### 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]*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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Rubi [A]  time = 0.0374351, 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, Rules used = {715, 112, 110} $\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{\sqrt{c} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}$

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

Rule 715

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(Sqrt[x]*Sqrt[b + c*x])/Sqrt[
b*x + c*x^2], Int[(d + e*x)^m/(Sqrt[x]*Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] &
& NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 112

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*x]*Sqrt[
1 + (d*x)/c])/(Sqrt[c + d*x]*Sqrt[1 + (f*x)/e]), Int[Sqrt[1 + (f*x)/e]/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]), x], x] /
; FreeQ[{b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] &&  !(GtQ[c, 0] && GtQ[e, 0])

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

Mathematica [A]  time = 0.563666, 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[ArcSin[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 = 0.262, size = 121, normalized size = 1.3 \begin{align*} -2\,{\frac{\sqrt{ex+d}\sqrt{x \left ( cx+b \right ) }b \left ( be-cd \right ) }{{c}^{2}x \left ( ce{x}^{2}+bxe+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 ) } \end{align*}

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. \begin{align*} \int \frac{\sqrt{e x + d}}{\sqrt{c x^{2} + b x}}\,{d x} \end{align*}

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

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, 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. \begin{align*} \int \frac{\sqrt{d + e x}}{\sqrt{x \left (b + c x\right )}}\, dx \end{align*}

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

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, algorithm="giac")

[Out]

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