Integrand size = 40, antiderivative size = 313 \[ \int \frac {A+B x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 B \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \left (A-\frac {a B}{b}\right ) \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}} \]
2*B*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)/b/d/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)-2*(A-a*B/b)*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 = 22.95 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.78 \[ \int \frac {A+B x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 i \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \left (b (-B c+A d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )+(-A b+a B) d \operatorname {EllipticPi}\left (-\frac {b c f-a d f}{b d e-b c f},i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )\right )}{b (-b c+a d) \sqrt {-c+\frac {d e}{f}} f \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {g+h x}} \]
((2*I)*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*(b*(-(B*c) + A*d)*E llipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e *h - c*f*h)] + (-(A*b) + a*B)*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*(-(b*c) + a*d)*Sqrt[-c + (d*e)/f]*f*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[g + h*x])
Time = 0.81 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {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.
\(\displaystyle \int \frac {A+B x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) |
\(\Big \downarrow \) 2110 |
\(\displaystyle \left (A-\frac {a B}{b}\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+\int \frac {B}{b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \left (A-\frac {a B}{b}\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+\frac {B \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b}\) |
\(\Big \downarrow \) 131 |
\(\displaystyle \left (A-\frac {a B}{b}\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+\frac {B \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {g+h x}}dx}{b \sqrt {e+f x}}\) |
\(\Big \downarrow \) 131 |
\(\displaystyle \left (A-\frac {a B}{b}\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+\frac {B \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}dx}{b \sqrt {e+f x} \sqrt {g+h x}}\) |
\(\Big \downarrow \) 130 |
\(\displaystyle \left (A-\frac {a B}{b}\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+\frac {2 B \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}\) |
\(\Big \downarrow \) 187 |
\(\displaystyle \frac {2 B \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-2 \left (A-\frac {a B}{b}\right ) \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {e-\frac {c f}{d}+\frac {f (c+d x)}{d}} \sqrt {g-\frac {c h}{d}+\frac {h (c+d x)}{d}}}d\sqrt {c+d x}\) |
\(\Big \downarrow \) 413 |
\(\displaystyle \frac {2 B \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \left (A-\frac {a B}{b}\right ) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {g-\frac {c h}{d}+\frac {h (c+d x)}{d}}}d\sqrt {c+d x}}{\sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e}}\) |
\(\Big \downarrow \) 413 |
\(\displaystyle \frac {2 B \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \left (A-\frac {a B}{b}\right ) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1} \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1}}d\sqrt {c+d x}}{\sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e} \sqrt {\frac {h (c+d x)}{d}-\frac {c h}{d}+g}}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle \frac {2 B \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \left (A-\frac {a B}{b}\right ) \sqrt {c f-d e} \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {f} (b c-a d) \sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e} \sqrt {\frac {h (c+d x)}{d}-\frac {c h}{d}+g}}\) |
(2*B*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))])/(b*d*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h* x]) - (2*(A - (a*B)/b)*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])
3.1.4.3.1 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[(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 = 2.96 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.53
method | result | size |
elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {2 B \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, F\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{b \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}+\frac {2 \left (A b -B a \right ) \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \Pi \left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {a}{b}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{b^{2} \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}\, \left (-\frac {g}{h}+\frac {a}{b}\right )}\right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\) | \(478\) |
default | \(-\frac {2 \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}\, \sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}\, \sqrt {\frac {\left (d x +c \right ) h}{c h -d g}}\, \sqrt {\frac {\left (f x +e \right ) h}{e h -f g}}\, \left (A \Pi \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \frac {\left (e h -f g \right ) b}{f \left (a h -g b \right )}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) b e \,h^{2}-A \Pi \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \frac {\left (e h -f g \right ) b}{f \left (a h -g b \right )}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) b f g h +B F\left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) a e \,h^{2}-B F\left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) a f g h -B F\left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) b e g h +B F\left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) b f \,g^{2}-B \Pi \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \frac {\left (e h -f g \right ) b}{f \left (a h -g b \right )}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) a e \,h^{2}+B \Pi \left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \frac {\left (e h -f g \right ) b}{f \left (a h -g b \right )}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) a f g h \right )}{f h b \left (a h -g b \right ) \left (d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g \right )}\) | \(668\) |
((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*B/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)*EllipticF(((x+g/h)/(g/h-e/f))^(1/2),((-g/h+e/f) /(-g/h+c/d))^(1/2))+2*(A*b-B*a)/b^2*(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)*Ellipti cPi(((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. \[ \int \frac {A+B x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {A + B x}{\left (a + b x\right ) \sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \]
\[ \int \frac {A+B x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {B x + A}{{\left (b x + a\right )} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
\[ \int \frac {A+B x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {B x + A}{{\left (b x + a\right )} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
Timed out. \[ \int \frac {A+B x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {A+B\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\left (a+b\,x\right )\,\sqrt {c+d\,x}} \,d x \]