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

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

[Out]

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

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Rubi [A]  time = 0.192396, 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 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{\sqrt{c} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.9861, size = 83, normalized size = 0.88 \[ \frac{2 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{\sqrt{c} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Mathematica [A]  time = 1.12414, size = 121, normalized size = 1.29 \[ -\frac{2 \sqrt{x} \left (\frac{b}{x}+c\right ) \sqrt{d+e x} \left (\frac{d \sqrt{\frac{d}{e x}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{d}{e}}}{\sqrt{x}}\right )|\frac{b e}{c d}\right )}{\sqrt{-\frac{d}{e}} \sqrt{\frac{b}{c x}+1} \left (\frac{d}{x}+e\right )}-\sqrt{x}\right )}{c \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(c + b/x)*Sqrt[x]*Sqrt[d + e*x]*(-Sqrt[x] + (d*Sqrt[1 + d/(e*x)]*EllipticE[A
rcSin[Sqrt[-(d/e)]/Sqrt[x]], (b*e)/(c*d)])/(Sqrt[-(d/e)]*Sqrt[1 + b/(c*x)]*(e +
d/x))))/(c*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.022, size = 121, normalized size = 1.3 \[ -2\,{\frac{\sqrt{ex+d}\sqrt{x \left ( cx+b \right ) }b \left ( be-cd \right ) }{{c}^{2}x \left ( ce{x}^{2}+bex+cdx+bd \right ) }\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2
)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*(b*e-c*d)/
c^2/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. \[ \int \frac{\sqrt{e x + d}}{\sqrt{c x^{2} + b x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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