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

Optimal. Leaf size=700 \[ -\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right ),\frac{h (d e-c f)}{f (d g-c h)}\right ) \left (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B (c h (f g-e h)+d g (e h+2 f g)))+b^2 \left (-\left (5 A d f h (c h (f g-e h)+d g (e h+2 f g))-B \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) \left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (c f h+d e h+d f g))+b^2 \left (-\left (10 A d f h (c f h+d e h+d f g)-B \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (7 a B d f h+b (5 A d f h-4 B (c f h+d e h+d f g)))}{15 d^2 f^2 h^2}+\frac{2 b B (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h} \]

[Out]

(2*b*(7*a*B*d*f*h + b*(5*A*d*f*h - 4*B*(d*f*g + d*e*h + c*f*h)))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(1
5*d^2*f^2*h^2) + (2*b*B*(a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(5*d*f*h) + (2*Sqrt[-(d*e) + c*f]
*(15*a^2*B*d^2*f^2*h^2 + 10*a*b*d*f*h*(3*A*d*f*h - 2*B*(d*f*g + d*e*h + c*f*h)) - b^2*(10*A*d*f*h*(d*f*g + d*e
*h + c*f*h) - B*(8*c^2*f^2*h^2 + 7*c*d*f*h*(f*g + e*h) + d^2*(8*f^2*g^2 + 7*e*f*g*h + 8*e^2*h^2))))*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))])/(15*d^3*f^(5/2)*h^3*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) - (2*Sqrt[-(d*e) + c*
f]*(15*a^2*d^2*f^2*h^2*(B*g - A*h) + 10*a*b*d*f*h*(3*A*d*f*g*h - B*(c*h*(f*g - e*h) + d*g*(2*f*g + e*h))) - b^
2*(5*A*d*f*h*(c*h*(f*g - e*h) + d*g*(2*f*g + e*h)) - B*(4*c^2*f*h^2*(f*g - e*h) + c*d*h*(3*f^2*g^2 + e*f*g*h -
4*e^2*h^2) + d^2*g*(8*f^2*g^2 + 3*e*f*g*h + 4*e^2*h^2))))*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))])/(
15*d^3*f^(5/2)*h^3*Sqrt[e + f*x]*Sqrt[g + h*x])

________________________________________________________________________________________

Rubi [A]  time = 2.33211, antiderivative size = 699, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1597, 1600, 1615, 158, 114, 113, 121, 120} \[ -\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) \left (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B c h (f g-e h)-B d g (e h+2 f g))+b^2 \left (-\left (5 A d f h (c h (f g-e h)+d g (e h+2 f g))-B \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) \left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (c f h+d e h+d f g))+b^2 \left (-\left (10 A d f h (c f h+d e h+d f g)-B \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (7 a B d f h+5 A b d f h-4 b B (c f h+d e h+d f g))}{15 d^2 f^2 h^2}+\frac{2 b B (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*(5*A*b*d*f*h + 7*a*B*d*f*h - 4*b*B*(d*f*g + d*e*h + c*f*h))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(1
5*d^2*f^2*h^2) + (2*b*B*(a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(5*d*f*h) + (2*Sqrt[-(d*e) + c*f]
*(15*a^2*B*d^2*f^2*h^2 + 10*a*b*d*f*h*(3*A*d*f*h - 2*B*(d*f*g + d*e*h + c*f*h)) - b^2*(10*A*d*f*h*(d*f*g + d*e
*h + c*f*h) - B*(8*c^2*f^2*h^2 + 7*c*d*f*h*(f*g + e*h) + d^2*(8*f^2*g^2 + 7*e*f*g*h + 8*e^2*h^2))))*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))])/(15*d^3*f^(5/2)*h^3*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) - (2*Sqrt[-(d*e) + c*
f]*(15*a^2*d^2*f^2*h^2*(B*g - A*h) + 10*a*b*d*f*h*(3*A*d*f*g*h - B*c*h*(f*g - e*h) - B*d*g*(2*f*g + e*h)) - b^
2*(5*A*d*f*h*(c*h*(f*g - e*h) + d*g*(2*f*g + e*h)) - B*(4*c^2*f*h^2*(f*g - e*h) + c*d*h*(3*f^2*g^2 + e*f*g*h -
4*e^2*h^2) + d^2*g*(8*f^2*g^2 + 3*e*f*g*h + 4*e^2*h^2))))*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))])/(
15*d^3*f^(5/2)*h^3*Sqrt[e + f*x]*Sqrt[g + h*x])

Rule 1597

Int[(((a_.) + (b_.)*(x_))^(m_.)*((A_.) + (B_.)*(x_)))/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[
(g_.) + (h_.)*(x_)]), x_Symbol] :> Dist[1/(d*f*h*(2*m + 3)), Int[((a + b*x)^(m - 1)/(Sqrt[c + d*x]*Sqrt[e + f*
x]*Sqrt[g + h*x]))*Simp[a*A*d*f*h*(2*m + 3) + (A*b + a*B)*d*f*h*(2*m + 3)*x + b*B*d*f*h*(2*m + 3)*x^2, x], x],
x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && IntegerQ[2*m] && GtQ[m, 0]

Rule 1600

Int[(((a_.) + (b_.)*(x_))^(m_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2))/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f
_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[(2*C*(a + b*x)^m*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h
*x])/(d*f*h*(2*m + 3)), x] + Dist[1/(d*f*h*(2*m + 3)), Int[((a + b*x)^(m - 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqr
t[g + h*x]))*Simp[a*A*d*f*h*(2*m + 3) - C*(a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*m) + ((A*b + a*B)*d*f*h*(2*m
+ 3) - C*(2*a*(d*f*g + d*e*h + c*f*h) + b*(2*m + 1)*(d*e*g + c*f*g + c*e*h)))*x + (b*B*d*f*h*(2*m + 3) + 2*C*(
a*d*f*h*m - b*(m + 1)*(d*f*g + d*e*h + c*f*h)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x]
&& IntegerQ[2*m] && GtQ[m, 0]

Rule 1615

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[
{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[(k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*(e + f*x)^
(p + 1))/(d*f*b^(q - 1)*(m + n + p + q + 1)), x] + Dist[1/(d*f*b^q*(m + n + p + q + 1)), Int[(a + b*x)^m*(c +
d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a +
b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*
(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; F
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && IntegersQ[2*m, 2*n, 2*p]

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+b x)^2 (A+B x)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=\frac{\int \frac{(a+b x) \left (7 a A d f h+7 (A b+a B) d f h x+7 b B d f h x^2\right )}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{7 d f h}\\ &=\frac{2 b B (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}+\frac{\int \frac{7 d f h \left (5 a^2 A d f h-b B (2 b c e g+a (d e g+c f g+c e h))\right )+7 d f h (5 a (2 A b+a B) d f h-b B (3 b (d e g+c f g+c e h)+2 a (d f g+d e h+c f h))) x+7 b d f h (5 A b d f h+7 a B d f h-4 b B (d f g+d e h+c f h)) x^2}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{35 d^2 f^2 h^2}\\ &=\frac{2 b (5 A b d f h+7 a B d f h-4 b B (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{15 d^2 f^2 h^2}+\frac{2 b B (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}+\frac{2 \int \frac{\frac{7}{2} d^2 f h \left (15 a^2 A d^2 f^2 h^2-10 a b B d f h (d e g+c f g+c e h)-b^2 \left (5 A d f h (d e g+c f g+c e h)-B \left (4 d^2 e g (f g+e h)+4 c^2 f h (f g+e h)+2 c d \left (2 f^2 g^2+3 e f g h+2 e^2 h^2\right )\right )\right )\right )+\frac{7}{2} d^2 f h \left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (d f g+d e h+c f h))-b^2 \left (10 A d f h (d f g+d e h+c f h)-B \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )\right ) x}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{105 d^4 f^3 h^3}\\ &=\frac{2 b (5 A b d f h+7 a B d f h-4 b B (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{15 d^2 f^2 h^2}+\frac{2 b B (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}-\frac{\left (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B c h (f g-e h)-B d g (2 f g+e h))-b^2 \left (5 A d f h (c h (f g-e h)+d g (2 f g+e h))-B \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )\right ) \int \frac{1}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{15 d^2 f^2 h^3}+\frac{\left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (d f g+d e h+c f h))-b^2 \left (10 A d f h (d f g+d e h+c f h)-B \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )\right ) \int \frac{\sqrt{g+h x}}{\sqrt{c+d x} \sqrt{e+f x}} \, dx}{15 d^2 f^2 h^3}\\ &=\frac{2 b (5 A b d f h+7 a B d f h-4 b B (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{15 d^2 f^2 h^2}+\frac{2 b B (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}-\frac{\left (\left (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B c h (f g-e h)-B d g (2 f g+e h))-b^2 \left (5 A d f h (c h (f g-e h)+d g (2 f g+e h))-B \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\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}{15 d^2 f^2 h^3 \sqrt{e+f x}}+\frac{\left (\left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (d f g+d e h+c f h))-b^2 \left (10 A d f h (d f g+d e h+c f h)-B \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )\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}{15 d^2 f^2 h^3 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}\\ &=\frac{2 b (5 A b d f h+7 a B d f h-4 b B (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{15 d^2 f^2 h^2}+\frac{2 b B (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}+\frac{2 \sqrt{-d e+c f} \left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (d f g+d e h+c f h))-b^2 \left (10 A d f h (d f g+d e h+c f h)-B \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )\right ) \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 )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{\left (\left (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B c h (f g-e h)-B d g (2 f g+e h))-b^2 \left (5 A d f h (c h (f g-e h)+d g (2 f g+e h))-B \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\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}{15 d^2 f^2 h^3 \sqrt{e+f x} \sqrt{g+h x}}\\ &=\frac{2 b (5 A b d f h+7 a B d f h-4 b B (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{15 d^2 f^2 h^2}+\frac{2 b B (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}+\frac{2 \sqrt{-d e+c f} \left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (d f g+d e h+c f h))-b^2 \left (10 A d f h (d f g+d e h+c f h)-B \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )\right ) \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 )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{2 \sqrt{-d e+c f} \left (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B c h (f g-e h)-B d g (2 f g+e h))-b^2 \left (5 A d f h (c h (f g-e h)+d g (2 f g+e h))-B \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )\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 )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{g+h x}}\\ \end{align*}

Mathematica [C]  time = 11.8427, size = 806, normalized size = 1.15 \[ -\frac{2 \left (-\sqrt{\frac{d e}{f}-c} \left (\left (B \left (\left (8 f^2 g^2+7 e f h g+8 e^2 h^2\right ) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right )-10 A d f h (d f g+d e h+c f h)\right ) b^2-10 a d f h (2 B (d f g+d e h+c f h)-3 A d f h) b+15 a^2 B d^2 f^2 h^2\right ) (e+f x) (g+h x) d^2+b \sqrt{\frac{d e}{f}-c} f h (c+d x) (e+f x) (g+h x) (-5 A b d f h-10 a B d f h+b B (4 c f h+d (4 f g+4 e h-3 f h x))) d^2-i h \left (-\left (B \left (e \left (4 f^2 g^2+3 e f h g+8 e^2 h^2\right ) d^2+c f \left (-4 f^2 g^2+e f h g+3 e^2 h^2\right ) d+4 c^2 f^2 h (e h-f g)\right )-5 A d f h (c f (e h-f g)+d e (f g+2 e h))\right ) b^2+10 a d f h (-3 A d e f h+B c f (e h-f g)+B d e (f g+2 e h)) b+15 a^2 d^2 f^2 (A f-B e) h^2\right ) (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right ),\frac{d f g-c f h}{d e h-c f h}\right ) d-i (d e-c f) h \left (\left (B \left (\left (8 f^2 g^2+7 e f h g+8 e^2 h^2\right ) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right )-10 A d f h (d f g+d e h+c f h)\right ) b^2-10 a d f h (2 B (d f g+d e h+c f h)-3 A d f h) b+15 a^2 B d^2 f^2 h^2\right ) (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right )|\frac{d f g-c f h}{d e h-c f h}\right )\right )}{15 d^4 \sqrt{\frac{d e}{f}-c} f^3 h^3 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-(d^2*Sqrt[-c + (d*e)/f]*(15*a^2*B*d^2*f^2*h^2 - 10*a*b*d*f*h*(-3*A*d*f*h + 2*B*(d*f*g + d*e*h + c*f*h))
+ b^2*(-10*A*d*f*h*(d*f*g + d*e*h + c*f*h) + B*(8*c^2*f^2*h^2 + 7*c*d*f*h*(f*g + e*h) + d^2*(8*f^2*g^2 + 7*e*f
*g*h + 8*e^2*h^2))))*(e + f*x)*(g + h*x)) + b*d^2*Sqrt[-c + (d*e)/f]*f*h*(c + d*x)*(e + f*x)*(g + h*x)*(-5*A*b
*d*f*h - 10*a*B*d*f*h + b*B*(4*c*f*h + d*(4*f*g + 4*e*h - 3*f*h*x))) - I*(d*e - c*f)*h*(15*a^2*B*d^2*f^2*h^2 -
10*a*b*d*f*h*(-3*A*d*f*h + 2*B*(d*f*g + d*e*h + c*f*h)) + b^2*(-10*A*d*f*h*(d*f*g + d*e*h + c*f*h) + B*(8*c^2
*f^2*h^2 + 7*c*d*f*h*(f*g + e*h) + d^2*(8*f^2*g^2 + 7*e*f*g*h + 8*e^2*h^2))))*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x
))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*
f*g - c*f*h)/(d*e*h - c*f*h)] - I*d*h*(15*a^2*d^2*f^2*(-(B*e) + A*f)*h^2 + 10*a*b*d*f*h*(-3*A*d*e*f*h + B*c*f*
(-(f*g) + e*h) + B*d*e*(f*g + 2*e*h)) - b^2*(-5*A*d*f*h*(c*f*(-(f*g) + e*h) + d*e*(f*g + 2*e*h)) + B*(4*c^2*f^
2*h*(-(f*g) + e*h) + c*d*f*(-4*f^2*g^2 + e*f*g*h + 3*e^2*h^2) + d^2*e*(4*f^2*g^2 + 3*e*f*g*h + 8*e^2*h^2))))*(
c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c
+ (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)]))/(15*d^4*Sqrt[-c + (d*e)/f]*f^3*h^3*Sqrt[c + d*x
]*Sqrt[e + f*x]*Sqrt[g + h*x])

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Maple [B]  time = 0.065, size = 8121, normalized size = 11.6 \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)^2*(B*x+A)/(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{{\left (B x + A\right )}{\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((b*x+a)^2*(B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B b^{2} x^{3} + A a^{2} +{\left (2 \, B a b + A b^{2}\right )} x^{2} +{\left (B a^{2} + 2 \, A a b\right )} x\right )} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}{d f h x^{3} + c e g +{\left (d f g +{\left (d e + c f\right )} h\right )} x^{2} +{\left (c e h +{\left (d e + c f\right )} g\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*b^2*x^3 + A*a^2 + (2*B*a*b + A*b^2)*x^2 + (B*a^2 + 2*A*a*b)*x)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*
x + g)/(d*f*h*x^3 + c*e*g + (d*f*g + (d*e + c*f)*h)*x^2 + (c*e*h + (d*e + c*f)*g)*x), x)

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Sympy [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)**2*(B*x+A)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

Timed out

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

[Out]

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