3.31 \(\int \frac {(a+b x)^{3/2} (A+C x^2)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\)

Optimal. Leaf size=1395 \[ \frac {C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (a+b x)^{3/2}}{3 d f h}-\frac {\sqrt {c h-d g} \left (4 b d f h \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (2 C (d f g+d e h+c f h) a^2-b (12 A d f h-5 C (d e g+c f g+c e h)) a+3 b^2 c C e g\right )\right )+(a d f h+b (d f g+d e h+c f h)) (C (3 a d f h-5 b (d f g+d e h+c f h)) (a d f h-3 b (d f g+d e h+c f h))+8 b d f h (3 A b d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h))))\right ) \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 ) (a+b x)}{24 b^2 d^3 \sqrt {b c-a d} f^3 h^4 \sqrt {c+d x} \sqrt {e+f x}}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} (C (3 a d f h-5 b (d f g+d e h+c f h)) (a d f h-3 b (d f g+d e h+c f h))+8 b d f h (3 A b d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) \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 ) \sqrt {a+b x}}{24 b d^3 f^3 h^3 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \sqrt {a+b x}}{12 d^2 f^2 h^2}+\frac {(C (3 a d f h-5 b (d f g+d e h+c f h)) (a d f h-3 b (d f g+d e h+c f h))+8 b d f h (3 A b d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) \sqrt {e+f x} \sqrt {g+h x} \sqrt {a+b x}}{24 b d^2 f^3 h^3 \sqrt {c+d x}}+\frac {(b e-a f) \sqrt {b g-a h} \left (-\left (\left (24 A d^2 f^2 h^2+C \left (\left (15 f^2 g^2+14 e f h g+15 e^2 h^2\right ) d^2+4 c f h (f g+e h) d+5 c^2 f^2 h^2\right )\right ) b^2\right )+6 a C d f h (c f h+2 d (f g+e h)) b+3 a^2 C d^2 f^2 h^2\right ) \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 )}{24 b^2 d^2 f^3 h^3 \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)}}} \]

[Out]

-1/24*(4*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))*(3*a*d*f*h-5*b*(c*f*h+d*e*h+d*f*g))+2*d*f*h*
(3*b^2*c*C*e*g+2*a^2*C*(c*f*h+d*e*h+d*f*g)-a*b*(12*A*d*f*h-5*C*(c*e*h+c*f*g+d*e*g))))+(a*d*f*h+b*(c*f*h+d*e*h+
d*f*g))*(C*(3*a*d*f*h-5*b*(c*f*h+d*e*h+d*f*g))*(a*d*f*h-3*b*(c*f*h+d*e*h+d*f*g))+8*b*d*f*h*(3*A*b*d*f*h-C*(2*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^3/f^3/h^4
/(-a*d+b*c)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)+1/24*(C*(3*a*d*f*h-5*b*(c*f*h+d*e*h+d*f*g))*(a*d*f*h-3*b*(c*f*h+
d*e*h+d*f*g))+8*b*d*f*h*(3*A*b*d*f*h-C*(2*b*(c*e*h+c*f*g+d*e*g)+a*(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^2/f^3/h^3/(d*x+c)^(1/2)+1/3*C*(b*x+a)^(3/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)
/d/f/h+1/12*C*(3*a*d*f*h-5*b*(c*f*h+d*e*h+d*f*g))*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/d^2/
f^2/h^2+1/24*(-a*f+b*e)*(3*a^2*C*d^2*f^2*h^2+6*a*b*C*d*f*h*(c*f*h+2*d*(e*h+f*g))-b^2*(24*A*d^2*f^2*h^2+C*(5*c^
2*f^2*h^2+4*c*d*f*h*(e*h+f*g)+d^2*(15*e^2*h^2+14*e*f*g*h+15*f^2*g^2))))*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^2/f^3/h^3/(-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)-1/24*(C*(3*a*d*f*h-5*b*(c*f*h+d*e*h+d*f*g))*(a*d*f*h-3*b*(c*f*h+d*e*h+d
*f*g))+8*b*d*f*h*(3*A*b*d*f*h-C*(2*b*(c*e*h+c*f*g+d*e*g)+a*(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^3/f^3/h^3/((-c*f+d*e)*(b
*x+a)/(-a*f+b*e)/(d*x+c))^(1/2)/(h*x+g)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 5.74, antiderivative size = 1376, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 10, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1601, 1600, 1602, 1598, 170, 419, 165, 537, 176, 424} \[ \frac {C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (a+b x)^{3/2}}{3 d f h}-\frac {\sqrt {c h-d g} \left ((a d f h+b (d f g+d e h+c f h)) \left (24 A d^2 f^2 h^2 b^2+15 C (d f g+d e h+c f h)^2 b^2-16 C d f h (d e g+c f g+c e h) b^2-22 a C d f h (d f g+d e h+c f h) b+3 a^2 C d^2 f^2 h^2\right )+4 b d f h \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (2 C (d f g+d e h+c f h) a^2-b (12 A d f h-5 C (d e g+c f g+c e h)) a+3 b^2 c C e g\right )\right )\right ) \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 ) (a+b x)}{24 b^2 d^3 \sqrt {b c-a d} f^3 h^4 \sqrt {c+d x} \sqrt {e+f x}}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} \left (24 A d^2 f^2 h^2 b^2+15 C (d f g+d e h+c f h)^2 b^2-16 C d f h (d e g+c f g+c e h) b^2-22 a C d f h (d f g+d e h+c f h) b+3 a^2 C d^2 f^2 h^2\right ) \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 ) \sqrt {a+b x}}{24 b d^3 f^3 h^3 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \sqrt {a+b x}}{12 d^2 f^2 h^2}+\frac {\left (24 A b f h d^2+\frac {3 a^2 C f h d^2}{b}-16 b C (d e g+c f g+c e h) d-22 a C (d f g+d e h+c f h) d+\frac {15 b C (d f g+d e h+c f h)^2}{f h}\right ) \sqrt {e+f x} \sqrt {g+h x} \sqrt {a+b x}}{24 d^2 f^2 h^2 \sqrt {c+d x}}+\frac {(b e-a f) \sqrt {b g-a h} \left (-\left (24 A d^2 f^2 h^2+C \left (\left (15 f^2 g^2+14 e f h g+15 e^2 h^2\right ) d^2+4 c f h (f g+e h) d+5 c^2 f^2 h^2\right )\right ) b^2+6 a C d f h (c f h+2 d (f g+e h)) b+3 a^2 C d^2 f^2 h^2\right ) \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 )}{24 b^2 d^2 f^3 h^3 \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)}}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(A + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

((24*A*b*d^2*f*h + (3*a^2*C*d^2*f*h)/b - 16*b*C*d*(d*e*g + c*f*g + c*e*h) - 22*a*C*d*(d*f*g + d*e*h + c*f*h) +
(15*b*C*(d*f*g + d*e*h + c*f*h)^2)/(f*h))*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(24*d^2*f^2*h^2*Sqrt[c +
d*x]) + (C*(3*a*d*f*h - 5*b*(d*f*g + d*e*h + c*f*h))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])
/(12*d^2*f^2*h^2) + (C*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*d*f*h) - (Sqrt[d*g - c*h]
*Sqrt[f*g - e*h]*(24*A*b^2*d^2*f^2*h^2 + 3*a^2*C*d^2*f^2*h^2 - 16*b^2*C*d*f*h*(d*e*g + c*f*g + c*e*h) - 22*a*b
*C*d*f*h*(d*f*g + d*e*h + c*f*h) + 15*b^2*C*(d*f*g + d*e*h + c*f*h)^2)*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))])/(24*b*d^3*f^3*h^3*Sqrt[((d*e - c*f)*(a + b*x))/((b*
e - a*f)*(c + d*x))]*Sqrt[g + h*x]) + ((b*e - a*f)*Sqrt[b*g - a*h]*(3*a^2*C*d^2*f^2*h^2 + 6*a*b*C*d*f*h*(c*f*h
+ 2*d*(f*g + e*h)) - b^2*(24*A*d^2*f^2*h^2 + C*(5*c^2*f^2*h^2 + 4*c*d*f*h*(f*g + e*h) + d^2*(15*f^2*g^2 + 14*
e*f*g*h + 15*e^2*h^2))))*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)))])/(24*b^2*d^2*f^3*h^3*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]*((a*d*f*h + b*(d*f*g + d*e*h + c*f*h))*(24*A*b^2*d^2*f^2*h^2 + 3*a^2*C*d^2*f^
2*h^2 - 16*b^2*C*d*f*h*(d*e*g + c*f*g + c*e*h) - 22*a*b*C*d*f*h*(d*f*g + d*e*h + c*f*h) + 15*b^2*C*(d*f*g + d*
e*h + c*f*h)^2) + 4*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))*(3*a*d*f*h - 5*b*(d*f*g
+ d*e*h + c*f*h)) + 2*d*f*h*(3*b^2*c*C*e*g + 2*a^2*C*(d*f*g + d*e*h + c*f*h) - a*b*(12*A*d*f*h - 5*C*(d*e*g +
c*f*g + c*e*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))])/(24*b^2*d^3*Sqr
t[b*c - a*d]*f^3*h^4*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 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 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 {(a+b x)^{3/2} \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx &=\frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}+\frac {\int \frac {\sqrt {a+b x} \left (-3 b c C e g+6 a A d f h-a C (d e g+c f g+c e h)+2 (3 A b d f h-2 b C (d e g+c f g+c e h)-a C (d f g+d e h+c f h)) x+C (3 a d f h-5 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}{6 d f h}\\ &=\frac {C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{12 d^2 f^2 h^2}+\frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}+\frac {\int \frac {-4 a d f h (3 b c C e g-6 a A d f h+a C (d e g+c f g+c e h))-C (b c e g+a (d e g+c f g+c e h)) (3 a d f h-5 b (d f g+d e h+c f h))-2 \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (3 b^2 c C e g+2 a^2 C (d f g+d e h+c f h)-a b (12 A d f h-5 C (d e g+c f g+c e h))\right )\right ) x+\left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right ) x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{24 d^2 f^2 h^2}\\ &=\frac {\left (24 A b d^2 f h+\frac {3 a^2 C d^2 f h}{b}-16 b C d (d e g+c f g+c e h)-22 a C d (d f g+d e h+c f h)+\frac {15 b C (d f g+d e h+c f h)^2}{f h}\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{24 d^2 f^2 h^2 \sqrt {c+d x}}+\frac {C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{12 d^2 f^2 h^2}+\frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}+\frac {\int \frac {-(b d e g+a c f h) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )-2 b d f h (4 a d f h (3 b c C e g-6 a A d f h+a C (d e g+c f g+c e h))+C (b c e g+a (d e g+c f g+c e h)) (3 a d f h-5 b (d f g+d e h+c f h)))-\left ((a d f h+b (d f g+d e h+c f h)) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )+4 b d f h \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (3 b^2 c C e g+2 a^2 C (d f g+d e h+c f h)-a b (12 A d f h-5 C (d e g+c f g+c e h))\right )\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{48 b d^3 f^3 h^3}+\frac {\left ((d e-c f) (d g-c h) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )\right ) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{48 b d^3 f^3 h^3}\\ &=\frac {\left (24 A b d^2 f h+\frac {3 a^2 C d^2 f h}{b}-16 b C d (d e g+c f g+c e h)-22 a C d (d f g+d e h+c f h)+\frac {15 b C (d f g+d e h+c f h)^2}{f h}\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{24 d^2 f^2 h^2 \sqrt {c+d x}}+\frac {C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{12 d^2 f^2 h^2}+\frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}+\frac {\left ((b e-a f) (b g-a h) \left (3 a^2 C d^2 f^2 h^2+6 a b C d f h (c f h+2 d (f g+e h))-b^2 \left (24 A d^2 f^2 h^2+C \left (5 c^2 f^2 h^2+4 c d f h (f g+e h)+d^2 \left (15 f^2 g^2+14 e f g h+15 e^2 h^2\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{48 b^2 d^2 f^3 h^3}-\frac {\left ((a d f h+b (d f g+d e h+c f h)) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )+4 b d f h \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (3 b^2 c C e g+2 a^2 C (d f g+d e h+c f h)-a b (12 A d f h-5 C (d e g+c f g+c e h))\right )\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{48 b^2 d^3 f^3 h^3}-\frac {\left ((d g-c h) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right ) \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 )}{24 b d^3 f^3 h^3 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}\\ &=\frac {\left (24 A b d^2 f h+\frac {3 a^2 C d^2 f h}{b}-16 b C d (d e g+c f g+c e h)-22 a C d (d f g+d e h+c f h)+\frac {15 b C (d f g+d e h+c f h)^2}{f h}\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{24 d^2 f^2 h^2 \sqrt {c+d x}}+\frac {C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{12 d^2 f^2 h^2}+\frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right ) \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 )}{24 b d^3 f^3 h^3 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {\left (\left ((a d f h+b (d f g+d e h+c f h)) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )+4 b d f h \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (3 b^2 c C e g+2 a^2 C (d f g+d e h+c f h)-a b (12 A d f h-5 C (d e g+c f g+c e h))\right )\right )\right ) (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 )}{24 b^2 d^3 f^3 h^3 \sqrt {c+d x} \sqrt {e+f x}}+\frac {\left ((b e-a f) (b g-a h) \left (3 a^2 C d^2 f^2 h^2+6 a b C d f h (c f h+2 d (f g+e h))-b^2 \left (24 A d^2 f^2 h^2+C \left (5 c^2 f^2 h^2+4 c d f h (f g+e h)+d^2 \left (15 f^2 g^2+14 e f g h+15 e^2 h^2\right )\right )\right )\right ) \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 )}{24 b^2 d^2 f^3 h^3 (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 {\left (24 A b d^2 f h+\frac {3 a^2 C d^2 f h}{b}-16 b C d (d e g+c f g+c e h)-22 a C d (d f g+d e h+c f h)+\frac {15 b C (d f g+d e h+c f h)^2}{f h}\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{24 d^2 f^2 h^2 \sqrt {c+d x}}+\frac {C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{12 d^2 f^2 h^2}+\frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right ) \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 )}{24 b d^3 f^3 h^3 \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} \left (3 a^2 C d^2 f^2 h^2+6 a b C d f h (c f h+2 d (f g+e h))-b^2 \left (24 A d^2 f^2 h^2+C \left (5 c^2 f^2 h^2+4 c d f h (f g+e h)+d^2 \left (15 f^2 g^2+14 e f g h+15 e^2 h^2\right )\right )\right )\right ) \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 )}{24 b^2 d^2 f^3 h^3 \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} \left ((a d f h+b (d f g+d e h+c f h)) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )+4 b d f h \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (3 b^2 c C e g+2 a^2 C (d f g+d e h+c f h)-a b (12 A d f h-5 C (d e g+c f g+c e h))\right )\right )\right ) (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 )}{24 b^2 d^3 \sqrt {b c-a d} f^3 h^4 \sqrt {c+d x} \sqrt {e+f x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 23.83, size = 38402, normalized size = 27.53 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((a + b*x)^(3/2)*(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)^(3/2)*(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 \[ \int \frac {{\left (C x^{2} + A\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

maple [B]  time = 0.26, size = 89496, normalized size = 64.15 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(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 \[ \int \frac {{\left (C x^{2} + A\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (C\,x^2+A\right )\,{\left (a+b\,x\right )}^{3/2}}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + C*x^2)*(a + b*x)^(3/2))/((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)^(3/2))/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(c + d*x)^(1/2)), x)

________________________________________________________________________________________

sympy [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)**(3/2)*(C*x**2+A)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________