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

Optimal. Leaf size=738 \[ \frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (a^2 C d f-2 a b C (c f+d e)+b^2 (2 c C e-A d f)\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right ),\frac{h (d e-c f)}{f (d g-c h)}\right )}{b^2 d \sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d) (b e-a f)}-\frac{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}-\frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a^3 b C (c f h+d e h+d f g)+a^4 C d f h-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)^2 (b e-a f) (b g-a h)}+\frac{\sqrt{f} \sqrt{g+h x} \left (\frac{a^2 C}{b}+A b\right ) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{e+f x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{d (g+h x)}{d g-c h}}} \]

[Out]

-(((A*b^2 + a^2*C)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)*(b*e - a*f)*(b*g - a*h)*(a + b*x)))
+ ((A*b + (a^2*C)/b)*Sqrt[f]*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[ArcSi
n[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/((b*c - a*d)*(b*e - a*f)*(b*g
- a*h)*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) + (Sqrt[-(d*e) + c*f]*(a^2*C*d*f - 2*a*b*C*(d*e + c*f)
+ b^2*(2*c*C*e - A*d*f))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticF[ArcSin[(Sqr
t[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(b^2*d*(b*c - a*d)*Sqrt[f]*(b*e - a
*f)*Sqrt[e + f*x]*Sqrt[g + h*x]) - (Sqrt[-(d*e) + c*f]*(a^4*C*d*f*h - A*b^4*(d*e*g + c*f*g + c*e*h) - 2*a^3*b*
C*(d*f*g + d*e*h + c*f*h) - 2*a*b^3*(2*c*C*e*g - A*d*f*g - A*d*e*h - A*c*f*h) - 3*a^2*b^2*(A*d*f*h - C*(d*e*g
+ c*f*g + c*e*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^2
*(b*c - a*d)^2*Sqrt[f]*(b*e - a*f)*(b*g - a*h)*Sqrt[e + f*x]*Sqrt[g + h*x])

________________________________________________________________________________________

Rubi [A]  time = 1.90697, antiderivative size = 738, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {1605, 1607, 169, 538, 537, 158, 114, 113, 121, 120} \[ -\frac{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}+\frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (a^2 C d f-2 a b C (c f+d e)+b^2 (2 c C e-A d f)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b^2 d \sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d) (b e-a f)}-\frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a^3 b C (c f h+d e h+d f g)+a^4 C d f h-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)^2 (b e-a f) (b g-a h)}+\frac{\sqrt{f} \sqrt{g+h x} \left (\frac{a^2 C}{b}+A b\right ) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{e+f x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{d (g+h x)}{d g-c h}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((A*b^2 + a^2*C)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)*(b*e - a*f)*(b*g - a*h)*(a + b*x)))
+ ((A*b + (a^2*C)/b)*Sqrt[f]*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[ArcSi
n[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/((b*c - a*d)*(b*e - a*f)*(b*g
- a*h)*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) + (Sqrt[-(d*e) + c*f]*(a^2*C*d*f - 2*a*b*C*(d*e + c*f)
+ b^2*(2*c*C*e - A*d*f))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticF[ArcSin[(Sqr
t[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(b^2*d*(b*c - a*d)*Sqrt[f]*(b*e - a
*f)*Sqrt[e + f*x]*Sqrt[g + h*x]) - (Sqrt[-(d*e) + c*f]*(a^4*C*d*f*h - A*b^4*(d*e*g + c*f*g + c*e*h) - 2*a^3*b*
C*(d*f*g + d*e*h + c*f*h) - 2*a*b^3*(2*c*C*e*g - A*d*f*g - A*d*e*h - A*c*f*h) - 3*a^2*b^2*(A*d*f*h - C*(d*e*g
+ c*f*g + c*e*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^2
*(b*c - a*d)^2*Sqrt[f]*(b*e - a*f)*(b*g - a*h)*Sqrt[e + f*x]*Sqrt[g + h*x])

Rule 1605

Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (C_.)*(x_)^2))/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt
[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[((A*b^2 + a^2*C)*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g
+ h*x])/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)), x] - Dist[1/(2*(m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g -
a*h)), Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*d*f*h*(m + 1) - 2*a*b
*(m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) + a*C*(a*(d*e*g + c*f*g + c*e*h) + 2
*b*c*e*g*(m + 1)) - 2*(A*b*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h)) - C*(a^2*(d*f*g + d*e*h + c*f
*h) - b^2*c*e*g*(m + 1) + a*b*(m + 1)*(d*e*g + c*f*g + c*e*h)))*x + d*f*h*(2*m + 5)*(A*b^2 + a^2*C)*x^2, x], x
], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, C}, 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{A+C x^2}{(a+b x)^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=-\frac{\left (A b^2+a^2 C\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac{\int \frac{-A b^2 (d e g+c f g+c e h)-2 a b (c C e g-A d f g-A d e h-A c f h)-a^2 (2 A d f h-C (d e g+c f g+c e h))+2 \left (b^2 c C e g+a^2 C (d f g+d e h+c f h)+a b (A d f h-C (d e g+c f g+c e h))\right ) x+\left (A b^2+a^2 C\right ) d f h x^2}{(a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{2 (b c-a d) (b e-a f) (b g-a h)}\\ &=-\frac{\left (A b^2+a^2 C\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac{\int \frac{2 b c C e g-2 a C d e g-2 a c C f g+\frac{2 a^2 C d f g}{b}-2 a c C e h+\frac{2 a^2 C d e h}{b}+\frac{2 a^2 c C f h}{b}+a A d f h-\frac{a^3 C d f h}{b^2}+\left (A b d f h+\frac{a^2 C d f h}{b}\right ) x}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{2 (b c-a d) (b e-a f) (b g-a h)}+\frac{\left (a^4 C d f h-A b^4 (d e g+c f g+c e h)-2 a^3 b C (d f g+d e h+c f h)-2 a b^3 (2 c C e g-A d f g-A d e h-A c f h)-3 a^2 b^2 (A d f h-C (d e g+c f g+c e h))\right ) \int \frac{1}{(a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{2 b^2 (b c-a d) (b e-a f) (b g-a h)}\\ &=-\frac{\left (A b^2+a^2 C\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac{\left (a^2 C d f-2 a b C (d e+c f)+b^2 (2 c C e-A d f)\right ) \int \frac{1}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{2 b^2 (b c-a d) (b e-a f)}+\frac{\left (\left (A b+\frac{a^2 C}{b}\right ) d f\right ) \int \frac{\sqrt{g+h x}}{\sqrt{c+d x} \sqrt{e+f x}} \, dx}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac{\left (a^4 C d f h-A b^4 (d e g+c f g+c e h)-2 a^3 b C (d f g+d e h+c f h)-2 a b^3 (2 c C e g-A d f g-A d e h-A c f h)-3 a^2 b^2 (A d f h-C (d e g+c f g+c e h))\right ) \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^2 (b c-a d) (b e-a f) (b g-a h)}\\ &=-\frac{\left (A b^2+a^2 C\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac{\left (\left (a^2 C d f-2 a b C (d e+c f)+b^2 (2 c C e-A d f)\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}{2 b^2 (b c-a d) (b e-a f) \sqrt{e+f x}}-\frac{\left (\left (a^4 C d f h-A b^4 (d e g+c f g+c e h)-2 a^3 b C (d f g+d e h+c f h)-2 a b^3 (2 c C e g-A d f g-A d e h-A c f h)-3 a^2 b^2 (A d f h-C (d e g+c f g+c e h))\right ) \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^2 (b c-a d) (b e-a f) (b g-a h) \sqrt{e+f x}}+\frac{\left (\left (A b+\frac{a^2 C}{b}\right ) d f \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}{2 (b c-a d) (b e-a f) (b g-a h) \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}\\ &=-\frac{\left (A b^2+a^2 C\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac{\left (A b+\frac{a^2 C}{b}\right ) \sqrt{f} \sqrt{-d e+c f} \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 )}{(b c-a d) (b e-a f) (b g-a h) \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{\left (\left (a^2 C d f-2 a b C (d e+c f)+b^2 (2 c C e-A d f)\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}{2 b^2 (b c-a d) (b e-a f) \sqrt{e+f x} \sqrt{g+h x}}-\frac{\left (\left (a^4 C d f h-A b^4 (d e g+c f g+c e h)-2 a^3 b C (d f g+d e h+c f h)-2 a b^3 (2 c C e g-A d f g-A d e h-A c f h)-3 a^2 b^2 (A d f h-C (d e g+c f g+c e h))\right ) \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^2 (b c-a d) (b e-a f) (b g-a h) \sqrt{e+f x} \sqrt{g+h x}}\\ &=-\frac{\left (A b^2+a^2 C\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac{\left (A b+\frac{a^2 C}{b}\right ) \sqrt{f} \sqrt{-d e+c f} \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 )}{(b c-a d) (b e-a f) (b g-a h) \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{\sqrt{-d e+c f} \left (a^2 C d f-2 a b C (d e+c f)+b^2 (2 c C e-A d f)\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 )}{b^2 d (b c-a d) \sqrt{f} (b e-a f) \sqrt{e+f x} \sqrt{g+h x}}-\frac{\sqrt{-d e+c f} \left (a^4 C d f h-A b^4 (d e g+c f g+c e h)-2 a^3 b C (d f g+d e h+c f h)-2 a b^3 (2 c C e g-A d f g-A d e h-A c f h)-3 a^2 b^2 (A d f h-C (d e g+c f g+c e h))\right ) \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^2 (b c-a d)^2 \sqrt{f} (b e-a f) (b g-a h) \sqrt{e+f x} \sqrt{g+h x}}\\ \end{align*}

Mathematica [C]  time = 15.7353, size = 636, normalized size = 0.86 \[ \frac{b^2 d (c+d x) (e+f x) (g+h x) \left (a^2 C+A b^2\right ) (a d-b c) \sqrt{\frac{d g}{h}-c}+(a+b x) \left (b d^2 (e+f x) (g+h x) \left (a^2 C+A b^2\right ) (b c-a d) \sqrt{\frac{d g}{h}-c}-i (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \left (-b (b e-a f) \left (a^2 C d (c h-d g)-2 a b h \left (A d^2+c^2 C\right )+b^2 \left (A c d h+A d^2 g+2 c^2 C g\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d g}{h}-c}}{\sqrt{c+d x}}\right ),\frac{d e h-c f h}{d f g-c f h}\right )+b f \left (a^2 C+A b^2\right ) (b c-a d) (c h-d g) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d g}{h}-c}}{\sqrt{c+d x}}\right )|\frac{d e h-c f h}{d f g-c f h}\right )-d \left (3 a^2 b^2 (C (c e h+c f g+d e g)-A d f h)-2 a^3 b C (c f h+d e h+d f g)+a^4 C d f h+2 a b^3 (A c f h+A d e h+A d f g-2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \Pi \left (\frac{(b c-a d) h}{b (c h-d g)};i \sinh ^{-1}\left (\frac{\sqrt{\frac{d g}{h}-c}}{\sqrt{c+d x}}\right )|\frac{d e h-c f h}{d f g-c f h}\right )\right )\right )}{b^2 d (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)^2 (b e-a f) (b g-a h) \sqrt{\frac{d g}{h}-c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*(A*b^2 + a^2*C)*d*(-(b*c) + a*d)*Sqrt[-c + (d*g)/h]*(c + d*x)*(e + f*x)*(g + h*x) + (a + b*x)*(b*(A*b^2 +
a^2*C)*d^2*(b*c - a*d)*Sqrt[-c + (d*g)/h]*(e + f*x)*(g + h*x) - I*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c +
d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*(b*(A*b^2 + a^2*C)*(b*c - a*d)*f*(-(d*g) + c*h)*EllipticE[I*ArcSinh[S
qrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)] - b*(b*e - a*f)*(-2*a*b*(c^2*C + A*d^2)*h +
a^2*C*d*(-(d*g) + c*h) + b^2*(2*c^2*C*g + A*d^2*g + A*c*d*h))*EllipticF[I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c +
d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)] - d*(a^4*C*d*f*h - A*b^4*(d*e*g + c*f*g + c*e*h) - 2*a^3*b*C*(d*f*g +
d*e*h + c*f*h) + 2*a*b^3*(-2*c*C*e*g + A*d*f*g + A*d*e*h + A*c*f*h) + 3*a^2*b^2*(-(A*d*f*h) + C*(d*e*g + c*f*
g + c*e*h)))*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)])))/(b^2*d*(b*c - a*d)^2*(b*e - a*f)*Sqrt[-c + (d*g)/h]*(b*g - a*h)*(a + b*x)*Sqrt[
c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])

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Maple [B]  time = 0.091, size = 17460, normalized size = 23.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C x^{2} + A}{{\left (b x + a\right )}^{2} \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((C*x^2+A)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C x^{2} + A}{{\left (b x + a\right )}^{2} \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((C*x^2+A)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="giac")

[Out]

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