Integrand size = 33, antiderivative size = 157 \[ \int \frac {(e x)^{3/2}}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {(e x)^{3/2}}{2 c d \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}+\frac {3 e \sqrt {e x} \sqrt {c+d x}}{4 c d^2 \sqrt {c^2-d^2 x^2}}-\frac {3 e^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {e x} \sqrt {c+d x}}{\sqrt {e} \sqrt {c^2-d^2 x^2}}\right )}{4 \sqrt {2} c d^{5/2}} \] Output:
-1/2*(e*x)^(3/2)/c/d/(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(1/2)+3/4*e*(e*x)^(1/2)* (d*x+c)^(1/2)/c/d^2/(-d^2*x^2+c^2)^(1/2)-3/8*e^(3/2)*arctan(1/e^(1/2)/(-d^ 2*x^2+c^2)^(1/2)*2^(1/2)*d^(1/2)*(e*x)^(1/2)*(d*x+c)^(1/2))*2^(1/2)/c/d^(5 /2)
Time = 5.31 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.75 \[ \int \frac {(e x)^{3/2}}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {(e x)^{3/2} \left (2 \sqrt {d} \sqrt {x} (3 c+d x)-3 \sqrt {2} \sqrt {c-d x} (c+d x) \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {x}}{\sqrt {c-d x}}\right )\right )}{8 c d^{5/2} x^{3/2} \sqrt {c+d x} \sqrt {c^2-d^2 x^2}} \] Input:
Integrate[(e*x)^(3/2)/(Sqrt[c + d*x]*(c^2 - d^2*x^2)^(3/2)),x]
Output:
((e*x)^(3/2)*(2*Sqrt[d]*Sqrt[x]*(3*c + d*x) - 3*Sqrt[2]*Sqrt[c - d*x]*(c + d*x)*ArcTan[(Sqrt[2]*Sqrt[d]*Sqrt[x])/Sqrt[c - d*x]]))/(8*c*d^(5/2)*x^(3/ 2)*Sqrt[c + d*x]*Sqrt[c^2 - d^2*x^2])
Time = 0.43 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {586, 105, 105, 104, 218}
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.
| \(\displaystyle \int \frac {(e x)^{3/2}}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 586 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \int \frac {(e x)^{3/2}}{(c-d x)^{3/2} (c+d x)^2}dx}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {(e x)^{3/2}}{c d \sqrt {c-d x} (c+d x)}-\frac {3 e \int \frac {\sqrt {e x}}{\sqrt {c-d x} (c+d x)^2}dx}{2 d}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {(e x)^{3/2}}{c d \sqrt {c-d x} (c+d x)}-\frac {3 e \left (\frac {e \int \frac {1}{\sqrt {e x} \sqrt {c-d x} (c+d x)}dx}{4 d}-\frac {\sqrt {e x} \sqrt {c-d x}}{2 c d (c+d x)}\right )}{2 d}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {(e x)^{3/2}}{c d \sqrt {c-d x} (c+d x)}-\frac {3 e \left (\frac {e \int \frac {1}{c e+\frac {2 c d x e}{c-d x}}d\frac {\sqrt {e x}}{\sqrt {c-d x}}}{2 d}-\frac {\sqrt {e x} \sqrt {c-d x}}{2 c d (c+d x)}\right )}{2 d}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {(e x)^{3/2}}{c d \sqrt {c-d x} (c+d x)}-\frac {3 e \left (\frac {\sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c-d x}}\right )}{2 \sqrt {2} c d^{3/2}}-\frac {\sqrt {e x} \sqrt {c-d x}}{2 c d (c+d x)}\right )}{2 d}\right )}{\sqrt {c^2-d^2 x^2}}\) |
Input:
Int[(e*x)^(3/2)/(Sqrt[c + d*x]*(c^2 - d^2*x^2)^(3/2)),x]
Output:
(Sqrt[c - d*x]*Sqrt[c + d*x]*((e*x)^(3/2)/(c*d*Sqrt[c - d*x]*(c + d*x)) - (3*e*(-1/2*(Sqrt[e*x]*Sqrt[c - d*x])/(c*d*(c + d*x)) + (Sqrt[e]*ArcTan[(Sq rt[2]*Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c - d*x])])/(2*Sqrt[2]*c*d^(3/2)))) /(2*d)))/Sqrt[c^2 - d^2*x^2]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_) , x_Symbol] :> Simp[(a + b*x^2)^FracPart[p]/((c + d*x)^FracPart[p]*(a/c + ( b*x)/d)^FracPart[p]) Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(245\) vs. \(2(121)=242\).
Time = 0.29 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.57
| method | result | size |
| default | \(\frac {\sqrt {2}\, \left (-3 \ln \left (\frac {3 c d x e +2 \sqrt {2}\, \sqrt {-\frac {e \,c^{2}}{d}}\, \sqrt {\left (-d x +c \right ) e x}\, d -e \,c^{2}}{d x +c}\right ) c \,d^{2} e \,x^{2}+2 x \sqrt {\left (-d x +c \right ) e x}\, \sqrt {2}\, \sqrt {-\frac {e \,c^{2}}{d}}\, d^{2}+6 \sqrt {\left (-d x +c \right ) e x}\, \sqrt {2}\, \sqrt {-\frac {e \,c^{2}}{d}}\, c d +3 \ln \left (\frac {3 c d x e +2 \sqrt {2}\, \sqrt {-\frac {e \,c^{2}}{d}}\, \sqrt {\left (-d x +c \right ) e x}\, d -e \,c^{2}}{d x +c}\right ) c^{3} e \right ) e \sqrt {-d^{2} x^{2}+c^{2}}\, \sqrt {e x}}{16 c \sqrt {-\frac {e \,c^{2}}{d}}\, \left (-d x +c \right ) \sqrt {\left (-d x +c \right ) e x}\, d^{3} \left (d x +c \right )^{\frac {3}{2}}}\) | \(246\) |
Input:
int((e*x)^(3/2)/(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(3/2),x,method=_RETURNVERBOSE )
Output:
1/16*2^(1/2)*(-3*ln((3*c*d*x*e+2*2^(1/2)*(-1/d*e*c^2)^(1/2)*((-d*x+c)*e*x) ^(1/2)*d-e*c^2)/(d*x+c))*c*d^2*e*x^2+2*x*((-d*x+c)*e*x)^(1/2)*2^(1/2)*(-1/ d*e*c^2)^(1/2)*d^2+6*((-d*x+c)*e*x)^(1/2)*2^(1/2)*(-1/d*e*c^2)^(1/2)*c*d+3 *ln((3*c*d*x*e+2*2^(1/2)*(-1/d*e*c^2)^(1/2)*((-d*x+c)*e*x)^(1/2)*d-e*c^2)/ (d*x+c))*c^3*e)/c*e*(-d^2*x^2+c^2)^(1/2)*(e*x)^(1/2)/(-1/d*e*c^2)^(1/2)/(- d*x+c)/((-d*x+c)*e*x)^(1/2)/d^3/(d*x+c)^(3/2)
Time = 0.15 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.71 \[ \int \frac {(e x)^{3/2}}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{2}} {\left (d^{3} e x^{3} + c d^{2} e x^{2} - c^{2} d e x - c^{3} e\right )} \sqrt {-\frac {e}{d}} \log \left (-\frac {17 \, d^{3} e x^{3} + 3 \, c d^{2} e x^{2} - 13 \, c^{2} d e x + c^{3} e - 8 \, \sqrt {\frac {1}{2}} \sqrt {-d^{2} x^{2} + c^{2}} {\left (3 \, d^{2} x - c d\right )} \sqrt {d x + c} \sqrt {e x} \sqrt {-\frac {e}{d}}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 4 \, \sqrt {-d^{2} x^{2} + c^{2}} {\left (d e x + 3 \, c e\right )} \sqrt {d x + c} \sqrt {e x}}{16 \, {\left (c d^{5} x^{3} + c^{2} d^{4} x^{2} - c^{3} d^{3} x - c^{4} d^{2}\right )}}, \frac {3 \, \sqrt {\frac {1}{2}} {\left (d^{3} e x^{3} + c d^{2} e x^{2} - c^{2} d e x - c^{3} e\right )} \sqrt {\frac {e}{d}} \arctan \left (\frac {4 \, \sqrt {\frac {1}{2}} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {e x} d \sqrt {\frac {e}{d}}}{3 \, d^{2} e x^{2} + 2 \, c d e x - c^{2} e}\right ) - 2 \, \sqrt {-d^{2} x^{2} + c^{2}} {\left (d e x + 3 \, c e\right )} \sqrt {d x + c} \sqrt {e x}}{8 \, {\left (c d^{5} x^{3} + c^{2} d^{4} x^{2} - c^{3} d^{3} x - c^{4} d^{2}\right )}}\right ] \] Input:
integrate((e*x)^(3/2)/(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(3/2),x, algorithm="fri cas")
Output:
[1/16*(3*sqrt(1/2)*(d^3*e*x^3 + c*d^2*e*x^2 - c^2*d*e*x - c^3*e)*sqrt(-e/d )*log(-(17*d^3*e*x^3 + 3*c*d^2*e*x^2 - 13*c^2*d*e*x + c^3*e - 8*sqrt(1/2)* sqrt(-d^2*x^2 + c^2)*(3*d^2*x - c*d)*sqrt(d*x + c)*sqrt(e*x)*sqrt(-e/d))/( d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) - 4*sqrt(-d^2*x^2 + c^2)*(d*e*x + 3*c*e)*sqrt(d*x + c)*sqrt(e*x))/(c*d^5*x^3 + c^2*d^4*x^2 - c^3*d^3*x - c ^4*d^2), 1/8*(3*sqrt(1/2)*(d^3*e*x^3 + c*d^2*e*x^2 - c^2*d*e*x - c^3*e)*sq rt(e/d)*arctan(4*sqrt(1/2)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c)*sqrt(e*x)*d* sqrt(e/d)/(3*d^2*e*x^2 + 2*c*d*e*x - c^2*e)) - 2*sqrt(-d^2*x^2 + c^2)*(d*e *x + 3*c*e)*sqrt(d*x + c)*sqrt(e*x))/(c*d^5*x^3 + c^2*d^4*x^2 - c^3*d^3*x - c^4*d^2)]
\[ \int \frac {(e x)^{3/2}}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (e x\right )^{\frac {3}{2}}}{\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {3}{2}} \sqrt {c + d x}}\, dx \] Input:
integrate((e*x)**(3/2)/(d*x+c)**(1/2)/(-d**2*x**2+c**2)**(3/2),x)
Output:
Integral((e*x)**(3/2)/((-(-c + d*x)*(c + d*x))**(3/2)*sqrt(c + d*x)), x)
\[ \int \frac {(e x)^{3/2}}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {3}{2}}}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} \sqrt {d x + c}} \,d x } \] Input:
integrate((e*x)^(3/2)/(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(3/2),x, algorithm="max ima")
Output:
integrate((e*x)^(3/2)/((-d^2*x^2 + c^2)^(3/2)*sqrt(d*x + c)), x)
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (121) = 242\).
Time = 0.26 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.12 \[ \int \frac {(e x)^{3/2}}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {\frac {8 \, \sqrt {-d e^{2} x + c e^{2}} \sqrt {e x} e^{4}}{{\left (d e^{2} x - c e^{2}\right )} c d^{2} {\left | e \right |}} - \frac {3 \, \sqrt {2} \sqrt {-d e} e^{3} \log \left (\frac {{\left | -4 \, \sqrt {2} e^{2} {\left | c \right |} - 6 \, c e^{2} + 2 \, {\left (\sqrt {-d e} \sqrt {e x} - \sqrt {-d e^{2} x + c e^{2}}\right )}^{2} \right |}}{{\left | 4 \, \sqrt {2} e^{2} {\left | c \right |} - 6 \, c e^{2} + 2 \, {\left (\sqrt {-d e} \sqrt {e x} - \sqrt {-d e^{2} x + c e^{2}}\right )}^{2} \right |}}\right )}{d^{3} {\left | c \right |} {\left | e \right |}} - \frac {8 \, {\left (\sqrt {-d e} c e^{7} - 3 \, \sqrt {-d e} {\left (\sqrt {-d e} \sqrt {e x} - \sqrt {-d e^{2} x + c e^{2}}\right )}^{2} e^{5}\right )}}{{\left (c^{2} e^{4} - 6 \, {\left (\sqrt {-d e} \sqrt {e x} - \sqrt {-d e^{2} x + c e^{2}}\right )}^{2} c e^{2} + {\left (\sqrt {-d e} \sqrt {e x} - \sqrt {-d e^{2} x + c e^{2}}\right )}^{4}\right )} d^{3} {\left | e \right |}}}{16 \, e} \] Input:
integrate((e*x)^(3/2)/(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(3/2),x, algorithm="gia c")
Output:
-1/16*(8*sqrt(-d*e^2*x + c*e^2)*sqrt(e*x)*e^4/((d*e^2*x - c*e^2)*c*d^2*abs (e)) - 3*sqrt(2)*sqrt(-d*e)*e^3*log(abs(-4*sqrt(2)*e^2*abs(c) - 6*c*e^2 + 2*(sqrt(-d*e)*sqrt(e*x) - sqrt(-d*e^2*x + c*e^2))^2)/abs(4*sqrt(2)*e^2*abs (c) - 6*c*e^2 + 2*(sqrt(-d*e)*sqrt(e*x) - sqrt(-d*e^2*x + c*e^2))^2))/(d^3 *abs(c)*abs(e)) - 8*(sqrt(-d*e)*c*e^7 - 3*sqrt(-d*e)*(sqrt(-d*e)*sqrt(e*x) - sqrt(-d*e^2*x + c*e^2))^2*e^5)/((c^2*e^4 - 6*(sqrt(-d*e)*sqrt(e*x) - sq rt(-d*e^2*x + c*e^2))^2*c*e^2 + (sqrt(-d*e)*sqrt(e*x) - sqrt(-d*e^2*x + c* e^2))^4)*d^3*abs(e)))/e
Timed out. \[ \int \frac {(e x)^{3/2}}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{3/2}}{{\left (c^2-d^2\,x^2\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \] Input:
int((e*x)^(3/2)/((c^2 - d^2*x^2)^(3/2)*(c + d*x)^(1/2)),x)
Output:
int((e*x)^(3/2)/((c^2 - d^2*x^2)^(3/2)*(c + d*x)^(1/2)), x)
Time = 0.25 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.68 \[ \int \frac {(e x)^{3/2}}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {e}\, e \left (-3 \sqrt {d}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}-\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) c i -3 \sqrt {d}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}-\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) d i x +3 \sqrt {d}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}+\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) c i +3 \sqrt {d}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}+\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) d i x +3 \sqrt {d}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}-\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) c i +3 \sqrt {d}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}-\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) d i x -3 \sqrt {d}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}+\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) c i -3 \sqrt {d}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}+\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) d i x +12 \sqrt {d}\, \sqrt {-d x +c}\, c i +12 \sqrt {d}\, \sqrt {-d x +c}\, d i x +12 \sqrt {x}\, c d +4 \sqrt {x}\, d^{2} x \right )}{16 \sqrt {-d x +c}\, c \,d^{3} \left (d x +c \right )} \] Input:
int((e*x)^(3/2)/(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(3/2),x)
Output:
(sqrt(e)*e*( - 3*sqrt(d)*sqrt(c - d*x)*sqrt(2)*log((sqrt(c - d*x) - sqrt(c )*sqrt(2) - sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*c*i - 3*sqrt(d)*sqrt(c - d*x)*sqrt(2)*log((sqrt(c - d*x) - sqrt(c)*sqrt(2) - sqrt(c) + sqrt(x)*sqr t(d)*i)/sqrt(c))*d*i*x + 3*sqrt(d)*sqrt(c - d*x)*sqrt(2)*log((sqrt(c - d*x ) - sqrt(c)*sqrt(2) + sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*c*i + 3*sqrt(d )*sqrt(c - d*x)*sqrt(2)*log((sqrt(c - d*x) - sqrt(c)*sqrt(2) + sqrt(c) + s qrt(x)*sqrt(d)*i)/sqrt(c))*d*i*x + 3*sqrt(d)*sqrt(c - d*x)*sqrt(2)*log((sq rt(c - d*x) + sqrt(c)*sqrt(2) - sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*c*i + 3*sqrt(d)*sqrt(c - d*x)*sqrt(2)*log((sqrt(c - d*x) + sqrt(c)*sqrt(2) - s qrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*d*i*x - 3*sqrt(d)*sqrt(c - d*x)*sqrt( 2)*log((sqrt(c - d*x) + sqrt(c)*sqrt(2) + sqrt(c) + sqrt(x)*sqrt(d)*i)/sqr t(c))*c*i - 3*sqrt(d)*sqrt(c - d*x)*sqrt(2)*log((sqrt(c - d*x) + sqrt(c)*s qrt(2) + sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*d*i*x + 12*sqrt(d)*sqrt(c - d*x)*c*i + 12*sqrt(d)*sqrt(c - d*x)*d*i*x + 12*sqrt(x)*c*d + 4*sqrt(x)*d* *2*x))/(16*sqrt(c - d*x)*c*d**3*(c + d*x))