Integrand size = 62, antiderivative size = 734 \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {b C \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{f h \sqrt {c+d x}}-\frac {b C \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{d f h \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {C (b e-a f) \sqrt {b g-a h} \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{f h \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac {\sqrt {-d g+c h} (a C d f h-b (2 B d f h-C (d f g+d e h+c f h))) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {-d g+c h} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{d \sqrt {b c-a d} f h^2 \sqrt {c+d x} \sqrt {e+f x}} \]
-(a*C*d*f*h-b*(2*B*d*f*h-C*(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)/d/f/h^2/(-a*d+b*c)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)+ b*C*(b*x+a)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/f/h/(d*x+c)^(1/2)-C*(-a*f+b* e)*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)/f/h/(-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)-b*C*Elli pticE((-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/f/h/ ((-c*f+d*e)*(b*x+a)/(-a*f+b*e)/(d*x+c))^(1/2)/(h*x+g)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(8107\) vs. \(2(734)=1468\).
Time = 42.83 (sec) , antiderivative size = 8107, normalized size of antiderivative = 11.04 \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \]
Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/(Sqrt[a + b*x]*Sqrt[c + d* x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
Time = 1.26 (sec) , antiderivative size = 732, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2004, 2099, 183, 188, 194, 321, 327, 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.
\(\displaystyle \int \frac {a^2 (-C)+a b B+b^2 B x+b^2 C x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) |
\(\Big \downarrow \) 2004 |
\(\displaystyle \int \frac {\sqrt {a+b x} \left (\frac {a b B-a^2 C}{a}+b C x\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx\) |
\(\Big \downarrow \) 2099 |
\(\displaystyle \frac {(-a C d f h+2 b B d f h-b C (c f h+d e h+d f g)) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 d f h}+\frac {b C (d e-c f) (d g-c h) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 d f h}-\frac {C (b e-a f) (b g-a h) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 f h}+\frac {b C \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{f h \sqrt {c+d x}}\) |
\(\Big \downarrow \) 183 |
\(\displaystyle \frac {(a+b x) \sqrt {\frac {(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt {\frac {(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} (-a C d f h+2 b B d f h-b C (c f h+d e h+d f g)) \int \frac {1}{\left (h-\frac {b (g+h x)}{a+b x}\right ) \sqrt {\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}+1} \sqrt {\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}+1}}d\frac {\sqrt {g+h x}}{\sqrt {a+b x}}}{d f h \sqrt {c+d x} \sqrt {e+f x}}+\frac {b C (d e-c f) (d g-c h) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 d f h}-\frac {C (b e-a f) (b g-a h) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 f h}+\frac {b C \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{f h \sqrt {c+d x}}\) |
\(\Big \downarrow \) 188 |
\(\displaystyle \frac {(a+b x) \sqrt {\frac {(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt {\frac {(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} (-a C d f h+2 b B d f h-b C (c f h+d e h+d f g)) \int \frac {1}{\left (h-\frac {b (g+h x)}{a+b x}\right ) \sqrt {\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}+1} \sqrt {\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}+1}}d\frac {\sqrt {g+h x}}{\sqrt {a+b x}}}{d f h \sqrt {c+d x} \sqrt {e+f x}}+\frac {b C (d e-c f) (d g-c h) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 d f h}-\frac {C \sqrt {g+h x} (b e-a f) (b g-a h) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \int \frac {1}{\sqrt {\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}+1} \sqrt {1-\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}}d\frac {\sqrt {e+f x}}{\sqrt {a+b x}}}{f h \sqrt {c+d x} (f g-e h) \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac {b C \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{f h \sqrt {c+d x}}\) |
\(\Big \downarrow \) 194 |
\(\displaystyle \frac {(a+b x) \sqrt {\frac {(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt {\frac {(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} (-a C d f h+2 b B d f h-b C (c f h+d e h+d f g)) \int \frac {1}{\left (h-\frac {b (g+h x)}{a+b x}\right ) \sqrt {\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}+1} \sqrt {\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}+1}}d\frac {\sqrt {g+h x}}{\sqrt {a+b x}}}{d f h \sqrt {c+d x} \sqrt {e+f x}}-\frac {C \sqrt {g+h x} (b e-a f) (b g-a h) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \int \frac {1}{\sqrt {\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}+1} \sqrt {1-\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}}d\frac {\sqrt {e+f x}}{\sqrt {a+b x}}}{f h \sqrt {c+d x} (f g-e h) \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac {b C \sqrt {a+b x} (d g-c h) \sqrt {-\frac {(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} \int \frac {\sqrt {1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}}}{\sqrt {1-\frac {(d g-c h) (e+f x)}{(f g-e h) (c+d x)}}}d\frac {\sqrt {e+f x}}{\sqrt {c+d x}}}{d f h \sqrt {g+h x} \sqrt {\frac {(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}}+\frac {b C \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{f h \sqrt {c+d x}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {(a+b x) \sqrt {\frac {(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt {\frac {(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} (-a C d f h+2 b B d f h-b C (c f h+d e h+d f g)) \int \frac {1}{\left (h-\frac {b (g+h x)}{a+b x}\right ) \sqrt {\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}+1} \sqrt {\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}+1}}d\frac {\sqrt {g+h x}}{\sqrt {a+b x}}}{d f h \sqrt {c+d x} \sqrt {e+f x}}-\frac {b C \sqrt {a+b x} (d g-c h) \sqrt {-\frac {(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} \int \frac {\sqrt {1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}}}{\sqrt {1-\frac {(d g-c h) (e+f x)}{(f g-e h) (c+d x)}}}d\frac {\sqrt {e+f x}}{\sqrt {c+d x}}}{d f h \sqrt {g+h x} \sqrt {\frac {(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}}-\frac {C \sqrt {g+h x} (b e-a f) \sqrt {b g-a h} \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{f h \sqrt {c+d x} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac {b C \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{f h \sqrt {c+d x}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {(a+b x) \sqrt {\frac {(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt {\frac {(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} (-a C d f h+2 b B d f h-b C (c f h+d e h+d f g)) \int \frac {1}{\left (h-\frac {b (g+h x)}{a+b x}\right ) \sqrt {\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}+1} \sqrt {\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}+1}}d\frac {\sqrt {g+h x}}{\sqrt {a+b x}}}{d f h \sqrt {c+d x} \sqrt {e+f x}}-\frac {C \sqrt {g+h x} (b e-a f) \sqrt {b g-a h} \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{f h \sqrt {c+d x} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac {b C \sqrt {a+b x} \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{d f h \sqrt {g+h x} \sqrt {\frac {(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}}+\frac {b C \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{f h \sqrt {c+d x}}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle \frac {(a+b x) \sqrt {c h-d g} \sqrt {\frac {(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt {\frac {(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} (-a C d f h+2 b B d f h-b C (c f h+d e h+d f g)) \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {c h-d g} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{d f h^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {b c-a d}}-\frac {C \sqrt {g+h x} (b e-a f) \sqrt {b g-a h} \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{f h \sqrt {c+d x} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac {b C \sqrt {a+b x} \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{d f h \sqrt {g+h x} \sqrt {\frac {(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}}+\frac {b C \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{f h \sqrt {c+d x}}\) |
Int[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqr t[e + f*x]*Sqrt[g + h*x]),x]
(b*C*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(f*h*Sqrt[c + d*x]) - (b*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))])/(d*f*h*Sqrt[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d *x))]*Sqrt[g + h*x]) - (C*(b*e - a*f)*Sqrt[b*g - a*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)))])/(f*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)))]) + (Sqrt[-(d*g) + c*h]*(2*b*B*d*f*h - a*C*d*f*h - b*C*(d*f*g + d*e*h + c*f*h))*(a + b*x)*S qrt[((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))])/(d*Sqrt[b*c - a*d]*f*h^2*Sqrt[c + d*x]*Sqrt[e + f*x])
3.1.22.3.1 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[Sqrt[(c_.) + (d_.)*(x_)]/(((a_.) + (b_.)*(x_))^(3/2)*Sqrt[(e_.) + (f_.) *(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-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]*Sq rt[(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]
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[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[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]
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[(Sqrt[(a_.) + (b_.)*(x_)]*((A_.) + (B_.)*(x_)))/(Sqrt[(c_.) + (d_.)*(x_ )]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[B* Sqrt[a + b*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/(f*h*Sqrt[c + d*x])), x] + (-Sim p[B*(b*e - a*f)*((b*g - a*h)/(2*b*f*h)) Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x ]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] + Simp[B*(d*e - c*f)*((d*g - c*h)/(2 *d*f*h)) Int[Sqrt[a + b*x]/((c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] + Simp[(2*A*b*d*f*h + B*(a*d*f*h - b*(d*f*g + d*e*h + c*f*h)))/(2*b *d*f*h) 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] && NeQ[2*A*d*f - B*(d*e + c*f), 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1551\) vs. \(2(669)=1338\).
Time = 4.90 (sec) , antiderivative size = 1552, normalized size of antiderivative = 2.11
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1552\) |
default | \(\text {Expression too large to display}\) | \(20101\) |
int((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(1/2)/(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*a*b-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*B*b^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)*Ellipti cPi(((-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)))+C*b^2*((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)*Elli...
Timed out. \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x +e)^(1/2)/(h*x+g)^(1/2),x, algorithm="fricas")
\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\sqrt {a + b x} \left (B b - C a + C b x\right )}{\sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \]
integrate((C*b**2*x**2+B*b**2*x+B*a*b-C*a**2)/(b*x+a)**(1/2)/(d*x+c)**(1/2 )/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)
\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(1/2)/(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)/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)
\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(1/2)/(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)/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)
Timed out. \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {-C\,a^2+B\,a\,b+C\,b^2\,x^2+B\,b^2\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}} \,d x \]
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)^(1/2)*(c + d*x)^(1/2)),x)