Integrand size = 33, antiderivative size = 588 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=-\frac {\sqrt [4]{c f^2-g (b f-a g)} (d+e x) \sqrt {\frac {(e f-d g)^2 \left (a+b x+c x^2\right )}{\left (c f^2-b f g+a g^2\right ) (d+e x)^2}} \left (1+\frac {\sqrt {c d^2-b d e+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}\right ) \sqrt {\frac {1-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b f g+a g^2\right ) (d+e x)}+\frac {\left (c d^2-b d e+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}}{\left (1+\frac {\sqrt {c d^2-b d e+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-b d e+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-b f g+a g^2} \sqrt {d+e x}}\right ),\frac {1}{4} \left (2+\frac {2 c d f+2 a e g-b (e f+d g)}{\sqrt {c d^2-e (b d-a e)} \sqrt {c f^2-g (b f-a g)}}\right )\right )}{\sqrt [4]{c d^2-b d e+a e^2} (e f-d g) \sqrt {a+b x+c x^2} \sqrt {1-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b f g+a g^2\right ) (d+e x)}+\frac {\left (c d^2-b d e+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}}} \]
-(c*f^2-g*(-a*g+b*f))^(1/4)*(e*x+d)*(cos(2*arctan((a*e^2-b*d*e+c*d^2)^(1/4 )*(g*x+f)^(1/2)/(a*g^2-b*f*g+c*f^2)^(1/4)/(e*x+d)^(1/2)))^2)^(1/2)/cos(2*a rctan((a*e^2-b*d*e+c*d^2)^(1/4)*(g*x+f)^(1/2)/(a*g^2-b*f*g+c*f^2)^(1/4)/(e *x+d)^(1/2)))*EllipticF(sin(2*arctan((a*e^2-b*d*e+c*d^2)^(1/4)*(g*x+f)^(1/ 2)/(a*g^2-b*f*g+c*f^2)^(1/4)/(e*x+d)^(1/2))),1/2*(2+(2*c*d*f+2*a*e*g-b*(d* g+e*f))/(c*d^2-e*(-a*e+b*d))^(1/2)/(c*f^2-g*(-a*g+b*f))^(1/2))^(1/2))*(1+( g*x+f)*(a*e^2-b*d*e+c*d^2)^(1/2)/(e*x+d)/(c*f^2-g*(-a*g+b*f))^(1/2))*((-d* g+e*f)^2*(c*x^2+b*x+a)/(a*g^2-b*f*g+c*f^2)/(e*x+d)^2)^(1/2)*((1-(2*c*d*f+2 *a*e*g-b*(d*g+e*f))*(g*x+f)/(a*g^2-b*f*g+c*f^2)/(e*x+d)+(a*e^2-b*d*e+c*d^2 )*(g*x+f)^2/(c*f^2-g*(-a*g+b*f))/(e*x+d)^2)/(1+(g*x+f)*(a*e^2-b*d*e+c*d^2) ^(1/2)/(e*x+d)/(c*f^2-g*(-a*g+b*f))^(1/2))^2)^(1/2)/(a*e^2-b*d*e+c*d^2)^(1 /4)/(-d*g+e*f)/(c*x^2+b*x+a)^(1/2)/(1-(2*c*d*f+2*a*e*g-b*(d*g+e*f))*(g*x+f )/(a*g^2-b*f*g+c*f^2)/(e*x+d)+(a*e^2-b*d*e+c*d^2)*(g*x+f)^2/(c*f^2-g*(-a*g +b*f))/(e*x+d)^2)^(1/2)
Time = 26.55 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {2} e \sqrt {-\frac {e \left (c d^2+e (-b d+a e)\right ) (f+g x)}{\left (-2 c d e f+e \sqrt {\left (b^2-4 a c\right ) e^2} f-2 a e^2 g-d \sqrt {\left (b^2-4 a c\right ) e^2} g+b e (e f+d g)\right ) (d+e x)}} \sqrt {a+x (b+c x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {2 a e^2-2 c d e x+b e (-d+e x)+\sqrt {\left (b^2-4 a c\right ) e^2} (d+e x)}{\sqrt {\left (b^2-4 a c\right ) e^2} (d+e x)}}}{\sqrt {2}}\right ),\frac {2 \sqrt {\left (b^2-4 a c\right ) e^2} (e f-d g)}{-2 c d e f+e \sqrt {\left (b^2-4 a c\right ) e^2} f-2 a e^2 g-d \sqrt {\left (b^2-4 a c\right ) e^2} g+b e (e f+d g)}\right )}{\sqrt {\left (b^2-4 a c\right ) e^2} \sqrt {d+e x} \sqrt {f+g x} \sqrt {-\frac {\left (c d^2+e (-b d+a e)\right ) (a+x (b+c x))}{\left (b^2-4 a c\right ) (d+e x)^2}}} \]
(2*Sqrt[2]*e*Sqrt[-((e*(c*d^2 + e*(-(b*d) + a*e))*(f + g*x))/((-2*c*d*e*f + e*Sqrt[(b^2 - 4*a*c)*e^2]*f - 2*a*e^2*g - d*Sqrt[(b^2 - 4*a*c)*e^2]*g + b*e*(e*f + d*g))*(d + e*x)))]*Sqrt[a + x*(b + c*x)]*EllipticF[ArcSin[Sqrt[ (2*a*e^2 - 2*c*d*e*x + b*e*(-d + e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x)) /(Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))]/Sqrt[2]], (2*Sqrt[(b^2 - 4*a*c)*e^2] *(e*f - d*g))/(-2*c*d*e*f + e*Sqrt[(b^2 - 4*a*c)*e^2]*f - 2*a*e^2*g - d*Sq rt[(b^2 - 4*a*c)*e^2]*g + b*e*(e*f + d*g))])/(Sqrt[(b^2 - 4*a*c)*e^2]*Sqrt [d + e*x]*Sqrt[f + g*x]*Sqrt[-(((c*d^2 + e*(-(b*d) + a*e))*(a + x*(b + c*x )))/((b^2 - 4*a*c)*(d + e*x)^2))])
Time = 0.72 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1280, 1416}
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.
| \(\displaystyle \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 1280 |
| \(\displaystyle -\frac {2 (d+e x) \sqrt {\frac {\left (a+b x+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2-b f g+c f^2\right )}} \int \frac {1}{\sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{\sqrt {a+b x+c x^2} (e f-d g)}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {(d+e x) \sqrt [4]{c f^2-g (b f-a g)} \sqrt {\frac {\left (a+b x+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2-b f g+c f^2\right )}} \left (\frac {(f+g x) \sqrt {a e^2-b d e+c d^2}}{(d+e x) \sqrt {c f^2-g (b f-a g)}}+1\right ) \sqrt {\frac {\frac {(f+g x)^2 \left (a e^2-b d e+c d^2\right )}{(d+e x)^2 \left (c f^2-g (b f-a g)\right )}-\frac {(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left (a g^2-b f g+c f^2\right )}+1}{\left (\frac {(f+g x) \sqrt {a e^2-b d e+c d^2}}{(d+e x) \sqrt {c f^2-g (b f-a g)}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-b e d+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-b g f+a g^2} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 c d f+2 a e g-b (e f+d g)}{\sqrt {c d^2-e (b d-a e)} \sqrt {c f^2-g (b f-a g)}}+2\right )\right )}{\sqrt {a+b x+c x^2} (e f-d g) \sqrt [4]{a e^2-b d e+c d^2} \sqrt {\frac {(f+g x)^2 \left (a e^2-b d e+c d^2\right )}{(d+e x)^2 \left (c f^2-g (b f-a g)\right )}-\frac {(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left (a g^2-b f g+c f^2\right )}+1}}\) |
-(((c*f^2 - g*(b*f - a*g))^(1/4)*(d + e*x)*Sqrt[((e*f - d*g)^2*(a + b*x + c*x^2))/((c*f^2 - b*f*g + a*g^2)*(d + e*x)^2)]*(1 + (Sqrt[c*d^2 - b*d*e + a*e^2]*(f + g*x))/(Sqrt[c*f^2 - g*(b*f - a*g)]*(d + e*x)))*Sqrt[(1 - ((2*c *d*f + 2*a*e*g - b*(e*f + d*g))*(f + g*x))/((c*f^2 - b*f*g + a*g^2)*(d + e *x)) + ((c*d^2 - b*d*e + a*e^2)*(f + g*x)^2)/((c*f^2 - g*(b*f - a*g))*(d + e*x)^2))/(1 + (Sqrt[c*d^2 - b*d*e + a*e^2]*(f + g*x))/(Sqrt[c*f^2 - g*(b* f - a*g)]*(d + e*x)))^2]*EllipticF[2*ArcTan[((c*d^2 - b*d*e + a*e^2)^(1/4) *Sqrt[f + g*x])/((c*f^2 - b*f*g + a*g^2)^(1/4)*Sqrt[d + e*x])], (2 + (2*c* d*f + 2*a*e*g - b*(e*f + d*g))/(Sqrt[c*d^2 - e*(b*d - a*e)]*Sqrt[c*f^2 - g *(b*f - a*g)]))/4])/((c*d^2 - b*d*e + a*e^2)^(1/4)*(e*f - d*g)*Sqrt[a + b* x + c*x^2]*Sqrt[1 - ((2*c*d*f + 2*a*e*g - b*(e*f + d*g))*(f + g*x))/((c*f^ 2 - b*f*g + a*g^2)*(d + e*x)) + ((c*d^2 - b*d*e + a*e^2)*(f + g*x)^2)/((c* f^2 - g*(b*f - a*g))*(d + e*x)^2)]))
3.10.19.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.) *(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[-2*(d + e*x)*(Sqrt[(e*f - d*g)^2* ((a + b*x + c*x^2)/((c*f^2 - b*f*g + a*g^2)*(d + e*x)^2))]/((e*f - d*g)*Sqr t[a + b*x + c*x^2])) Subst[Int[1/Sqrt[1 - (2*c*d*f - b*e*f - b*d*g + 2*a* e*g)*(x^2/(c*f^2 - b*f*g + a*g^2)) + (c*d^2 - b*d*e + a*e^2)*(x^4/(c*f^2 - b*f*g + a*g^2))], x], x, Sqrt[f + g*x]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c , d, e, f, g}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Time = 6.06 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.02
| method | result | size |
| default | \(\frac {8 \left (-2 c^{2} d g \,x^{2}+2 c^{2} e f \,x^{2}+2 \sqrt {-4 a c +b^{2}}\, c d g x -2 \sqrt {-4 a c +b^{2}}\, c e f x -2 b c d g x +2 b c e f x +\sqrt {-4 a c +b^{2}}\, b d g -\sqrt {-4 a c +b^{2}}\, b e f +2 a c d g -2 a c e f -b^{2} d g +b^{2} e f \right ) F\left (\sqrt {-\frac {\left (e \sqrt {-4 a c +b^{2}}-b e +2 c d \right ) \left (g x +f \right )}{\left (d g -e f \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right )}}, 2 \sqrt {\frac {\sqrt {-4 a c +b^{2}}\, \left (d g -e f \right ) c}{\left (g \sqrt {-4 a c +b^{2}}+b g -2 c f \right ) \left (e \sqrt {-4 a c +b^{2}}-b e +2 c d \right )}}\right ) \sqrt {\frac {\left (2 c f -b g +g \sqrt {-4 a c +b^{2}}\right ) \left (e x +d \right )}{\left (d g -e f \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right )}}\, \sqrt {\frac {\left (2 c f -b g +g \sqrt {-4 a c +b^{2}}\right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}{\left (g \sqrt {-4 a c +b^{2}}+b g -2 c f \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right )}}\, \sqrt {-\frac {\left (e \sqrt {-4 a c +b^{2}}-b e +2 c d \right ) \left (g x +f \right )}{\left (d g -e f \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right )}}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {g x +f}\, \sqrt {e x +d}}{\sqrt {-\frac {\left (g x +f \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right ) \left (e x +d \right )}{c}}\, \left (b g -2 c f -g \sqrt {-4 a c +b^{2}}\right ) \left (-e \sqrt {-4 a c +b^{2}}+b e -2 c d \right ) \sqrt {\left (g x +f \right ) \left (c \,x^{2}+b x +a \right ) \left (e x +d \right )}}\) | \(601\) |
| elliptic | \(\frac {2 \sqrt {\left (g x +f \right ) \left (c \,x^{2}+b x +a \right ) \left (e x +d \right )}\, \left (-\frac {f}{g}+\frac {d}{e}\right ) \sqrt {\frac {\left (-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x +\frac {f}{g}\right )}{\left (-\frac {d}{e}+\frac {f}{g}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}\, {\left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}^{2} \sqrt {\frac {\left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}\right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}{\left (\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}\, \sqrt {\frac {\left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}\right ) \left (x +\frac {d}{e}\right )}{\left (-\frac {d}{e}+\frac {f}{g}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}\, F\left (\sqrt {\frac {\left (-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x +\frac {f}{g}\right )}{\left (-\frac {d}{e}+\frac {f}{g}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}, \sqrt {\frac {\left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (-\frac {f}{g}+\frac {d}{e}\right )}{\left (-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}\right )}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {e x +d}\, \left (-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}\right ) \sqrt {c e g \left (x +\frac {f}{g}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x +\frac {d}{e}\right )}}\) | \(621\) |
8*(-2*c^2*d*g*x^2+2*c^2*e*f*x^2+2*(-4*a*c+b^2)^(1/2)*c*d*g*x-2*(-4*a*c+b^2 )^(1/2)*c*e*f*x-2*b*c*d*g*x+2*b*c*e*f*x+(-4*a*c+b^2)^(1/2)*b*d*g-(-4*a*c+b ^2)^(1/2)*b*e*f+2*a*c*d*g-2*a*c*e*f-b^2*d*g+b^2*e*f)*EllipticF((-(e*(-4*a* c+b^2)^(1/2)-b*e+2*c*d)*(g*x+f)/(d*g-e*f)/(-b-2*c*x+(-4*a*c+b^2)^(1/2)))^( 1/2),2*((-4*a*c+b^2)^(1/2)*(d*g-e*f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(e *(-4*a*c+b^2)^(1/2)-b*e+2*c*d))^(1/2))*((2*c*f-b*g+g*(-4*a*c+b^2)^(1/2))*( e*x+d)/(d*g-e*f)/(-b-2*c*x+(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*f-b*g+g*(-4*a* c+b^2)^(1/2))*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f )/(-b-2*c*x+(-4*a*c+b^2)^(1/2)))^(1/2)*(-(e*(-4*a*c+b^2)^(1/2)-b*e+2*c*d)* (g*x+f)/(d*g-e*f)/(-b-2*c*x+(-4*a*c+b^2)^(1/2)))^(1/2)*(c*x^2+b*x+a)^(1/2) *(g*x+f)^(1/2)*(e*x+d)^(1/2)/(-1/c*(g*x+f)*(-b-2*c*x+(-4*a*c+b^2)^(1/2))*( b+2*c*x+(-4*a*c+b^2)^(1/2))*(e*x+d))^(1/2)/(b*g-2*c*f-g*(-4*a*c+b^2)^(1/2) )/(-e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/((g*x+f)*(c*x^2+b*x+a)*(e*x+d))^(1/2)
\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {e x + d} \sqrt {g x + f}} \,d x } \]
integral(sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)*sqrt(g*x + f)/(c*e*g*x^4 + (c *e*f + (c*d + b*e)*g)*x^3 + a*d*f + ((c*d + b*e)*f + (b*d + a*e)*g)*x^2 + (a*d*g + (b*d + a*e)*f)*x), x)
\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\sqrt {d + e x} \sqrt {f + g x} \sqrt {a + b x + c x^{2}}}\, dx \]
\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {e x + d} \sqrt {g x + f}} \,d x } \]
\[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {e x + d} \sqrt {g x + f}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\sqrt {f+g\,x}\,\sqrt {d+e\,x}\,\sqrt {c\,x^2+b\,x+a}} \,d x \]