Integrand size = 37, antiderivative size = 124 \[ \int \frac {(c e+d e x)^{7/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {10 e^3 \sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}}{21 d}-\frac {2 e (c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{7 d}+\frac {10 e^{7/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right ),-1\right )}{21 d} \] Output:
-10/21*e^3*(d*e*x+c*e)^(1/2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)/d-2/7*e*(d*e*x +c*e)^(5/2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)/d+10/21*e^(7/2)*EllipticF((d*e* x+c*e)^(1/2)/e^(1/2),I)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.69 \[ \int \frac {(c e+d e x)^{7/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 e^3 \sqrt {e (c+d x)} \left (\sqrt {1-c^2-2 c d x-d^2 x^2} \left (5+3 c^2+6 c d x+3 d^2 x^2\right )-5 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},(c+d x)^2\right )\right )}{21 d} \] Input:
Integrate[(c*e + d*e*x)^(7/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]
Output:
(-2*e^3*Sqrt[e*(c + d*x)]*(Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]*(5 + 3*c^2 + 6*c*d*x + 3*d^2*x^2) - 5*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2]))/( 21*d)
Time = 0.30 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {1116, 1116, 1113, 762}
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.
\(\displaystyle \int \frac {(c e+d e x)^{7/2}}{\sqrt {-c^2-2 c d x-d^2 x^2+1}} \, dx\) |
\(\Big \downarrow \) 1116 |
\(\displaystyle \frac {5}{7} e^2 \int \frac {(c e+d x e)^{3/2}}{\sqrt {-c^2-2 d x c-d^2 x^2+1}}dx-\frac {2 e \sqrt {-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{7 d}\) |
\(\Big \downarrow \) 1116 |
\(\displaystyle \frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {c e+d x e} \sqrt {-c^2-2 d x c-d^2 x^2+1}}dx-\frac {2 e \sqrt {-c^2-2 c d x-d^2 x^2+1} \sqrt {c e+d e x}}{3 d}\right )-\frac {2 e \sqrt {-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{7 d}\) |
\(\Big \downarrow \) 1113 |
\(\displaystyle \frac {5}{7} e^2 \left (\frac {2 e \int \frac {1}{\sqrt {1-\frac {(c e+d x e)^2}{e^2}}}d\sqrt {c e+d x e}}{3 d}-\frac {2 e \sqrt {-c^2-2 c d x-d^2 x^2+1} \sqrt {c e+d e x}}{3 d}\right )-\frac {2 e \sqrt {-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{7 d}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {5}{7} e^2 \left (\frac {2 e^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right ),-1\right )}{3 d}-\frac {2 e \sqrt {-c^2-2 c d x-d^2 x^2+1} \sqrt {c e+d e x}}{3 d}\right )-\frac {2 e \sqrt {-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{7 d}\) |
Input:
Int[(c*e + d*e*x)^(7/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]
Output:
(-2*e*(c*e + d*e*x)^(5/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(7*d) + (5*e^ 2*((-2*e*Sqrt[c*e + d*e*x]*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(3*d) + (2*e ^(3/2)*EllipticF[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(3*d)))/7
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_ Symbol] :> Simp[(4/e)*Sqrt[-c/(b^2 - 4*a*c)] Subst[Int[1/Sqrt[Simp[1 - b^ 2*(x^4/(d^2*(b^2 - 4*a*c))), x]], x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && LtQ[c/(b^2 - 4*a*c), 0]
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])
Leaf count of result is larger than twice the leaf count of optimal. \(211\) vs. \(2(104)=208\).
Time = 2.57 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.71
method | result | size |
default | \(\frac {\sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, e^{3} \left (-6 x^{5} d^{5}-30 x^{4} c \,d^{4}-60 c^{2} d^{3} x^{3}-60 c^{3} d^{2} x^{2}-30 c^{4} d x -4 d^{3} x^{3}-6 c^{5}-12 c \,d^{2} x^{2}+5 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, \operatorname {EllipticF}\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right )-12 c^{2} d x -4 c^{3}+10 d x +10 c \right )}{21 d \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}-d x -c \right )}\) | \(212\) |
risch | \(\frac {2 \left (3 d^{2} x^{2}+6 c d x +3 c^{2}+5\right ) \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) \sqrt {e \left (d x +c \right ) \left (-d^{2} x^{2}-2 c d x -c^{2}+1\right )}\, e^{4}}{21 d \sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}+\frac {10 \left (-\frac {c -1}{d}+\frac {c +1}{d}\right ) \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c +1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}, \sqrt {\frac {-\frac {c +1}{d}+\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c}{d}}}\right ) \sqrt {e \left (d x +c \right ) \left (-d^{2} x^{2}-2 c d x -c^{2}+1\right )}\, e^{4}}{21 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d e x +c e}\, \sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}\) | \(415\) |
elliptic | \(\frac {\sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \sqrt {e \left (d x +c \right )}\, \left (-\frac {2 d \,e^{3} x^{2} \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d e x +c e}}{7}-\frac {4 c \,e^{3} x \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d e x +c e}}{7}-\frac {2 \left (\frac {18 c^{2} d^{2} e^{4}}{7}+\frac {2 d \,e^{3} \left (-\frac {15}{2} c^{2} d e +\frac {5}{2} d e \right )}{7}\right ) \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d e x +c e}}{3 d^{3} e}+\frac {2 \left (c^{4} e^{4}+\frac {4 c \,e^{3} \left (-c^{3} e +c e \right )}{7}+\frac {2 \left (\frac {18 c^{2} d^{2} e^{4}}{7}+\frac {2 d \,e^{3} \left (-\frac {15}{2} c^{2} d e +\frac {5}{2} d e \right )}{7}\right ) \left (-\frac {3}{2} c^{2} d e +\frac {1}{2} d e \right )}{3 d^{3} e}\right ) \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{\sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d e x +c e}}+\frac {2 \left (4 c^{3} d \,e^{4}+\frac {2 d \,e^{3} \left (-2 c^{3} e +2 c e \right )}{7}+\frac {4 c \,e^{3} \left (-\frac {9}{2} c^{2} d e +\frac {3}{2} d e \right )}{7}-\frac {2 \left (\frac {18 c^{2} d^{2} e^{4}}{7}+\frac {2 d \,e^{3} \left (-\frac {15}{2} c^{2} d e +\frac {5}{2} d e \right )}{7}\right ) c}{d}\right ) \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \left (\left (-\frac {c -1}{d}+\frac {c}{d}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )-\frac {c \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{d}\right )}{\sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d e x +c e}}\right )}{\left (d x +c \right ) \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, e}\) | \(923\) |
Input:
int((d*e*x+c*e)^(7/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x,method=_RETURNVERBO SE)
Output:
1/21*(e*(d*x+c))^(1/2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)*e^3*(-6*x^5*d^5-30*x ^4*c*d^4-60*c^2*d^3*x^3-60*c^3*d^2*x^2-30*c^4*d*x-4*d^3*x^3-6*c^5-12*c*d^2 *x^2+5*(-2*d*x-2*c+2)^(1/2)*(d*x+c)^(1/2)*(2*d*x+2*c+2)^(1/2)*EllipticF(1/ 2*(-2*d*x-2*c+2)^(1/2),2^(1/2))-12*c^2*d*x-4*c^3+10*d*x+10*c)/d/(d^3*x^3+3 *c*d^2*x^2+3*c^2*d*x+c^3-d*x-c)
Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.84 \[ \int \frac {(c e+d e x)^{7/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 \, {\left (5 \, \sqrt {-d^{3} e} e^{3} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) + {\left (3 \, d^{4} e^{3} x^{2} + 6 \, c d^{3} e^{3} x + {\left (3 \, c^{2} + 5\right )} d^{2} e^{3}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}\right )}}{21 \, d^{3}} \] Input:
integrate((d*e*x+c*e)^(7/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm="f ricas")
Output:
-2/21*(5*sqrt(-d^3*e)*e^3*weierstrassPInverse(4/d^2, 0, (d*x + c)/d) + (3* d^4*e^3*x^2 + 6*c*d^3*e^3*x + (3*c^2 + 5)*d^2*e^3)*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*sqrt(d*e*x + c*e))/d^3
\[ \int \frac {(c e+d e x)^{7/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {\left (e \left (c + d x\right )\right )^{\frac {7}{2}}}{\sqrt {- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \] Input:
integrate((d*e*x+c*e)**(7/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)
Output:
Integral((e*(c + d*x))**(7/2)/sqrt(-(c + d*x - 1)*(c + d*x + 1)), x)
\[ \int \frac {(c e+d e x)^{7/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{\frac {7}{2}}}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}} \,d x } \] Input:
integrate((d*e*x+c*e)^(7/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm="m axima")
Output:
integrate((d*e*x + c*e)^(7/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1), x)
\[ \int \frac {(c e+d e x)^{7/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{\frac {7}{2}}}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}} \,d x } \] Input:
integrate((d*e*x+c*e)^(7/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm="g iac")
Output:
integrate((d*e*x + c*e)^(7/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1), x)
Timed out. \[ \int \frac {(c e+d e x)^{7/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^{7/2}}{\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}} \,d x \] Input:
int((c*e + d*e*x)^(7/2)/(1 - d^2*x^2 - 2*c*d*x - c^2)^(1/2),x)
Output:
int((c*e + d*e*x)^(7/2)/(1 - d^2*x^2 - 2*c*d*x - c^2)^(1/2), x)
\[ \int \frac {(c e+d e x)^{7/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\frac {\sqrt {e}\, e^{3} \left (-6 \sqrt {d x +c}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, c^{4}-12 \sqrt {d x +c}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, c^{3} d x -6 \sqrt {d x +c}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, c^{2} d^{2} x^{2}-8 \sqrt {d x +c}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, c^{2}+4 \sqrt {d x +c}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, c d x +2 \sqrt {d x +c}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, d^{2} x^{2}+15 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, x^{2}}{3 c^{2} d^{3} x^{3}+9 c^{3} d^{2} x^{2}+9 c^{4} d x -d^{3} x^{3}+3 c^{5}-3 c \,d^{2} x^{2}-6 c^{2} d x -4 c^{3}+d x +c}d x \right ) c^{2} d^{3}-5 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, x^{2}}{3 c^{2} d^{3} x^{3}+9 c^{3} d^{2} x^{2}+9 c^{4} d x -d^{3} x^{3}+3 c^{5}-3 c \,d^{2} x^{2}-6 c^{2} d x -4 c^{3}+d x +c}d x \right ) d^{3}+30 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, x}{3 c^{2} d^{3} x^{3}+9 c^{3} d^{2} x^{2}+9 c^{4} d x -d^{3} x^{3}+3 c^{5}-3 c \,d^{2} x^{2}-6 c^{2} d x -4 c^{3}+d x +c}d x \right ) c^{3} d^{2}-10 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, x}{3 c^{2} d^{3} x^{3}+9 c^{3} d^{2} x^{2}+9 c^{4} d x -d^{3} x^{3}+3 c^{5}-3 c \,d^{2} x^{2}-6 c^{2} d x -4 c^{3}+d x +c}d x \right ) c \,d^{2}\right )}{7 d \left (3 c^{2}-1\right )} \] Input:
int((d*e*x+c*e)^(7/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x)
Output:
(sqrt(e)*e**3*( - 6*sqrt(c + d*x)*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)* c**4 - 12*sqrt(c + d*x)*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*c**3*d*x - 6*sqrt(c + d*x)*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*c**2*d**2*x**2 - 8*sqrt(c + d*x)*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*c**2 + 4*sqrt(c + d*x)*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*c*d*x + 2*sqrt(c + d*x)*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*d**2*x**2 + 15*int((sqrt(c + d*x)*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*x**2)/(3*c**5 + 9*c**4*d*x + 9*c**3*d** 2*x**2 - 4*c**3 + 3*c**2*d**3*x**3 - 6*c**2*d*x - 3*c*d**2*x**2 + c - d**3 *x**3 + d*x),x)*c**2*d**3 - 5*int((sqrt(c + d*x)*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*x**2)/(3*c**5 + 9*c**4*d*x + 9*c**3*d**2*x**2 - 4*c**3 + 3* c**2*d**3*x**3 - 6*c**2*d*x - 3*c*d**2*x**2 + c - d**3*x**3 + d*x),x)*d**3 + 30*int((sqrt(c + d*x)*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*x)/(3*c** 5 + 9*c**4*d*x + 9*c**3*d**2*x**2 - 4*c**3 + 3*c**2*d**3*x**3 - 6*c**2*d*x - 3*c*d**2*x**2 + c - d**3*x**3 + d*x),x)*c**3*d**2 - 10*int((sqrt(c + d* x)*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*x)/(3*c**5 + 9*c**4*d*x + 9*c** 3*d**2*x**2 - 4*c**3 + 3*c**2*d**3*x**3 - 6*c**2*d*x - 3*c*d**2*x**2 + c - d**3*x**3 + d*x),x)*c*d**2))/(7*d*(3*c**2 - 1))