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

Optimal. Leaf size=293 \[ \frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right ),\frac{h (d e-c f)}{f (d g-c h)}\right )}{b \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}} \]

[Out]

(2*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticF[ArcSin[(Sqrt[f
]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(b*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h*x]
) - (2*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticPi[-((b*(d*e
- c*f))/((b*c - a*d)*f)), ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))
])/(b*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h*x])

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Rubi [A]  time = 0.504431, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {175, 121, 120, 169, 538, 537} \[ \frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticF[ArcSin[(Sqrt[f
]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(b*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h*x]
) - (2*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticPi[-((b*(d*e
- c*f))/((b*c - a*d)*f)), ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))
])/(b*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h*x])

Rule 175

Int[Sqrt[(c_.) + (d_.)*(x_)]/(((a_.) + (b_.)*(x_))*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbo
l] :> Dist[d/b, Int[1/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] + Dist[(b*c - a*d)/b, Int[1/((a + b*
x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 121

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

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(
b*Sqrt[(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] &
& SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x] && (PosQ[-((b*c - a*d)/d)] || NegQ[-((b*e - a*f)/f)
])

Rule 169

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] &&  !SimplerQ[e
+ f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]

Rule 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rubi steps

\begin{align*} \int \frac{\sqrt{c+d x}}{(a+b x) \sqrt{e+f x} \sqrt{g+h x}} \, dx &=\frac{d \int \frac{1}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{b}+\frac{(b c-a d) \int \frac{1}{(a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{b}\\ &=-\frac{(2 (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{\left (b c-a d-b x^2\right ) \sqrt{e-\frac{c f}{d}+\frac{f x^2}{d}} \sqrt{g-\frac{c h}{d}+\frac{h x^2}{d}}} \, dx,x,\sqrt{c+d x}\right )}{b}+\frac{\left (d \sqrt{\frac{d (e+f x)}{d e-c f}}\right ) \int \frac{1}{\sqrt{c+d x} \sqrt{\frac{d e}{d e-c f}+\frac{d f x}{d e-c f}} \sqrt{g+h x}} \, dx}{b \sqrt{e+f x}}\\ &=-\frac{\left (2 (b c-a d) \sqrt{\frac{d (e+f x)}{d e-c f}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (b c-a d-b x^2\right ) \sqrt{1+\frac{f x^2}{d \left (e-\frac{c f}{d}\right )}} \sqrt{g-\frac{c h}{d}+\frac{h x^2}{d}}} \, dx,x,\sqrt{c+d x}\right )}{b \sqrt{e+f x}}+\frac{\left (d \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}}\right ) \int \frac{1}{\sqrt{c+d x} \sqrt{\frac{d e}{d e-c f}+\frac{d f x}{d e-c f}} \sqrt{\frac{d g}{d g-c h}+\frac{d h x}{d g-c h}}} \, dx}{b \sqrt{e+f x} \sqrt{g+h x}}\\ &=\frac{2 \sqrt{-d e+c f} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{-d e+c f}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{\left (2 (b c-a d) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (b c-a d-b x^2\right ) \sqrt{1+\frac{f x^2}{d \left (e-\frac{c f}{d}\right )}} \sqrt{1+\frac{h x^2}{d \left (g-\frac{c h}{d}\right )}}} \, dx,x,\sqrt{c+d x}\right )}{b \sqrt{e+f x} \sqrt{g+h x}}\\ &=\frac{2 \sqrt{-d e+c f} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{-d e+c f}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 \sqrt{-d e+c f} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{-d e+c f}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}\\ \end{align*}

Mathematica [C]  time = 1.71334, size = 202, normalized size = 0.69 \[ -\frac{2 i \sqrt{c+d x} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{f (c+d x)}{d e-c f}}\right ),\frac{d e h-c f h}{d f g-c f h}\right )-\Pi \left (\frac{b (c f-d e)}{(b c-a d) f};i \sinh ^{-1}\left (\sqrt{\frac{f (c+d x)}{d e-c f}}\right )|\frac{d e h-c f h}{d f g-c f h}\right )\right )}{b \sqrt{e+f x} \sqrt{g+h x} \sqrt{\frac{f (c+d x)}{d (e+f x)}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)*Sqrt[c + d*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*(EllipticF[I*ArcSinh[Sqrt[(f*(c + d*x))/(d*e - c*f)]], (
d*e*h - c*f*h)/(d*f*g - c*f*h)] - EllipticPi[(b*(-(d*e) + c*f))/((b*c - a*d)*f), I*ArcSinh[Sqrt[(f*(c + d*x))/
(d*e - c*f)]], (d*e*h - c*f*h)/(d*f*g - c*f*h)]))/(b*Sqrt[(f*(c + d*x))/(d*(e + f*x))]*Sqrt[e + f*x]*Sqrt[g +
h*x])

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Maple [A]  time = 0.043, size = 382, normalized size = 1.3 \begin{align*} 2\,{\frac{\sqrt{dx+c}\sqrt{fx+e}\sqrt{hx+g}}{fb \left ( dfh{x}^{3}+cfh{x}^{2}+deh{x}^{2}+dfg{x}^{2}+cehx+cfgx+degx+ceg \right ) }\sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}}\sqrt{-{\frac{ \left ( hx+g \right ) d}{ch-dg}}}\sqrt{-{\frac{ \left ( fx+e \right ) d}{cf-de}}} \left ({\it EllipticF} \left ( \sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) cf-{\it EllipticF} \left ( \sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) de-{\it EllipticPi} \left ( \sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}},-{\frac{ \left ( cf-de \right ) b}{f \left ( ad-bc \right ) }},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) cf+{\it EllipticPi} \left ( \sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}},-{\frac{ \left ( cf-de \right ) b}{f \left ( ad-bc \right ) }},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) de \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2)/(b*x+a)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)

[Out]

2*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/f/b*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*(-(f*
x+e)*d/(c*f-d*e))^(1/2)*(EllipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*c*f-EllipticF(
((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*d*e-EllipticPi(((d*x+c)*f/(c*f-d*e))^(1/2),-(c*f-
d*e)*b/f/(a*d-b*c),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*c*f+EllipticPi(((d*x+c)*f/(c*f-d*e))^(1/2),-(c*f-d*e)*b/f/
(a*d-b*c),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*d*e)/(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g
*x+c*e*g)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(1/2)/(b*x+a)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="giac")

[Out]

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