3.9.68 \(\int \frac {\sqrt {1+c x}}{\sqrt {b x} \sqrt {1-d x}} \, dx\) [868]

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

[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 = 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 = {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. \(102\) vs. \(2(38)=76\).
time = 1.99, size = 102, normalized size = 2.68 \begin {gather*} \frac {2 \sqrt {1-d x} \left (-1-c x+\frac {\sqrt {1+\frac {1}{c x}} \sqrt {x} E\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {1}{c}}}{\sqrt {x}}\right )|-\frac {c}{d}\right )}{\sqrt {-\frac {1}{c}} \sqrt {1-\frac {1}{d x}}}\right )}{d \sqrt {b x} \sqrt {1+c x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(128\) vs. \(2(29)=58\).
time = 0.09, size = 129, normalized size = 3.39

method	result	size
default	\(-\frac {2 \left (\EllipticF \left (\sqrt {c x +1}, \sqrt {\frac {d}{c +d}}\right ) c +\EllipticF \left (\sqrt {c x +1}, \sqrt {\frac {d}{c +d}}\right ) d -\EllipticE \left (\sqrt {c x +1}, \sqrt {\frac {d}{c +d}}\right ) c -\EllipticE \left (\sqrt {c x +1}, \sqrt {\frac {d}{c +d}}\right ) d \right ) \sqrt {-c x}\, \sqrt {-\frac {\left (d x -1\right ) c}{c +d}}\, \sqrt {-d x +1}}{\left (d x -1\right ) \sqrt {b x}\, c d}\)	\(129\)
elliptic	\(\frac {\sqrt {-b x \left (d x -1\right ) \left (c x +1\right )}\, \left (\frac {2 \sqrt {c \left (x +\frac {1}{c}\right )}\, \sqrt {\frac {x -\frac {1}{d}}{-\frac {1}{d}-\frac {1}{c}}}\, \sqrt {-c x}\, \EllipticF \left (\sqrt {c \left (x +\frac {1}{c}\right )}, \sqrt {-\frac {1}{c \left (-\frac {1}{d}-\frac {1}{c}\right )}}\right )}{c \sqrt {-b c d \,x^{3}+b c \,x^{2}-b d \,x^{2}+b x}}+\frac {2 \sqrt {c \left (x +\frac {1}{c}\right )}\, \sqrt {\frac {x -\frac {1}{d}}{-\frac {1}{d}-\frac {1}{c}}}\, \sqrt {-c x}\, \left (\left (-\frac {1}{d}-\frac {1}{c}\right ) \EllipticE \left (\sqrt {c \left (x +\frac {1}{c}\right )}, \sqrt {-\frac {1}{c \left (-\frac {1}{d}-\frac {1}{c}\right )}}\right )+\frac {\EllipticF \left (\sqrt {c \left (x +\frac {1}{c}\right )}, \sqrt {-\frac {1}{c \left (-\frac {1}{d}-\frac {1}{c}\right )}}\right )}{d}\right )}{\sqrt {-b c d \,x^{3}+b c \,x^{2}-b d \,x^{2}+b x}}\right )}{\sqrt {b x}\, \sqrt {c x +1}\, \sqrt {-d x +1}}\)	\(283\)

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

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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 = 0.14, size = 223, normalized size = 5.87 \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 = -0.03 \begin {gather*} \int \frac {\sqrt {c\,x+1}}{\sqrt {b\,x}\,\sqrt {1-d\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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