3.2626 \(\int \frac{\sqrt{e+f x}}{\sqrt{a+b x} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=134 \[ \frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt{d} \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}} \]

[Out]

(2*Sqrt[-(b*c) + a*d]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[Ar
cSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f)
)])/(b*Sqrt[d]*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)])

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Rubi [A]  time = 0.35832, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt{d} \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]),x]

[Out]

(2*Sqrt[-(b*c) + a*d]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[Ar
cSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f)
)])/(b*Sqrt[d]*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)])

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Rubi in Sympy [A]  time = 42.8359, size = 114, normalized size = 0.85 \[ \frac{2 \sqrt{\frac{b \left (- c - d x\right )}{a d - b c}} \sqrt{e + f x} \sqrt{a d - b c} E\left (\operatorname{asin}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{a d - b c}} \right )}\middle | \frac{f \left (a d - b c\right )}{d \left (a f - b e\right )}\right )}{b \sqrt{d} \sqrt{\frac{b \left (- e - f x\right )}{a f - b e}} \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(b*(-c - d*x)/(a*d - b*c))*sqrt(e + f*x)*sqrt(a*d - b*c)*elliptic_e(asin(s
qrt(d)*sqrt(a + b*x)/sqrt(a*d - b*c)), f*(a*d - b*c)/(d*(a*f - b*e)))/(b*sqrt(d)
*sqrt(b*(-e - f*x)/(a*f - b*e))*sqrt(c + d*x))

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Mathematica [A]  time = 2.26468, size = 154, normalized size = 1.15 \[ \frac{2 \sqrt{c+d x} \left (\frac{(a f-b e) \sqrt{\frac{b (e+f x)}{f (a+b x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{a-\frac{b e}{f}}}{\sqrt{a+b x}}\right )|\frac{b c f-a d f}{b d e-a d f}\right )}{b \sqrt{a-\frac{b e}{f}} \sqrt{\frac{b (c+d x)}{d (a+b x)}}}+\frac{e+f x}{\sqrt{a+b x}}\right )}{d \sqrt{e+f x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]),x]

[Out]

(2*Sqrt[c + d*x]*((e + f*x)/Sqrt[a + b*x] + ((-(b*e) + a*f)*Sqrt[(b*(e + f*x))/(
f*(a + b*x))]*EllipticE[ArcSin[Sqrt[a - (b*e)/f]/Sqrt[a + b*x]], (b*c*f - a*d*f)
/(b*d*e - a*d*f)])/(b*Sqrt[a - (b*e)/f]*Sqrt[(b*(c + d*x))/(d*(a + b*x))])))/(d*
Sqrt[e + f*x])

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Maple [A]  time = 0.027, size = 209, normalized size = 1.6 \[ -2\,{\frac{ \left ({a}^{2}df-abcf-bead+{b}^{2}ce \right ) \sqrt{dx+c}\sqrt{bx+a}\sqrt{fx+e}}{{b}^{2}d \left ( bdf{x}^{3}+adf{x}^{2}+bcf{x}^{2}+bde{x}^{2}+acfx+adex+bcex+ace \right ) }{\it EllipticE} \left ( \sqrt{{\frac{d \left ( bx+a \right ) }{ad-bc}}},\sqrt{{\frac{ \left ( ad-bc \right ) f}{d \left ( af-be \right ) }}} \right ) \sqrt{-{\frac{ \left ( dx+c \right ) b}{ad-bc}}}\sqrt{-{\frac{ \left ( fx+e \right ) b}{af-be}}}\sqrt{{\frac{d \left ( bx+a \right ) }{ad-bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d
-b*c)*f/d/(a*f-b*e))^(1/2))*(-(d*x+c)*b/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^
(1/2)*(d*(b*x+a)/(a*d-b*c))^(1/2)/d/b^2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*(f*x+e)^(1/2
)/(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(f*x + e)/(sqrt(b*x + a)*sqrt(d*x + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(f*x + e)/(sqrt(b*x + a)*sqrt(d*x + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(f*x + e)/(sqrt(b*x + a)*sqrt(d*x + c)), x)