\(\int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+d x}} \, dx\) [864]

Optimal result

Integrand size = 27, antiderivative size = 38 \[ \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+d x}} \, dx=\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right ),-\frac {d}{c}\right )}{\sqrt {b} \sqrt {c}} \]

[Out]

2*EllipticF(c^(1/2)*(b*x)^(1/2)/b^(1/2),(-d/c)^(1/2))/b^(1/2)/c^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {117} \[ \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+d x}} \, dx=\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right ),-\frac {d}{c}\right )}{\sqrt {b} \sqrt {c}} \]

[In]

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

[Out]

(2*EllipticF[ArcSin[(Sqrt[c]*Sqrt[b*x])/Sqrt[b]], -(d/c)])/(Sqrt[b]*Sqrt[c])

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2/(b*Sqrt[e]))*Rt
[-b/d, 2]*EllipticF[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] && GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])

Rubi steps \begin{align*} \text {integral}& = \frac {2 F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )|-\frac {d}{c}\right )}{\sqrt {b} \sqrt {c}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(38)=76\).

Time = 3.37 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.34 \[ \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+d x}} \, dx=-\frac {2 \sqrt {\frac {c-\frac {1}{x}}{c}} \sqrt {\frac {d+\frac {1}{x}}{d}} x^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {1}{c}}}{\sqrt {x}}\right ),-\frac {c}{d}\right )}{\sqrt {\frac {1}{c}} \sqrt {b x} \sqrt {1-c x} \sqrt {1+d x}} \]

[In]

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

[Out]

(-2*Sqrt[(c - x^(-1))/c]*Sqrt[(d + x^(-1))/d]*x^(3/2)*EllipticF[ArcSin[Sqrt[c^(-1)]/Sqrt[x]], -(c/d)])/(Sqrt[c
^(-1)]*Sqrt[b*x]*Sqrt[1 - c*x]*Sqrt[1 + d*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(29)=58\).

Time = 1.82 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.68

method	result	size
default	\(-\frac {2 \sqrt {-c x +1}\, \sqrt {-\frac {\left (c x -1\right ) d}{c +d}}\, \sqrt {-d x}\, F\left (\sqrt {d x +1}, \sqrt {\frac {c}{c +d}}\right )}{\sqrt {b x}\, \left (c x -1\right ) d}\)	\(64\)
elliptic	\(\frac {2 \sqrt {-b x \left (c x -1\right ) \left (d x +1\right )}\, \sqrt {\left (x +\frac {1}{d}\right ) d}\, \sqrt {\frac {x -\frac {1}{c}}{-\frac {1}{d}-\frac {1}{c}}}\, \sqrt {-d x}\, F\left (\sqrt {\left (x +\frac {1}{d}\right ) d}, \sqrt {-\frac {1}{d \left (-\frac {1}{d}-\frac {1}{c}\right )}}\right )}{\sqrt {b x}\, \sqrt {-c x +1}\, \sqrt {d x +1}\, d \sqrt {-b c d \,x^{3}-c b \,x^{2}+b d \,x^{2}+b x}}\)	\(137\)

[In]

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

[Out]

-2*(-c*x+1)^(1/2)*(-(c*x-1)*d/(c+d))^(1/2)*(-d*x)^(1/2)*EllipticF((d*x+1)^(1/2),(c/(c+d))^(1/2))/(b*x)^(1/2)/(
c*x-1)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.26 \[ \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+d x}} \, dx=-\frac {2 \, \sqrt {-b c d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} + c d + d^{2}\right )}}{3 \, c^{2} d^{2}}, -\frac {4 \, {\left (2 \, c^{3} + 3 \, c^{2} d - 3 \, c d^{2} - 2 \, d^{3}\right )}}{27 \, c^{3} d^{3}}, \frac {3 \, c d x + c - d}{3 \, c d}\right )}{b c d} \]

[In]

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

[Out]

-2*sqrt(-b*c*d)*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))/(b*c*d)

Sympy [F]

\[ \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+d x}} \, dx=\int \frac {1}{\sqrt {b x} \sqrt {- c x + 1} \sqrt {d x + 1}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+d x}} \, dx=\int { \frac {1}{\sqrt {b x} \sqrt {-c x + 1} \sqrt {d x + 1}} \,d x } \]

[In]

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

[Out]

integrate(1/(sqrt(b*x)*sqrt(-c*x + 1)*sqrt(d*x + 1)), x)

Giac [F]

\[ \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+d x}} \, dx=\int { \frac {1}{\sqrt {b x} \sqrt {-c x + 1} \sqrt {d x + 1}} \,d x } \]

[In]

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

[Out]

integrate(1/(sqrt(b*x)*sqrt(-c*x + 1)*sqrt(d*x + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+d x}} \, dx=\int \frac {1}{\sqrt {b\,x}\,\sqrt {1-c\,x}\,\sqrt {d\,x+1}} \,d x \]

[In]

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

[Out]

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