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

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

[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 = 0.0404162, 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, 117, 116} \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{\sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}} \]

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])

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 117

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

Rule 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (PosQ[-(b/d)] || NegQ[-(b/f)])

Rubi steps

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

Mathematica [A]  time = 0.121854, size = 94, normalized size = 1. \[ -\frac{2 x^{3/2} \sqrt{\frac{\frac{b}{x}+c}{c}} \sqrt{\frac{\frac{d}{x}+e}{e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )}{\sqrt{-\frac{b}{c}} \sqrt{x (b+c x)} \sqrt{d+e x}} \]

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

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Maple [A]  time = 0.268, size = 113, normalized size = 1.2 \begin{align*} 2\,{\frac{b\sqrt{ex+d}\sqrt{x \left ( cx+b \right ) }}{cx \left ( ce{x}^{2}+bxe+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}}}} \end{align*}

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. \begin{align*} \int \frac{1}{\sqrt{c x^{2} + b x} \sqrt{e x + d}}\,{d x} \end{align*}

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, 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. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + b x} \sqrt{e x + d}}{c e x^{3} + b d x +{\left (c d + b e\right )} x^{2}}, x\right ) \end{align*}

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, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + b*x)*sqrt(e*x + d)/(c*e*x^3 + b*d*x + (c*d + b*e)*x^2), x)

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

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

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, algorithm="giac")

[Out]

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