Integrand size = 22, antiderivative size = 132 \[ \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=-\frac {2 \sqrt {a} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {b} \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-2*a^(1/2)*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*(1-b*x^2/a)^(1/2) *EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^( 1/2)*c+a^(1/2)*d))^(1/2))/b^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {2 i \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \sqrt {a-b x^2}} \] Input:
Integrate[1/(Sqrt[c + d*x]*Sqrt[a - b*x^2]),x]
Output:
((2*I)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[ b] - d*x)/(c + d*x))]*(c + d*x)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/ Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)]) /(d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*Sqrt[a - b*x^2])
Time = 0.24 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {512, 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.
\(\displaystyle \int \frac {1}{\sqrt {a-b x^2} \sqrt {c+d x}} \, dx\) |
\(\Big \downarrow \) 512 |
\(\displaystyle \frac {\sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}\) |
\(\Big \downarrow \) 511 |
\(\displaystyle -\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle -\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}\) |
Input:
Int[1/(Sqrt[c + d*x]*Sqrt[a - b*x^2]),x]
Output:
(-2*Sqrt[a]*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b* x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((S qrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*Sqrt[c + d*x]*Sqrt[a - b*x^2])
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[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[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Time = 1.59 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.47
method | result | size |
default | \(\frac {2 \left (b c -d \sqrt {a b}\right ) \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}{b d \left (-b d \,x^{3}-b c \,x^{2}+a d x +a c \right )}\) | \(194\) |
elliptic | \(\frac {2 \sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\) | \(237\) |
Input:
int(1/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
2*(b*c-d*(a*b)^(1/2))*EllipticF((-(d*x+c)*b/(d*(a*b)^(1/2)-b*c))^(1/2),(-( d*(a*b)^(1/2)-b*c)/(d*(a*b)^(1/2)+b*c))^(1/2))*((b*x+(a*b)^(1/2))*d/(d*(a* b)^(1/2)-b*c))^(1/2)*((-b*x+(a*b)^(1/2))*d/(d*(a*b)^(1/2)+b*c))^(1/2)*(-(d *x+c)*b/(d*(a*b)^(1/2)-b*c))^(1/2)*(-b*x^2+a)^(1/2)*(d*x+c)^(1/2)/b/d/(-b* d*x^3-b*c*x^2+a*d*x+a*c)
Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.51 \[ \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=-\frac {2 \, \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )}{b d} \] Input:
integrate(1/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
-2*sqrt(-b*d)*weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b* c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d)/(b*d)
\[ \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {1}{\sqrt {a - b x^{2}} \sqrt {c + d x}}\, dx \] Input:
integrate(1/(d*x+c)**(1/2)/(-b*x**2+a)**(1/2),x)
Output:
Integral(1/(sqrt(a - b*x**2)*sqrt(c + d*x)), x)
\[ \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {1}{\sqrt {-b x^{2} + a} \sqrt {d x + c}} \,d x } \] Input:
integrate(1/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(-b*x^2 + a)*sqrt(d*x + c)), x)
\[ \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {1}{\sqrt {-b x^{2} + a} \sqrt {d x + c}} \,d x } \] Input:
integrate(1/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(-b*x^2 + a)*sqrt(d*x + c)), x)
Timed out. \[ \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {1}{\sqrt {a-b\,x^2}\,\sqrt {c+d\,x}} \,d x \] Input:
int(1/((a - b*x^2)^(1/2)*(c + d*x)^(1/2)),x)
Output:
int(1/((a - b*x^2)^(1/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \] Input:
int(1/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x)
Output:
int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x )