\(\int \frac {A+C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [34]

Optimal result

Integrand size = 44, antiderivative size = 867 \[ \int \frac {A+C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 \left (A b^2+a^2 C\right ) d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{b (b c-a d) (b e-a f) (b g-a h) \sqrt {c+d x}}-\frac {2 \left (A b^2+a^2 C\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {2 \left (A b^2+a^2 C\right ) \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{b (b c-a d) (b e-a f) (b g-a h) \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 \left (2 a b c C+A b^2 d-a^2 C d\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{b^2 (b c-a d) \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {2 C \sqrt {-d g+c h} (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {-d g+c h} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{b^2 \sqrt {b c-a d} h \sqrt {c+d x} \sqrt {e+f x}} \]

[Out]

2*C*(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/h/(-a*d+b*c)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)+2*(A*b^
2+C*a^2)*d*(b*x+a)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/b/(-a*d+b*c)/(-a*f+b*e)/(-a*h+b*g)/(d*x+c)^(1/2)-2*(A*b^2
+C*a^2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)/(-a*f+b*e)/(-a*h+b*g)/(b*x+a)^(1/2)-2*(A*b^2*d-C*
a^2*d+2*C*a*b*c)*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*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^(1/2)*(h*x+g)^(1/2)/b^2/(-a*d+b*c)/(-
a*h+b*g)^(1/2)/(-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)-2*(A*b^2+C*a^2)*E
llipticE((-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
/(-a*d+b*c)/(-a*f+b*e)/(-a*h+b*g)/((-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 = 1.23 (sec) , antiderivative size = 867, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {1619, 1616, 1612, 176, 430, 171, 551, 182, 435} \[ \int \frac {A+C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right ) \left (C a^2+A b^2\right )}{b (b c-a d) (b e-a f) (b g-a h) \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (C a^2+A b^2\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}+\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (C a^2+A b^2\right )}{b (b c-a d) (b e-a f) (b g-a h) \sqrt {c+d x}}-\frac {2 \left (-C d a^2+2 b c C a+A b^2 d\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{b^2 (b c-a d) \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {2 C \sqrt {c h-d g} (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {c h-d g} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{b^2 \sqrt {b c-a d} h \sqrt {c+d x} \sqrt {e+f x}} \]

[In]

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

[Out]

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

Rule 1619

Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (C_.)*(x_)^2))/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt
[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[(A*b^2 + a^2*C)*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g
+ h*x]/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))), x] - Dist[1/(2*(m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g -
a*h)), Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*d*f*h*(m + 1) - 2*a*b
*(m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) + a*C*(a*(d*e*g + c*f*g + c*e*h) + 2
*b*c*e*g*(m + 1)) - 2*(A*b*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h)) - C*(a^2*(d*f*g + d*e*h + c*f
*h) - b^2*c*e*g*(m + 1) + a*b*(m + 1)*(d*e*g + c*f*g + c*e*h)))*x + d*f*h*(2*m + 5)*(A*b^2 + a^2*C)*x^2, x], x
], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, C}, x] && IntegerQ[2*m] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (A b^2+a^2 C\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}+\frac {\int \frac {-a (a A d f h-a C (d e g+c f g+c e h)+b (c C e g-A d f g-A d e h-A c f h))+\left (2 a^2 C (d f g+d e h+c f h)+b^2 (c C e g+A d f g+A d e h+A c f h)+a b (A d f h-C (d e g+c f g+c e h))\right ) x+2 \left (A b^2+a^2 C\right ) d f h x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{(b c-a d) (b e-a f) (b g-a h)} \\ & = \frac {2 \left (A b^2+a^2 C\right ) d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{b (b c-a d) (b e-a f) (b g-a h) \sqrt {c+d x}}-\frac {2 \left (A b^2+a^2 C\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}+\frac {\int \frac {-2 d (a c C+A b d) f (b e-a f) h (b g-a h)+2 C d (b c-a d) f (b e-a f) h (b g-a h) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{2 b d (b c-a d) f (b e-a f) h (b g-a h)}+\frac {\left (\left (A b^2+a^2 C\right ) (d e-c f) (d g-c h)\right ) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b (b c-a d) (b e-a f) (b g-a h)} \\ & = \frac {2 \left (A b^2+a^2 C\right ) d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{b (b c-a d) (b e-a f) (b g-a h) \sqrt {c+d x}}-\frac {2 \left (A b^2+a^2 C\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}+\frac {C \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b^2}-\frac {\left (2 a b c C+A b^2 d-a^2 C d\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b^2 (b c-a d)}-\frac {\left (2 \left (A b^2+a^2 C\right ) (d g-c 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 )}{b (b c-a d) (b e-a f) (b g-a h) \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}} \\ & = \frac {2 \left (A b^2+a^2 C\right ) d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{b (b c-a d) (b e-a f) (b g-a h) \sqrt {c+d x}}-\frac {2 \left (A b^2+a^2 C\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {2 \left (A b^2+a^2 C\right ) \sqrt {d g-c h} \sqrt {f g-e 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 )}{b (b c-a d) (b e-a f) (b g-a h) \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {\left (2 C (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 )}{b^2 \sqrt {c+d x} \sqrt {e+f x}}-\frac {\left (2 \left (2 a b c C+A b^2 d-a^2 C d\right ) \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 )}{b^2 (b c-a d) (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 {2 \left (A b^2+a^2 C\right ) d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{b (b c-a d) (b e-a f) (b g-a h) \sqrt {c+d x}}-\frac {2 \left (A b^2+a^2 C\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {2 \left (A b^2+a^2 C\right ) \sqrt {d g-c h} \sqrt {f g-e 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 )}{b (b c-a d) (b e-a f) (b g-a h) \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 \left (2 a b c C+A b^2 d-a^2 C d\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 )}{b^2 (b c-a d) \sqrt {b g-a h} \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 {2 C \sqrt {-d g+c 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 )}{b^2 \sqrt {b c-a d} h \sqrt {c+d x} \sqrt {e+f x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 31.92 (sec) , antiderivative size = 721, normalized size of antiderivative = 0.83 \[ \int \frac {A+C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 (b e-a f) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} (e+f x)^{3/2} (g+h x)^{3/2} \left (2 a C (-b c+a d) h (-b g+a h) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )-A b^2 h \left (b (d g-c h) E\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right )|\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )+d (-b g+a h) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )\right )-a^2 C h \left (b (d g-c h) E\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right )|\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )+d (-b g+a h) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )\right )+C (b c-a d) (b g-a h)^2 \operatorname {EllipticPi}\left (\frac {b (-f g+e h)}{(b e-a f) h},\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )\right )}{b^2 (b c-a d) h (f g-e h)^3 (a+b x)^{5/2} \sqrt {c+d x} \left (-\frac {(b e-a f) (b g-a h) (e+f x) (g+h x)}{(f g-e h)^2 (a+b x)^2}\right )^{3/2}} \]

[In]

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

[Out]

(-2*(b*e - a*f)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*(e + f*x)^(3/2)*(g + h*x)^(3/2)*(2*a*C*(
-(b*c) + a*d)*h*(-(b*g) + a*h)*EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-
(b*c) + a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))] - A*b^2*h*(b*(d*g - c*h)*EllipticE[ArcSin[Sqrt[((-(b*e
) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))] + d*
(-(b*g) + a*h)*EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(
f*g) + e*h))/((b*e - a*f)*(d*g - c*h))]) - a^2*C*h*(b*(d*g - c*h)*EllipticE[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h
*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))] + d*(-(b*g) + a*h)*
EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((
b*e - a*f)*(d*g - c*h))]) + C*(b*c - a*d)*(b*g - a*h)^2*EllipticPi[(b*(-(f*g) + e*h))/((b*e - a*f)*h), ArcSin[
Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g -
c*h))]))/(b^2*(b*c - a*d)*h*(f*g - e*h)^3*(a + b*x)^(5/2)*Sqrt[c + d*x]*(-(((b*e - a*f)*(b*g - a*h)*(e + f*x)
*(g + h*x))/((f*g - e*h)^2*(a + b*x)^2)))^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2285\) vs. \(2(794)=1588\).

Time = 7.83 (sec) , antiderivative size = 2286, normalized size of antiderivative = 2.64

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2286\)
default	\(\text {Expression too large to display}\)	\(33894\)

[In]

int((C*x^2+A)/(b*x+a)^(3/2)/(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)*(2*(b*d*f*h*x^
3+b*c*f*h*x^2+b*d*e*h*x^2+b*d*f*g*x^2+b*c*e*h*x+b*c*f*g*x+b*d*e*g*x+b*c*e*g)/b/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*
e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)*(A*b^2+C*a^2)/((x+a/b)*(b*d*f*h*x^3+b*c*f*h*x^2
+b*d*e*h*x^2+b*d*f*g*x^2+b*c*e*h*x+b*c*f*g*x+b*d*e*g*x+b*c*e*g))^(1/2)+2*(-C*a/b^2+1/b^2*(a^2*d*f*h-a*b*c*f*h-
a*b*d*e*h-a*b*d*f*g+b^2*c*e*h+b^2*c*f*g+b^2*d*e*g)*(A*b^2+C*a^2)/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*
g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)-(b*c*e*h+b*c*f*g+b*d*e*g)/b/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*
h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)*(A*b^2+C*a^2))*(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*(C/b-1/b*(a*d*f*h-b*c*f*h-b
*d*e*h-b*d*f*g)*(A*b^2+C*a^2)/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e
*g-b^3*c*e*g)-(2*b*c*f*h+2*b*d*e*h+2*b*d*f*g)/b/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b
^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)*(A*b^2+C*a^2))*(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)))-2*d*f*h*(A*b^2+C*a^2)
/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*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. \[ \int \frac {A+C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((C*x^2+A)/(b*x+a)^(3/2)/(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)

Giac [F(-2)]

Exception generated. \[ \int \frac {A+C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m operator + Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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