Integrand size = 60, antiderivative size = 309 \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 C \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 (b B-2 a C) \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}} \]
2*C*EllipticF(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),((-c*f+d*e)*h/f/(-c*h+ d*g))^(1/2))*(c*f-d*e)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*(d*(h*x+g)/(-c*h +d*g))^(1/2)/d/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)-2*(B*b-2*C*a)*EllipticP i(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),-b*(-c*f+d*e)/(-a*d+b*c)/f,((-c*f+ d*e)*h/f/(-c*h+d*g))^(1/2))*(c*f-d*e)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*( d*(h*x+g)/(-c*h+d*g))^(1/2)/(-a*d+b*c)/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)
Result contains complex when optimal does not.
Time = 22.06 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.81 \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 i \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \left (-\left ((b c C-b B d+a C d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )\right )+(-b B+2 a C) d \operatorname {EllipticPi}\left (-\frac {b c f-a d f}{b d e-b c f},i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )\right )}{(-b c+a d) \sqrt {-c+\frac {d e}{f}} f \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {g+h x}} \]
Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^2*Sqrt[c + d*x] *Sqrt[e + f*x]*Sqrt[g + h*x]),x]
((2*I)*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*(-((b*c*C - b*B*d + a*C*d)*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c* f*h)/(d*e*h - c*f*h)]) + (-(b*B) + 2*a*C)*d*EllipticPi[-((b*c*f - a*d*f)/( b*d*e - b*c*f)), I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f *h)/(d*e*h - c*f*h)]))/((-(b*c) + a*d)*Sqrt[-c + (d*e)/f]*f*Sqrt[(d*(e + f *x))/(f*(c + d*x))]*Sqrt[g + h*x])
Time = 0.85 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2004, 2110, 27, 131, 131, 130, 187, 413, 413, 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.
| \(\displaystyle \int \frac {a^2 (-C)+a b B+b^2 B x+b^2 C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) |
| \(\Big \downarrow \) 2004 |
| \(\displaystyle \int \frac {\frac {a b B-a^2 C}{a}+b C x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx\) |
| \(\Big \downarrow \) 2110 |
| \(\displaystyle (b B-2 a C) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+\int \frac {C}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle (b B-2 a C) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+C \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle (b B-2 a C) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+\frac {C \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {g+h x}}dx}{\sqrt {e+f x}}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle (b B-2 a C) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+\frac {C \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}dx}{\sqrt {e+f x} \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 130 |
| \(\displaystyle (b B-2 a C) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+\frac {2 C \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 187 |
| \(\displaystyle \frac {2 C \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-2 (b B-2 a C) \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {e-\frac {c f}{d}+\frac {f (c+d x)}{d}} \sqrt {g-\frac {c h}{d}+\frac {h (c+d x)}{d}}}d\sqrt {c+d x}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {2 C \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 (b B-2 a C) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {g-\frac {c h}{d}+\frac {h (c+d x)}{d}}}d\sqrt {c+d x}}{\sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {2 C \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 (b B-2 a C) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1} \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1}}d\sqrt {c+d x}}{\sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e} \sqrt {\frac {h (c+d x)}{d}-\frac {c h}{d}+g}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {2 C \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 (b B-2 a C) \sqrt {c f-d e} \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {f} (b c-a d) \sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e} \sqrt {\frac {h (c+d x)}{d}-\frac {c h}{d}+g}}\) |
Int[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^2*Sqrt[c + d*x]*Sqrt[ e + f*x]*Sqrt[g + h*x]),x]
(2*C*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x)) /(d*g - c*h)]*EllipticF[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]] , ((d*e - c*f)*h)/(f*(d*g - c*h))])/(d*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h*x] ) - (2*(b*B - 2*a*C)*Sqrt[-(d*e) + c*f]*Sqrt[1 + (f*(c + d*x))/(d*e - c*f) ]*Sqrt[1 + (h*(c + d*x))/(d*g - c*h)]*EllipticPi[-((b*(d*e - c*f))/((b*c - a*d)*f)), ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f )*h)/(f*(d*g - c*h))])/((b*c - a*d)*Sqrt[f]*Sqrt[e - (c*f)/d + (f*(c + d*x ))/d]*Sqrt[g - (c*h)/d + (h*(c + d*x))/d])
3.1.19.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ [b/(b*e - a*f), 0] && SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f *x] && (PosQ[-(b*c - a*d)/d] || NegQ[-(b*e - a*f)/f])
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[Sqrt[b*((c + d*x)/(b*c - a*d))]/Sqrt[c + d*x] Int[1/(Sq rt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e + f*x]), x ], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[(b*c - a*d)/b, 0] && Simpler Q[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x]
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && !SimplerQ[e + f*x, c + d*x] && !SimplerQ[g + h*x, c + d*x]
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[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
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[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.)*((g_.) + (h_.)*(x_))^(q_.), x_Symbol] :> Simp[PolynomialRem ainder[Px, a + b*x, x] Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^ q, x], x] + Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d *x)^n*(e + f*x)^p*(g + h*x)^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p , q}, x] && PolyQ[Px, x] && EqQ[m, -1]
Time = 3.03 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.54
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {2 C \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, F\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}+\frac {2 \left (B b -2 C a \right ) \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \Pi \left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {a}{b}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{b \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}\, \left (-\frac {g}{h}+\frac {a}{b}\right )}\right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\) | \(475\) |
| default | \(-\frac {2 \sqrt {h x +g}\, \sqrt {f x +e}\, \sqrt {d x +c}\, \sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}\, \sqrt {\frac {\left (d x +c \right ) h}{c h -d g}}\, \sqrt {\frac {\left (f x +e \right ) h}{e h -f g}}\, \left (B \Pi \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \frac {\left (e h -f g \right ) b}{f \left (a h -g b \right )}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) b e \,h^{2}-B \Pi \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \frac {\left (e h -f g \right ) b}{f \left (a h -g b \right )}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) b f g h +C F\left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) a e \,h^{2}-C F\left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) a f g h -C F\left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) b e g h +C F\left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) b f \,g^{2}-2 C \Pi \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \frac {\left (e h -f g \right ) b}{f \left (a h -g b \right )}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) a e \,h^{2}+2 C \Pi \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \frac {\left (e h -f g \right ) b}{f \left (a h -g b \right )}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) a f g h \right )}{h f \left (a h -g b \right ) \left (d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g \right )}\) | \(666\) |
int((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2)/ (h*x+g)^(1/2),x,method=_RETURNVERBOSE)
((d*x+c)*(f*x+e)*(h*x+g))^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)* (2*C*(g/h-e/f)*((x+g/h)/(g/h-e/f))^(1/2)*((x+c/d)/(-g/h+c/d))^(1/2)*((x+e/ f)/(-g/h+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f* g*x+d*e*g*x+c*e*g)^(1/2)*EllipticF(((x+g/h)/(g/h-e/f))^(1/2),((-g/h+e/f)/( -g/h+c/d))^(1/2))+2*(B*b-2*C*a)/b*(g/h-e/f)*((x+g/h)/(g/h-e/f))^(1/2)*((x+ c/d)/(-g/h+c/d))^(1/2)*((x+e/f)/(-g/h+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+d*e *h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)/(-g/h+a/b)*EllipticP i(((x+g/h)/(g/h-e/f))^(1/2),(-g/h+e/f)/(-g/h+a/b),((-g/h+e/f)/(-g/h+c/d))^ (1/2)))
Timed out. \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^ (1/2)/(h*x+g)^(1/2),x, algorithm="fricas")
Timed out. \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
integrate((C*b**2*x**2+B*b**2*x+B*a*b-C*a**2)/(b*x+a)**2/(d*x+c)**(1/2)/(f *x+e)**(1/2)/(h*x+g)**(1/2),x)
\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{2} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^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)^2*sqrt(d*x + c) *sqrt(f*x + e)*sqrt(h*x + g)), x)
\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{2} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^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)^2*sqrt(d*x + c) *sqrt(f*x + e)*sqrt(h*x + g)), x)
Timed out. \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Hanged} \]