\(\int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx\) [252]

Optimal result

Integrand size = 17, antiderivative size = 147 \[ \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx=-\frac {\sqrt {c+d x}}{3 (b c-a d) (a+b x)^3}+\frac {5 d \sqrt {c+d x}}{12 (b c-a d)^2 (a+b x)^2}-\frac {5 d^2 \sqrt {c+d x}}{8 (b c-a d)^3 (a+b x)}+\frac {5 d^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 \sqrt {b} (b c-a d)^{7/2}} \] Output:

-1/3*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^3+5/12*d*(d*x+c)^(1/2)/(-a*d+b*c)^2/ 
(b*x+a)^2-5/8*d^2*(d*x+c)^(1/2)/(-a*d+b*c)^3/(b*x+a)+5/8*d^3*arctanh(b^(1/ 
2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(1/2)/(-a*d+b*c)^(7/2)
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx=\frac {\sqrt {c+d x} \left (33 a^2 d^2+2 a b d (-13 c+20 d x)+b^2 \left (8 c^2-10 c d x+15 d^2 x^2\right )\right )}{24 (-b c+a d)^3 (a+b x)^3}+\frac {5 d^3 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{8 \sqrt {b} (-b c+a d)^{7/2}} \] Input:

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

Output:

(Sqrt[c + d*x]*(33*a^2*d^2 + 2*a*b*d*(-13*c + 20*d*x) + b^2*(8*c^2 - 10*c* 
d*x + 15*d^2*x^2)))/(24*(-(b*c) + a*d)^3*(a + b*x)^3) + (5*d^3*ArcTan[(Sqr 
t[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(8*Sqrt[b]*(-(b*c) + a*d)^(7/2))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {52, 52, 52, 73, 221}

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.

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

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(\frac {33 \left (\left (\frac {5}{11} d^{2} x^{2}-\frac {10}{33} c d x +\frac {8}{33} c^{2}\right ) b^{2}-\frac {26 \left (-\frac {20 x d}{13}+c \right ) a d b}{33}+a^{2} d^{2}\right ) \sqrt {x d +c}\, \sqrt {\left (a d -b c \right ) b}+15 d^{3} \left (b x +a \right )^{3} \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{24 \sqrt {\left (a d -b c \right ) b}\, \left (a d -b c \right )^{3} \left (b x +a \right )^{3}}\)	\(130\)
derivativedivides	\(2 d^{3} \left (\frac {\sqrt {x d +c}}{6 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )^{3}}+\frac {\frac {5 \sqrt {x d +c}}{24 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {x d +c}}{8 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )}+\frac {3 \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )}{6 \left (a d -b c \right )}}{a d -b c}\right )\)	\(187\)
default	\(2 d^{3} \left (\frac {\sqrt {x d +c}}{6 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )^{3}}+\frac {\frac {5 \sqrt {x d +c}}{24 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {x d +c}}{8 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )}+\frac {3 \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )}{6 \left (a d -b c \right )}}{a d -b c}\right )\)	\(187\)

Input:

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

Output:

1/24*(33*((5/11*d^2*x^2-10/33*c*d*x+8/33*c^2)*b^2-26/33*(-20/13*x*d+c)*a*d 
*b+a^2*d^2)*(d*x+c)^(1/2)*((a*d-b*c)*b)^(1/2)+15*d^3*(b*x+a)^3*arctan(b*(d 
*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)))/((a*d-b*c)*b)^(1/2)/(a*d-b*c)^3/(b*x+a)^ 
3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (123) = 246\).

Time = 0.10 (sec) , antiderivative size = 884, normalized size of antiderivative = 6.01 \[ \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/48*(15*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*sqrt( 
b^2*c - a*b*d)*log((b*d*x + 2*b*c - a*d - 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + 
 c))/(b*x + a)) + 2*(8*b^4*c^3 - 34*a*b^3*c^2*d + 59*a^2*b^2*c*d^2 - 33*a^ 
3*b*d^3 + 15*(b^4*c*d^2 - a*b^3*d^3)*x^2 - 10*(b^4*c^2*d - 5*a*b^3*c*d^2 + 
 4*a^2*b^2*d^3)*x)*sqrt(d*x + c))/(a^3*b^5*c^4 - 4*a^4*b^4*c^3*d + 6*a^5*b 
^3*c^2*d^2 - 4*a^6*b^2*c*d^3 + a^7*b*d^4 + (b^8*c^4 - 4*a*b^7*c^3*d + 6*a^ 
2*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4)*x^3 + 3*(a*b^7*c^4 - 4*a^2* 
b^6*c^3*d + 6*a^3*b^5*c^2*d^2 - 4*a^4*b^4*c*d^3 + a^5*b^3*d^4)*x^2 + 3*(a^ 
2*b^6*c^4 - 4*a^3*b^5*c^3*d + 6*a^4*b^4*c^2*d^2 - 4*a^5*b^3*c*d^3 + a^6*b^ 
2*d^4)*x), -1/24*(15*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3* 
d^3)*sqrt(-b^2*c + a*b*d)*arctan(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c)/(b*d*x 
 + b*c)) + (8*b^4*c^3 - 34*a*b^3*c^2*d + 59*a^2*b^2*c*d^2 - 33*a^3*b*d^3 + 
 15*(b^4*c*d^2 - a*b^3*d^3)*x^2 - 10*(b^4*c^2*d - 5*a*b^3*c*d^2 + 4*a^2*b^ 
2*d^3)*x)*sqrt(d*x + c))/(a^3*b^5*c^4 - 4*a^4*b^4*c^3*d + 6*a^5*b^3*c^2*d^ 
2 - 4*a^6*b^2*c*d^3 + a^7*b*d^4 + (b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2 
*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4)*x^3 + 3*(a*b^7*c^4 - 4*a^2*b^6*c^3*d 
 + 6*a^3*b^5*c^2*d^2 - 4*a^4*b^4*c*d^3 + a^5*b^3*d^4)*x^2 + 3*(a^2*b^6*c^4 
 - 4*a^3*b^5*c^3*d + 6*a^4*b^4*c^2*d^2 - 4*a^5*b^3*c*d^3 + a^6*b^2*d^4)*x) 
]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(1/(b*x+a)^4/(d*x+c)^(1/2),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(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.57 \[ \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx=-\frac {5 \, d^{3} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{8 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-b^{2} c + a b d}} - \frac {15 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d^{3} - 40 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c d^{3} + 33 \, \sqrt {d x + c} b^{2} c^{2} d^{3} + 40 \, {\left (d x + c\right )}^{\frac {3}{2}} a b d^{4} - 66 \, \sqrt {d x + c} a b c d^{4} + 33 \, \sqrt {d x + c} a^{2} d^{5}}{24 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{3}} \] Input:

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

Output:

-5/8*d^3*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^3*c^3 - 3*a*b^2* 
c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-b^2*c + a*b*d)) - 1/24*(15*(d*x + c 
)^(5/2)*b^2*d^3 - 40*(d*x + c)^(3/2)*b^2*c*d^3 + 33*sqrt(d*x + c)*b^2*c^2* 
d^3 + 40*(d*x + c)^(3/2)*a*b*d^4 - 66*sqrt(d*x + c)*a*b*c*d^4 + 33*sqrt(d* 
x + c)*a^2*d^5)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*((d*x 
 + c)*b - b*c + a*d)^3)
 

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.48 \[ \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx=\frac {\frac {11\,d^3\,\sqrt {c+d\,x}}{8\,\left (a\,d-b\,c\right )}+\frac {5\,b^2\,d^3\,{\left (c+d\,x\right )}^{5/2}}{8\,{\left (a\,d-b\,c\right )}^3}+\frac {5\,b\,d^3\,{\left (c+d\,x\right )}^{3/2}}{3\,{\left (a\,d-b\,c\right )}^2}}{\left (c+d\,x\right )\,\left (3\,a^2\,b\,d^2-6\,a\,b^2\,c\,d+3\,b^3\,c^2\right )+b^3\,{\left (c+d\,x\right )}^3-\left (3\,b^3\,c-3\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^2+a^3\,d^3-b^3\,c^3+3\,a\,b^2\,c^2\,d-3\,a^2\,b\,c\,d^2}+\frac {5\,d^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{8\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{7/2}} \] Input:

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

Output:

((11*d^3*(c + d*x)^(1/2))/(8*(a*d - b*c)) + (5*b^2*d^3*(c + d*x)^(5/2))/(8 
*(a*d - b*c)^3) + (5*b*d^3*(c + d*x)^(3/2))/(3*(a*d - b*c)^2))/((c + d*x)* 
(3*b^3*c^2 + 3*a^2*b*d^2 - 6*a*b^2*c*d) + b^3*(c + d*x)^3 - (3*b^3*c - 3*a 
*b^2*d)*(c + d*x)^2 + a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2) + 
 (5*d^3*atan((b^(1/2)*(c + d*x)^(1/2))/(a*d - b*c)^(1/2)))/(8*b^(1/2)*(a*d 
 - b*c)^(7/2))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 597, normalized size of antiderivative = 4.06 \[ \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx=\frac {15 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{3} d^{3}+45 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{2} b \,d^{3} x +45 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a \,b^{2} d^{3} x^{2}+15 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) b^{3} d^{3} x^{3}+33 \sqrt {d x +c}\, a^{3} b \,d^{3}-59 \sqrt {d x +c}\, a^{2} b^{2} c \,d^{2}+40 \sqrt {d x +c}\, a^{2} b^{2} d^{3} x +34 \sqrt {d x +c}\, a \,b^{3} c^{2} d -50 \sqrt {d x +c}\, a \,b^{3} c \,d^{2} x +15 \sqrt {d x +c}\, a \,b^{3} d^{3} x^{2}-8 \sqrt {d x +c}\, b^{4} c^{3}+10 \sqrt {d x +c}\, b^{4} c^{2} d x -15 \sqrt {d x +c}\, b^{4} c \,d^{2} x^{2}}{24 b \left (a^{4} b^{3} d^{4} x^{3}-4 a^{3} b^{4} c \,d^{3} x^{3}+6 a^{2} b^{5} c^{2} d^{2} x^{3}-4 a \,b^{6} c^{3} d \,x^{3}+b^{7} c^{4} x^{3}+3 a^{5} b^{2} d^{4} x^{2}-12 a^{4} b^{3} c \,d^{3} x^{2}+18 a^{3} b^{4} c^{2} d^{2} x^{2}-12 a^{2} b^{5} c^{3} d \,x^{2}+3 a \,b^{6} c^{4} x^{2}+3 a^{6} b \,d^{4} x -12 a^{5} b^{2} c \,d^{3} x +18 a^{4} b^{3} c^{2} d^{2} x -12 a^{3} b^{4} c^{3} d x +3 a^{2} b^{5} c^{4} x +a^{7} d^{4}-4 a^{6} b c \,d^{3}+6 a^{5} b^{2} c^{2} d^{2}-4 a^{4} b^{3} c^{3} d +a^{3} b^{4} c^{4}\right )} \] Input:

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

Output:

(15*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c 
)))*a**3*d**3 + 45*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b) 
*sqrt(a*d - b*c)))*a**2*b*d**3*x + 45*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c 
 + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b**2*d**3*x**2 + 15*sqrt(b)*sqrt(a 
*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*b**3*d**3*x**3 
 + 33*sqrt(c + d*x)*a**3*b*d**3 - 59*sqrt(c + d*x)*a**2*b**2*c*d**2 + 40*s 
qrt(c + d*x)*a**2*b**2*d**3*x + 34*sqrt(c + d*x)*a*b**3*c**2*d - 50*sqrt(c 
 + d*x)*a*b**3*c*d**2*x + 15*sqrt(c + d*x)*a*b**3*d**3*x**2 - 8*sqrt(c + d 
*x)*b**4*c**3 + 10*sqrt(c + d*x)*b**4*c**2*d*x - 15*sqrt(c + d*x)*b**4*c*d 
**2*x**2)/(24*b*(a**7*d**4 - 4*a**6*b*c*d**3 + 3*a**6*b*d**4*x + 6*a**5*b* 
*2*c**2*d**2 - 12*a**5*b**2*c*d**3*x + 3*a**5*b**2*d**4*x**2 - 4*a**4*b**3 
*c**3*d + 18*a**4*b**3*c**2*d**2*x - 12*a**4*b**3*c*d**3*x**2 + a**4*b**3* 
d**4*x**3 + a**3*b**4*c**4 - 12*a**3*b**4*c**3*d*x + 18*a**3*b**4*c**2*d** 
2*x**2 - 4*a**3*b**4*c*d**3*x**3 + 3*a**2*b**5*c**4*x - 12*a**2*b**5*c**3* 
d*x**2 + 6*a**2*b**5*c**2*d**2*x**3 + 3*a*b**6*c**4*x**2 - 4*a*b**6*c**3*d 
*x**3 + b**7*c**4*x**3))