Integrand size = 31, antiderivative size = 74 \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^2 x^2}} \, dx=-\frac {2 \sqrt {\frac {f (c+d x)}{d+c f}} \operatorname {EllipticPi}\left (\frac {2 b}{b+a f},\arcsin \left (\frac {\sqrt {1-f x}}{\sqrt {2}}\right ),\frac {2 d}{d+c f}\right )}{(b+a f) \sqrt {c+d x}} \]
-2*EllipticPi(1/2*(-f*x+1)^(1/2)*2^(1/2),2*b/(a*f+b),2^(1/2)*(d/(c*f+d))^( 1/2))*(f*(d*x+c)/(c*f+d))^(1/2)/(a*f+b)/(d*x+c)^(1/2)
Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.74 \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^2 x^2}} \, dx=\frac {2 i (c+d x) \sqrt {\frac {d (-1+f x)}{f (c+d x)}} \sqrt {\frac {d+d f x}{c f+d f x}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {d+c f}{f}}}{\sqrt {c+d x}}\right ),\frac {-d+c f}{d+c f}\right )-\operatorname {EllipticPi}\left (\frac {b c f-a d f}{b d+b c f},i \text {arcsinh}\left (\frac {\sqrt {-\frac {d+c f}{f}}}{\sqrt {c+d x}}\right ),\frac {-d+c f}{d+c f}\right )\right )}{(-b c+a d) \sqrt {-\frac {d+c f}{f}} \sqrt {1-f^2 x^2}} \]
((2*I)*(c + d*x)*Sqrt[(d*(-1 + f*x))/(f*(c + d*x))]*Sqrt[(d + d*f*x)/(c*f + d*f*x)]*(EllipticF[I*ArcSinh[Sqrt[-((d + c*f)/f)]/Sqrt[c + d*x]], (-d + c*f)/(d + c*f)] - EllipticPi[(b*c*f - a*d*f)/(b*d + b*c*f), I*ArcSinh[Sqrt [-((d + c*f)/f)]/Sqrt[c + d*x]], (-d + c*f)/(d + c*f)]))/((-(b*c) + a*d)*S qrt[-((d + c*f)/f)]*Sqrt[1 - f^2*x^2])
Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {730, 186, 413, 412}
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.
\(\displaystyle \int \frac {1}{\sqrt {1-f^2 x^2} (a+b x) \sqrt {c+d x}} \, dx\) |
\(\Big \downarrow \) 730 |
\(\displaystyle \int \frac {1}{\sqrt {1-f x} \sqrt {f x+1} (a+b x) \sqrt {c+d x}}dx\) |
\(\Big \downarrow \) 186 |
\(\displaystyle -2 \int \frac {1}{\sqrt {f x+1} (-((1-f x) b)+b+a f) \sqrt {c-\frac {d (1-f x)}{f}+\frac {d}{f}}}d\sqrt {1-f x}\) |
\(\Big \downarrow \) 413 |
\(\displaystyle -\frac {2 \sqrt {1-\frac {d (1-f x)}{c f+d}} \int \frac {1}{\sqrt {f x+1} (-((1-f x) b)+b+a f) \sqrt {1-\frac {d (1-f x)}{d+c f}}}d\sqrt {1-f x}}{\sqrt {c-\frac {d (1-f x)}{f}+\frac {d}{f}}}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle -\frac {2 \sqrt {1-\frac {d (1-f x)}{c f+d}} \operatorname {EllipticPi}\left (\frac {2 b}{b+a f},\arcsin \left (\frac {\sqrt {1-f x}}{\sqrt {2}}\right ),\frac {2 d}{d+c f}\right )}{(a f+b) \sqrt {c-\frac {d (1-f x)}{f}+\frac {d}{f}}}\) |
(-2*Sqrt[1 - (d*(1 - f*x))/(d + c*f)]*EllipticPi[(2*b)/(b + a*f), ArcSin[S qrt[1 - f*x]/Sqrt[2]], (2*d)/(d + c*f)])/((b + a*f)*Sqrt[c + d/f - (d*(1 - f*x))/f])
3.1.74.3.1 Defintions of rubi rules used
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[1/(Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))*Sqrt[(a_) + (b_.)*(x_) ^2]), x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a] Int[1/((e + f*x )*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NegQ[b/a] && GtQ[a, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(180\) vs. \(2(71)=142\).
Time = 2.08 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.45
method | result | size |
default | \(-\frac {2 \left (c f -d \right ) \Pi \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d}}, -\frac {\left (c f -d \right ) b}{f \left (a d -b c \right )}, \sqrt {\frac {c f -d}{c f +d}}\right ) \sqrt {-\frac {\left (f x +1\right ) d}{c f -d}}\, \sqrt {-\frac {\left (f x -1\right ) d}{c f +d}}\, \sqrt {\frac {\left (d x +c \right ) f}{c f -d}}\, \sqrt {-f^{2} x^{2}+1}\, \sqrt {d x +c}}{f \left (a d -b c \right ) \left (d \,f^{2} x^{3}+c \,f^{2} x^{2}-d x -c \right )}\) | \(181\) |
elliptic | \(\frac {2 \sqrt {-\left (f^{2} x^{2}-1\right ) \left (d x +c \right )}\, \left (\frac {c}{d}-\frac {1}{f}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {1}{f}}}\, \sqrt {\frac {x -\frac {1}{f}}{-\frac {c}{d}-\frac {1}{f}}}\, \sqrt {\frac {x +\frac {1}{f}}{-\frac {c}{d}+\frac {1}{f}}}\, \Pi \left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {1}{f}}}, \frac {-\frac {c}{d}+\frac {1}{f}}{-\frac {c}{d}+\frac {a}{b}}, \sqrt {\frac {-\frac {c}{d}+\frac {1}{f}}{-\frac {c}{d}-\frac {1}{f}}}\right )}{\sqrt {-f^{2} x^{2}+1}\, \sqrt {d x +c}\, b \sqrt {-d \,f^{2} x^{3}-c \,f^{2} x^{2}+d x +c}\, \left (-\frac {c}{d}+\frac {a}{b}\right )}\) | \(236\) |
-2*(c*f-d)*EllipticPi(((d*x+c)*f/(c*f-d))^(1/2),-(c*f-d)*b/f/(a*d-b*c),((c *f-d)/(c*f+d))^(1/2))*(-(f*x+1)*d/(c*f-d))^(1/2)*(-(f*x-1)*d/(c*f+d))^(1/2 )*((d*x+c)*f/(c*f-d))^(1/2)*(-f^2*x^2+1)^(1/2)*(d*x+c)^(1/2)/f/(a*d-b*c)/( d*f^2*x^3+c*f^2*x^2-d*x-c)
Timed out. \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^2 x^2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^2 x^2}} \, dx=\int \frac {1}{\sqrt {- \left (f x - 1\right ) \left (f x + 1\right )} \left (a + b x\right ) \sqrt {c + d x}}\, dx \]
\[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-f^{2} x^{2} + 1} {\left (b x + a\right )} \sqrt {d x + c}} \,d x } \]
\[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-f^{2} x^{2} + 1} {\left (b x + a\right )} \sqrt {d x + c}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^2 x^2}} \, dx=\int \frac {1}{\sqrt {1-f^2\,x^2}\,\left (a+b\,x\right )\,\sqrt {c+d\,x}} \,d x \]