Integrand size = 32, antiderivative size = 91 \[ \int \frac {\sqrt {4-d^2 x^2}}{(2-d x) \sqrt {e+f x}} \, dx=-\frac {4 \sqrt {d e+2 f} \sqrt {\frac {d (e+f x)}{d e+2 f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {2-d x}}{\sqrt {d e+2 f}}\right )|\frac {1}{4} \left (2+\frac {d e}{f}\right )\right )}{d \sqrt {f} \sqrt {e+f x}} \] Output:
-4*(d*e+2*f)^(1/2)*(d*(f*x+e)/(d*e+2*f))^(1/2)*EllipticE(f^(1/2)*(-d*x+2)^ (1/2)/(d*e+2*f)^(1/2),1/2*(2+d*e/f)^(1/2))/d/f^(1/2)/(f*x+e)^(1/2)
Result contains complex when optimal does not.
Time = 11.15 (sec) , antiderivative size = 287, normalized size of antiderivative = 3.15 \[ \int \frac {\sqrt {4-d^2 x^2}}{(2-d x) \sqrt {e+f x}} \, dx=\frac {2 \left (f^2 \sqrt {-e-\frac {2 f}{d}} \left (-4+d^2 x^2\right )-i d (d e+2 f) \sqrt {\frac {f (-2+d x)}{d (e+f x)}} \sqrt {\frac {f (2+d x)}{d (e+f x)}} (e+f x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-e-\frac {2 f}{d}}}{\sqrt {e+f x}}\right )|\frac {d e-2 f}{d e+2 f}\right )+4 i d f \sqrt {\frac {f (-2+d x)}{d (e+f x)}} \sqrt {\frac {f (2+d x)}{d (e+f x)}} (e+f x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-e-\frac {2 f}{d}}}{\sqrt {e+f x}}\right ),\frac {d e-2 f}{d e+2 f}\right )\right )}{d f^2 \sqrt {-e-\frac {2 f}{d}} \sqrt {e+f x} \sqrt {4-d^2 x^2}} \] Input:
Integrate[Sqrt[4 - d^2*x^2]/((2 - d*x)*Sqrt[e + f*x]),x]
Output:
(2*(f^2*Sqrt[-e - (2*f)/d]*(-4 + d^2*x^2) - I*d*(d*e + 2*f)*Sqrt[(f*(-2 + d*x))/(d*(e + f*x))]*Sqrt[(f*(2 + d*x))/(d*(e + f*x))]*(e + f*x)^(3/2)*Ell ipticE[I*ArcSinh[Sqrt[-e - (2*f)/d]/Sqrt[e + f*x]], (d*e - 2*f)/(d*e + 2*f )] + (4*I)*d*f*Sqrt[(f*(-2 + d*x))/(d*(e + f*x))]*Sqrt[(f*(2 + d*x))/(d*(e + f*x))]*(e + f*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-e - (2*f)/d]/Sqrt[e + f*x]], (d*e - 2*f)/(d*e + 2*f)]))/(d*f^2*Sqrt[-e - (2*f)/d]*Sqrt[e + f*x]* Sqrt[4 - d^2*x^2])
Time = 0.31 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.48, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {667, 600, 508, 327, 511, 321}
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.
\(\displaystyle \int \frac {\sqrt {4-d^2 x^2}}{(2-d x) \sqrt {e+f x}} \, dx\) |
\(\Big \downarrow \) 667 |
\(\displaystyle \int \frac {d x+2}{\sqrt {4-d^2 x^2} \sqrt {e+f x}}dx\) |
\(\Big \downarrow \) 600 |
\(\displaystyle \frac {d \int \frac {\sqrt {e+f x}}{\sqrt {4-d^2 x^2}}dx}{f}-\frac {(d e-2 f) \int \frac {1}{\sqrt {e+f x} \sqrt {4-d^2 x^2}}dx}{f}\) |
\(\Big \downarrow \) 508 |
\(\displaystyle -\frac {(d e-2 f) \int \frac {1}{\sqrt {e+f x} \sqrt {4-d^2 x^2}}dx}{f}-\frac {2 \sqrt {e+f x} \int \frac {\sqrt {1-\frac {f (2-d x)}{d e+2 f}}}{\sqrt {\frac {1}{4} (d x-2)+1}}d\left (\frac {1}{2} \sqrt {2-d x}\right )}{f \sqrt {\frac {d (e+f x)}{d e+2 f}}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle -\frac {(d e-2 f) \int \frac {1}{\sqrt {e+f x} \sqrt {4-d^2 x^2}}dx}{f}-\frac {2 \sqrt {e+f x} E\left (\arcsin \left (\frac {1}{2} \sqrt {2-d x}\right )|\frac {4 f}{d e+2 f}\right )}{f \sqrt {\frac {d (e+f x)}{d e+2 f}}}\) |
\(\Big \downarrow \) 511 |
\(\displaystyle \frac {2 (d e-2 f) \sqrt {\frac {d (e+f x)}{d e+2 f}} \int \frac {1}{\sqrt {1-\frac {f (2-d x)}{d e+2 f}} \sqrt {\frac {1}{4} (d x-2)+1}}d\left (\frac {1}{2} \sqrt {2-d x}\right )}{d f \sqrt {e+f x}}-\frac {2 \sqrt {e+f x} E\left (\arcsin \left (\frac {1}{2} \sqrt {2-d x}\right )|\frac {4 f}{d e+2 f}\right )}{f \sqrt {\frac {d (e+f x)}{d e+2 f}}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {2 (d e-2 f) \sqrt {\frac {d (e+f x)}{d e+2 f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {2-d x}\right ),\frac {4 f}{d e+2 f}\right )}{d f \sqrt {e+f x}}-\frac {2 \sqrt {e+f x} E\left (\arcsin \left (\frac {1}{2} \sqrt {2-d x}\right )|\frac {4 f}{d e+2 f}\right )}{f \sqrt {\frac {d (e+f x)}{d e+2 f}}}\) |
Input:
Int[Sqrt[4 - d^2*x^2]/((2 - d*x)*Sqrt[e + f*x]),x]
Output:
(-2*Sqrt[e + f*x]*EllipticE[ArcSin[Sqrt[2 - d*x]/2], (4*f)/(d*e + 2*f)])/( f*Sqrt[(d*(e + f*x))/(d*e + 2*f)]) + (2*(d*e - 2*f)*Sqrt[(d*(e + f*x))/(d* e + 2*f)]*EllipticF[ArcSin[Sqrt[2 - d*x]/2], (4*f)/(d*e + 2*f)])/(d*f*Sqrt [e + f*x])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[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] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
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]
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]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[(((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*( x_)), x_Symbol] :> Int[(a/d + c*(x/e))*(f + g*x)^n*(a + c*x^2)^(p - 1), x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(160\) vs. \(2(76)=152\).
Time = 2.23 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.77
method | result | size |
default | \(\frac {2 \left (d^{2} e^{2}-4 f^{2}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}, \sqrt {\frac {d e +2 f}{d e -2 f}}\right ) \sqrt {-\frac {f \left (d x -2\right )}{d e +2 f}}\, \sqrt {-\frac {\left (d x +2\right ) f}{d e -2 f}}\, \sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}\, \sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+4}}{f^{2} d \left (d^{2} f \,x^{3}+d^{2} e \,x^{2}-4 f x -4 e \right )}\) | \(161\) |
elliptic | \(\frac {\sqrt {-\left (f x +e \right ) \left (d^{2} x^{2}-4\right )}\, \left (\frac {4 \left (\frac {e}{f}+\frac {2}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}\, \sqrt {\frac {x +\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\, \sqrt {\frac {x -\frac {2}{d}}{-\frac {e}{f}-\frac {2}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}, \sqrt {\frac {-\frac {e}{f}-\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+4 f x +4 e}}+\frac {2 d \left (\frac {e}{f}+\frac {2}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}\, \sqrt {\frac {x +\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\, \sqrt {\frac {x -\frac {2}{d}}{-\frac {e}{f}-\frac {2}{d}}}\, \left (\left (-\frac {e}{f}+\frac {2}{d}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}, \sqrt {\frac {-\frac {e}{f}-\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\right )-\frac {2 \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}, \sqrt {\frac {-\frac {e}{f}-\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\right )}{d}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+4 f x +4 e}}\right )}{\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+4}}\) | \(443\) |
Input:
int((-d^2*x^2+4)^(1/2)/(-d*x+2)/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)
Output:
2*(d^2*e^2-4*f^2)*EllipticE((d*(f*x+e)/(d*e+2*f))^(1/2),((d*e+2*f)/(d*e-2* f))^(1/2))*(-f*(d*x-2)/(d*e+2*f))^(1/2)*(-(d*x+2)*f/(d*e-2*f))^(1/2)*(d*(f *x+e)/(d*e+2*f))^(1/2)*(f*x+e)^(1/2)*(-d^2*x^2+4)^(1/2)/f^2/d/(d^2*f*x^3+d ^2*e*x^2-4*f*x-4*e)
Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (76) = 152\).
Time = 0.08 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.10 \[ \int \frac {\sqrt {4-d^2 x^2}}{(2-d x) \sqrt {e+f x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {-d^{2} f} d f {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} e^{2} + 12 \, f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 36 \, e f^{2}\right )}}{27 \, d^{2} f^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 12 \, f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 36 \, e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right )\right ) + \sqrt {-d^{2} f} {\left (d e - 6 \, f\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 12 \, f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 36 \, e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right )\right )}}{3 \, d^{2} f^{2}} \] Input:
integrate((-d^2*x^2+4)^(1/2)/(-d*x+2)/(f*x+e)^(1/2),x, algorithm="fricas")
Output:
2/3*(3*sqrt(-d^2*f)*d*f*weierstrassZeta(4/3*(d^2*e^2 + 12*f^2)/(d^2*f^2), -8/27*(d^2*e^3 - 36*e*f^2)/(d^2*f^3), weierstrassPInverse(4/3*(d^2*e^2 + 1 2*f^2)/(d^2*f^2), -8/27*(d^2*e^3 - 36*e*f^2)/(d^2*f^3), 1/3*(3*f*x + e)/f) ) + sqrt(-d^2*f)*(d*e - 6*f)*weierstrassPInverse(4/3*(d^2*e^2 + 12*f^2)/(d ^2*f^2), -8/27*(d^2*e^3 - 36*e*f^2)/(d^2*f^3), 1/3*(3*f*x + e)/f))/(d^2*f^ 2)
\[ \int \frac {\sqrt {4-d^2 x^2}}{(2-d x) \sqrt {e+f x}} \, dx=- \int \frac {\sqrt {- d^{2} x^{2} + 4}}{d x \sqrt {e + f x} - 2 \sqrt {e + f x}}\, dx \] Input:
integrate((-d**2*x**2+4)**(1/2)/(-d*x+2)/(f*x+e)**(1/2),x)
Output:
-Integral(sqrt(-d**2*x**2 + 4)/(d*x*sqrt(e + f*x) - 2*sqrt(e + f*x)), x)
\[ \int \frac {\sqrt {4-d^2 x^2}}{(2-d x) \sqrt {e+f x}} \, dx=\int { -\frac {\sqrt {-d^{2} x^{2} + 4}}{{\left (d x - 2\right )} \sqrt {f x + e}} \,d x } \] Input:
integrate((-d^2*x^2+4)^(1/2)/(-d*x+2)/(f*x+e)^(1/2),x, algorithm="maxima")
Output:
-integrate(sqrt(-d^2*x^2 + 4)/((d*x - 2)*sqrt(f*x + e)), x)
Exception generated. \[ \int \frac {\sqrt {4-d^2 x^2}}{(2-d x) \sqrt {e+f x}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-d^2*x^2+4)^(1/2)/(-d*x+2)/(f*x+e)^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\sqrt {4-d^2 x^2}}{(2-d x) \sqrt {e+f x}} \, dx=-\int \frac {\sqrt {4-d^2\,x^2}}{\sqrt {e+f\,x}\,\left (d\,x-2\right )} \,d x \] Input:
int(-(4 - d^2*x^2)^(1/2)/((e + f*x)^(1/2)*(d*x - 2)),x)
Output:
-int((4 - d^2*x^2)^(1/2)/((e + f*x)^(1/2)*(d*x - 2)), x)
\[ \int \frac {\sqrt {4-d^2 x^2}}{(2-d x) \sqrt {e+f x}} \, dx=-\left (\int \frac {\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+4}}{d f \,x^{2}+d e x -2 f x -2 e}d x \right ) \] Input:
int((-d^2*x^2+4)^(1/2)/(-d*x+2)/(f*x+e)^(1/2),x)
Output:
- int((sqrt(e + f*x)*sqrt( - d**2*x**2 + 4))/(d*e*x + d*f*x**2 - 2*e - 2* f*x),x)