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

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 = {110} \[ \frac {2 E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {b x}}{\sqrt {b}}\right )|-\frac {c}{d}\right )}{\sqrt {b} \sqrt {d}} \]

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 110

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Sqrt[e]*Rt[-(b/d)
, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/b, 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]  time = 0.39, size = 102, normalized size = 2.68 \[ \frac {2 \sqrt {1-d x} \left (\frac {\sqrt {x} \sqrt {\frac {1}{c x}+1} E\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {1}{c}}}{\sqrt {x}}\right )|-\frac {c}{d}\right )}{\sqrt {-\frac {1}{c}} \sqrt {1-\frac {1}{d x}}}-c x-1\right )}{d \sqrt {b x} \sqrt {c x+1}} \]

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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fricas [F]  time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b x} \sqrt {c x + 1} \sqrt {-d x + 1}}{b d x^{2} - b x}, x\right ) \]

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]

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

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

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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maple [B]  time = 0.05, size = 129, normalized size = 3.39 \[ -\frac {2 \left (-c \EllipticE \left (\sqrt {c x +1}, \sqrt {\frac {d}{c +d}}\right )+c \EllipticF \left (\sqrt {c x +1}, \sqrt {\frac {d}{c +d}}\right )-d \EllipticE \left (\sqrt {c x +1}, \sqrt {\frac {d}{c +d}}\right )+d \EllipticF \left (\sqrt {c x +1}, \sqrt {\frac {d}{c +d}}\right )\right ) \sqrt {-c x}\, \sqrt {-\frac {\left (d x -1\right ) c}{c +d}}\, \sqrt {-d x +1}}{\left (d x -1\right ) \sqrt {b x}\, c d} \]

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)

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

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

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

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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