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

Optimal. Leaf size=1097 \[ \frac{2 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (a+b x)^2}{7 d f h}+\frac{4 C (2 a d f h-3 b (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (a+b x)}{35 d^2 f^2 h^2}-\frac{4 \sqrt{c f-d e} \left (\left (35 A d^2 f^2 (d f g+d e h+c f h) h^2+2 C \left (2 \left (6 f^3 g^3+5 e f^2 h g^2+5 e^2 f h^2 g+6 e^3 h^3\right ) d^3+c f h \left (10 f^2 g^2+9 e f h g+10 e^2 h^2\right ) d^2+10 c^2 f^2 h^2 (f g+e h) d+12 c^3 f^3 h^3\right )\right ) b^2-7 a d f h \left (15 A d^2 f^2 h^2+C \left (\left (8 f^2 g^2+7 e f h g+8 e^2 h^2\right ) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right )\right ) b+35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)\right ) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{g+h x} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{105 d^4 f^{7/2} h^4 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 \sqrt{c f-d e} \left (\left (35 A d^2 f^2 (c h (f g-e h)+d g (2 f g+e h)) h^2+C \left (g \left (48 f^3 g^3+16 e f^2 h g^2+17 e^2 f h^2 g+24 e^3 h^3\right ) d^3+2 c h \left (8 f^3 g^3+e f^2 h g^2+3 e^2 f h^2 g-12 e^3 h^3\right ) d^2+c^2 f h^2 \left (17 f^2 g^2+6 e f h g-23 e^2 h^2\right ) d+24 c^3 f^2 h^3 (f g-e h)\right )\right ) b^2-14 a d f h \left (15 A d^2 f^2 g h^2+C \left (g \left (8 f^2 g^2+3 e f h g+4 e^2 h^2\right ) d^2+c h \left (3 f^2 g^2+e f h g-4 e^2 h^2\right ) d+4 c^2 f h^2 (f g-e h)\right )\right ) b+35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+C (c h (f g-e h)+d g (2 f g+e h))\right )\right ) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right ),\frac{(d e-c f) h}{f (d g-c h)}\right )}{105 d^4 f^{7/2} h^4 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 (4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a d f h-2 b (d f g+d e h+c f h))+5 b d f h (7 A b d f h-C (5 b (d e g+c f g+c e h)+2 a (d f g+d e h+c f h)))) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{105 d^3 f^3 h^3} \]

[Out]

(2*(4*C*(2*a*d*f*h - 3*b*(d*f*g + d*e*h + c*f*h))*(a*d*f*h - 2*b*(d*f*g + d*e*h + c*f*h)) + 5*b*d*f*h*(7*A*b*d
*f*h - C*(5*b*(d*e*g + c*f*g + c*e*h) + 2*a*(d*f*g + d*e*h + c*f*h))))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*
x])/(105*d^3*f^3*h^3) + (4*C*(2*a*d*f*h - 3*b*(d*f*g + d*e*h + c*f*h))*(a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*S
qrt[g + h*x])/(35*d^2*f^2*h^2) + (2*C*(a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(7*d*f*h) - (4*Sq
rt[-(d*e) + c*f]*(35*a^2*C*d^2*f^2*h^2*(d*f*g + d*e*h + c*f*h) - 7*a*b*d*f*h*(15*A*d^2*f^2*h^2 + C*(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))) + b^2*(35*A*d^2*f^2*h^2*(d*f*g + d*e*h
+ c*f*h) + 2*C*(12*c^3*f^3*h^3 + 10*c^2*d*f^2*h^2*(f*g + e*h) + c*d^2*f*h*(10*f^2*g^2 + 9*e*f*g*h + 10*e^2*h^
2) + 2*d^3*(6*f^3*g^3 + 5*e*f^2*g^2*h + 5*e^2*f*g*h^2 + 6*e^3*h^3))))*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))])/(105*d^4
*f^(7/2)*h^4*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) + (2*Sqrt[-(d*e) + c*f]*(35*a^2*d^2*f^2*h^2*(3*A*d
*f*h^2 + C*(c*h*(f*g - e*h) + d*g*(2*f*g + e*h))) - 14*a*b*d*f*h*(15*A*d^2*f^2*g*h^2 + C*(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))) + b^2*(35*A*d^2*f^
2*h^2*(c*h*(f*g - e*h) + d*g*(2*f*g + e*h)) + C*(24*c^3*f^2*h^3*(f*g - e*h) + c^2*d*f*h^2*(17*f^2*g^2 + 6*e*f*
g*h - 23*e^2*h^2) + 2*c*d^2*h*(8*f^3*g^3 + e*f^2*g^2*h + 3*e^2*f*g*h^2 - 12*e^3*h^3) + d^3*g*(48*f^3*g^3 + 16*
e*f^2*g^2*h + 17*e^2*f*g*h^2 + 24*e^3*h^3))))*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))])/(105*d^4*f^(7/
2)*h^4*Sqrt[e + f*x]*Sqrt[g + h*x])

________________________________________________________________________________________

Rubi [A]  time = 3.3008, antiderivative size = 1083, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {1601, 1600, 1615, 158, 114, 113, 121, 120} \[ \frac{2 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (a+b x)^2}{7 d f h}+\frac{4 C (2 a d f h-3 b (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (a+b x)}{35 d^2 f^2 h^2}-\frac{4 \sqrt{c f-d e} \left (\left (35 A d^2 f^2 (d f g+d e h+c f h) h^2+2 C \left (2 \left (6 f^3 g^3+5 e f^2 h g^2+5 e^2 f h^2 g+6 e^3 h^3\right ) d^3+c f h \left (10 f^2 g^2+9 e f h g+10 e^2 h^2\right ) d^2+10 c^2 f^2 h^2 (f g+e h) d+12 c^3 f^3 h^3\right )\right ) b^2-7 a d f h \left (15 A d^2 f^2 h^2+C \left (\left (8 f^2 g^2+7 e f h g+8 e^2 h^2\right ) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right )\right ) b+35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)\right ) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{g+h x} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{105 d^4 f^{7/2} h^4 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 \sqrt{c f-d e} \left (\left (35 A d^2 f^2 (c h (f g-e h)+d g (2 f g+e h)) h^2+C \left (g \left (48 f^3 g^3+16 e f^2 h g^2+17 e^2 f h^2 g+24 e^3 h^3\right ) d^3+2 c h \left (8 f^3 g^3+e f^2 h g^2+3 e^2 f h^2 g-12 e^3 h^3\right ) d^2+c^2 f h^2 \left (17 f^2 g^2+6 e f h g-23 e^2 h^2\right ) d+24 c^3 f^2 h^3 (f g-e h)\right )\right ) b^2-14 a d f h \left (15 A d^2 f^2 g h^2+C \left (g \left (8 f^2 g^2+3 e f h g+4 e^2 h^2\right ) d^2+c h \left (3 f^2 g^2+e f h g-4 e^2 h^2\right ) d+4 c^2 f h^2 (f g-e h)\right )\right ) b+35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C (f g-e h) h+C d g (2 f g+e h)\right )\right ) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{105 d^4 f^{7/2} h^4 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 \left (8 C d f h a^2-38 b C (d f g+d e h+c f h) a+\frac{24 b^2 C (d f g+d e h+c f h)^2}{d f h}+35 A b^2 d f h-25 b^2 C (d e g+c f g+c e h)\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{105 d^2 f^2 h^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(35*A*b^2*d*f*h + 8*a^2*C*d*f*h - 25*b^2*C*(d*e*g + c*f*g + c*e*h) - 38*a*b*C*(d*f*g + d*e*h + c*f*h) + (24
*b^2*C*(d*f*g + d*e*h + c*f*h)^2)/(d*f*h))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(105*d^2*f^2*h^2) + (4*C
*(2*a*d*f*h - 3*b*(d*f*g + d*e*h + c*f*h))*(a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(35*d^2*f^2*h^
2) + (2*C*(a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(7*d*f*h) - (4*Sqrt[-(d*e) + c*f]*(35*a^2*C*d
^2*f^2*h^2*(d*f*g + d*e*h + c*f*h) - 7*a*b*d*f*h*(15*A*d^2*f^2*h^2 + C*(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))) + b^2*(35*A*d^2*f^2*h^2*(d*f*g + d*e*h + c*f*h) + 2*C*(12*c^3*f^3*
h^3 + 10*c^2*d*f^2*h^2*(f*g + e*h) + c*d^2*f*h*(10*f^2*g^2 + 9*e*f*g*h + 10*e^2*h^2) + 2*d^3*(6*f^3*g^3 + 5*e*
f^2*g^2*h + 5*e^2*f*g*h^2 + 6*e^3*h^3))))*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))])/(105*d^4*f^(7/2)*h^4*Sqrt[e + f*x]*S
qrt[(d*(g + h*x))/(d*g - c*h)]) + (2*Sqrt[-(d*e) + c*f]*(35*a^2*d^2*f^2*h^2*(3*A*d*f*h^2 + c*C*h*(f*g - e*h) +
C*d*g*(2*f*g + e*h)) - 14*a*b*d*f*h*(15*A*d^2*f^2*g*h^2 + C*(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))) + b^2*(35*A*d^2*f^2*h^2*(c*h*(f*g - e*h) + d*g
*(2*f*g + e*h)) + C*(24*c^3*f^2*h^3*(f*g - e*h) + c^2*d*f*h^2*(17*f^2*g^2 + 6*e*f*g*h - 23*e^2*h^2) + 2*c*d^2*
h*(8*f^3*g^3 + e*f^2*g^2*h + 3*e^2*f*g*h^2 - 12*e^3*h^3) + d^3*g*(48*f^3*g^3 + 16*e*f^2*g^2*h + 17*e^2*f*g*h^2
+ 24*e^3*h^3))))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticF[ArcSin[(Sqrt[f]*Sq
rt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(105*d^4*f^(7/2)*h^4*Sqrt[e + f*x]*Sqrt[g
+ h*x])

Rule 1601

Int[(((a_.) + (b_.)*(x_))^(m_.)*((A_.) + (C_.)*(x_)^2))/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqr
t[(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]*Sqrt[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*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 + 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, C}, 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 \left (A+C x^2\right )}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=\frac{2 C (a+b x)^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{7 d f h}+\frac{\int \frac{(a+b x) \left (-4 b c C e g+7 a A d f h-a C (d e g+c f g+c e h)+(7 A b d f h-5 b C (d e g+c f g+c e h)-2 a C (d f g+d e h+c f h)) x+2 C (2 a d f h-3 b (d f g+d e h+c f h)) x^2\right )}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{7 d f h}\\ &=\frac{4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{35 d^2 f^2 h^2}+\frac{2 C (a+b x)^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{7 d f h}+\frac{\int \frac{-5 a d f h (4 b c C e g-7 a A d f h+a C (d e g+c f g+c e h))-2 C (2 b c e g+a (d e g+c f g+c e h)) (2 a d f h-3 b (d f g+d e h+c f h))-2 \left (C (3 b (d e g+c f g+c e h)+2 a (d f g+d e h+c f h)) (2 a d f h-3 b (d f g+d e h+c f h))+5 d f h \left (2 b^2 c C e g+a^2 C (d f g+d e h+c f h)-a b (7 A d f h-3 C (d e g+c f g+c e h))\right )\right ) x+(4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a d f h-2 b (d f g+d e h+c f h))+5 b d f h (7 A b d f h-5 b C (d e g+c f g+c e h)-2 a C (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 \left (35 A b^2 d f h+8 a^2 C d f h-25 b^2 C (d e g+c f g+c e h)-38 a b C (d f g+d e h+c f h)+\frac{24 b^2 C (d f g+d e h+c f h)^2}{d f h}\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{105 d^2 f^2 h^2}+\frac{4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{35 d^2 f^2 h^2}+\frac{2 C (a+b x)^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{7 d f h}+\frac{2 \int \frac{\frac{1}{2} d \left (35 a^2 d^2 f^2 h^2 (3 A d f h-C (d e g+c f g+c e h))+28 a b C d f h \left (2 d^2 e g (f g+e h)+2 c^2 f h (f g+e h)+c d \left (2 f^2 g^2+3 e f g h+2 e^2 h^2\right )\right )-b^2 \left (35 A d^2 f^2 h^2 (d e g+c f g+c e h)+C \left (24 c^3 f^2 h^2 (f g+e h)+c^2 d f h \left (23 f^2 g^2+34 e f g h+23 e^2 h^2\right )+d^3 e g \left (24 f^2 g^2+23 e f g h+24 e^2 h^2\right )+2 c d^2 \left (12 f^3 g^3+17 e f^2 g^2 h+17 e^2 f g h^2+12 e^3 h^3\right )\right )\right )\right )-d \left (35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)-7 a b d f h \left (15 A d^2 f^2 h^2+C \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 )+b^2 \left (35 A d^2 f^2 h^2 (d f g+d e h+c f h)+2 C \left (12 c^3 f^3 h^3+10 c^2 d f^2 h^2 (f g+e h)+c d^2 f h \left (10 f^2 g^2+9 e f g h+10 e^2 h^2\right )+2 d^3 \left (6 f^3 g^3+5 e f^2 g^2 h+5 e^2 f g h^2+6 e^3 h^3\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 \left (35 A b^2 d f h+8 a^2 C d f h-25 b^2 C (d e g+c f g+c e h)-38 a b C (d f g+d e h+c f h)+\frac{24 b^2 C (d f g+d e h+c f h)^2}{d f h}\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{105 d^2 f^2 h^2}+\frac{4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{35 d^2 f^2 h^2}+\frac{2 C (a+b x)^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{7 d f h}-\frac{\left (2 \left (35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)-7 a b d f h \left (15 A d^2 f^2 h^2+C \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 )+b^2 \left (35 A d^2 f^2 h^2 (d f g+d e h+c f h)+2 C \left (12 c^3 f^3 h^3+10 c^2 d f^2 h^2 (f g+e h)+c d^2 f h \left (10 f^2 g^2+9 e f g h+10 e^2 h^2\right )+2 d^3 \left (6 f^3 g^3+5 e f^2 g^2 h+5 e^2 f g h^2+6 e^3 h^3\right )\right )\right )\right )\right ) \int \frac{\sqrt{g+h x}}{\sqrt{c+d x} \sqrt{e+f x}} \, dx}{105 d^3 f^3 h^4}+\frac{\left (35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-14 a b d f h \left (15 A d^2 f^2 g h^2+C \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 )+b^2 \left (35 A d^2 f^2 h^2 (c h (f g-e h)+d g (2 f g+e h))+C \left (24 c^3 f^2 h^3 (f g-e h)+c^2 d f h^2 \left (17 f^2 g^2+6 e f g h-23 e^2 h^2\right )+2 c d^2 h \left (8 f^3 g^3+e f^2 g^2 h+3 e^2 f g h^2-12 e^3 h^3\right )+d^3 g \left (48 f^3 g^3+16 e f^2 g^2 h+17 e^2 f g h^2+24 e^3 h^3\right )\right )\right )\right ) \int \frac{1}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{105 d^3 f^3 h^4}\\ &=\frac{2 \left (35 A b^2 d f h+8 a^2 C d f h-25 b^2 C (d e g+c f g+c e h)-38 a b C (d f g+d e h+c f h)+\frac{24 b^2 C (d f g+d e h+c f h)^2}{d f h}\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{105 d^2 f^2 h^2}+\frac{4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{35 d^2 f^2 h^2}+\frac{2 C (a+b x)^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{7 d f h}+\frac{\left (\left (35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-14 a b d f h \left (15 A d^2 f^2 g h^2+C \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 )+b^2 \left (35 A d^2 f^2 h^2 (c h (f g-e h)+d g (2 f g+e h))+C \left (24 c^3 f^2 h^3 (f g-e h)+c^2 d f h^2 \left (17 f^2 g^2+6 e f g h-23 e^2 h^2\right )+2 c d^2 h \left (8 f^3 g^3+e f^2 g^2 h+3 e^2 f g h^2-12 e^3 h^3\right )+d^3 g \left (48 f^3 g^3+16 e f^2 g^2 h+17 e^2 f g h^2+24 e^3 h^3\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}{105 d^3 f^3 h^4 \sqrt{e+f x}}-\frac{\left (2 \left (35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)-7 a b d f h \left (15 A d^2 f^2 h^2+C \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 )+b^2 \left (35 A d^2 f^2 h^2 (d f g+d e h+c f h)+2 C \left (12 c^3 f^3 h^3+10 c^2 d f^2 h^2 (f g+e h)+c d^2 f h \left (10 f^2 g^2+9 e f g h+10 e^2 h^2\right )+2 d^3 \left (6 f^3 g^3+5 e f^2 g^2 h+5 e^2 f g h^2+6 e^3 h^3\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}{105 d^3 f^3 h^4 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}\\ &=\frac{2 \left (35 A b^2 d f h+8 a^2 C d f h-25 b^2 C (d e g+c f g+c e h)-38 a b C (d f g+d e h+c f h)+\frac{24 b^2 C (d f g+d e h+c f h)^2}{d f h}\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{105 d^2 f^2 h^2}+\frac{4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{35 d^2 f^2 h^2}+\frac{2 C (a+b x)^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{7 d f h}-\frac{4 \sqrt{-d e+c f} \left (35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)-7 a b d f h \left (15 A d^2 f^2 h^2+C \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 )+b^2 \left (35 A d^2 f^2 h^2 (d f g+d e h+c f h)+2 C \left (12 c^3 f^3 h^3+10 c^2 d f^2 h^2 (f g+e h)+c d^2 f h \left (10 f^2 g^2+9 e f g h+10 e^2 h^2\right )+2 d^3 \left (6 f^3 g^3+5 e f^2 g^2 h+5 e^2 f g h^2+6 e^3 h^3\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 )}{105 d^4 f^{7/2} h^4 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{\left (\left (35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-14 a b d f h \left (15 A d^2 f^2 g h^2+C \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 )+b^2 \left (35 A d^2 f^2 h^2 (c h (f g-e h)+d g (2 f g+e h))+C \left (24 c^3 f^2 h^3 (f g-e h)+c^2 d f h^2 \left (17 f^2 g^2+6 e f g h-23 e^2 h^2\right )+2 c d^2 h \left (8 f^3 g^3+e f^2 g^2 h+3 e^2 f g h^2-12 e^3 h^3\right )+d^3 g \left (48 f^3 g^3+16 e f^2 g^2 h+17 e^2 f g h^2+24 e^3 h^3\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}{105 d^3 f^3 h^4 \sqrt{e+f x} \sqrt{g+h x}}\\ &=\frac{2 \left (35 A b^2 d f h+8 a^2 C d f h-25 b^2 C (d e g+c f g+c e h)-38 a b C (d f g+d e h+c f h)+\frac{24 b^2 C (d f g+d e h+c f h)^2}{d f h}\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{105 d^2 f^2 h^2}+\frac{4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{35 d^2 f^2 h^2}+\frac{2 C (a+b x)^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{7 d f h}-\frac{4 \sqrt{-d e+c f} \left (35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)-7 a b d f h \left (15 A d^2 f^2 h^2+C \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 )+b^2 \left (35 A d^2 f^2 h^2 (d f g+d e h+c f h)+2 C \left (12 c^3 f^3 h^3+10 c^2 d f^2 h^2 (f g+e h)+c d^2 f h \left (10 f^2 g^2+9 e f g h+10 e^2 h^2\right )+2 d^3 \left (6 f^3 g^3+5 e f^2 g^2 h+5 e^2 f g h^2+6 e^3 h^3\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 )}{105 d^4 f^{7/2} h^4 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 \sqrt{-d e+c f} \left (35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-14 a b d f h \left (15 A d^2 f^2 g h^2+C \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 )+b^2 \left (35 A d^2 f^2 h^2 (c h (f g-e h)+d g (2 f g+e h))+C \left (24 c^3 f^2 h^3 (f g-e h)+c^2 d f h^2 \left (17 f^2 g^2+6 e f g h-23 e^2 h^2\right )+2 c d^2 h \left (8 f^3 g^3+e f^2 g^2 h+3 e^2 f g h^2-12 e^3 h^3\right )+d^3 g \left (48 f^3 g^3+16 e f^2 g^2 h+17 e^2 f g h^2+24 e^3 h^3\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 )}{105 d^4 f^{7/2} h^4 \sqrt{e+f x} \sqrt{g+h x}}\\ \end{align*}

Mathematica [C]  time = 18.6955, size = 18383, normalized size = 16.76 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.076, size = 12279, normalized size = 11.2 \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*(C*x^2+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 (C x^{2} + 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*(C*x^2+A)/(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]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C b^{2} x^{4} + 2 \, C a b x^{3} + 2 \, A a b x + A a^{2} +{\left (C a^{2} + A b^{2}\right )} x^{2}\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*(C*x^2+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="fricas")

[Out]

integral((C*b^2*x^4 + 2*C*a*b*x^3 + 2*A*a*b*x + A*a^2 + (C*a^2 + A*b^2)*x^2)*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*(C*x**2+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 (C x^{2} + 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*(C*x^2+A)/(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)