\(\int \frac {\sqrt {1+c x}}{\sqrt {b x} \sqrt {1-c x}} \, dx\) [867]

Optimal result

Integrand size = 27, antiderivative size = 33 \[ \int \frac {\sqrt {1+c x}}{\sqrt {b x} \sqrt {1-c x}} \, dx=\frac {2 E\left (\left .\arcsin \left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {b} \sqrt {c}} \]

[Out]

2*EllipticE(c^(1/2)*(b*x)^(1/2)/b^(1/2),I)/b^(1/2)/c^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, 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 = {111} \[ \int \frac {\sqrt {1+c x}}{\sqrt {b x} \sqrt {1-c x}} \, dx=\frac {2 E\left (\left .\arcsin \left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {b} \sqrt {c}} \]

[In]

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

[Out]

(2*EllipticE[ArcSin[(Sqrt[c]*Sqrt[b*x])/Sqrt[b]], -1])/(Sqrt[b]*Sqrt[c])

Rule 111

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> 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] && NeQ[
d*e - c*f, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !LtQ[-b/d, 0]

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.71 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.58 \[ \int \frac {\sqrt {1+c x}}{\sqrt {b x} \sqrt {1-c x}} \, dx=\frac {2 x \left (3 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},c^2 x^2\right )+c x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},c^2 x^2\right )\right )}{3 \sqrt {b x}} \]

[In]

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

[Out]

(2*x*(3*Hypergeometric2F1[1/4, 1/2, 5/4, c^2*x^2] + c*x*Hypergeometric2F1[1/2, 3/4, 7/4, c^2*x^2]))/(3*Sqrt[b*
x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(23)=46\).

Time = 0.62 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48

method	result	size
default	\(\frac {2 \sqrt {2}\, \sqrt {-c x}\, \left (F\left (\sqrt {c x +1}, \frac {\sqrt {2}}{2}\right )-E\left (\sqrt {c x +1}, \frac {\sqrt {2}}{2}\right )\right )}{c \sqrt {b x}}\)	\(49\)
elliptic	\(\frac {\sqrt {-b x \left (c^{2} x^{2}-1\right )}\, \left (\frac {\sqrt {c \left (x +\frac {1}{c}\right )}\, \sqrt {-2 c \left (x -\frac {1}{c}\right )}\, \sqrt {-c x}\, F\left (\sqrt {c \left (x +\frac {1}{c}\right )}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {-b \,c^{2} x^{3}+b x}}+\frac {\sqrt {c \left (x +\frac {1}{c}\right )}\, \sqrt {-2 c \left (x -\frac {1}{c}\right )}\, \sqrt {-c x}\, \left (-\frac {2 E\left (\sqrt {c \left (x +\frac {1}{c}\right )}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {F\left (\sqrt {c \left (x +\frac {1}{c}\right )}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {-b \,c^{2} x^{3}+b x}}\right )}{\sqrt {b x}\, \sqrt {-c x +1}\, \sqrt {c x +1}}\)	\(182\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.58 \[ \int \frac {\sqrt {1+c x}}{\sqrt {b x} \sqrt {1-c x}} \, dx=\frac {2 \, {\left (\sqrt {-b c^{2}} c {\rm weierstrassZeta}\left (\frac {4}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right )\right ) - \sqrt {-b c^{2}} {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right )\right )}}{b c^{2}} \]

[In]

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

[Out]

2*(sqrt(-b*c^2)*c*weierstrassZeta(4/c^2, 0, weierstrassPInverse(4/c^2, 0, x)) - sqrt(-b*c^2)*weierstrassPInver
se(4/c^2, 0, x))/(b*c^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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