\(\int \frac {1}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx\) [1517]

Optimal result

Integrand size = 22, antiderivative size = 185 \[ \int \frac {1}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\frac {2 d \sqrt {a-b x^2}}{\left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.79 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.69 \[ \int \frac {1}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\frac {2 \left (\frac {d^2 \left (a-b x^2\right )}{b c^2-a d^2}-i \sqrt {\frac {d \left (\sqrt {a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {a} d}} \sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \left (E\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )\right )}{d \sqrt {c+d x} \sqrt {a-b x^2}} \] Input:

Integrate[1/((c + d*x)^(3/2)*Sqrt[a - b*x^2]),x]
 

Output:

(2*((d^2*(a - b*x^2))/(b*c^2 - a*d^2) - I*Sqrt[(d*(Sqrt[a] - Sqrt[b]*x))/( 
Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[(d*(Sqrt[a] + Sqrt[b]*x))/(-(Sqrt[b]*c) + Sqr 
t[a]*d)]*Sqrt[-((Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d))]*(EllipticE[I 
*ArcSinh[Sqrt[-((Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d))]], (Sqrt[b]*c 
 + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - EllipticF[I*ArcSinh[Sqrt[-((Sqrt[ 
b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d))]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b] 
*c - Sqrt[a]*d)])))/(d*Sqrt[c + d*x]*Sqrt[a - b*x^2])
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {498, 27, 509, 508, 327}

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.

\(\displaystyle \int \frac {1}{\sqrt {a-b x^2} (c+d x)^{3/2}} \, dx\)

\(\Big \downarrow \) 498

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 d \sqrt {a-b x^2}}{\sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\)

Input:

Int[1/((c + d*x)^(3/2)*Sqrt[a - b*x^2]),x]
 

Output:

(2*d*Sqrt[a - b*x^2])/((b*c^2 - a*d^2)*Sqrt[c + d*x]) - (2*Sqrt[a]*Sqrt[b] 
*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/S 
qrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/((b*c^2 - a*d^2)*Sqrt[ 
(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2])
 

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 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 498

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S 
imp[b/((n + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n 
 + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n 
, -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp 
lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 0])
 

rule 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 509

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq 
rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], 
x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(616\) vs. \(2(154)=308\).

Time = 2.68 (sec) , antiderivative size = 617, normalized size of antiderivative = 3.34

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 \left (-b d \,x^{2}+a d \right )}{\left (a \,d^{2}-b \,c^{2}\right ) \sqrt {\left (x +\frac {c}{d}\right ) \left (-b d \,x^{2}+a d \right )}}-\frac {2 b c \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\left (a \,d^{2}-b \,c^{2}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {2 b d \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\left (a \,d^{2}-b \,c^{2}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(617\)
default	\(-\frac {2 \left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) a \,d^{2}-\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b \,c^{2}-a \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) d^{2}+\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b \,c^{2}-b \,x^{2} d^{2}+a \,d^{2}\right ) \sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}{d \left (a \,d^{2}-b \,c^{2}\right ) \left (-b d \,x^{3}-b c \,x^{2}+a d x +a c \right )}\)	\(626\)

Input:

int(1/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(-2*(-b*d*x^2+a* 
d)/(a*d^2-b*c^2)/((x+c/d)*(-b*d*x^2+a*d))^(1/2)-2*b*c/(a*d^2-b*c^2)*(c/d-1 
/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2)) 
/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)) 
)^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a* 
b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))-2* 
b*d/(a*d^2-b*c^2)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1 
/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2) 
)/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((-c/d- 
1/b*(a*b)^(1/2))*EllipticE(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/ 
b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+1/b*(a*b)^(1/2)*EllipticF((( 
x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b 
)^(1/2)))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.29 \[ \int \frac {1}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {-b x^{2} + a} \sqrt {d x + c} d^{2} - 2 \, {\left (c d x + c^{2}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (d^{2} x + c d\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right )\right )}}{3 \, {\left (b c^{3} d - a c d^{3} + {\left (b c^{2} d^{2} - a d^{4}\right )} x\right )}} \] Input:

integrate(1/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

2/3*(3*sqrt(-b*x^2 + a)*sqrt(d*x + c)*d^2 - 2*(c*d*x + c^2)*sqrt(-b*d)*wei 
erstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/ 
(b*d^3), 1/3*(3*d*x + c)/d) + 3*(d^2*x + c*d)*sqrt(-b*d)*weierstrassZeta(4 
/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), weierstra 
ssPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3 
), 1/3*(3*d*x + c)/d)))/(b*c^3*d - a*c*d^3 + (b*c^2*d^2 - a*d^4)*x)
 

Sympy [F]

\[ \int \frac {1}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int \frac {1}{\sqrt {a - b x^{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(d*x+c)**(3/2)/(-b*x**2+a)**(1/2),x)
 

Output:

Integral(1/(sqrt(a - b*x**2)*(c + d*x)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {1}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int { \frac {1}{\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/(sqrt(-b*x^2 + a)*(d*x + c)^(3/2)), x)
 

Giac [F]

\[ \int \frac {1}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int { \frac {1}{\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/(sqrt(-b*x^2 + a)*(d*x + c)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int \frac {1}{\sqrt {a-b\,x^2}\,{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:

int(1/((a - b*x^2)^(1/2)*(c + d*x)^(3/2)),x)
 

Output:

int(1/((a - b*x^2)^(1/2)*(c + d*x)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{2} x^{4}-2 b c d \,x^{3}+a \,d^{2} x^{2}-b \,c^{2} x^{2}+2 a c d x +a \,c^{2}}d x \] Input:

int(1/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x)
 

Output:

int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c**2 + 2*a*c*d*x + a*d**2*x**2 - b 
*c**2*x**2 - 2*b*c*d*x**3 - b*d**2*x**4),x)