Integrand size = 27, antiderivative size = 86 \[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx=-\frac {2 \sqrt {c} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )|-\frac {d}{c}\right )}{\sqrt {b} d}+\frac {2 (c+d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right ),-\frac {d}{c}\right )}{\sqrt {b} \sqrt {c} d} \] Output:
-2*c^(1/2)*EllipticE(c^(1/2)*(b*x)^(1/2)/b^(1/2),(-d/c)^(1/2))/b^(1/2)/d+2 *(c+d)*EllipticF(c^(1/2)*(b*x)^(1/2)/b^(1/2),(-d/c)^(1/2))/b^(1/2)/c^(1/2) /d
Time = 3.75 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx=\frac {-\frac {2 \sqrt {\frac {1}{c}} (-1+c x) (1+d x)}{d}-2 \sqrt {1-\frac {1}{c x}} \sqrt {1+\frac {1}{d x}} x^{3/2} E\left (\arcsin \left (\frac {\sqrt {\frac {1}{c}}}{\sqrt {x}}\right )|-\frac {c}{d}\right )}{\sqrt {\frac {1}{c}} \sqrt {b x} \sqrt {1-c x} \sqrt {1+d x}} \] Input:
Integrate[Sqrt[1 - c*x]/(Sqrt[b*x]*Sqrt[1 + d*x]),x]
Output:
((-2*Sqrt[c^(-1)]*(-1 + c*x)*(1 + d*x))/d - 2*Sqrt[1 - 1/(c*x)]*Sqrt[1 + 1 /(d*x)]*x^(3/2)*EllipticE[ArcSin[Sqrt[c^(-1)]/Sqrt[x]], -(c/d)])/(Sqrt[c^( -1)]*Sqrt[b*x]*Sqrt[1 - c*x]*Sqrt[1 + d*x])
Time = 0.16 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.49, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {120}
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.
\(\displaystyle \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {d x+1}} \, dx\) |
\(\Big \downarrow \) 120 |
\(\displaystyle -\frac {2 E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {b x}}{\sqrt {-b}}\right )|-\frac {c}{d}\right )}{\sqrt {-b} \sqrt {d}}\) |
Input:
Int[Sqrt[1 - c*x]/(Sqrt[b*x]*Sqrt[1 + d*x]),x]
Output:
(-2*EllipticE[ArcSin[(Sqrt[d]*Sqrt[b*x])/Sqrt[-b]], -(c/d)])/(Sqrt[-b]*Sqr t[d])
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt Q[e, 0] && !LtQ[-b/d, 0]
Time = 0.64 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.70
method | result | size |
default | \(\frac {2 \left (c +d \right ) \operatorname {EllipticE}\left (\sqrt {x d +1}, \sqrt {\frac {c}{c +d}}\right ) \sqrt {-x d}\, \sqrt {-\frac {\left (c x -1\right ) d}{c +d}}}{\sqrt {-c x +1}\, \sqrt {b x}\, d^{2}}\) | \(60\) |
elliptic | \(\frac {\sqrt {-b x \left (c x -1\right ) \left (x d +1\right )}\, \left (\frac {2 \sqrt {\left (x +\frac {1}{d}\right ) d}\, \sqrt {\frac {x -\frac {1}{c}}{-\frac {1}{d}-\frac {1}{c}}}\, \sqrt {-x d}\, \operatorname {EllipticF}\left (\sqrt {\left (x +\frac {1}{d}\right ) d}, \sqrt {-\frac {1}{d \left (-\frac {1}{d}-\frac {1}{c}\right )}}\right )}{d \sqrt {-b c d \,x^{3}-x^{2} b c +b d \,x^{2}+b x}}-\frac {2 c \sqrt {\left (x +\frac {1}{d}\right ) d}\, \sqrt {\frac {x -\frac {1}{c}}{-\frac {1}{d}-\frac {1}{c}}}\, \sqrt {-x d}\, \left (\left (-\frac {1}{d}-\frac {1}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\left (x +\frac {1}{d}\right ) d}, \sqrt {-\frac {1}{d \left (-\frac {1}{d}-\frac {1}{c}\right )}}\right )+\frac {\operatorname {EllipticF}\left (\sqrt {\left (x +\frac {1}{d}\right ) d}, \sqrt {-\frac {1}{d \left (-\frac {1}{d}-\frac {1}{c}\right )}}\right )}{c}\right )}{d \sqrt {-b c d \,x^{3}-x^{2} b c +b d \,x^{2}+b x}}\right )}{\sqrt {b x}\, \sqrt {-c x +1}\, \sqrt {x d +1}}\) | \(287\) |
Input:
int((-c*x+1)^(1/2)/(b*x)^(1/2)/(d*x+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
2*(c+d)/(-c*x+1)^(1/2)*EllipticE((d*x+1)^(1/2),(c/(c+d))^(1/2))*(-x*d)^(1/ 2)*(-(c*x-1)*d/(c+d))^(1/2)/(b*x)^(1/2)/d^2
Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (68) = 136\).
Time = 0.10 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.58 \[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {-b c d} c d {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} + c d + d^{2}\right )}}{3 \, c^{2} d^{2}}, -\frac {4 \, {\left (2 \, c^{3} + 3 \, c^{2} d - 3 \, c d^{2} - 2 \, d^{3}\right )}}{27 \, c^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} + c d + d^{2}\right )}}{3 \, c^{2} d^{2}}, -\frac {4 \, {\left (2 \, c^{3} + 3 \, c^{2} d - 3 \, c d^{2} - 2 \, d^{3}\right )}}{27 \, c^{3} d^{3}}, \frac {3 \, c d x + c - d}{3 \, c d}\right )\right ) + \sqrt {-b c d} {\left (c + 2 \, d\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} + c d + d^{2}\right )}}{3 \, c^{2} d^{2}}, -\frac {4 \, {\left (2 \, c^{3} + 3 \, c^{2} d - 3 \, c d^{2} - 2 \, d^{3}\right )}}{27 \, c^{3} d^{3}}, \frac {3 \, c d x + c - d}{3 \, c d}\right )\right )}}{3 \, b c d^{2}} \] Input:
integrate((-c*x+1)^(1/2)/(b*x)^(1/2)/(d*x+1)^(1/2),x, algorithm="fricas")
Output:
-2/3*(3*sqrt(-b*c*d)*c*d*weierstrassZeta(4/3*(c^2 + c*d + d^2)/(c^2*d^2), -4/27*(2*c^3 + 3*c^2*d - 3*c*d^2 - 2*d^3)/(c^3*d^3), weierstrassPInverse(4 /3*(c^2 + c*d + d^2)/(c^2*d^2), -4/27*(2*c^3 + 3*c^2*d - 3*c*d^2 - 2*d^3)/ (c^3*d^3), 1/3*(3*c*d*x + c - d)/(c*d))) + sqrt(-b*c*d)*(c + 2*d)*weierstr assPInverse(4/3*(c^2 + c*d + d^2)/(c^2*d^2), -4/27*(2*c^3 + 3*c^2*d - 3*c* d^2 - 2*d^3)/(c^3*d^3), 1/3*(3*c*d*x + c - d)/(c*d)))/(b*c*d^2)
\[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx=\int \frac {\sqrt {- c x + 1}}{\sqrt {b x} \sqrt {d x + 1}}\, dx \] Input:
integrate((-c*x+1)**(1/2)/(b*x)**(1/2)/(d*x+1)**(1/2),x)
Output:
Integral(sqrt(-c*x + 1)/(sqrt(b*x)*sqrt(d*x + 1)), x)
\[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx=\int { \frac {\sqrt {-c x + 1}}{\sqrt {b x} \sqrt {d x + 1}} \,d x } \] Input:
integrate((-c*x+1)^(1/2)/(b*x)^(1/2)/(d*x+1)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-c*x + 1)/(sqrt(b*x)*sqrt(d*x + 1)), x)
\[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx=\int { \frac {\sqrt {-c x + 1}}{\sqrt {b x} \sqrt {d x + 1}} \,d x } \] Input:
integrate((-c*x+1)^(1/2)/(b*x)^(1/2)/(d*x+1)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-c*x + 1)/(sqrt(b*x)*sqrt(d*x + 1)), x)
Timed out. \[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx=\int \frac {\sqrt {1-c\,x}}{\sqrt {b\,x}\,\sqrt {d\,x+1}} \,d x \] Input:
int((1 - c*x)^(1/2)/((b*x)^(1/2)*(d*x + 1)^(1/2)),x)
Output:
int((1 - c*x)^(1/2)/((b*x)^(1/2)*(d*x + 1)^(1/2)), x)
\[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx=\frac {\sqrt {b}\, \left (\int \frac {\sqrt {d x +1}\, \sqrt {-c x +1}}{\sqrt {x}\, d x +\sqrt {x}}d x \right )}{b} \] Input:
int((-c*x+1)^(1/2)/(b*x)^(1/2)/(d*x+1)^(1/2),x)
Output:
(sqrt(b)*int((sqrt(d*x + 1)*sqrt( - c*x + 1))/(sqrt(x)*d*x + sqrt(x)),x))/ b