Input:

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

Output:

(2*(3*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**4 + 18*sq 
rt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*c*x + 36*sqrt(4 
*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c**2*x**2 + 24*sqrt 
(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b*c**3*x**3 - 16*a**2* 
b*c**2 + 16*a*b**3*c + 24*a*b**2*c**2*x - 32*a*c**4*x**3 - 3*b**5 - 6*b**4 
*c*x + 8*b**2*c**3*x**3))/(3*b*d**4*(64*a**3*b**3*c**3 + 384*a**3*b**2*c** 
4*x + 768*a**3*b*c**5*x**2 + 512*a**3*c**6*x**3 - 48*a**2*b**5*c**2 - 288* 
a**2*b**4*c**3*x - 576*a**2*b**3*c**4*x**2 - 384*a**2*b**2*c**5*x**3 + 12* 
a*b**7*c + 72*a*b**6*c**2*x + 144*a*b**5*c**3*x**2 + 96*a*b**4*c**4*x**3 - 
 b**9 - 6*b**8*c*x - 12*b**7*c**2*x**2 - 8*b**6*c**3*x**3))