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

Optimal. Leaf size=570 \[ \frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (3 a^2 d f h^2-3 a b h^2 (c f+d e)+b^2 (-(d g (f g-e h)-c h (2 e h+f g)))\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right ),\frac{h (d e-c f)}{f (d g-c h)}\right )}{3 b^3 d \sqrt{f} h \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} (3 a d f h-b (c f h+d e h+d f g)) E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{3 b^2 d \sqrt{f} h \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{2 (b e-a f) (b g-a h) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b^3 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.28584, antiderivative size = 570, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {161, 1607, 169, 538, 537, 158, 114, 113, 121, 120} \[ \frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (3 a^2 d f h^2-3 a b h^2 (c f+d e)+b^2 (-(d g (f g-e h)-c h (2 e h+f g)))\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{3 b^3 d \sqrt{f} h \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} (3 a d f h-b (c f h+d e h+d f g)) E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{3 b^2 d \sqrt{f} h \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{2 (b e-a f) (b g-a h) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b^3 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 161

Int[((a_.) + (b_.)*(x_))^(m_)*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)], x_Sy
mbol] :> Simp[(2*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(b*(2*m + 5)), x] + Dist[1/(b*(2
*m + 5)), Int[((a + b*x)^m*Simp[3*b*c*e*g - a*(d*e*g + c*f*g + c*e*h) + 2*(b*(d*e*g + c*f*g + c*e*h) - a*(d*f*
g + d*e*h + c*f*h))*x - (3*a*d*f*h - b*(d*f*g + d*e*h + c*f*h))*x^2, x])/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g +
h*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] &&  !LtQ[m, -1]

Rule 1607

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.)*((g_.) + (h_.)*(x_)
)^(q_.), x_Symbol] :> Dist[PolynomialRemainder[Px, a + b*x, x], Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h
*x)^q, x], x] + Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q,
x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && PolyQ[Px, x] && EqQ[m, -1]

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)
])

Rule 158

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

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])

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)
])

Rubi steps

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

Mathematica [C]  time = 13.8408, size = 1250, normalized size = 2.19 \[ \frac{2 \sqrt{c+d x} \left (\frac{b^2 f h c^3}{d^2 (c+d x)}-\frac{b^2 f h c^2}{d^2}-\frac{3 a b f h c^2}{d (c+d x)}+\frac{3 a b f h c}{d}+\frac{b^2 f h x c}{d}-\frac{b^2 e g c}{c+d x}+\frac{3 a b f g c}{c+d x}+\frac{3 a b e h c}{c+d x}-\frac{b^2 e^2 h c}{c f+d x f}-\frac{b^2 f g^2 c}{c h+d x h}+b^2 f h x^2+3 b^2 e g-3 a b f g-3 a b e h+\frac{b^2 e^2 h}{f}+2 b^2 f g x+2 b^2 e h x-3 a b f h x-\frac{i b \sqrt{\frac{d g}{h}-c} (3 a d f h-b (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d g}{h}-c}}{\sqrt{c+d x}}\right )|\frac{d e h-c f h}{d f g-c f h}\right )}{d^2}+\frac{i b \sqrt{\frac{d g}{h}-c} (-b f g-2 b e h+3 a f h) \sqrt{c+d x} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d g}{h}-c}}{\sqrt{c+d x}}\right ),\frac{d e h-c f h}{d f g-c f h}\right )}{d}+\frac{3 i a^2 f \sqrt{\frac{d g}{h}-c} h^2 \sqrt{c+d x} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \Pi \left (\frac{(b c-a d) h}{b (c h-d g)};i \sinh ^{-1}\left (\frac{\sqrt{\frac{d g}{h}-c}}{\sqrt{c+d x}}\right )|\frac{d e h-c f h}{d f g-c f h}\right )}{d g-c h}+\frac{3 i b^2 e g \sqrt{\frac{d g}{h}-c} h \sqrt{c+d x} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \Pi \left (\frac{(b c-a d) h}{b (c h-d g)};i \sinh ^{-1}\left (\frac{\sqrt{\frac{d g}{h}-c}}{\sqrt{c+d x}}\right )|\frac{d e h-c f h}{d f g-c f h}\right )}{d g-c h}+\frac{3 i a b e \sqrt{\frac{d g}{h}-c} h^2 \sqrt{c+d x} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \Pi \left (\frac{(b c-a d) h}{b (c h-d g)};i \sinh ^{-1}\left (\frac{\sqrt{\frac{d g}{h}-c}}{\sqrt{c+d x}}\right )|\frac{d e h-c f h}{d f g-c f h}\right )}{c h-d g}+\frac{3 i a b f g \sqrt{\frac{d g}{h}-c} h \sqrt{c+d x} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \Pi \left (\frac{(b c-a d) h}{b (c h-d g)};i \sinh ^{-1}\left (\frac{\sqrt{\frac{d g}{h}-c}}{\sqrt{c+d x}}\right )|\frac{d e h-c f h}{d f g-c f h}\right )}{c h-d g}+\frac{b^2 f g^2}{h}-\frac{3 a b d e g}{c+d x}+\frac{b^2 d e^2 g}{c f+d x f}+\frac{b^2 d e g^2}{c h+d x h}\right )}{3 b^3 \sqrt{e+f x} \sqrt{g+h x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.059, size = 3678, normalized size = 6.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(((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)*EllipticE(((d*x+c)*
f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*e*f*g*h+3*((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)*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))*a*b*c*d^2*e*f*h^2+3*((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)*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))*a*b*c*d^2*f^2*g*h+3*((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)*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))
*a^2*d^3*e*f*h^2-3*((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)*Ellip
ticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a^2*d^3*e*f*h^2+3*((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))*a*b*d^3*e^2*h^2-2*((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))*b^2*c*d^2*e^2*h^2-3*((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)*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))*a^2*c*d^2*f^2*h^2-3*((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))*a*b*c^2*d*f^2*h^2+2*((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))*b^2*c^2*d*e*f*h^2
+((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))*b^2*c^2*d*f^2*g*h+3*((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)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1
/2))*a*b*c^2*d*f^2*h^2+x^2*b^2*c*d^2*f^2*h^2+x^2*b^2*d^3*e*f*h^2+3*((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))*a^2*c*d^2*f^2*h^2-3*((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)*
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))*a*b*d^3*e^2*h
^2+3*((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)*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))*b^2*d^3*e^2*g*h+((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)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((
c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*e^2*h^2+((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)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*f^2*
g^2-((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)*EllipticE(((d*x+c)*f
/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*d^3*e^2*g*h+b^2*c*d^2*e*f*g*h-((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)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*
h/f/(c*h-d*g))^(1/2))*b^2*d^3*e*f*g^2+x^2*b^2*d^3*f^2*g*h+x^3*b^2*d^3*f^2*h^2-((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)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*
h-d*g))^(1/2))*b^2*c^3*f^2*h^2+x*b^2*c*d^2*f^2*g*h+x*b^2*d^3*e*f*g*h+x*b^2*c*d^2*e*f*h^2-3*((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)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f
-d*e)*h/f/(c*h-d*g))^(1/2))*a*b*c*d^2*e*f*h^2-3*((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)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a*b*c*d^2*f^2*g
*h+3*((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)*EllipticE(((d*x+c)*
f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a*b*d^3*e*f*g*h-3*((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)*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))*a*b*d^3*e*f*g*h-3*((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)*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))*b^2*c*d^2*e*f*g*h-((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))*b^2*c*d^2*f^2*g^2-((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))*b^2*d^3*e^2*g*h+((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))*b^2*d^3*e*f*g
^2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/b^3/h/f/d^2/(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} \sqrt{f x + e} \sqrt{h x + g}}{b x + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)/(b*x + a), 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)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(b*x+a),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} \sqrt{e + f x} \sqrt{g + h x}}{a + b x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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