\(\int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [6]

Optimal result

Integrand size = 42, antiderivative size = 981 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {(5 a B d f h+b (4 A d f h-3 B (d f g+d e h+c f h))) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 d f^2 h^2 \sqrt {c+d x}}+\frac {b B \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} (5 a B d f h+b (4 A d f h-3 B (d f g+d e h+c f h))) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{4 d^2 f^2 h^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {(b e-a f) \sqrt {b g-a h} (3 a B d f h+b (4 A d f h-B (c f h+3 d (f g+e h)))) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{4 b d f^2 h^2 \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {\sqrt {-d g+c h} (4 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))-(a d f h+b (d f g+d e h+c f h)) (5 a B d f h+b (4 A d f h-3 B (d f g+d e h+c f h)))) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {-d g+c h} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{4 b d^2 \sqrt {b c-a d} f^2 h^3 \sqrt {c+d x} \sqrt {e+f x}} \]

[Out]

1/4*(4*d*f*h*(2*a*(2*A*b+B*a)*d*f*h-b*B*(b*(c*e*h+c*f*g+d*e*g)+a*(c*f*h+d*e*h+d*f*g)))-(a*d*f*h+b*(c*f*h+d*e*h
+d*f*g))*(5*a*B*d*f*h+b*(4*A*d*f*h-3*B*(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/d^2/f^2/h^3/(-a*d+b*c)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)+1/4*(5*a*B*d*f*h+b*(4*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)/d/f^2/h^2/(d*x+c)^(1/2)+1/2*b*B*(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*(-a*f+b*e)*(3*a*B*d*f*h+b*(4*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/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*(5*a*B*d*f*h+b*(4*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)/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] (warning: unable to verify)

Time = 2.30 (sec) , antiderivative size = 976, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1611, 1614, 1616, 1612, 176, 430, 171, 551, 182, 435} \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {b \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} B}{2 d f h}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{4 d^2 f^2 h^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {(b e-a f) \sqrt {b g-a h} (4 A b d f h+3 a B d f h-b B (c f h+3 d (f g+e h))) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{4 b d f^2 h^2 \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac {\sqrt {c h-d g} ((a d f h+b (d f g+d e h+c f h)) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-4 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {c h-d g} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{4 b d^2 \sqrt {b c-a d} f^2 h^3 \sqrt {c+d x} \sqrt {e+f x}}+\frac {(4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 d f^2 h^2 \sqrt {c+d x}} \]

[In]

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

[Out]

((4*A*b*d*f*h + 5*a*B*d*f*h - 3*b*B*(d*f*g + d*e*h + c*f*h))*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(4*d*f
^2*h^2*Sqrt[c + d*x]) + (b*B*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(2*d*f*h) - (Sqrt[d*g -
c*h]*Sqrt[f*g - e*h]*(4*A*b*d*f*h + 5*a*B*d*f*h - 3*b*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]*Sq
rt[c + d*x])], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))])/(4*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]) - ((b*e - a*f)*Sqrt[b*g - a*h]*(4*A*b*d*f*h + 3*a*B*d*f*h - b*B*(c*
f*h + 3*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*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]*((a*d*f*h + b*(d*f*g + d*e*h + c*f*h))*(4*A*b*d*f*h + 5*a*B*d*f*h - 3*b*B*(d*f*g +
d*e*h + c*f*h)) - 4*d*f*h*(2*a*(2*A*b + a*B)*d*f*h - b*B*(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*d^2*Sqrt[b*c - a*d]*f^2*h
^3*Sqrt[c + d*x]*Sqrt[e + f*x])

Rule 171

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 176

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 182

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 430

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

Rule 435

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

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], 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 1611

Int[(((a_.) + (b_.)*(x_))^(m_.)*((A_.) + (B_.)*(x_)))/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[
(g_.) + (h_.)*(x_)]), x_Symbol] :> 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) + (A*b + a*B)*d*f*h*(2*m + 3)*x + b*B*d*f*h*(2*m + 3)*x^2, x], x],
x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && IntegerQ[2*m] && GtQ[m, 0]

Rule 1612

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 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 1616

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/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*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], 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*} \text {integral}& = \frac {\int \frac {\sqrt {a+b x} \left (6 a A d f h+6 (A b+a B) d f h x+6 b B d f h x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{6 d f h} \\ & = \frac {b B \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}+\frac {\int \frac {6 d f h \left (4 a^2 A d f h-b B (b c e g+a (d e g+c f g+c e h))\right )+12 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (d f g+d e h+c f h))) x+6 b d f h (4 A b d f h+5 a B d f h-3 b 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}{24 d^2 f^2 h^2} \\ & = \frac {(4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 d f^2 h^2 \sqrt {c+d x}}+\frac {b B \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}+\frac {\int \frac {-6 b d f h \left ((b d e g+a c f h) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-2 d f h \left (4 a^2 A d f h-b B (b c e g+a (d e g+c f g+c e h))\right )\right )-6 b d f h ((a d f h+b (d f g+d e h+c f h)) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-4 d f h (2 a (2 A b+a B) d f h-b B (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}{48 b d^3 f^3 h^3}+\frac {((d e-c f) (d g-c h) (4 A b d f h+5 a B d f h-3 b 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 d^2 f^2 h^2} \\ & = \frac {(4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 d f^2 h^2 \sqrt {c+d x}}+\frac {b B \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}-\frac {((b e-a f) (b g-a h) (4 A b d f h+3 a B d f h-b B (c f h+3 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 d f^2 h^2}-\frac {((a d f h+b (d f g+d e h+c f h)) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-4 d f h (2 a (2 A b+a B) d f h-b B (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 d^2 f^2 h^2}-\frac {\left ((d g-c h) (4 A b d f h+5 a B d f h-3 b 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 ) \text {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 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 {(4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 d f^2 h^2 \sqrt {c+d x}}+\frac {b B \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} (4 A b d f h+5 a B d f h-3 b 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 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 (((a d f h+b (d f g+d e h+c f h)) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-4 d f h (2 a (2 A b+a B) d f h-b B (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 ) \text {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 d^2 f^2 h^2 \sqrt {c+d x} \sqrt {e+f x}}-\frac {\left ((b e-a f) (b g-a h) (4 A b d f h+3 a B d f h-b 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}\right ) \text {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 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 {(4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 d f^2 h^2 \sqrt {c+d x}}+\frac {b B \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} (4 A b d f h+5 a B d f h-3 b 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 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} (4 A b d f h+3 a B d f h-b 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} 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 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} ((a d f h+b (d f g+d e h+c f h)) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-4 d f h (2 a (2 A b+a B) d f h-b B (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 d^2 \sqrt {b c-a d} f^2 h^3 \sqrt {c+d x} \sqrt {e+f x}} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(21961\) vs. \(2(981)=1962\).

Time = 36.59 (sec) , antiderivative size = 21961, normalized size of antiderivative = 22.39 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1813\) vs. \(2(898)=1796\).

Time = 5.18 (sec) , antiderivative size = 1814, normalized size of antiderivative = 1.85

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1814\)
default	\(\text {Expression too large to display}\)	\(55936\)

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

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

[In]

integrate((b*x+a)**(3/2)*(B*x+A)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

Integral((A + B*x)*(a + b*x)**(3/2)/(sqrt(c + d*x)*sqrt(e + f*x)*sqrt(g + h*x)), x)

Maxima [F]

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

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2}}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \]

[In]

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

[Out]

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