\(\int \frac {x^5}{(c+d x) (c^2-d^2 x^2)^{7/2}} \, dx\) [488]

Optimal result

Integrand size = 27, antiderivative size = 126 \[ \int \frac {x^5}{(c+d x) \left (c^2-d^2 x^2\right )^{7/2}} \, dx=-\frac {c^2 (14 c-15 d x)}{35 d^6 \left (c^2-d^2 x^2\right )^{5/2}}+\frac {c^4}{7 d^6 (c+d x) \left (c^2-d^2 x^2\right )^{5/2}}+\frac {7 c-9 d x}{21 d^6 \left (c^2-d^2 x^2\right )^{3/2}}+\frac {x}{7 c^2 d^5 \sqrt {c^2-d^2 x^2}} \] Output:

-1/35*c^2*(-15*d*x+14*c)/d^6/(-d^2*x^2+c^2)^(5/2)+1/7*c^4/d^6/(d*x+c)/(-d^ 
2*x^2+c^2)^(5/2)+1/21*(-9*d*x+7*c)/d^6/(-d^2*x^2+c^2)^(3/2)+1/7*x/c^2/d^5/ 
(-d^2*x^2+c^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.83 \[ \int \frac {x^5}{(c+d x) \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (8 c^6+8 c^5 d x-20 c^4 d^2 x^2-20 c^3 d^3 x^3+15 c^2 d^4 x^4+15 c d^5 x^5+15 d^6 x^6\right )}{105 c^2 d^6 (c-d x)^3 (c+d x)^4} \] Input:

Integrate[x^5/((c + d*x)*(c^2 - d^2*x^2)^(7/2)),x]
 

Output:

(Sqrt[c^2 - d^2*x^2]*(8*c^6 + 8*c^5*d*x - 20*c^4*d^2*x^2 - 20*c^3*d^3*x^3 
+ 15*c^2*d^4*x^4 + 15*c*d^5*x^5 + 15*d^6*x^6))/(105*c^2*d^6*(c - d*x)^3*(c 
 + d*x)^4)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {568, 530, 27, 2345, 27, 208}

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.

\(\displaystyle \int \frac {x^5}{(c+d x) \left (c^2-d^2 x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 568

\(\displaystyle \frac {x^4}{7 d^2 (c+d x) \left (c^2-d^2 x^2\right )^{5/2}}-\frac {\int \frac {x^3 (4 c-5 d x)}{\left (c^2-d^2 x^2\right )^{7/2}}dx}{7 d^2}\)

\(\Big \downarrow \) 530

\(\displaystyle \frac {x^4}{7 d^2 (c+d x) \left (c^2-d^2 x^2\right )^{5/2}}-\frac {\frac {c^2 (4 c-5 d x)}{5 d^4 \left (c^2-d^2 x^2\right )^{5/2}}-\frac {\int -\frac {5 \left (\frac {c^4}{d^3}-\frac {4 x c^3}{d^2}+\frac {5 x^2 c^2}{d}\right )}{\left (c^2-d^2 x^2\right )^{5/2}}dx}{5 c^2}}{7 d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^4}{7 d^2 (c+d x) \left (c^2-d^2 x^2\right )^{5/2}}-\frac {\frac {\int \frac {\frac {c^4}{d^3}-\frac {4 x c^3}{d^2}+\frac {5 x^2 c^2}{d}}{\left (c^2-d^2 x^2\right )^{5/2}}dx}{c^2}+\frac {c^2 (4 c-5 d x)}{5 d^4 \left (c^2-d^2 x^2\right )^{5/2}}}{7 d^2}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {x^4}{7 d^2 (c+d x) \left (c^2-d^2 x^2\right )^{5/2}}-\frac {\frac {-\frac {\int \frac {3 c^4}{d^3 \left (c^2-d^2 x^2\right )^{3/2}}dx}{3 c^2}-\frac {2 c^2 (2 c-3 d x)}{3 d^4 \left (c^2-d^2 x^2\right )^{3/2}}}{c^2}+\frac {c^2 (4 c-5 d x)}{5 d^4 \left (c^2-d^2 x^2\right )^{5/2}}}{7 d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^4}{7 d^2 (c+d x) \left (c^2-d^2 x^2\right )^{5/2}}-\frac {\frac {-\frac {c^2 \int \frac {1}{\left (c^2-d^2 x^2\right )^{3/2}}dx}{d^3}-\frac {2 c^2 (2 c-3 d x)}{3 d^4 \left (c^2-d^2 x^2\right )^{3/2}}}{c^2}+\frac {c^2 (4 c-5 d x)}{5 d^4 \left (c^2-d^2 x^2\right )^{5/2}}}{7 d^2}\)

\(\Big \downarrow \) 208

\(\displaystyle \frac {x^4}{7 d^2 (c+d x) \left (c^2-d^2 x^2\right )^{5/2}}-\frac {\frac {c^2 (4 c-5 d x)}{5 d^4 \left (c^2-d^2 x^2\right )^{5/2}}+\frac {-\frac {2 c^2 (2 c-3 d x)}{3 d^4 \left (c^2-d^2 x^2\right )^{3/2}}-\frac {x}{d^3 \sqrt {c^2-d^2 x^2}}}{c^2}}{7 d^2}\)

Input:

Int[x^5/((c + d*x)*(c^2 - d^2*x^2)^(7/2)),x]
 

Output:

x^4/(7*d^2*(c + d*x)*(c^2 - d^2*x^2)^(5/2)) - ((c^2*(4*c - 5*d*x))/(5*d^4* 
(c^2 - d^2*x^2)^(5/2)) + ((-2*c^2*(2*c - 3*d*x))/(3*d^4*(c^2 - d^2*x^2)^(3 
/2)) - x/(d^3*Sqrt[c^2 - d^2*x^2]))/c^2)/(7*d^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 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 530

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symb 
ol] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Co 
eff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Po 
lynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x 
)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(a 
 + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Qx + e*(2*p + 3), x], x], x]] /; 
FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[n, 
1] && IntegerQ[2*p]
 

rule 568

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

rule 2345

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b 
*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.73

method	result	size
gosper	\(\frac {\left (-d x +c \right ) \left (15 d^{6} x^{6}+15 c \,d^{5} x^{5}+15 c^{2} d^{4} x^{4}-20 d^{3} c^{3} x^{3}-20 c^{4} d^{2} x^{2}+8 c^{5} d x +8 c^{6}\right )}{105 d^{6} c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}\)	\(92\)
orering	\(\frac {\left (-d x +c \right ) \left (15 d^{6} x^{6}+15 c \,d^{5} x^{5}+15 c^{2} d^{4} x^{4}-20 d^{3} c^{3} x^{3}-20 c^{4} d^{2} x^{2}+8 c^{5} d x +8 c^{6}\right )}{105 d^{6} c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}\)	\(92\)
trager	\(\frac {\left (15 d^{6} x^{6}+15 c \,d^{5} x^{5}+15 c^{2} d^{4} x^{4}-20 d^{3} c^{3} x^{3}-20 c^{4} d^{2} x^{2}+8 c^{5} d x +8 c^{6}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{105 c^{2} d^{6} \left (d x +c \right )^{4} \left (-d x +c \right )^{3}}\)	\(101\)
default	\(\frac {c^{4} \left (\frac {x}{5 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 c^{4} \sqrt {-d^{2} x^{2}+c^{2}}}}{c^{2}}\right )}{d^{5}}+\frac {\frac {x^{3}}{2 d^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}-\frac {3 c^{2} \left (\frac {x}{4 d^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}-\frac {c^{2} \left (\frac {x}{5 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 c^{4} \sqrt {-d^{2} x^{2}+c^{2}}}}{c^{2}}\right )}{4 d^{2}}\right )}{2 d^{2}}}{d}+\frac {c^{2} \left (\frac {x}{4 d^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}-\frac {c^{2} \left (\frac {x}{5 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 c^{4} \sqrt {-d^{2} x^{2}+c^{2}}}}{c^{2}}\right )}{4 d^{2}}\right )}{d^{3}}-\frac {c \left (\frac {x^{2}}{3 d^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}-\frac {2 c^{2}}{15 d^{4} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}\right )}{d^{2}}-\frac {c^{3}}{5 d^{6} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}-\frac {c^{5} \left (-\frac {1}{7 c d \left (x +\frac {c}{d}\right ) \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}+\frac {6 d \left (-\frac {-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d}{10 c^{2} d^{2} \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d \right )}{15 c^{2} d^{2} \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d \right )}{15 d^{2} c^{4} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}}{c^{2}}\right )}{7 c}\right )}{d^{6}}\)	\(609\)

Input:

int(x^5/(d*x+c)/(-d^2*x^2+c^2)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

1/105*(-d*x+c)*(15*d^6*x^6+15*c*d^5*x^5+15*c^2*d^4*x^4-20*c^3*d^3*x^3-20*c 
^4*d^2*x^2+8*c^5*d*x+8*c^6)/d^6/c^2/(-d^2*x^2+c^2)^(7/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (110) = 220\).

Time = 0.15 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.90 \[ \int \frac {x^5}{(c+d x) \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\frac {8 \, d^{7} x^{7} + 8 \, c d^{6} x^{6} - 24 \, c^{2} d^{5} x^{5} - 24 \, c^{3} d^{4} x^{4} + 24 \, c^{4} d^{3} x^{3} + 24 \, c^{5} d^{2} x^{2} - 8 \, c^{6} d x - 8 \, c^{7} - {\left (15 \, d^{6} x^{6} + 15 \, c d^{5} x^{5} + 15 \, c^{2} d^{4} x^{4} - 20 \, c^{3} d^{3} x^{3} - 20 \, c^{4} d^{2} x^{2} + 8 \, c^{5} d x + 8 \, c^{6}\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{105 \, {\left (c^{2} d^{13} x^{7} + c^{3} d^{12} x^{6} - 3 \, c^{4} d^{11} x^{5} - 3 \, c^{5} d^{10} x^{4} + 3 \, c^{6} d^{9} x^{3} + 3 \, c^{7} d^{8} x^{2} - c^{8} d^{7} x - c^{9} d^{6}\right )}} \] Input:

integrate(x^5/(d*x+c)/(-d^2*x^2+c^2)^(7/2),x, algorithm="fricas")
 

Output:

1/105*(8*d^7*x^7 + 8*c*d^6*x^6 - 24*c^2*d^5*x^5 - 24*c^3*d^4*x^4 + 24*c^4* 
d^3*x^3 + 24*c^5*d^2*x^2 - 8*c^6*d*x - 8*c^7 - (15*d^6*x^6 + 15*c*d^5*x^5 
+ 15*c^2*d^4*x^4 - 20*c^3*d^3*x^3 - 20*c^4*d^2*x^2 + 8*c^5*d*x + 8*c^6)*sq 
rt(-d^2*x^2 + c^2))/(c^2*d^13*x^7 + c^3*d^12*x^6 - 3*c^4*d^11*x^5 - 3*c^5* 
d^10*x^4 + 3*c^6*d^9*x^3 + 3*c^7*d^8*x^2 - c^8*d^7*x - c^9*d^6)
 

Sympy [F]

\[ \int \frac {x^5}{(c+d x) \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\int \frac {x^{5}}{\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {7}{2}} \left (c + d x\right )}\, dx \] Input:

integrate(x**5/(d*x+c)/(-d**2*x**2+c**2)**(7/2),x)
 

Output:

Integral(x**5/((-(-c + d*x)*(c + d*x))**(7/2)*(c + d*x)), x)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.43 \[ \int \frac {x^5}{(c+d x) \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\frac {c^{4}}{7 \, {\left ({\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} d^{7} x + {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} c d^{6}\right )}} + \frac {x^{3}}{2 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} d^{3}} - \frac {c x^{2}}{3 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} d^{4}} - \frac {c^{2} x}{14 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} d^{5}} - \frac {c^{3}}{15 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} d^{6}} + \frac {x}{14 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d^{5}} + \frac {x}{7 \, \sqrt {-d^{2} x^{2} + c^{2}} c^{2} d^{5}} \] Input:

integrate(x^5/(d*x+c)/(-d^2*x^2+c^2)^(7/2),x, algorithm="maxima")
 

Output:

1/7*c^4/((-d^2*x^2 + c^2)^(5/2)*d^7*x + (-d^2*x^2 + c^2)^(5/2)*c*d^6) + 1/ 
2*x^3/((-d^2*x^2 + c^2)^(5/2)*d^3) - 1/3*c*x^2/((-d^2*x^2 + c^2)^(5/2)*d^4 
) - 1/14*c^2*x/((-d^2*x^2 + c^2)^(5/2)*d^5) - 1/15*c^3/((-d^2*x^2 + c^2)^( 
5/2)*d^6) + 1/14*x/((-d^2*x^2 + c^2)^(3/2)*d^5) + 1/7*x/(sqrt(-d^2*x^2 + c 
^2)*c^2*d^5)
 

Giac [F]

\[ \int \frac {x^5}{(c+d x) \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\int { \frac {x^{5}}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {7}{2}} {\left (d x + c\right )}} \,d x } \] Input:

integrate(x^5/(d*x+c)/(-d^2*x^2+c^2)^(7/2),x, algorithm="giac")
 

Output:

integrate(x^5/((-d^2*x^2 + c^2)^(7/2)*(d*x + c)), x)
 

Mupad [B] (verification not implemented)

Time = 7.67 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.25 \[ \int \frac {x^5}{(c+d x) \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {c^2-d^2\,x^2}\,\left (\frac {53\,c}{168\,d^6}-\frac {3\,x}{7\,d^5}\right )}{{\left (c+d\,x\right )}^2\,{\left (c-d\,x\right )}^2}-\frac {\sqrt {c^2-d^2\,x^2}\,\left (\frac {9\,c^3}{35\,d^6}-\frac {5\,c^2\,x}{14\,d^5}\right )}{{\left (c+d\,x\right )}^3\,{\left (c-d\,x\right )}^3}+\frac {c\,\sqrt {c^2-d^2\,x^2}}{56\,d^6\,{\left (c+d\,x\right )}^4}+\frac {x\,\sqrt {c^2-d^2\,x^2}}{7\,c^2\,d^5\,\left (c+d\,x\right )\,\left (c-d\,x\right )} \] Input:

int(x^5/((c^2 - d^2*x^2)^(7/2)*(c + d*x)),x)
 

Output:

((c^2 - d^2*x^2)^(1/2)*((53*c)/(168*d^6) - (3*x)/(7*d^5)))/((c + d*x)^2*(c 
 - d*x)^2) - ((c^2 - d^2*x^2)^(1/2)*((9*c^3)/(35*d^6) - (5*c^2*x)/(14*d^5) 
))/((c + d*x)^3*(c - d*x)^3) + (c*(c^2 - d^2*x^2)^(1/2))/(56*d^6*(c + d*x) 
^4) + (x*(c^2 - d^2*x^2)^(1/2))/(7*c^2*d^5*(c + d*x)*(c - d*x))
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.10 \[ \int \frac {x^5}{(c+d x) \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\frac {8 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5}+8 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d x -16 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{2} x^{2}-16 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{3} x^{3}+8 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{4} x^{4}+8 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{5} x^{5}+8 c^{6}+8 c^{5} d x -20 c^{4} d^{2} x^{2}-20 c^{3} d^{3} x^{3}+15 c^{2} d^{4} x^{4}+15 c \,d^{5} x^{5}+15 d^{6} x^{6}}{105 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{6} \left (d^{5} x^{5}+c \,d^{4} x^{4}-2 c^{2} d^{3} x^{3}-2 c^{3} d^{2} x^{2}+c^{4} d x +c^{5}\right )} \] Input:

int(x^5/(d*x+c)/(-d^2*x^2+c^2)^(7/2),x)
 

Output:

(8*sqrt(c**2 - d**2*x**2)*c**5 + 8*sqrt(c**2 - d**2*x**2)*c**4*d*x - 16*sq 
rt(c**2 - d**2*x**2)*c**3*d**2*x**2 - 16*sqrt(c**2 - d**2*x**2)*c**2*d**3* 
x**3 + 8*sqrt(c**2 - d**2*x**2)*c*d**4*x**4 + 8*sqrt(c**2 - d**2*x**2)*d** 
5*x**5 + 8*c**6 + 8*c**5*d*x - 20*c**4*d**2*x**2 - 20*c**3*d**3*x**3 + 15* 
c**2*d**4*x**4 + 15*c*d**5*x**5 + 15*d**6*x**6)/(105*sqrt(c**2 - d**2*x**2 
)*c**2*d**6*(c**5 + c**4*d*x - 2*c**3*d**2*x**2 - 2*c**2*d**3*x**3 + c*d** 
4*x**4 + d**5*x**5))