Integrand size = 62, antiderivative size = 436 \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 (b B-2 a C) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {b g-a h} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {2 C \sqrt {-d g+c h} (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {-d g+c h} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{\sqrt {b c-a d} h \sqrt {c+d x} \sqrt {e+f x}} \]
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)/h/(-a*d+b*c)^(1/2)/(d*x+ c)^(1/2)/(f*x+e)^(1/2)+2*(B*b-2*C*a)*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) /(-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)
Time = 25.18 (sec) , antiderivative size = 583, normalized size of antiderivative = 1.34 \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 (a+b x)^{3/2} \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \left (-\frac {b B \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} (g+h x) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )}{(b g-a h) (a+b x) \sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac {2 a C \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} (g+h x) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )}{(-b g+a h) (a+b x) \sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {C (-f g+e h) \sqrt {-\frac {(b e-a f) (b g-a h) (e+f x) (g+h x)}{(f g-e h)^2 (a+b x)^2}} \operatorname {EllipticPi}\left (\frac {b (-f g+e h)}{(b e-a f) h},\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )}{(b e-a f) h}\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \]
Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
(2*(a + b*x)^(3/2)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*( -((b*B*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x))]*(g + h*x)*Ell ipticF[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*g - a*h)*(a + b*x)*Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))])) - (2*a* C*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x))]*(g + h*x)*Elliptic F[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*g) + a*h)*(a + b*x)*Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]) + (C*(-(f* g) + e*h)*Sqrt[-(((b*e - a*f)*(b*g - a*h)*(e + f*x)*(g + h*x))/((f*g - e*h )^2*(a + b*x)^2))]*EllipticPi[(b*(-(f*g) + e*h))/((b*e - a*f)*h), ArcSin[S qrt[((-(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*e - a*f)*h)))/(Sqrt[c + d* x]*Sqrt[e + f*x]*Sqrt[g + h*x])
Time = 0.83 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2004, 2101, 183, 188, 321, 412}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {a^2 (-C)+a b B+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) |
\(\Big \downarrow \) 2004 |
\(\displaystyle \int \frac {\frac {a b B-a^2 C}{a}+b C x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx\) |
\(\Big \downarrow \) 2101 |
\(\displaystyle (b B-2 a C) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+C \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx\) |
\(\Big \downarrow \) 183 |
\(\displaystyle (b B-2 a C) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+\frac {2 C (a+b x) \sqrt {\frac {(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt {\frac {(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \int \frac {1}{\left (h-\frac {b (g+h x)}{a+b x}\right ) \sqrt {\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}+1} \sqrt {\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}+1}}d\frac {\sqrt {g+h x}}{\sqrt {a+b x}}}{\sqrt {c+d x} \sqrt {e+f x}}\) |
\(\Big \downarrow \) 188 |
\(\displaystyle \frac {2 \sqrt {g+h x} (b B-2 a C) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \int \frac {1}{\sqrt {\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}+1} \sqrt {1-\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}}d\frac {\sqrt {e+f x}}{\sqrt {a+b x}}}{\sqrt {c+d x} (f g-e h) \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac {2 C (a+b x) \sqrt {\frac {(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt {\frac {(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \int \frac {1}{\left (h-\frac {b (g+h x)}{a+b x}\right ) \sqrt {\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}+1} \sqrt {\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}+1}}d\frac {\sqrt {g+h x}}{\sqrt {a+b x}}}{\sqrt {c+d x} \sqrt {e+f x}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {2 C (a+b x) \sqrt {\frac {(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt {\frac {(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \int \frac {1}{\left (h-\frac {b (g+h x)}{a+b x}\right ) \sqrt {\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}+1} \sqrt {\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}+1}}d\frac {\sqrt {g+h x}}{\sqrt {a+b x}}}{\sqrt {c+d x} \sqrt {e+f x}}+\frac {2 \sqrt {g+h x} (b B-2 a C) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {c+d x} \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle \frac {2 \sqrt {g+h x} (b B-2 a C) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {c+d x} \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac {2 C (a+b x) \sqrt {c h-d g} \sqrt {\frac {(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt {\frac {(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {c h-d g} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{h \sqrt {c+d x} \sqrt {e+f x} \sqrt {b c-a d}}\) |
Int[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^(3/2)*Sqrt[c + d*x]*S qrt[e + f*x]*Sqrt[g + h*x]),x]
(2*(b*B - 2*a*C)*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqr t[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))) ])/(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))])/(Sqrt[b*c - a*d ]*h*Sqrt[c + d*x]*Sqrt[e + f*x])
3.1.23.3.1 Defintions of rubi rules used
Int[Sqrt[(a_.) + (b_.)*(x_)]/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*( x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[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)*Sq rt[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]
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.) *(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[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]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(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] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[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] && S implerSqrtQ[-f/e, -d/c])
Int[(u_)*((d_) + (e_.)*(x_))^(q_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.) , x_Symbol] :> Int[u*(d + e*x)^(p + q)*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, b , c, d, e, q}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)] *Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[(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] + Simp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(855\) vs. \(2(398)=796\).
Time = 6.31 (sec) , antiderivative size = 856, normalized size of antiderivative = 1.96
method | result | size |
elliptic | \(\frac {\sqrt {\left (b x +a \right ) \left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {2 \left (B b -C a \right ) \left (\frac {g}{h}-\frac {a}{b}\right ) \sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \left (x +\frac {c}{d}\right )^{2} \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {e}{f}\right )}{\left (-\frac {e}{f}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {g}{h}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, F\left (\sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}, \sqrt {\frac {\left (\frac {e}{f}-\frac {c}{d}\right ) \left (\frac {g}{h}-\frac {a}{b}\right )}{\left (-\frac {a}{b}+\frac {e}{f}\right ) \left (-\frac {c}{d}+\frac {g}{h}\right )}}\right )}{\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (-\frac {c}{d}+\frac {a}{b}\right ) \sqrt {b d f h \left (x +\frac {a}{b}\right ) \left (x +\frac {c}{d}\right ) \left (x +\frac {e}{f}\right ) \left (x +\frac {g}{h}\right )}}+\frac {2 C b \left (\frac {g}{h}-\frac {a}{b}\right ) \sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \left (x +\frac {c}{d}\right )^{2} \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {e}{f}\right )}{\left (-\frac {e}{f}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {g}{h}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \left (-\frac {c F\left (\sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}, \sqrt {\frac {\left (\frac {e}{f}-\frac {c}{d}\right ) \left (\frac {g}{h}-\frac {a}{b}\right )}{\left (-\frac {a}{b}+\frac {e}{f}\right ) \left (-\frac {c}{d}+\frac {g}{h}\right )}}\right )}{d}+\left (\frac {c}{d}-\frac {a}{b}\right ) \Pi \left (\sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}, \frac {-\frac {g}{h}+\frac {a}{b}}{-\frac {g}{h}+\frac {c}{d}}, \sqrt {\frac {\left (\frac {e}{f}-\frac {c}{d}\right ) \left (\frac {g}{h}-\frac {a}{b}\right )}{\left (-\frac {a}{b}+\frac {e}{f}\right ) \left (-\frac {c}{d}+\frac {g}{h}\right )}}\right )\right )}{\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (-\frac {c}{d}+\frac {a}{b}\right ) \sqrt {b d f h \left (x +\frac {a}{b}\right ) \left (x +\frac {c}{d}\right ) \left (x +\frac {e}{f}\right ) \left (x +\frac {g}{h}\right )}}\right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\) | \(856\) |
default | \(\text {Expression too large to display}\) | \(2955\) |
int((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1 /2)/(h*x+g)^(1/2),x,method=_RETURNVERBOSE)
((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*b-C*a)*(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*(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))))
Timed out. \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x +e)^(1/2)/(h*x+g)^(1/2),x, algorithm="fricas")
\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {B b - C a + C b x}{\sqrt {a + b x} \sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \]
integrate((C*b**2*x**2+B*b**2*x+B*a*b-C*a**2)/(b*x+a)**(3/2)/(d*x+c)**(1/2 )/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)
\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x +e)^(1/2)/(h*x+g)^(1/2),x, algorithm="maxima")
integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)
\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x +e)^(1/2)/(h*x+g)^(1/2),x, algorithm="giac")
integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)
Timed out. \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {-C\,a^2+B\,a\,b+C\,b^2\,x^2+B\,b^2\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \]
int((C*b^2*x^2 - C*a^2 + B*a*b + B*b^2*x)/((e + f*x)^(1/2)*(g + h*x)^(1/2) *(a + b*x)^(3/2)*(c + d*x)^(1/2)),x)