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

Optimal. Leaf size=937 \[ -\frac {\sqrt {d g-c h} \sqrt {f g-e h} (a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right ) C}{4 b d^2 f^2 h^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {(b e-a f) \sqrt {b g-a h} (a d f h+b (c f h+3 d (f g+e h))) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right ) C}{4 b^2 d f^2 h^2 \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} C}{2 d f h}+\frac {(a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} C}{4 b d f^2 h^2 \sqrt {c+d x}}-\frac {\sqrt {c h-d g} (C (a d f h-3 b (d f g+d e h+c f h)) (a d f h+b (d f g+d e h+c f h))-4 b d f h (2 A b d f h-C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left (-\frac {b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {c h-d g} \sqrt {a+b x}}\right )|\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{4 b^2 d^2 \sqrt {b c-a d} f^2 h^3 \sqrt {c+d x} \sqrt {e+f x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 2.28, antiderivative size = 936, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.204, Rules used = {1601, 1602, 1598, 170, 419, 165, 537, 176, 424} \[ -\frac {\sqrt {d g-c h} \sqrt {f g-e h} (a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right ) C}{4 b d^2 f^2 h^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {(b e-a f) \sqrt {b g-a h} (b c f h+a d f h+3 b d (f g+e h)) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} F\left (\sin ^{-1}\left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right ) C}{4 b^2 d f^2 h^2 \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} C}{2 d f h}+\frac {(a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} C}{4 b d f^2 h^2 \sqrt {c+d x}}-\frac {\sqrt {c h-d g} (C (a d f h-3 b (d f g+d e h+c f h)) (a d f h+b (d f g+d e h+c f h))-4 b d f h (2 A b d f h-C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left (-\frac {b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {c h-d g} \sqrt {a+b x}}\right )|\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{4 b^2 d^2 \sqrt {b c-a d} f^2 h^3 \sqrt {c+d x} \sqrt {e+f x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 165

Int[Sqrt[(a_.) + (b_.)*(x_)]/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_S
ymbol] :> Dist[(2*(a + b*x)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*Sqrt[((b*g - a*h)*(e + f*x))
/((f*g - e*h)*(a + b*x))])/(Sqrt[c + d*x]*Sqrt[e + f*x]), Subst[Int[1/((h - b*x^2)*Sqrt[1 + ((b*c - a*d)*x^2)/
(d*g - c*h)]*Sqrt[1 + ((b*e - a*f)*x^2)/(f*g - e*h)]), x], x, Sqrt[g + h*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b,
c, d, e, f, g, h}, x]

Rule 170

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Dist[(2*Sqrt[g + h*x]*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/((f*g - e*h)*Sqrt[c +
d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))]), Subst[Int[1/(Sqrt[1 + ((b*c - a*d)*x^2)/(d*e
- c*f)]*Sqrt[1 - ((b*g - a*h)*x^2)/(f*g - e*h)]), x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d
, e, f, g, h}, x]

Rule 176

Int[Sqrt[(c_.) + (d_.)*(x_)]/(((a_.) + (b_.)*(x_))^(3/2)*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Dist[(-2*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))])/((b*e - a*f)*Sqrt
[g + h*x]*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]), Subst[Int[Sqrt[1 + ((b*c - a*d)*x^2)/(d*e -
c*f)]/Sqrt[1 - ((b*g - a*h)*x^2)/(f*g - e*h)], x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e
, f, g, h}, x]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 537

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

Rule 1598

Int[((A_.) + (B_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.
) + (h_.)*(x_)]), x_Symbol] :> Dist[(A*b - a*B)/b, Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h
*x]), x], x] + Dist[B/b, Int[Sqrt[a + b*x]/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] /; FreeQ[{a, b,
c, d, e, f, g, h, A, B}, 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 1602

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

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b x} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx &=\frac {C \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}+\frac {\int \frac {4 a A d f h-C (b c e g+a (d e g+c f g+c e h))+2 (2 A b d f h-C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h))) x+C (a d f h-3 b (d f g+d e h+c f h)) x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{4 d f h}\\ &=\frac {C (a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 b d f^2 h^2 \sqrt {c+d x}}+\frac {C \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}+\frac {\int \frac {-C (b d e g+a c f h) (a d f h-3 b (d f g+d e h+c f h))+2 b d f h (4 a A d f h-C (b c e g+a (d e g+c f g+c e h)))-(C (a d f h-3 b (d f g+d e h+c f h)) (a d f h+b (d f g+d e h+c f h))-4 b d f h (2 A b d f h-C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{8 b d^2 f^2 h^2}+\frac {(C (d e-c f) (d g-c h) (a d f h-3 b (d f g+d e h+c f h))) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{8 b d^2 f^2 h^2}\\ &=\frac {C (a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 b d f^2 h^2 \sqrt {c+d x}}+\frac {C \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}+\frac {(C (b e-a f) (b g-a h) (b c f h+a d f h+3 b d (f g+e h))) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{8 b^2 d f^2 h^2}-\frac {(C (a d f h-3 b (d f g+d e h+c f h)) (a d f h+b (d f g+d e h+c f h))-4 b d f h (2 A b d f h-C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{8 b^2 d^2 f^2 h^2}-\frac {\left (C (d g-c h) (a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {(-b c+a d) x^2}{b e-a f}}}{\sqrt {1-\frac {(d g-c h) x^2}{f g-e h}}} \, dx,x,\frac {\sqrt {e+f x}}{\sqrt {c+d x}}\right )}{4 b d^2 f^2 h^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}\\ &=\frac {C (a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 b d f^2 h^2 \sqrt {c+d x}}+\frac {C \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}-\frac {C \sqrt {d g-c h} \sqrt {f g-e h} (a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{4 b d^2 f^2 h^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {\left ((C (a d f h-3 b (d f g+d e h+c f h)) (a d f h+b (d f g+d e h+c f h))-4 b d f h (2 A b d f h-C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (h-b x^2\right ) \sqrt {1+\frac {(b c-a d) x^2}{d g-c h}} \sqrt {1+\frac {(b e-a f) x^2}{f g-e h}}} \, dx,x,\frac {\sqrt {g+h x}}{\sqrt {a+b x}}\right )}{4 b^2 d^2 f^2 h^2 \sqrt {c+d x} \sqrt {e+f x}}+\frac {\left (C (b e-a f) (b g-a h) (b c f h+a d f h+3 b d (f g+e h)) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {(b c-a d) x^2}{d e-c f}} \sqrt {1-\frac {(b g-a h) x^2}{f g-e h}}} \, dx,x,\frac {\sqrt {e+f x}}{\sqrt {a+b x}}\right )}{4 b^2 d f^2 h^2 (f g-e h) \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}\\ &=\frac {C (a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 b d f^2 h^2 \sqrt {c+d x}}+\frac {C \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}-\frac {C \sqrt {d g-c h} \sqrt {f g-e h} (a d f h-3 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{4 b d^2 f^2 h^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {C (b e-a f) \sqrt {b g-a h} (b c f h+a d f h+3 b d (f g+e h)) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} F\left (\sin ^{-1}\left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{4 b^2 d f^2 h^2 \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac {\sqrt {-d g+c h} (C (a d f h-3 b (d f g+d e h+c f h)) (a d f h+b (d f g+d e h+c f h))-4 b d f h (2 A b d f h-C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left (-\frac {b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {-d g+c h} \sqrt {a+b x}}\right )|\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{4 b^2 d^2 \sqrt {b c-a d} f^2 h^3 \sqrt {c+d x} \sqrt {e+f x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 17.32, size = 16699, normalized size = 17.82 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

maple [B]  time = 0.13, size = 42545, normalized size = 45.41 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________