3.2643 \(\int \frac{\sqrt{e+f x}}{\sqrt{a+b x} \sqrt{c+d x}} \, dx\)

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[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/S
qrt[-(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 = 0.0494805, 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, Rules used = {114, 113} \[ \frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt{d} \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}} \]

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[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/S
qrt[-(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)]
)

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*
x]*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]), Int[Sqrt[(b*e)/(b*e - a*f
) + (b*f*x)/(b*e - a*f)]/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-((b*c - a*d)/d), 0]

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x
] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-((b*c - a*d)/d),
0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)
/b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{e+f x}}{\sqrt{a+b x} \sqrt{c+d x}} \, dx &=\frac{\left (\sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x}\right ) \int \frac{\sqrt{\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}}}{\sqrt{a+b x} \sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}}} \, dx}{\sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}\\ &=\frac{2 \sqrt{-b c+a d} \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{-b c+a d}}\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}}}\\ \end{align*}

Mathematica [A]  time = 1.31774, 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 = 0.018, size = 209, normalized size = 1.6 \begin{align*} -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 ) }\sqrt{{\frac{d \left ( bx+a \right ) }{ad-bc}}}{\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}}}} \end{align*}

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)*(d*(b*x+a)/(a*d-b*c))^(1/2)*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^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. \begin{align*} \int \frac{\sqrt{f x + e}}{\sqrt{b x + a} \sqrt{d x + c}}\,{d x} \end{align*}

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, 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. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e}}{b d x^{2} + a c +{\left (b c + a d\right )} x}, x\right ) \end{align*}

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, algorithm="fricas")

[Out]

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

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

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

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, algorithm="giac")

[Out]

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