3.9.71 \(\int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+c x}} \, dx\) [871]

Optimal. Leaf size=37 \[ -\frac {2 E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {-b}}\right )\right |-1\right )}{\sqrt {-b} \sqrt {c}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {111} \begin {gather*} -\frac {2 E\left (\left .\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {-b}}\right )\right |-1\right )}{\sqrt {-b} \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.84, size = 52, normalized size = 1.41 \begin {gather*} -\frac {2 x \left (-3 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};c^2 x^2\right )+c x \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};c^2 x^2\right )\right )}{3 \sqrt {b x}} \end {gather*}

Warning: Unable to verify antiderivative.

[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])

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Maple [A]
time = 0.08, size = 33, normalized size = 0.89

method	result	size
default	\(\frac {2 \sqrt {2}\, \sqrt {-c x}\, \EllipticE \left (\sqrt {c x +1}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {b x}}\)	\(33\)
elliptic	\(\frac {\sqrt {-b x \left (c^{2} x^{2}-1\right )}\, \left (\frac {\sqrt {c \left (x +\frac {1}{c}\right )}\, \sqrt {-2 c \left (x -\frac {1}{c}\right )}\, \sqrt {-c x}\, \EllipticF \left (\sqrt {c \left (x +\frac {1}{c}\right )}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {-b \,c^{2} x^{3}+b x}}-\frac {\sqrt {c \left (x +\frac {1}{c}\right )}\, \sqrt {-2 c \left (x -\frac {1}{c}\right )}\, \sqrt {-c x}\, \left (-\frac {2 \EllipticE \left (\sqrt {c \left (x +\frac {1}{c}\right )}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\EllipticF \left (\sqrt {c \left (x +\frac {1}{c}\right )}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {-b \,c^{2} x^{3}+b x}}\right )}{\sqrt {b x}\, \sqrt {-c x +1}\, \sqrt {c x +1}}\)	\(183\)

Verification of antiderivative is not currently implemented for this CAS.

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.13, size = 51, normalized size = 1.38 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)*weierstrassPInve
rse(4/c^2, 0, x))/(b*c^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- c x + 1}}{\sqrt {b x} \sqrt {c x + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {1-c\,x}}{\sqrt {b\,x}\,\sqrt {c\,x+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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