3.410 \(\int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=94 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{\sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}} \]

[Out]

(2*Sqrt[-b]*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c
]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.217371, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{\sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[d + e*x]*Sqrt[b*x + c*x^2]),x]

[Out]

(2*Sqrt[-b]*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c
]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 23.6042, size = 83, normalized size = 0.88 \[ \frac{2 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{\sqrt{c} \sqrt{d + e x} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(1/2)/(c*x**2+b*x)**(1/2),x)

[Out]

2*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqrt(1 + e*x/d)*elliptic_f(asin(sqrt(c)*sqrt(
x)/sqrt(-b)), b*e/(c*d))/(sqrt(c)*sqrt(d + e*x)*sqrt(b*x + c*x**2))

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Mathematica [A]  time = 0.193965, size = 94, normalized size = 1. \[ -\frac{2 x^{3/2} \sqrt{\frac{\frac{b}{x}+c}{c}} \sqrt{\frac{\frac{d}{x}+e}{e}} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )}{\sqrt{-\frac{b}{c}} \sqrt{x (b+c x)} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[d + e*x]*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*Sqrt[(c + b/x)/c]*Sqrt[(e + d/x)/e]*x^(3/2)*EllipticF[ArcSin[Sqrt[-(b/c)]/Sq
rt[x]], (c*d)/(b*e)])/(Sqrt[-(b/c)]*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])

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Maple [A]  time = 0.024, size = 113, normalized size = 1.2 \[ 2\,{\frac{b\sqrt{ex+d}\sqrt{x \left ( cx+b \right ) }}{cx \left ( ce{x}^{2}+bex+cdx+bd \right ) }{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{-{\frac{cx}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{{\frac{cx+b}{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2} + b x} \sqrt{e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*x^2 + b*x)*sqrt(e*x + d)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x \left (b + c x\right )} \sqrt{d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(1/2)/(c*x**2+b*x)**(1/2),x)

[Out]

Integral(1/(sqrt(x*(b + c*x))*sqrt(d + e*x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2} + b x} \sqrt{e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*sqrt(e*x + d)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^2 + b*x)*sqrt(e*x + d)), x)