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

Optimal. Leaf size=38 \[ \frac {2 \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right ),-\frac {d}{c}\right )}{\sqrt {b} \sqrt {c}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 38, 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 = {116} \[ \frac {2 F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )|-\frac {d}{c}\right )}{\sqrt {b} \sqrt {c}} \]

Antiderivative was successfully verified.

[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 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[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*} \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+d x}} \, dx &=\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*}

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Mathematica [B]  time = 0.13, size = 89, normalized size = 2.34 \[ -\frac {2 x^{3/2} \sqrt {\frac {c-\frac {1}{x}}{c}} \sqrt {\frac {d+\frac {1}{x}}{d}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {\frac {1}{c}}}{\sqrt {x}}\right ),-\frac {c}{d}\right )}{\sqrt {\frac {1}{c}} \sqrt {b x} \sqrt {1-c x} \sqrt {d x+1}} \]

Antiderivative was successfully verified.

[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])

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fricas [F]  time = 1.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b x} \sqrt {-c x + 1} \sqrt {d x + 1}}{b c d x^{3} + {\left (b c - b d\right )} x^{2} - b x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x} \sqrt {-c x + 1} \sqrt {d x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.07, size = 64, normalized size = 1.68 \[ -\frac {2 \sqrt {-c x +1}\, \sqrt {-\frac {\left (c x -1\right ) d}{c +d}}\, \sqrt {-d x}\, \EllipticF \left (\sqrt {d x +1}, \sqrt {\frac {c}{c +d}}\right )}{\sqrt {b x}\, \left (c x -1\right ) d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x} \sqrt {-c x + 1} \sqrt {d x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {b\,x}\,\sqrt {1-c\,x}\,\sqrt {d\,x+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x} \sqrt {- c x + 1} \sqrt {d x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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