3.12 \(\int \frac{\sqrt{a+b x} (d e+c f+2 d f x)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx\)

Optimal. Leaf size=472 \[ -\frac{2 d (e+f x) (b g-a h)^{3/2} \sqrt{\frac{(a+b x) (f g-e h)}{(e+f x) (b g-a h)}} \sqrt{\frac{(c+d x) (f g-e h)}{(e+f x) (d g-c h)}} \Pi \left (\frac{f (b g-a h)}{(b e-a f) h};\sin ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{g+h x}}{\sqrt{b g-a h} \sqrt{e+f x}}\right )|\frac{(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{h^2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{b e-a f}}+\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{h \sqrt{a+b x}}-\frac{2 \sqrt{c+d x} \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}} E\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{h \sqrt{g+h x} \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}}} \]

[Out]

(2*b*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(h*Sqrt[a + b*x]) - (2*Sqrt[b*g - a*h]*Sqrt[f*g - e*h]*Sqrt[c
+ d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))]*EllipticE[ArcSin[(Sqrt[b*g - a*h]*Sqrt[e + f*x
])/(Sqrt[f*g - e*h]*Sqrt[a + b*x])], -(((b*c - a*d)*(f*g - e*h))/((d*e - c*f)*(b*g - a*h)))])/(h*Sqrt[((b*e -
a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]) - (2*d*(b*g - a*h)^(3/2)*Sqrt[((f*g - e*h)*(a + b*x))/
((b*g - a*h)*(e + f*x))]*Sqrt[((f*g - e*h)*(c + d*x))/((d*g - c*h)*(e + f*x))]*(e + f*x)*EllipticPi[(f*(b*g -
a*h))/((b*e - a*f)*h), ArcSin[(Sqrt[b*e - a*f]*Sqrt[g + h*x])/(Sqrt[b*g - a*h]*Sqrt[e + f*x])], ((d*e - c*f)*(
b*g - a*h))/((b*e - a*f)*(d*g - c*h))])/(Sqrt[b*e - a*f]*h^2*Sqrt[a + b*x]*Sqrt[c + d*x])

________________________________________________________________________________________

Rubi [A]  time = 0.519671, antiderivative size = 472, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.102, Rules used = {1595, 165, 537, 176, 424} \[ -\frac{2 d (e+f x) (b g-a h)^{3/2} \sqrt{\frac{(a+b x) (f g-e h)}{(e+f x) (b g-a h)}} \sqrt{\frac{(c+d x) (f g-e h)}{(e+f x) (d g-c h)}} \Pi \left (\frac{f (b g-a h)}{(b e-a f) h};\sin ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{g+h x}}{\sqrt{b g-a h} \sqrt{e+f x}}\right )|\frac{(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{h^2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{b e-a f}}+\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{h \sqrt{a+b x}}-\frac{2 \sqrt{c+d x} \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}} E\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{h \sqrt{g+h x} \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x]*(d*e + c*f + 2*d*f*x))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(2*b*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(h*Sqrt[a + b*x]) - (2*Sqrt[b*g - a*h]*Sqrt[f*g - e*h]*Sqrt[c
+ d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))]*EllipticE[ArcSin[(Sqrt[b*g - a*h]*Sqrt[e + f*x
])/(Sqrt[f*g - e*h]*Sqrt[a + b*x])], -(((b*c - a*d)*(f*g - e*h))/((d*e - c*f)*(b*g - a*h)))])/(h*Sqrt[((b*e -
a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]) - (2*d*(b*g - a*h)^(3/2)*Sqrt[((f*g - e*h)*(a + b*x))/
((b*g - a*h)*(e + f*x))]*Sqrt[((f*g - e*h)*(c + d*x))/((d*g - c*h)*(e + f*x))]*(e + f*x)*EllipticPi[(f*(b*g -
a*h))/((b*e - a*f)*h), ArcSin[(Sqrt[b*e - a*f]*Sqrt[g + h*x])/(Sqrt[b*g - a*h]*Sqrt[e + f*x])], ((d*e - c*f)*(
b*g - a*h))/((b*e - a*f)*(d*g - c*h))])/(Sqrt[b*e - a*f]*h^2*Sqrt[a + b*x]*Sqrt[c + d*x])

Rule 1595

Int[(Sqrt[(a_.) + (b_.)*(x_)]*((A_.) + (B_.)*(x_)))/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g
_.) + (h_.)*(x_)]), x_Symbol] :> Simp[(b*B*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(d*f*h*Sqrt[a + b*x]), x
] + (-Dist[(B*(b*g - a*h))/(2*f*h), Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[g + h*x]), x], x] + Di
st[(B*(b*e - a*f)*(b*g - a*h))/(2*d*f*h), Int[Sqrt[c + d*x]/((a + b*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x],
x]) /; FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && EqQ[2*A*d*f - B*(d*e + c*f), 0]

Rule 165

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

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)
])

Rule 176

Int[Sqrt[(c_.) + (d_.)*(x_)]/(((a_.) + (b_.)*(x_))^(3/2)*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Dist[(-2*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))])/((b*e - a*f)*Sqrt
[g + h*x]*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]), Subst[Int[Sqrt[1 + ((b*c - a*d)*x^2)/(d*e -
c*f)]/Sqrt[1 - ((b*g - a*h)*x^2)/(f*g - e*h)], x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e
, f, g, h}, x]

Rule 424

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

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b x} (d e+c f+2 d f x)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{h \sqrt{a+b x}}-\frac{(d (b g-a h)) \int \frac{\sqrt{e+f x}}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{g+h x}} \, dx}{h}+\frac{((b e-a f) (b g-a h)) \int \frac{\sqrt{c+d x}}{(a+b x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{h}\\ &=\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{h \sqrt{a+b x}}-\frac{\left (2 d (b g-a h) \sqrt{\frac{(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} \sqrt{\frac{(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\left (h-f x^2\right ) \sqrt{1+\frac{(-b e+a f) x^2}{b g-a h}} \sqrt{1+\frac{(-d e+c f) x^2}{d g-c h}}} \, dx,x,\frac{\sqrt{g+h x}}{\sqrt{e+f x}}\right )}{h \sqrt{a+b x} \sqrt{c+d x}}-\frac{\left (2 (b g-a h) \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{(b c-a d) x^2}{d e-c f}}}{\sqrt{1-\frac{(b g-a h) x^2}{f g-e h}}} \, dx,x,\frac{\sqrt{e+f x}}{\sqrt{a+b x}}\right )}{h \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x}}\\ &=\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{h \sqrt{a+b x}}-\frac{2 \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{h \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x}}-\frac{2 d (b g-a h)^{3/2} \sqrt{\frac{(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} \sqrt{\frac{(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} (e+f x) \Pi \left (\frac{f (b g-a h)}{(b e-a f) h};\sin ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{g+h x}}{\sqrt{b g-a h} \sqrt{e+f x}}\right )|\frac{(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{\sqrt{b e-a f} h^2 \sqrt{a+b x} \sqrt{c+d x}}\\ \end{align*}

Mathematica [B]  time = 15.4652, size = 6583, normalized size = 13.95 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[a + b*x]*(d*e + c*f + 2*d*f*x))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

Result too large to show

________________________________________________________________________________________

Maple [B]  time = 0.064, size = 13177, normalized size = 27.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (2 \, d f x + d e + c f\right )} \sqrt{b x + a}}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b x} \left (c f + d e + 2 d f x\right )}{\sqrt{c + d x} \sqrt{e + f x} \sqrt{g + h x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (2 \, d f x + d e + c f\right )} \sqrt{b x + a}}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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