3.61 \(\int \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (A+B x+C x^2) \, dx\)

Optimal. Leaf size=1182 \[ \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {2 (2 a C d f-b (3 B d f-2 C (d e+c f))) \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{21 b d^2 f^2}-\frac {2 (7 b d f (b c C e+a C d e+a c C f-3 A b d f)+(a d f-4 b (d e+c f)) (2 a C d f-b (3 B d f-2 C (d e+c f)))) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{105 b^2 d^2 f^3}-\frac {2 \sqrt {a d-b c} \left (\left (2 C \left (8 d^4 e^4-4 c d^3 f e^3-3 c^2 d^2 f^2 e^2-4 c^3 d f^3 e+8 c^4 f^4\right )+3 d f \left (14 A d f \left (d^2 e^2-c d f e+c^2 f^2\right )-B \left (8 d^3 e^3-5 c d^2 f e^2-5 c^2 d f^2 e+8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (8 d^3 e^3-6 c d^2 f e^2-6 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (14 A d f (d e+c f)-B \left (5 d^2 e^2-6 c d f e+5 c^2 f^2\right )\right )\right ) b^3+3 a^2 d^2 f^2 \left (d f (5 B d e+5 B c f+14 A d f)-2 C \left (d^2 e^2-c d f e+c^2 f^2\right )\right ) b^2-8 a^3 d^3 f^3 (C d e+c C f+3 B d f) b+16 a^4 C d^4 f^4\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right ) \sqrt {e+f x}}{315 b^4 d^{7/2} f^4 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{315 b^3 d^3 f^3}-\frac {2 \sqrt {a d-b c} (b e-a f) (d e-c f) \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{315 b^4 d^{7/2} f^4 \sqrt {c+d x} \sqrt {e+f x}} \]

[Out]

2/9*C*(b*x+a)^(3/2)*(d*x+c)^(3/2)*(f*x+e)^(3/2)/b/d/f-2/21*(2*a*C*d*f-b*(3*B*d*f-2*C*(c*f+d*e)))*(d*x+c)^(3/2)
*(f*x+e)^(3/2)*(b*x+a)^(1/2)/b/d^2/f^2-2/105*(7*b*d*f*(-3*A*b*d*f+C*a*c*f+C*a*d*e+C*b*c*e)+(a*d*f-4*b*(c*f+d*e
))*(2*a*C*d*f-b*(3*B*d*f-2*C*(c*f+d*e))))*(f*x+e)^(3/2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^2/d^2/f^3+2/315*(8*a^3*C
*d^3*f^3+3*a^2*b*d^2*f^2*(-4*B*d*f-C*c*f+C*d*e)-3*a*b^2*d*f^2*((-7*A*d^2+C*c^2)*f+B*d*(-2*c*f+d*e))-b^3*(C*(-8
*c^3*f^3-3*c^2*d*e*f^2+16*d^3*e^3)+3*d*f*(7*A*d*f*(-c*f+2*d*e)-B*(-4*c^2*f^2-c*d*e*f+8*d^2*e^2))))*(b*x+a)^(1/
2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^3/d^3/f^3-2/315*(16*a^4*C*d^4*f^4-8*a^3*b*d^3*f^3*(3*B*d*f+C*c*f+C*d*e)+3*a^2
*b^2*d^2*f^2*(d*f*(14*A*d*f+5*B*c*f+5*B*d*e)-2*C*(c^2*f^2-c*d*e*f+d^2*e^2))-a*b^3*d*f*(C*(8*c^3*f^3-6*c^2*d*e*
f^2-6*c*d^2*e^2*f+8*d^3*e^3)+3*d*f*(14*A*d*f*(c*f+d*e)-B*(5*c^2*f^2-6*c*d*e*f+5*d^2*e^2)))+b^4*(2*C*(8*c^4*f^4
-4*c^3*d*e*f^3-3*c^2*d^2*e^2*f^2-4*c*d^3*e^3*f+8*d^4*e^4)+3*d*f*(14*A*d*f*(c^2*f^2-c*d*e*f+d^2*e^2)-B*(8*c^3*f
^3-5*c^2*d*e*f^2-5*c*d^2*e^2*f+8*d^3*e^3))))*EllipticE(d^(1/2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/(
-a*f+b*e))^(1/2))*(a*d-b*c)^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(f*x+e)^(1/2)/b^4/d^(7/2)/f^4/(d*x+c)^(1/2)/(b*
(f*x+e)/(-a*f+b*e))^(1/2)-2/315*(-a*f+b*e)*(-c*f+d*e)*(8*a^3*C*d^3*f^3+3*a^2*b*d^2*f^2*(-4*B*d*f-C*c*f+C*d*e)-
3*a*b^2*d*f^2*((-7*A*d^2+C*c^2)*f+B*d*(-2*c*f+d*e))-b^3*(C*(-8*c^3*f^3-3*c^2*d*e*f^2+16*d^3*e^3)+3*d*f*(7*A*d*
f*(-c*f+2*d*e)-B*(-4*c^2*f^2-c*d*e*f+8*d^2*e^2))))*EllipticF(d^(1/2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)
*f/d/(-a*f+b*e))^(1/2))*(a*d-b*c)^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(b*(f*x+e)/(-a*f+b*e))^(1/2)/b^4/d^(7/2)/
f^4/(d*x+c)^(1/2)/(f*x+e)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 4.17, antiderivative size = 1154, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 7, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {1615, 154, 158, 114, 113, 121, 120} \[ \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}+\frac {2 (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{21 b d^2 f^2}-\frac {2 (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{105 b^2 d^2 f^3}-\frac {2 \sqrt {a d-b c} \left (\left (2 C \left (8 d^4 e^4-4 c d^3 f e^3-3 c^2 d^2 f^2 e^2-4 c^3 d f^3 e+8 c^4 f^4\right )+3 d f \left (14 A d f \left (d^2 e^2-c d f e+c^2 f^2\right )-B \left (8 d^3 e^3-5 c d^2 f e^2-5 c^2 d f^2 e+8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (8 d^3 e^3-6 c d^2 f e^2-6 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (14 A d f (d e+c f)-B \left (5 d^2 e^2-6 c d f e+5 c^2 f^2\right )\right )\right ) b^3+3 a^2 d^2 f^2 \left (d f (5 B d e+5 B c f+14 A d f)-2 C \left (d^2 e^2-c d f e+c^2 f^2\right )\right ) b^2-8 a^3 d^3 f^3 (C d e+c C f+3 B d f) b+16 a^4 C d^4 f^4\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right ) \sqrt {e+f x}}{315 b^4 d^{7/2} f^4 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \left (\frac {8 C d f a^3}{b}+3 (C d e-c C f-4 B d f) a^2-3 b \left (\frac {C f c^2}{d}-2 B f c+B d e-7 A d f\right ) a+b^2 \left (\frac {8 C f c^3}{d^2}+\frac {3 C e c^2}{d}+21 A f c-42 A d e-B \left (\frac {12 f c^2}{d}+3 e c-\frac {24 d e^2}{f}\right )-\frac {16 C d e^3}{f^2}\right )\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{315 b^2 d f}-\frac {2 \sqrt {a d-b c} (b e-a f) (d e-c f) \left (-\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{315 b^4 d^{7/2} f^4 \sqrt {c+d x} \sqrt {e+f x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((8*a^3*C*d*f)/b - 3*a*b*(B*d*e - 2*B*c*f + (c^2*C*f)/d - 7*A*d*f) + 3*a^2*(C*d*e - c*C*f - 4*B*d*f) + b^2*
((3*c^2*C*e)/d - 42*A*d*e - (16*C*d*e^3)/f^2 + 21*A*c*f + (8*c^3*C*f)/d^2 - B*(3*c*e - (24*d*e^2)/f + (12*c^2*
f)/d)))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(315*b^2*d*f) - (2*(7*b*d*f*(b*c*C*e + a*C*d*e + a*c*C*f -
3*A*b*d*f) - (a*d*f - 4*b*(d*e + c*f))*(3*b*B*d*f - 2*a*C*d*f - 2*b*C*(d*e + c*f)))*Sqrt[a + b*x]*Sqrt[c + d*x
]*(e + f*x)^(3/2))/(105*b^2*d^2*f^3) + (2*(3*b*B*d*f - 2*a*C*d*f - 2*b*C*(d*e + c*f))*Sqrt[a + b*x]*(c + d*x)^
(3/2)*(e + f*x)^(3/2))/(21*b*d^2*f^2) + (2*C*(a + b*x)^(3/2)*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(9*b*d*f) - (2*S
qrt[-(b*c) + a*d]*(16*a^4*C*d^4*f^4 - 8*a^3*b*d^3*f^3*(C*d*e + c*C*f + 3*B*d*f) + 3*a^2*b^2*d^2*f^2*(d*f*(5*B*
d*e + 5*B*c*f + 14*A*d*f) - 2*C*(d^2*e^2 - c*d*e*f + c^2*f^2)) - a*b^3*d*f*(C*(8*d^3*e^3 - 6*c*d^2*e^2*f - 6*c
^2*d*e*f^2 + 8*c^3*f^3) + 3*d*f*(14*A*d*f*(d*e + c*f) - B*(5*d^2*e^2 - 6*c*d*e*f + 5*c^2*f^2))) + b^4*(2*C*(8*
d^4*e^4 - 4*c*d^3*e^3*f - 3*c^2*d^2*e^2*f^2 - 4*c^3*d*e*f^3 + 8*c^4*f^4) + 3*d*f*(14*A*d*f*(d^2*e^2 - c*d*e*f
+ c^2*f^2) - B*(8*d^3*e^3 - 5*c*d^2*e^2*f - 5*c^2*d*e*f^2 + 8*c^3*f^3))))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt
[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(315
*b^4*d^(7/2)*f^4*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) - (2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*(d*e - c*f
)*(8*a^3*C*d^3*f^3 + 3*a^2*b*d^2*f^2*(C*d*e - c*C*f - 4*B*d*f) - 3*a*b^2*d*f^2*((c^2*C - 7*A*d^2)*f + B*d*(d*e
- 2*c*f)) - b^3*(C*(16*d^3*e^3 - 3*c^2*d*e*f^2 - 8*c^3*f^3) + 3*d*f*(7*A*d*f*(2*d*e - c*f) - B*(8*d^2*e^2 - c
*d*e*f - 4*c^2*f^2))))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[
d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(315*b^4*d^(7/2)*f^4*Sqrt[c + d*x]*Sq
rt[e + f*x])

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 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 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)
])

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 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && 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 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]

Rubi steps

\begin {align*} \int \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx &=\frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}+\frac {2 \int \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (-\frac {3}{2} b (b c C e+a C d e+a c C f-3 A b d f)+\frac {3}{2} b (3 b B d f-2 a C d f-2 b C (d e+c f)) x\right ) \, dx}{9 b^2 d f}\\ &=\frac {2 (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{21 b d^2 f^2}+\frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}+\frac {4 \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (-\frac {3}{4} b (7 a d f (b c C e+a C d e+a c C f-3 A b d f)+(b c e+3 a (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f)))-\frac {3}{4} b (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) x\right )}{\sqrt {a+b x}} \, dx}{63 b^2 d^2 f^2}\\ &=-\frac {2 (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{105 b^2 d^2 f^3}+\frac {2 (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{21 b d^2 f^2}+\frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}+\frac {8 \int \frac {\sqrt {e+f x} \left (-\frac {3}{8} b (5 b c f (7 a d f (b c C e+a C d e+a c C f-3 A b d f)+(b c e+3 a (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f)))-(b c e+a d e+3 a c f) (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))))+\frac {3}{8} b \left (8 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (C d e-c C f-4 B d f)-3 a b^2 d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right )-b^3 \left (C \left (16 d^3 e^3-3 c^2 d e f^2-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d e f-4 c^2 f^2\right )\right )\right )\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{315 b^3 d^2 f^3}\\ &=\frac {2 \left (8 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (C d e-c C f-4 B d f)-3 a b^2 d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right )-b^3 \left (C \left (16 d^3 e^3-3 c^2 d e f^2-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d e f-4 c^2 f^2\right )\right )\right )\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{315 b^3 d^3 f^3}-\frac {2 (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{105 b^2 d^2 f^3}+\frac {2 (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{21 b d^2 f^2}+\frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}+\frac {16 \int \frac {-\frac {3}{16} b \left (8 a^4 C d^3 f^3 (d e+c f)-a^3 b d^2 f^2 \left (12 B d f (d e+c f)+C \left (3 d^2 e^2+10 c d e f+3 c^2 f^2\right )\right )-3 a^2 b^2 d f \left (C \left (d^3 e^3+c^3 f^3\right )-d f \left (7 A d f (d e+c f)+2 B \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right )\right )+b^4 c e \left (C \left (8 d^3 e^3-3 c d^2 e^2 f-3 c^2 d e f^2+8 c^3 f^3\right )+3 d f \left (7 A d f (d e+c f)-B \left (4 d^2 e^2-2 c d e f+4 c^2 f^2\right )\right )\right )+a b^3 \left (2 C \left (4 d^4 e^4-5 c d^3 e^3 f-5 c^3 d e f^3+4 c^4 f^4\right )+3 d f \left (7 A d f \left (d^2 e^2-6 c d e f+c^2 f^2\right )-B \left (4 d^3 e^3-6 c d^2 e^2 f-6 c^2 d e f^2+4 c^3 f^3\right )\right )\right )\right )-\frac {3}{16} b \left (16 a^4 C d^4 f^4-8 a^3 b d^3 f^3 (C d e+c C f+3 B d f)+3 a^2 b^2 d^2 f^2 \left (d f (5 B d e+5 B c f+14 A d f)-2 C \left (d^2 e^2-c d e f+c^2 f^2\right )\right )-a b^3 d f \left (C \left (8 d^3 e^3-6 c d^2 e^2 f-6 c^2 d e f^2+8 c^3 f^3\right )+3 d f \left (14 A d f (d e+c f)-B \left (5 d^2 e^2-6 c d e f+5 c^2 f^2\right )\right )\right )+b^4 \left (C \left (16 d^4 e^4-8 c d^3 e^3 f-6 c^2 d^2 e^2 f^2-8 c^3 d e f^3+16 c^4 f^4\right )+3 d f \left (14 A d f \left (d^2 e^2-c d e f+c^2 f^2\right )-B \left (8 d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+8 c^3 f^3\right )\right )\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{945 b^4 d^3 f^3}\\ &=\frac {2 \left (8 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (C d e-c C f-4 B d f)-3 a b^2 d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right )-b^3 \left (C \left (16 d^3 e^3-3 c^2 d e f^2-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d e f-4 c^2 f^2\right )\right )\right )\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{315 b^3 d^3 f^3}-\frac {2 (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{105 b^2 d^2 f^3}+\frac {2 (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{21 b d^2 f^2}+\frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {\left ((b e-a f) (d e-c f) \left (8 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (C d e-c C f-4 B d f)-3 a b^2 d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right )-b^3 \left (C \left (16 d^3 e^3-3 c^2 d e f^2-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d e f-4 c^2 f^2\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{315 b^3 d^3 f^4}-\frac {\left (16 a^4 C d^4 f^4-8 a^3 b d^3 f^3 (C d e+c C f+3 B d f)+3 a^2 b^2 d^2 f^2 \left (d f (5 B d e+5 B c f+14 A d f)-2 C \left (d^2 e^2-c d e f+c^2 f^2\right )\right )-a b^3 d f \left (C \left (8 d^3 e^3-6 c d^2 e^2 f-6 c^2 d e f^2+8 c^3 f^3\right )+3 d f \left (14 A d f (d e+c f)-B \left (5 d^2 e^2-6 c d e f+5 c^2 f^2\right )\right )\right )+b^4 \left (2 C \left (8 d^4 e^4-4 c d^3 e^3 f-3 c^2 d^2 e^2 f^2-4 c^3 d e f^3+8 c^4 f^4\right )+3 d f \left (14 A d f \left (d^2 e^2-c d e f+c^2 f^2\right )-B \left (8 d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+8 c^3 f^3\right )\right )\right )\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{315 b^3 d^3 f^4}\\ &=\frac {2 \left (8 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (C d e-c C f-4 B d f)-3 a b^2 d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right )-b^3 \left (C \left (16 d^3 e^3-3 c^2 d e f^2-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d e f-4 c^2 f^2\right )\right )\right )\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{315 b^3 d^3 f^3}-\frac {2 (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{105 b^2 d^2 f^3}+\frac {2 (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{21 b d^2 f^2}+\frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {\left ((b e-a f) (d e-c f) \left (8 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (C d e-c C f-4 B d f)-3 a b^2 d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right )-b^3 \left (C \left (16 d^3 e^3-3 c^2 d e f^2-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d e f-4 c^2 f^2\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}}\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}} \, dx}{315 b^3 d^3 f^4 \sqrt {c+d x}}-\frac {\left (\left (16 a^4 C d^4 f^4-8 a^3 b d^3 f^3 (C d e+c C f+3 B d f)+3 a^2 b^2 d^2 f^2 \left (d f (5 B d e+5 B c f+14 A d f)-2 C \left (d^2 e^2-c d e f+c^2 f^2\right )\right )-a b^3 d f \left (C \left (8 d^3 e^3-6 c d^2 e^2 f-6 c^2 d e f^2+8 c^3 f^3\right )+3 d f \left (14 A d f (d e+c f)-B \left (5 d^2 e^2-6 c d e f+5 c^2 f^2\right )\right )\right )+b^4 \left (2 C \left (8 d^4 e^4-4 c d^3 e^3 f-3 c^2 d^2 e^2 f^2-4 c^3 d e f^3+8 c^4 f^4\right )+3 d f \left (14 A d f \left (d^2 e^2-c d e f+c^2 f^2\right )-B \left (8 d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+8 c^3 f^3\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x}\right ) \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}} \, dx}{315 b^3 d^3 f^4 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}\\ &=\frac {2 \left (8 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (C d e-c C f-4 B d f)-3 a b^2 d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right )-b^3 \left (C \left (16 d^3 e^3-3 c^2 d e f^2-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d e f-4 c^2 f^2\right )\right )\right )\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{315 b^3 d^3 f^3}-\frac {2 (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{105 b^2 d^2 f^3}+\frac {2 (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{21 b d^2 f^2}+\frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {2 \sqrt {-b c+a d} \left (16 a^4 C d^4 f^4-8 a^3 b d^3 f^3 (C d e+c C f+3 B d f)+3 a^2 b^2 d^2 f^2 \left (d f (5 B d e+5 B c f+14 A d f)-2 C \left (d^2 e^2-c d e f+c^2 f^2\right )\right )-a b^3 d f \left (C \left (8 d^3 e^3-6 c d^2 e^2 f-6 c^2 d e f^2+8 c^3 f^3\right )+3 d f \left (14 A d f (d e+c f)-B \left (5 d^2 e^2-6 c d e f+5 c^2 f^2\right )\right )\right )+b^4 \left (2 C \left (8 d^4 e^4-4 c d^3 e^3 f-3 c^2 d^2 e^2 f^2-4 c^3 d e f^3+8 c^4 f^4\right )+3 d f \left (14 A d f \left (d^2 e^2-c d e f+c^2 f^2\right )-B \left (8 d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+8 c^3 f^3\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{315 b^4 d^{7/2} f^4 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {\left ((b e-a f) (d e-c f) \left (8 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (C d e-c C f-4 B d f)-3 a b^2 d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right )-b^3 \left (C \left (16 d^3 e^3-3 c^2 d e f^2-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d e f-4 c^2 f^2\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}}\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}} \, dx}{315 b^3 d^3 f^4 \sqrt {c+d x} \sqrt {e+f x}}\\ &=\frac {2 \left (8 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (C d e-c C f-4 B d f)-3 a b^2 d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right )-b^3 \left (C \left (16 d^3 e^3-3 c^2 d e f^2-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d e f-4 c^2 f^2\right )\right )\right )\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{315 b^3 d^3 f^3}-\frac {2 (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{105 b^2 d^2 f^3}+\frac {2 (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{21 b d^2 f^2}+\frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {2 \sqrt {-b c+a d} \left (16 a^4 C d^4 f^4-8 a^3 b d^3 f^3 (C d e+c C f+3 B d f)+3 a^2 b^2 d^2 f^2 \left (d f (5 B d e+5 B c f+14 A d f)-2 C \left (d^2 e^2-c d e f+c^2 f^2\right )\right )-a b^3 d f \left (C \left (8 d^3 e^3-6 c d^2 e^2 f-6 c^2 d e f^2+8 c^3 f^3\right )+3 d f \left (14 A d f (d e+c f)-B \left (5 d^2 e^2-6 c d e f+5 c^2 f^2\right )\right )\right )+b^4 \left (2 C \left (8 d^4 e^4-4 c d^3 e^3 f-3 c^2 d^2 e^2 f^2-4 c^3 d e f^3+8 c^4 f^4\right )+3 d f \left (14 A d f \left (d^2 e^2-c d e f+c^2 f^2\right )-B \left (8 d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+8 c^3 f^3\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{315 b^4 d^{7/2} f^4 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {-b c+a d} (b e-a f) (d e-c f) \left (8 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (C d e-c C f-4 B d f)-3 a b^2 d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right )-b^3 \left (C \left (16 d^3 e^3-3 c^2 d e f^2-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d e f-4 c^2 f^2\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{315 b^4 d^{7/2} f^4 \sqrt {c+d x} \sqrt {e+f x}}\\ \end {align*}

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Mathematica [C]  time = 17.30, size = 11933, normalized size = 10.10 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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fricas [F]  time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C x^{2} + B x + A\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((C*x^2 + B*x + A)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C x^{2} + B x + A\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.09, size = 14778, normalized size = 12.50 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C x^{2} + B x + A\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {e+f\,x}\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}\,\left (C\,x^2+B\,x+A\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b x} \sqrt {c + d x} \sqrt {e + f x} \left (A + B x + C x^{2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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