\(\int \frac {(a+b x)^2 (A+C x^2)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [26]

Optimal result

Integrand size = 42, antiderivative size = 1097 \[ \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 (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}+\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 (\arcsin \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)+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}} \operatorname {EllipticF}\left (\arcsin \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}} \]

[Out]

2/105*(4*C*(2*a*d*f*h-3*b*(c*f*h+d*e*h+d*f*g))*(a*d*f*h-2*b*(c*f*h+d*e*h+d*f*g))+5*b*d*f*h*(7*A*b*d*f*h-C*(5*b
*(c*e*h+c*f*g+d*e*g)+2*a*(c*f*h+d*e*h+d*f*g))))*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/d^3/f^3/h^3+4/35*C*(
2*a*d*f*h-3*b*(c*f*h+d*e*h+d*f*g))*(b*x+a)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/d^2/f^2/h^2+2/7*C*(b*x+a)
^2*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/d/f/h-4/105*(35*a^2*C*d^2*f^2*h^2*(c*f*h+d*e*h+d*f*g)-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*(e*h+f*g)+d^2*(8*e^2*h^2+7*e*f*g*h+8*f^2*g^2)))+b^2*(35*A*d^2*f^
2*h^2*(c*f*h+d*e*h+d*f*g)+2*C*(12*c^3*f^3*h^3+10*c^2*d*f^2*h^2*(e*h+f*g)+c*d^2*f*h*(10*e^2*h^2+9*e*f*g*h+10*f^
2*g^2)+2*d^3*(6*e^3*h^3+5*e^2*f*g*h^2+5*e*f^2*g^2*h+6*f^3*g^3))))*EllipticE(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1
/2),((-c*f+d*e)*h/f/(-c*h+d*g))^(1/2))*(c*f-d*e)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*(h*x+g)^(1/2)/d^4/f^(7/2)/
h^4/(f*x+e)^(1/2)/(d*(h*x+g)/(-c*h+d*g))^(1/2)+2/105*(35*a^2*d^2*f^2*h^2*(3*A*d*f*h^2+C*(c*h*(-e*h+f*g)+d*g*(e
*h+2*f*g)))-14*a*b*d*f*h*(15*A*d^2*f^2*g*h^2+C*(4*c^2*f*h^2*(-e*h+f*g)+c*d*h*(-4*e^2*h^2+e*f*g*h+3*f^2*g^2)+d^
2*g*(4*e^2*h^2+3*e*f*g*h+8*f^2*g^2)))+b^2*(35*A*d^2*f^2*h^2*(c*h*(-e*h+f*g)+d*g*(e*h+2*f*g))+C*(24*c^3*f^2*h^3
*(-e*h+f*g)+c^2*d*f*h^2*(-23*e^2*h^2+6*e*f*g*h+17*f^2*g^2)+2*c*d^2*h*(-12*e^3*h^3+3*e^2*f*g*h^2+e*f^2*g^2*h+8*
f^3*g^3)+d^3*g*(24*e^3*h^3+17*e^2*f*g*h^2+16*e*f^2*g^2*h+48*f^3*g^3))))*EllipticF(f^(1/2)*(d*x+c)^(1/2)/(c*f-d
*e)^(1/2),((-c*f+d*e)*h/f/(-c*h+d*g))^(1/2))*(c*f-d*e)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*(d*(h*x+g)/(-c*h+d*g
))^(1/2)/d^4/f^(7/2)/h^4/(f*x+e)^(1/2)/(h*x+g)^(1/2)

Rubi [A] (verified)

Time = 2.13 (sec) , antiderivative size = 1083, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1615, 1614, 1629, 164, 115, 114, 122, 121} \[ \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 \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 (\arcsin \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}} \operatorname {EllipticF}\left (\arcsin \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} \]

[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 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*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)))], 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 115

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 121

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*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)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && Si
mplerQ[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])

Rule 122

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 164

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 1614

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[(((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 1629

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Result contains complex when optimal does not.

Time = 32.54 (sec) , antiderivative size = 1291, normalized size of antiderivative = 1.18 \[ \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 \left (-2 d^2 \sqrt {-c+\frac {d e}{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 ) (e+f x) (g+h x)+d^2 \sqrt {-c+\frac {d e}{f}} f h (c+d x) (e+f x) (g+h x) \left (35 a^2 C d^2 f^2 h^2-14 a b C d f h (4 c f h+d (4 f g+4 e h-3 f h x))+b^2 \left (35 A d^2 f^2 h^2+C \left (24 c^2 f^2 h^2+c d f h (23 f g+23 e h-18 f h x)+d^2 \left (24 e^2 h^2+e f h (23 g-18 h x)+3 f^2 \left (8 g^2-6 g h x+5 h^2 x^2\right )\right )\right )\right )\right )-2 i (d e-c f) h \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 ) (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 \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )+i d h \left (35 a^2 d^2 f^2 h^2 \left (3 A d f^2 h+c C f (-f g+e h)+C d e (f g+2 e h)\right )-14 a b d f h \left (15 A d^2 e f^2 h^2+C \left (4 c^2 f^2 h (-f g+e h)+c d f \left (-4 f^2 g^2+e f g h+3 e^2 h^2\right )+d^2 e \left (4 f^2 g^2+3 e f g h+8 e^2 h^2\right )\right )\right )+b^2 \left (35 A d^2 f^2 h^2 (c f (-f g+e h)+d e (f g+2 e h))+C \left (24 c^3 f^3 h^2 (-f g+e h)+c^2 d f^2 h \left (-23 f^2 g^2+6 e f g h+17 e^2 h^2\right )+2 c d^2 f \left (-12 f^3 g^3+3 e f^2 g^2 h+e^2 f g h^2+8 e^3 h^3\right )+d^3 e \left (24 f^3 g^3+17 e f^2 g^2 h+16 e^2 f g h^2+48 e^3 h^3\right )\right )\right )\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)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )\right )}{105 d^5 \sqrt {-c+\frac {d e}{f}} f^4 h^4 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \]

[In]

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

[Out]

(2*(-2*d^2*Sqrt[-c + (d*e)/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))))*(e + f*x)*(g + h*x) + d^2*S
qrt[-c + (d*e)/f]*f*h*(c + d*x)*(e + f*x)*(g + h*x)*(35*a^2*C*d^2*f^2*h^2 - 14*a*b*C*d*f*h*(4*c*f*h + d*(4*f*g
+ 4*e*h - 3*f*h*x)) + b^2*(35*A*d^2*f^2*h^2 + C*(24*c^2*f^2*h^2 + c*d*f*h*(23*f*g + 23*e*h - 18*f*h*x) + d^2*
(24*e^2*h^2 + e*f*h*(23*g - 18*h*x) + 3*f^2*(8*g^2 - 6*g*h*x + 5*h^2*x^2))))) - (2*I)*(d*e - c*f)*h*(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))))*(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*(35*a^2*d^2*f^2*h^2*(3*A*d*f^2*h + c*C*f*(-(f*g) + e*h) + C*d*e*(f*g + 2*e*h)) - 14*a*b*d*f*h*(15*A*d^2*e*f
^2*h^2 + C*(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))) + b^2*(35*A*d^2*f^2*h^2*(c*f*(-(f*g) + e*h) + d*e*(f*g + 2*e*h)) + C*(24*c^3*f^3*h^2*(-(f*g)
+ e*h) + c^2*d*f^2*h*(-23*f^2*g^2 + 6*e*f*g*h + 17*e^2*h^2) + 2*c*d^2*f*(-12*f^3*g^3 + 3*e*f^2*g^2*h + e^2*f*
g*h^2 + 8*e^3*h^3) + d^3*e*(24*f^3*g^3 + 17*e*f^2*g^2*h + 16*e^2*f*g*h^2 + 48*e^3*h^3))))*(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)]))/(105*d^5*Sqrt[-c + (d*e)/f]*f^4*h^4*Sqrt[c + d*x]*Sqrt[e + f*x]*Sq
rt[g + h*x])

Maple [A] (verified)

Time = 3.37 (sec) , antiderivative size = 1238, normalized size of antiderivative = 1.13

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1238\)
default	\(\text {Expression too large to display}\)	\(12279\)

[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,method=_RETURNVERBOSE)

[Out]

((d*x+c)*(f*x+e)*(h*x+g))^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)*(2/7*C*b^2/d/f/h*x^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)^(1/2)+2/5*(2*C*a*b-2/7*C*b^2/d/f/h*(3*c*f*h+3*d*e*h+3
*d*f*g))/d/f/h*x*(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)^(1/2)+2/3*(b^2*A+C*a^
2-2/7*C*b^2/d/f/h*(5/2*c*e*h+5/2*c*f*g+5/2*d*e*g)-2/5*(2*C*a*b-2/7*C*b^2/d/f/h*(3*c*f*h+3*d*e*h+3*d*f*g))/d/f/
h*(2*c*f*h+2*d*e*h+2*d*f*g))/d/f/h*(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)^(1/
2)+2*(a^2*A-2/5*(2*C*a*b-2/7*C*b^2/d/f/h*(3*c*f*h+3*d*e*h+3*d*f*g))/d/f/h*c*e*g-2/3*(b^2*A+C*a^2-2/7*C*b^2/d/f
/h*(5/2*c*e*h+5/2*c*f*g+5/2*d*e*g)-2/5*(2*C*a*b-2/7*C*b^2/d/f/h*(3*c*f*h+3*d*e*h+3*d*f*g))/d/f/h*(2*c*f*h+2*d*
e*h+2*d*f*g))/d/f/h*(1/2*c*e*h+1/2*c*f*g+1/2*d*e*g))*(g/h-e/f)*((x+g/h)/(g/h-e/f))^(1/2)*((x+c/d)/(-g/h+c/d))^
(1/2)*((x+e/f)/(-g/h+e/f))^(1/2)/(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)^(1/2)
*EllipticF(((x+g/h)/(g/h-e/f))^(1/2),((-g/h+e/f)/(-g/h+c/d))^(1/2))+2*(2*a*b*A-4/7*C*b^2/d/f/h*c*e*g-2/5*(2*C*
a*b-2/7*C*b^2/d/f/h*(3*c*f*h+3*d*e*h+3*d*f*g))/d/f/h*(3/2*c*e*h+3/2*c*f*g+3/2*d*e*g)-2/3*(b^2*A+C*a^2-2/7*C*b^
2/d/f/h*(5/2*c*e*h+5/2*c*f*g+5/2*d*e*g)-2/5*(2*C*a*b-2/7*C*b^2/d/f/h*(3*c*f*h+3*d*e*h+3*d*f*g))/d/f/h*(2*c*f*h
+2*d*e*h+2*d*f*g))/d/f/h*(c*f*h+d*e*h+d*f*g))*(g/h-e/f)*((x+g/h)/(g/h-e/f))^(1/2)*((x+c/d)/(-g/h+c/d))^(1/2)*(
(x+e/f)/(-g/h+e/f))^(1/2)/(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)^(1/2)*((-g/h
+c/d)*EllipticE(((x+g/h)/(g/h-e/f))^(1/2),((-g/h+e/f)/(-g/h+c/d))^(1/2))-c/d*EllipticF(((x+g/h)/(g/h-e/f))^(1/
2),((-g/h+e/f)/(-g/h+c/d))^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.17 (sec) , antiderivative size = 1665, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b x)^2 \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Too large to display} \]

[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]

2/315*(3*(15*C*b^2*d^4*f^4*h^4*x^2 + 24*C*b^2*d^4*f^4*g^2*h^2 + (23*C*b^2*d^4*e*f^3 + (23*C*b^2*c*d^3 - 56*C*a
*b*d^4)*f^4)*g*h^3 + (24*C*b^2*d^4*e^2*f^2 + (23*C*b^2*c*d^3 - 56*C*a*b*d^4)*e*f^3 + (24*C*b^2*c^2*d^2 - 56*C*
a*b*c*d^3 + 35*(C*a^2 + A*b^2)*d^4)*f^4)*h^4 - 6*(3*C*b^2*d^4*f^4*g*h^3 + (3*C*b^2*d^4*e*f^3 + (3*C*b^2*c*d^3
- 7*C*a*b*d^4)*f^4)*h^4)*x)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g) + (48*C*b^2*d^4*f^4*g^4 + 16*(C*b^2*d^4*
e*f^3 + (C*b^2*c*d^3 - 7*C*a*b*d^4)*f^4)*g^3*h + (11*C*b^2*d^4*e^2*f^2 + 14*(C*b^2*c*d^3 - 3*C*a*b*d^4)*e*f^3
+ (11*C*b^2*c^2*d^2 - 42*C*a*b*c*d^3 + 70*(C*a^2 + A*b^2)*d^4)*f^4)*g^2*h^2 + (16*C*b^2*d^4*e^3*f + 14*(C*b^2*
c*d^3 - 3*C*a*b*d^4)*e^2*f^2 + 7*(2*C*b^2*c^2*d^2 - 6*C*a*b*c*d^3 + 5*(C*a^2 + A*b^2)*d^4)*e*f^3 + (16*C*b^2*c
^3*d - 42*C*a*b*c^2*d^2 - 210*A*a*b*d^4 + 35*(C*a^2 + A*b^2)*c*d^3)*f^4)*g*h^3 + (48*C*b^2*d^4*e^4 + 16*(C*b^2
*c*d^3 - 7*C*a*b*d^4)*e^3*f + (11*C*b^2*c^2*d^2 - 42*C*a*b*c*d^3 + 70*(C*a^2 + A*b^2)*d^4)*e^2*f^2 + (16*C*b^2
*c^3*d - 42*C*a*b*c^2*d^2 - 210*A*a*b*d^4 + 35*(C*a^2 + A*b^2)*c*d^3)*e*f^3 + (48*C*b^2*c^4 - 112*C*a*b*c^3*d
- 210*A*a*b*c*d^3 + 315*A*a^2*d^4 + 70*(C*a^2 + A*b^2)*c^2*d^2)*f^4)*h^4)*sqrt(d*f*h)*weierstrassPInverse(4/3*
(d^2*f^2*g^2 - (d^2*e*f + c*d*f^2)*g*h + (d^2*e^2 - c*d*e*f + c^2*f^2)*h^2)/(d^2*f^2*h^2), -4/27*(2*d^3*f^3*g^
3 - 3*(d^3*e*f^2 + c*d^2*f^3)*g^2*h - 3*(d^3*e^2*f - 4*c*d^2*e*f^2 + c^2*d*f^3)*g*h^2 + (2*d^3*e^3 - 3*c*d^2*e
^2*f - 3*c^2*d*e*f^2 + 2*c^3*f^3)*h^3)/(d^3*f^3*h^3), 1/3*(3*d*f*h*x + d*f*g + (d*e + c*f)*h)/(d*f*h)) + 6*(24
*C*b^2*d^4*f^4*g^3*h + 4*(5*C*b^2*d^4*e*f^3 + (5*C*b^2*c*d^3 - 14*C*a*b*d^4)*f^4)*g^2*h^2 + (20*C*b^2*d^4*e^2*
f^2 + (18*C*b^2*c*d^3 - 49*C*a*b*d^4)*e*f^3 + (20*C*b^2*c^2*d^2 - 49*C*a*b*c*d^3 + 35*(C*a^2 + A*b^2)*d^4)*f^4
)*g*h^3 + (24*C*b^2*d^4*e^3*f + 4*(5*C*b^2*c*d^3 - 14*C*a*b*d^4)*e^2*f^2 + (20*C*b^2*c^2*d^2 - 49*C*a*b*c*d^3
+ 35*(C*a^2 + A*b^2)*d^4)*e*f^3 + (24*C*b^2*c^3*d - 56*C*a*b*c^2*d^2 - 105*A*a*b*d^4 + 35*(C*a^2 + A*b^2)*c*d^
3)*f^4)*h^4)*sqrt(d*f*h)*weierstrassZeta(4/3*(d^2*f^2*g^2 - (d^2*e*f + c*d*f^2)*g*h + (d^2*e^2 - c*d*e*f + c^2
*f^2)*h^2)/(d^2*f^2*h^2), -4/27*(2*d^3*f^3*g^3 - 3*(d^3*e*f^2 + c*d^2*f^3)*g^2*h - 3*(d^3*e^2*f - 4*c*d^2*e*f^
2 + c^2*d*f^3)*g*h^2 + (2*d^3*e^3 - 3*c*d^2*e^2*f - 3*c^2*d*e*f^2 + 2*c^3*f^3)*h^3)/(d^3*f^3*h^3), weierstrass
PInverse(4/3*(d^2*f^2*g^2 - (d^2*e*f + c*d*f^2)*g*h + (d^2*e^2 - c*d*e*f + c^2*f^2)*h^2)/(d^2*f^2*h^2), -4/27*
(2*d^3*f^3*g^3 - 3*(d^3*e*f^2 + c*d^2*f^3)*g^2*h - 3*(d^3*e^2*f - 4*c*d^2*e*f^2 + c^2*d*f^3)*g*h^2 + (2*d^3*e^
3 - 3*c*d^2*e^2*f - 3*c^2*d*e*f^2 + 2*c^3*f^3)*h^3)/(d^3*f^3*h^3), 1/3*(3*d*f*h*x + d*f*g + (d*e + c*f)*h)/(d*
f*h))))/(d^5*f^5*h^5)

Sympy [F]

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

[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]

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

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^2 \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\left (C\,x^2+A\right )\,{\left (a+b\,x\right )}^2}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \]

[In]

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

[Out]

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