3.9.72 \(\int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx\) [872]

Optimal. Leaf size=42 \[ -\frac {2 E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {b x}}{\sqrt {-b}}\right )|-\frac {c}{d}\right )}{\sqrt {-b} \sqrt {d}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 42, 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 (\text {ArcSin}\left (\frac {\sqrt {d} \sqrt {b x}}{\sqrt {-b}}\right )|-\frac {c}{d}\right )}{\sqrt {-b} \sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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+d x}} \, dx &=-\frac {2 E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {b x}}{\sqrt {-b}}\right )|-\frac {c}{d}\right )}{\sqrt {-b} \sqrt {d}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(112\) vs. \(2(42)=84\).
time = 2.07, size = 112, normalized size = 2.67 \begin {gather*} \frac {-\frac {2 \sqrt {\frac {1}{c}} (-1+c x) (1+d x)}{d}-2 \sqrt {1-\frac {1}{c x}} \sqrt {1+\frac {1}{d x}} x^{3/2} E\left (\sin ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.08, size = 67, normalized size = 1.60

method	result	size
default	\(-\frac {2 \left (c +d \right ) \EllipticE \left (\sqrt {d x +1}, \sqrt {\frac {c}{c +d}}\right ) \sqrt {-d x}\, \sqrt {-\frac {\left (c x -1\right ) d}{c +d}}\, \sqrt {-c x +1}}{\left (c x -1\right ) \sqrt {b x}\, d^{2}}\)	\(67\)
elliptic	\(\frac {\sqrt {-b x \left (c x -1\right ) \left (d x +1\right )}\, \left (\frac {2 \sqrt {\left (x +\frac {1}{d}\right ) d}\, \sqrt {\frac {x -\frac {1}{c}}{-\frac {1}{d}-\frac {1}{c}}}\, \sqrt {-d x}\, \EllipticF \left (\sqrt {\left (x +\frac {1}{d}\right ) d}, \sqrt {-\frac {1}{d \left (-\frac {1}{d}-\frac {1}{c}\right )}}\right )}{d \sqrt {-b c d \,x^{3}-b c \,x^{2}+b d \,x^{2}+b x}}-\frac {2 c \sqrt {\left (x +\frac {1}{d}\right ) d}\, \sqrt {\frac {x -\frac {1}{c}}{-\frac {1}{d}-\frac {1}{c}}}\, \sqrt {-d x}\, \left (\left (-\frac {1}{d}-\frac {1}{c}\right ) \EllipticE \left (\sqrt {\left (x +\frac {1}{d}\right ) d}, \sqrt {-\frac {1}{d \left (-\frac {1}{d}-\frac {1}{c}\right )}}\right )+\frac {\EllipticF \left (\sqrt {\left (x +\frac {1}{d}\right ) d}, \sqrt {-\frac {1}{d \left (-\frac {1}{d}-\frac {1}{c}\right )}}\right )}{c}\right )}{d \sqrt {-b c d \,x^{3}-b c \,x^{2}+b d \,x^{2}+b x}}\right )}{\sqrt {b x}\, \sqrt {-c x +1}\, \sqrt {d x +1}}\)	\(287\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(sqrt(-c*x + 1)/(sqrt(b*x)*sqrt(d*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 = 222, normalized size = 5.29 \begin {gather*} -\frac {2 \, {\left (3 \, \sqrt {-b c d} c d {\rm weierstrassZeta}\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}}, {\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 )\right ) + \sqrt {-b c d} {\left (c + 2 \, d\right )} {\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 )\right )}}{3 \, b c d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*sqrt(-b*c*d)*c*d*weierstrassZeta(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), 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))) + sqrt(-b*c*d)*(c + 2*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^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 {d 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)/(d*x+1)**(1/2),x)

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {1-c\,x}}{\sqrt {b\,x}\,\sqrt {d\,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)*(d*x + 1)^(1/2)),x)

[Out]

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

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