\(\int \frac {(b d+2 c d x)^6}{(a+b x+c x^2)^3} \, dx\) [73]

Optimal result

Integrand size = 24, antiderivative size = 108 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^3} \, dx=60 c^2 d^6 (b+2 c x)-\frac {d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}-\frac {5 c d^6 (b+2 c x)^3}{a+b x+c x^2}-60 c^2 \sqrt {b^2-4 a c} d^6 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \] Output:

60*c^2*d^6*(2*c*x+b)-1/2*d^6*(2*c*x+b)^5/(c*x^2+b*x+a)^2-5*c*d^6*(2*c*x+b) 
^3/(c*x^2+b*x+a)-60*c^2*(-4*a*c+b^2)^(1/2)*d^6*arctanh((2*c*x+b)/(-4*a*c+b 
^2)^(1/2))
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.05 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^3} \, dx=d^6 \left (64 c^3 x-\frac {\left (b^2-4 a c\right )^2 (b+2 c x)}{2 (a+x (b+c x))^2}+\frac {9 c \left (-b^2+4 a c\right ) (b+2 c x)}{a+x (b+c x)}-60 c^2 \sqrt {-b^2+4 a c} \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )\right ) \] Input:

Integrate[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^3,x]
 

Output:

d^6*(64*c^3*x - ((b^2 - 4*a*c)^2*(b + 2*c*x))/(2*(a + x*(b + c*x))^2) + (9 
*c*(-b^2 + 4*a*c)*(b + 2*c*x))/(a + x*(b + c*x)) - 60*c^2*Sqrt[-b^2 + 4*a* 
c]*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1110, 27, 1110, 1116, 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.

\(\displaystyle \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1110

\(\displaystyle 5 c d^2 \int \frac {d^4 (b+2 c x)^4}{\left (c x^2+b x+a\right )^2}dx-\frac {d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle 5 c d^6 \int \frac {(b+2 c x)^4}{\left (c x^2+b x+a\right )^2}dx-\frac {d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1110

\(\displaystyle 5 c d^6 \left (6 c \int \frac {(b+2 c x)^2}{c x^2+b x+a}dx-\frac {(b+2 c x)^3}{a+b x+c x^2}\right )-\frac {d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1116

\(\displaystyle 5 c d^6 \left (6 c \left (\left (b^2-4 a c\right ) \int \frac {1}{c x^2+b x+a}dx+2 (b+2 c x)\right )-\frac {(b+2 c x)^3}{a+b x+c x^2}\right )-\frac {d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1083

\(\displaystyle 5 c d^6 \left (6 c \left (2 (b+2 c x)-2 \left (b^2-4 a c\right ) \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)\right )-\frac {(b+2 c x)^3}{a+b x+c x^2}\right )-\frac {d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 219

\(\displaystyle 5 c d^6 \left (6 c \left (2 (b+2 c x)-2 \sqrt {b^2-4 a c} \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right )-\frac {(b+2 c x)^3}{a+b x+c x^2}\right )-\frac {d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}\)

Input:

Int[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^3,x]
 

Output:

-1/2*(d^6*(b + 2*c*x)^5)/(a + b*x + c*x^2)^2 + 5*c*d^6*(-((b + 2*c*x)^3/(a 
 + b*x + c*x^2)) + 6*c*(2*(b + 2*c*x) - 2*Sqrt[b^2 - 4*a*c]*ArcTanh[(b + 2 
*c*x)/Sqrt[b^2 - 4*a*c]]))
 

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 1110

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy 
mbol] :> Simp[d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), 
x] - Simp[d*e*((m - 1)/(b*(p + 1)))   Int[(d + e*x)^(m - 2)*(a + b*x + c*x^ 
2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && N 
eQ[m + 2*p + 3, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]
 

rule 1116

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

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.40

method	result	size
default	\(d^{6} \left (64 c^{3} x -\frac {\left (-72 a \,c^{4}+18 c^{3} b^{2}\right ) x^{3}-27 b \,c^{2} \left (4 a c -b^{2}\right ) x^{2}-2 c \left (28 a^{2} c^{2}+13 c a \,b^{2}-5 b^{4}\right ) x -\frac {b \left (56 a^{2} c^{2}-10 c a \,b^{2}-b^{4}\right )}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}-60 c^{2} \sqrt {4 a c -b^{2}}\, \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )\right )\)	\(151\)
risch	\(64 d^{6} c^{3} x +\frac {\left (72 a \,d^{6} c^{4}-18 b^{2} d^{6} c^{3}\right ) x^{3}+27 b \,d^{6} c^{2} \left (4 a c -b^{2}\right ) x^{2}+2 d^{6} c \left (28 a^{2} c^{2}+13 c a \,b^{2}-5 b^{4}\right ) x +\frac {b \,d^{6} \left (56 a^{2} c^{2}-10 c a \,b^{2}-b^{4}\right )}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+30 \sqrt {-4 a c +b^{2}}\, d^{6} c^{2} \ln \left (-2 \sqrt {-4 a c +b^{2}}\, c x -b \sqrt {-4 a c +b^{2}}-4 a c +b^{2}\right )-30 \sqrt {-4 a c +b^{2}}\, d^{6} c^{2} \ln \left (2 \sqrt {-4 a c +b^{2}}\, c x +b \sqrt {-4 a c +b^{2}}-4 a c +b^{2}\right )\)	\(234\)

Input:

int((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

d^6*(64*c^3*x-((-72*a*c^4+18*b^2*c^3)*x^3-27*b*c^2*(4*a*c-b^2)*x^2-2*c*(28 
*a^2*c^2+13*a*b^2*c-5*b^4)*x-1/2*b*(56*a^2*c^2-10*a*b^2*c-b^4))/(c*x^2+b*x 
+a)^2-60*c^2*(4*a*c-b^2)^(1/2)*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. 275 vs. \(2 (102) = 204\).

Time = 0.09 (sec) , antiderivative size = 571, normalized size of antiderivative = 5.29 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^3} \, dx=\left [\frac {128 \, c^{5} d^{6} x^{5} + 256 \, b c^{4} d^{6} x^{4} + 4 \, {\left (23 \, b^{2} c^{3} + 100 \, a c^{4}\right )} d^{6} x^{3} - 2 \, {\left (27 \, b^{3} c^{2} - 236 \, a b c^{3}\right )} d^{6} x^{2} - 4 \, {\left (5 \, b^{4} c - 13 \, a b^{2} c^{2} - 60 \, a^{2} c^{3}\right )} d^{6} x - {\left (b^{5} + 10 \, a b^{3} c - 56 \, a^{2} b c^{2}\right )} d^{6} + 60 \, {\left (c^{4} d^{6} x^{4} + 2 \, b c^{3} d^{6} x^{3} + 2 \, a b c^{2} d^{6} x + a^{2} c^{2} d^{6} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} x^{2}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right )}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}, \frac {128 \, c^{5} d^{6} x^{5} + 256 \, b c^{4} d^{6} x^{4} + 4 \, {\left (23 \, b^{2} c^{3} + 100 \, a c^{4}\right )} d^{6} x^{3} - 2 \, {\left (27 \, b^{3} c^{2} - 236 \, a b c^{3}\right )} d^{6} x^{2} - 4 \, {\left (5 \, b^{4} c - 13 \, a b^{2} c^{2} - 60 \, a^{2} c^{3}\right )} d^{6} x - {\left (b^{5} + 10 \, a b^{3} c - 56 \, a^{2} b c^{2}\right )} d^{6} - 120 \, {\left (c^{4} d^{6} x^{4} + 2 \, b c^{3} d^{6} x^{3} + 2 \, a b c^{2} d^{6} x + a^{2} c^{2} d^{6} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} x^{2}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}\right ] \] Input:

integrate((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^3,x, algorithm="fricas")
 

Output:

[1/2*(128*c^5*d^6*x^5 + 256*b*c^4*d^6*x^4 + 4*(23*b^2*c^3 + 100*a*c^4)*d^6 
*x^3 - 2*(27*b^3*c^2 - 236*a*b*c^3)*d^6*x^2 - 4*(5*b^4*c - 13*a*b^2*c^2 - 
60*a^2*c^3)*d^6*x - (b^5 + 10*a*b^3*c - 56*a^2*b*c^2)*d^6 + 60*(c^4*d^6*x^ 
4 + 2*b*c^3*d^6*x^3 + 2*a*b*c^2*d^6*x + a^2*c^2*d^6 + (b^2*c^2 + 2*a*c^3)* 
d^6*x^2)*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)))/(c^2*x^4 + 2*b*c*x^3 + 2*a*b* 
x + (b^2 + 2*a*c)*x^2 + a^2), 1/2*(128*c^5*d^6*x^5 + 256*b*c^4*d^6*x^4 + 4 
*(23*b^2*c^3 + 100*a*c^4)*d^6*x^3 - 2*(27*b^3*c^2 - 236*a*b*c^3)*d^6*x^2 - 
 4*(5*b^4*c - 13*a*b^2*c^2 - 60*a^2*c^3)*d^6*x - (b^5 + 10*a*b^3*c - 56*a^ 
2*b*c^2)*d^6 - 120*(c^4*d^6*x^4 + 2*b*c^3*d^6*x^3 + 2*a*b*c^2*d^6*x + a^2* 
c^2*d^6 + (b^2*c^2 + 2*a*c^3)*d^6*x^2)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^ 
2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)))/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b 
^2 + 2*a*c)*x^2 + a^2)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (105) = 210\).

Time = 1.61 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.77 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^3} \, dx=64 c^{3} d^{6} x + c^{2} d^{6} \sqrt {- 3600 a c + 900 b^{2}} \log {\left (x + \frac {30 b c^{2} d^{6} - c^{2} d^{6} \sqrt {- 3600 a c + 900 b^{2}}}{60 c^{3} d^{6}} \right )} - c^{2} d^{6} \sqrt {- 3600 a c + 900 b^{2}} \log {\left (x + \frac {30 b c^{2} d^{6} + c^{2} d^{6} \sqrt {- 3600 a c + 900 b^{2}}}{60 c^{3} d^{6}} \right )} + \frac {56 a^{2} b c^{2} d^{6} - 10 a b^{3} c d^{6} - b^{5} d^{6} + x^{3} \cdot \left (144 a c^{4} d^{6} - 36 b^{2} c^{3} d^{6}\right ) + x^{2} \cdot \left (216 a b c^{3} d^{6} - 54 b^{3} c^{2} d^{6}\right ) + x \left (112 a^{2} c^{3} d^{6} + 52 a b^{2} c^{2} d^{6} - 20 b^{4} c d^{6}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \cdot \left (4 a c + 2 b^{2}\right )} \] Input:

integrate((2*c*d*x+b*d)**6/(c*x**2+b*x+a)**3,x)
 

Output:

64*c**3*d**6*x + c**2*d**6*sqrt(-3600*a*c + 900*b**2)*log(x + (30*b*c**2*d 
**6 - c**2*d**6*sqrt(-3600*a*c + 900*b**2))/(60*c**3*d**6)) - c**2*d**6*sq 
rt(-3600*a*c + 900*b**2)*log(x + (30*b*c**2*d**6 + c**2*d**6*sqrt(-3600*a* 
c + 900*b**2))/(60*c**3*d**6)) + (56*a**2*b*c**2*d**6 - 10*a*b**3*c*d**6 - 
 b**5*d**6 + x**3*(144*a*c**4*d**6 - 36*b**2*c**3*d**6) + x**2*(216*a*b*c* 
*3*d**6 - 54*b**3*c**2*d**6) + x*(112*a**2*c**3*d**6 + 52*a*b**2*c**2*d**6 
 - 20*b**4*c*d**6))/(2*a**2 + 4*a*b*x + 4*b*c*x**3 + 2*c**2*x**4 + x**2*(4 
*a*c + 2*b**2))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^3,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 = 0.15 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.81 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^3} \, dx=64 \, c^{3} d^{6} x + \frac {60 \, {\left (b^{2} c^{2} d^{6} - 4 \, a c^{3} d^{6}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} - \frac {36 \, b^{2} c^{3} d^{6} x^{3} - 144 \, a c^{4} d^{6} x^{3} + 54 \, b^{3} c^{2} d^{6} x^{2} - 216 \, a b c^{3} d^{6} x^{2} + 20 \, b^{4} c d^{6} x - 52 \, a b^{2} c^{2} d^{6} x - 112 \, a^{2} c^{3} d^{6} x + b^{5} d^{6} + 10 \, a b^{3} c d^{6} - 56 \, a^{2} b c^{2} d^{6}}{2 \, {\left (c x^{2} + b x + a\right )}^{2}} \] Input:

integrate((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^3,x, algorithm="giac")
 

Output:

64*c^3*d^6*x + 60*(b^2*c^2*d^6 - 4*a*c^3*d^6)*arctan((2*c*x + b)/sqrt(-b^2 
 + 4*a*c))/sqrt(-b^2 + 4*a*c) - 1/2*(36*b^2*c^3*d^6*x^3 - 144*a*c^4*d^6*x^ 
3 + 54*b^3*c^2*d^6*x^2 - 216*a*b*c^3*d^6*x^2 + 20*b^4*c*d^6*x - 52*a*b^2*c 
^2*d^6*x - 112*a^2*c^3*d^6*x + b^5*d^6 + 10*a*b^3*c*d^6 - 56*a^2*b*c^2*d^6 
)/(c*x^2 + b*x + a)^2
 

Mupad [B] (verification not implemented)

Time = 5.82 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.35 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^3} \, dx=\frac {x\,\left (56\,a^2\,c^3\,d^6+26\,a\,b^2\,c^2\,d^6-10\,b^4\,c\,d^6\right )+x^3\,\left (72\,a\,c^4\,d^6-18\,b^2\,c^3\,d^6\right )-\frac {b^5\,d^6}{2}-x^2\,\left (27\,b^3\,c^2\,d^6-108\,a\,b\,c^3\,d^6\right )+28\,a^2\,b\,c^2\,d^6-5\,a\,b^3\,c\,d^6}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}+64\,c^3\,d^6\,x-60\,c^2\,d^6\,\mathrm {atan}\left (\frac {30\,b\,c^2\,d^6\,\sqrt {4\,a\,c-b^2}+60\,c^3\,d^6\,x\,\sqrt {4\,a\,c-b^2}}{120\,a\,c^3\,d^6-30\,b^2\,c^2\,d^6}\right )\,\sqrt {4\,a\,c-b^2} \] Input:

int((b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^3,x)
 

Output:

(x*(56*a^2*c^3*d^6 - 10*b^4*c*d^6 + 26*a*b^2*c^2*d^6) + x^3*(72*a*c^4*d^6 
- 18*b^2*c^3*d^6) - (b^5*d^6)/2 - x^2*(27*b^3*c^2*d^6 - 108*a*b*c^3*d^6) + 
 28*a^2*b*c^2*d^6 - 5*a*b^3*c*d^6)/(x^2*(2*a*c + b^2) + a^2 + c^2*x^4 + 2* 
a*b*x + 2*b*c*x^3) + 64*c^3*d^6*x - 60*c^2*d^6*atan((30*b*c^2*d^6*(4*a*c - 
 b^2)^(1/2) + 60*c^3*d^6*x*(4*a*c - b^2)^(1/2))/(120*a*c^3*d^6 - 30*b^2*c^ 
2*d^6))*(4*a*c - b^2)^(1/2)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 421, normalized size of antiderivative = 3.90 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^3} \, dx=\frac {d^{6} \left (-120 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b \,c^{2}-240 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} c^{2} x -240 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a b \,c^{3} x^{2}-120 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{3} c^{2} x^{2}-240 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c^{3} x^{3}-120 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b \,c^{4} x^{4}-200 a^{3} c^{3}+10 a^{2} b^{2} c^{2}-160 a^{2} b \,c^{3} x -400 a^{2} c^{4} x^{2}-10 a \,b^{4} c -40 a \,b^{3} c^{2} x +180 a \,b^{2} c^{3} x^{2}-200 a \,c^{5} x^{4}-b^{6}-20 b^{5} c x -100 b^{4} c^{2} x^{2}+210 b^{2} c^{4} x^{4}+128 b \,c^{5} x^{5}\right )}{2 b \left (c^{2} x^{4}+2 b c \,x^{3}+2 a c \,x^{2}+b^{2} x^{2}+2 a b x +a^{2}\right )} \] Input:

int((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^3,x)
 

Output:

(d**6*( - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2 
*b*c**2 - 240*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b* 
*2*c**2*x - 240*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a* 
b*c**3*x**2 - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))* 
b**3*c**2*x**2 - 240*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2 
))*b**2*c**3*x**3 - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b 
**2))*b*c**4*x**4 - 200*a**3*c**3 + 10*a**2*b**2*c**2 - 160*a**2*b*c**3*x 
- 400*a**2*c**4*x**2 - 10*a*b**4*c - 40*a*b**3*c**2*x + 180*a*b**2*c**3*x* 
*2 - 200*a*c**5*x**4 - b**6 - 20*b**5*c*x - 100*b**4*c**2*x**2 + 210*b**2* 
c**4*x**4 + 128*b*c**5*x**5))/(2*b*(a**2 + 2*a*b*x + 2*a*c*x**2 + b**2*x** 
2 + 2*b*c*x**3 + c**2*x**4))