\(\int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^2 x^2}} \, dx\) [74]

Optimal result

Integrand size = 31, antiderivative size = 74 \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^2 x^2}} \, dx=-\frac {2 \sqrt {\frac {f (c+d x)}{d+c f}} \operatorname {EllipticPi}\left (\frac {2 b}{b+a f},\arcsin \left (\frac {\sqrt {1-f x}}{\sqrt {2}}\right ),\frac {2 d}{d+c f}\right )}{(b+a f) \sqrt {c+d x}} \]

[Out]

-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)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {946, 174, 552, 551} \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^2 x^2}} \, dx=-\frac {2 \sqrt {\frac {f (c+d x)}{c f+d}} \operatorname {EllipticPi}\left (\frac {2 b}{b+a f},\arcsin \left (\frac {\sqrt {1-f x}}{\sqrt {2}}\right ),\frac {2 d}{d+c f}\right )}{(a f+b) \sqrt {c+d x}} \]

[In]

Int[1/((a + b*x)*Sqrt[c + d*x]*Sqrt[1 - f^2*x^2]),x]

[Out]

(-2*Sqrt[(f*(c + d*x))/(d + c*f)]*EllipticPi[(2*b)/(b + a*f), ArcSin[Sqrt[1 - f*x]/Sqrt[2]], (2*d)/(d + c*f)])
/((b + a*f)*Sqrt[c + d*x])

Rule 174

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-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]

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[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] && SimplerSqrtQ[-f/e, -d/c])

Rule 552

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[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]

Rule 946

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-c/
a, 2]}, Dist[1/Sqrt[a], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, c, d
, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f x} \sqrt {1+f x}} \, dx \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (b+a f-b x^2\right ) \sqrt {c+\frac {d}{f}-\frac {d x^2}{f}}} \, dx,x,\sqrt {1-f x}\right )\right ) \\ & = -\frac {\left (2 \sqrt {\frac {f (c+d x)}{d+c f}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (b+a f-b x^2\right ) \sqrt {1-\frac {d x^2}{\left (c+\frac {d}{f}\right ) f}}} \, dx,x,\sqrt {1-f x}\right )}{\sqrt {c+d x}} \\ & = -\frac {2 \sqrt {\frac {f (c+d x)}{d+c f}} \Pi \left (\frac {2 b}{b+a f};\sin ^{-1}\left (\frac {\sqrt {1-f x}}{\sqrt {2}}\right )|\frac {2 d}{d+c f}\right )}{(b+a f) \sqrt {c+d x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.05 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.74 \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^2 x^2}} \, dx=\frac {2 i (c+d x) \sqrt {\frac {d (-1+f x)}{f (c+d x)}} \sqrt {\frac {d+d f x}{c f+d f x}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {d+c f}{f}}}{\sqrt {c+d x}}\right ),\frac {-d+c f}{d+c f}\right )-\operatorname {EllipticPi}\left (\frac {b c f-a d f}{b d+b c f},i \text {arcsinh}\left (\frac {\sqrt {-\frac {d+c f}{f}}}{\sqrt {c+d x}}\right ),\frac {-d+c f}{d+c f}\right )\right )}{(-b c+a d) \sqrt {-\frac {d+c f}{f}} \sqrt {1-f^2 x^2}} \]

[In]

Integrate[1/((a + b*x)*Sqrt[c + d*x]*Sqrt[1 - f^2*x^2]),x]

[Out]

((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[Sq
rt[-((d + c*f)/f)]/Sqrt[c + d*x]], (-d + c*f)/(d + c*f)]))/((-(b*c) + a*d)*Sqrt[-((d + c*f)/f)]*Sqrt[1 - f^2*x
^2])

Maple [B] (verified)

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	\(\frac {2 \sqrt {-\left (f^{2} x^{2}-1\right ) \left (d x +c \right )}\, \left (\frac {c}{d}-\frac {1}{f}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {1}{f}}}\, \sqrt {\frac {x -\frac {1}{f}}{-\frac {c}{d}-\frac {1}{f}}}\, \sqrt {\frac {x +\frac {1}{f}}{-\frac {c}{d}+\frac {1}{f}}}\, \Pi \left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {1}{f}}}, \frac {-\frac {c}{d}+\frac {1}{f}}{-\frac {c}{d}+\frac {a}{b}}, \sqrt {\frac {-\frac {c}{d}+\frac {1}{f}}{-\frac {c}{d}-\frac {1}{f}}}\right )}{\sqrt {-f^{2} x^{2}+1}\, \sqrt {d x +c}\, b \sqrt {-d \,f^{2} x^{3}-c \,f^{2} x^{2}+d x +c}\, \left (-\frac {c}{d}+\frac {a}{b}\right )}\)	\(236\)

[In]

int(1/(b*x+a)/(d*x+c)^(1/2)/(-f^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-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)

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^2 x^2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(b*x+a)/(d*x+c)^(1/2)/(-f^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^2 x^2}} \, dx=\int \frac {1}{\sqrt {- \left (f x - 1\right ) \left (f x + 1\right )} \left (a + b x\right ) \sqrt {c + d x}}\, dx \]

[In]

integrate(1/(b*x+a)/(d*x+c)**(1/2)/(-f**2*x**2+1)**(1/2),x)

[Out]

Integral(1/(sqrt(-(f*x - 1)*(f*x + 1))*(a + b*x)*sqrt(c + d*x)), x)

Maxima [F]

\[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-f^{2} x^{2} + 1} {\left (b x + a\right )} \sqrt {d x + c}} \,d x } \]

[In]

integrate(1/(b*x+a)/(d*x+c)^(1/2)/(-f^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-f^2*x^2 + 1)*(b*x + a)*sqrt(d*x + c)), x)

Giac [F]

\[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-f^{2} x^{2} + 1} {\left (b x + a\right )} \sqrt {d x + c}} \,d x } \]

[In]

integrate(1/(b*x+a)/(d*x+c)^(1/2)/(-f^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-f^2*x^2 + 1)*(b*x + a)*sqrt(d*x + c)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^2 x^2}} \, dx=\int \frac {1}{\sqrt {1-f^2\,x^2}\,\left (a+b\,x\right )\,\sqrt {c+d\,x}} \,d x \]

[In]

int(1/((1 - f^2*x^2)^(1/2)*(a + b*x)*(c + d*x)^(1/2)),x)

[Out]

int(1/((1 - f^2*x^2)^(1/2)*(a + b*x)*(c + d*x)^(1/2)), x)