Integrand size = 26, antiderivative size = 1077 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx=-\frac {\sqrt [4]{b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)} \left (b+\sqrt {b^2-4 a c}+2 c x\right )^{3/2} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {\frac {\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right )^2 \left (d+f x^2\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}} \left (1+\frac {\sqrt {2 c^2 d-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right ) \sqrt {\frac {1-\frac {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}+\frac {\left (4 c^2 d+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}}{\left (1+\frac {\sqrt {2 c^2 d-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{2 c^2 d-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt [4]{b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}}\right ),\frac {1}{2} \left (1+\frac {\left (b+\sqrt {b^2-4 a c}\right ) (c d+a f)}{\sqrt {2 c^2 d-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)}}\right )\right )}{\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt [4]{2 c^2 d-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {a+b x+c x^2} \sqrt {d+f x^2} \sqrt {1-\frac {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}+\frac {\left (4 c^2 d+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}}} \]
-(cos(2*arctan((2*c^2*d-2*a*c*f+b*f*(b+(-4*a*c+b^2)^(1/2)))^(1/4)*(2*a+x*( b+(-4*a*c+b^2)^(1/2)))^(1/2)/(b^2*d-2*a*(-a*f+c*d)+b*d*(-4*a*c+b^2)^(1/2)) ^(1/4)/(b+2*c*x+(-4*a*c+b^2)^(1/2))^(1/2)))^2)^(1/2)/cos(2*arctan((2*c^2*d -2*a*c*f+b*f*(b+(-4*a*c+b^2)^(1/2)))^(1/4)*(2*a+x*(b+(-4*a*c+b^2)^(1/2)))^ (1/2)/(b^2*d-2*a*(-a*f+c*d)+b*d*(-4*a*c+b^2)^(1/2))^(1/4)/(b+2*c*x+(-4*a*c +b^2)^(1/2))^(1/2)))*EllipticF(sin(2*arctan((2*c^2*d-2*a*c*f+b*f*(b+(-4*a* c+b^2)^(1/2)))^(1/4)*(2*a+x*(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(b^2*d-2*a*(-a*f +c*d)+b*d*(-4*a*c+b^2)^(1/2))^(1/4)/(b+2*c*x+(-4*a*c+b^2)^(1/2))^(1/2))),1 /2*(2+2*(a*f+c*d)*(b+(-4*a*c+b^2)^(1/2))/(b^2*d-2*a*(-a*f+c*d)+b*d*(-4*a*c +b^2)^(1/2))^(1/2)/(2*c^2*d-2*a*c*f+b*f*(b+(-4*a*c+b^2)^(1/2)))^(1/2))^(1/ 2))*(b+2*c*x+(-4*a*c+b^2)^(1/2))^(3/2)*(b^2*d-2*a*(-a*f+c*d)+b*d*(-4*a*c+b ^2)^(1/2))^(1/4)*(1+(2*a+x*(b+(-4*a*c+b^2)^(1/2)))*(2*c^2*d-2*a*c*f+b*f*(b +(-4*a*c+b^2)^(1/2)))^(1/2)/(b+2*c*x+(-4*a*c+b^2)^(1/2))/(b^2*d-2*a*(-a*f+ c*d)+b*d*(-4*a*c+b^2)^(1/2))^(1/2))*(2*a+x*(b+(-4*a*c+b^2)^(1/2)))^(1/2)*( (f*x^2+d)*(4*a*c-(b+(-4*a*c+b^2)^(1/2))^2)^2/(b+2*c*x+(-4*a*c+b^2)^(1/2))^ 2/(4*a^2*f+d*(b+(-4*a*c+b^2)^(1/2))^2))^(1/2)*((1-4*(a*f+c*d)*(b+(-4*a*c+b ^2)^(1/2))*(2*a+x*(b+(-4*a*c+b^2)^(1/2)))/(b+2*c*x+(-4*a*c+b^2)^(1/2))/(4* a^2*f+d*(b+(-4*a*c+b^2)^(1/2))^2)+(2*a+x*(b+(-4*a*c+b^2)^(1/2)))^2*(4*c^2* d+f*(b+(-4*a*c+b^2)^(1/2))^2)/(b+2*c*x+(-4*a*c+b^2)^(1/2))^2/(4*a^2*f+d*(b +(-4*a*c+b^2)^(1/2))^2))/(1+(2*a+x*(b+(-4*a*c+b^2)^(1/2)))*(2*c^2*d-2*a...
Result contains complex when optimal does not.
Time = 4.03 (sec) , antiderivative size = 600, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx=-\frac {2 \sqrt {2} \left (-b+\sqrt {b^2-4 a c}-2 c x\right ) \left (-i \sqrt {d}+\sqrt {f} x\right ) \sqrt {-\frac {c \sqrt {b^2-4 a c} \left (i \sqrt {d}+\sqrt {f} x\right )}{\left (-2 i c \sqrt {d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} \sqrt {\frac {c \left (-i \sqrt {d} \left (\sqrt {b^2-4 a c}+2 c x\right )+\sqrt {f} \left (-2 a+\sqrt {b^2-4 a c} x\right )+b \left (-i \sqrt {d}-\sqrt {f} x\right )\right )}{\left (2 i c \sqrt {d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\left (-2 i c \sqrt {d}+\left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 i c \sqrt {d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}}\right ),\frac {c d-i \sqrt {b^2-4 a c} \sqrt {d} \sqrt {f}+a f}{c d+i \sqrt {b^2-4 a c} \sqrt {d} \sqrt {f}+a f}\right )}{\left (-2 i c \sqrt {d}+\left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \sqrt {\frac {i c \sqrt {b^2-4 a c} \left (\sqrt {d}+i \sqrt {f} x\right )}{\left (2 i c \sqrt {d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} \sqrt {d+f x^2} \sqrt {a+x (b+c x)}} \]
(-2*Sqrt[2]*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)*((-I)*Sqrt[d] + Sqrt[f]*x)*Sq rt[-((c*Sqrt[b^2 - 4*a*c]*(I*Sqrt[d] + Sqrt[f]*x))/(((-2*I)*c*Sqrt[d] + (b + Sqrt[b^2 - 4*a*c])*Sqrt[f])*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)))]*Sqrt[(c *((-I)*Sqrt[d]*(Sqrt[b^2 - 4*a*c] + 2*c*x) + Sqrt[f]*(-2*a + Sqrt[b^2 - 4* a*c]*x) + b*((-I)*Sqrt[d] - Sqrt[f]*x)))/(((2*I)*c*Sqrt[d] + (b + Sqrt[b^2 - 4*a*c])*Sqrt[f])*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))]*EllipticF[ArcSin[Sq rt[(((-2*I)*c*Sqrt[d] + (-b + Sqrt[b^2 - 4*a*c])*Sqrt[f])*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(((2*I)*c*Sqrt[d] + (b + Sqrt[b^2 - 4*a*c])*Sqrt[f])*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))]], (c*d - I*Sqrt[b^2 - 4*a*c]*Sqrt[d]*Sqrt[f ] + a*f)/(c*d + I*Sqrt[b^2 - 4*a*c]*Sqrt[d]*Sqrt[f] + a*f)])/(((-2*I)*c*Sq rt[d] + (-b + Sqrt[b^2 - 4*a*c])*Sqrt[f])*Sqrt[(I*c*Sqrt[b^2 - 4*a*c]*(Sqr t[d] + I*Sqrt[f]*x))/(((2*I)*c*Sqrt[d] + (b + Sqrt[b^2 - 4*a*c])*Sqrt[f])* (-b + Sqrt[b^2 - 4*a*c] - 2*c*x))]*Sqrt[d + f*x^2]*Sqrt[a + x*(b + c*x)])
Time = 1.22 (sec) , antiderivative size = 1077, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1324, 732, 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.
\(\displaystyle \int \frac {1}{\sqrt {d+f x^2} \sqrt {a+b x+c x^2}} \, dx\) |
\(\Big \downarrow \) 1324 |
\(\displaystyle \frac {\sqrt {\sqrt {b^2-4 a c}+b+2 c x} \sqrt {x \left (\sqrt {b^2-4 a c}+b\right )+2 a} \int \frac {1}{\sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {f x^2+d}}dx}{\sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 732 |
\(\displaystyle -\frac {2 \left (\sqrt {b^2-4 a c}+b+2 c x\right )^{3/2} \sqrt {x \left (\sqrt {b^2-4 a c}+b\right )+2 a} \sqrt {\frac {\left (4 a c-\left (\sqrt {b^2-4 a c}+b\right )^2\right )^2 \left (d+f x^2\right )}{\left (\sqrt {b^2-4 a c}+b+2 c x\right )^2 \left (4 a^2 f+d \left (\sqrt {b^2-4 a c}+b\right )^2\right )}} \int \frac {1}{\sqrt {\frac {\left (4 d c^2+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}-\frac {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1}}d\frac {\sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt {b+2 c x+\sqrt {b^2-4 a c}}}}{\left (4 a c-\left (\sqrt {b^2-4 a c}+b\right )^2\right ) \sqrt {d+f x^2} \sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle -\frac {\sqrt [4]{d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)} \left (b+2 c x+\sqrt {b^2-4 a c}\right )^{3/2} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {\frac {\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right )^2 \left (f x^2+d\right )}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}} \left (\frac {\sqrt {2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)} \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1\right ) \sqrt {\frac {\frac {\left (4 d c^2+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}-\frac {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1}{\left (\frac {\sqrt {2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)} \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt [4]{d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)} \sqrt {b+2 c x+\sqrt {b^2-4 a c}}}\right ),\frac {1}{2} \left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) (c d+a f)}{\sqrt {2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)}}+1\right )\right )}{\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt [4]{2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {c x^2+b x+a} \sqrt {f x^2+d} \sqrt {\frac {\left (4 d c^2+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}-\frac {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1}}\) |
-(((b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 2*a*(c*d - a*f))^(1/4)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)^(3/2)*Sqrt[2*a + (b + Sqrt[b^2 - 4*a*c])*x]*Sqrt[((4*a*c - (b + Sqrt[b^2 - 4*a*c])^2)^2*(d + f*x^2))/(((b + Sqrt[b^2 - 4*a*c])^2*d + 4*a^2*f)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)^2)]*(1 + (Sqrt[2*c^2*d - 2*a*c* f + b*(b + Sqrt[b^2 - 4*a*c])*f]*(2*a + (b + Sqrt[b^2 - 4*a*c])*x))/(Sqrt[ b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 2*a*(c*d - a*f)]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)))*Sqrt[(1 - (4*(b + Sqrt[b^2 - 4*a*c])*(c*d + a*f)*(2*a + (b + Sqrt [b^2 - 4*a*c])*x))/(((b + Sqrt[b^2 - 4*a*c])^2*d + 4*a^2*f)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)) + ((4*c^2*d + (b + Sqrt[b^2 - 4*a*c])^2*f)*(2*a + (b + Sqrt[b^2 - 4*a*c])*x)^2)/(((b + Sqrt[b^2 - 4*a*c])^2*d + 4*a^2*f)*(b + Sqr t[b^2 - 4*a*c] + 2*c*x)^2))/(1 + (Sqrt[2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4*a*c])*f]*(2*a + (b + Sqrt[b^2 - 4*a*c])*x))/(Sqrt[b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 2*a*(c*d - a*f)]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)))^2]*Ellipti cF[2*ArcTan[((2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4*a*c])*f)^(1/4)*Sqrt[ 2*a + (b + Sqrt[b^2 - 4*a*c])*x])/((b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 2*a*(c *d - a*f))^(1/4)*Sqrt[b + Sqrt[b^2 - 4*a*c] + 2*c*x])], (1 + ((b + Sqrt[b^ 2 - 4*a*c])*(c*d + a*f))/(Sqrt[2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4*a*c ])*f]*Sqrt[b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 2*a*(c*d - a*f)]))/2])/((4*a*c - (b + Sqrt[b^2 - 4*a*c])^2)*(2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4*a*c] )*f)^(1/4)*Sqrt[a + b*x + c*x^2]*Sqrt[d + f*x^2]*Sqrt[1 - (4*(b + Sqrt[...
3.1.13.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(a_) + (b_.)* (x_)^2]), x_Symbol] :> Simp[-2*(c + d*x)*(Sqrt[(d*e - c*f)^2*((a + b*x^2)/( (b*e^2 + a*f^2)*(c + d*x)^2))]/((d*e - c*f)*Sqrt[a + b*x^2])) Subst[Int[1 /Sqrt[Simp[1 - (2*b*c*e + 2*a*d*f)*(x^2/(b*e^2 + a*f^2)) + (b*c^2 + a*d^2)* (x^4/(b*e^2 + a*f^2)), x]], x], x, Sqrt[e + f*x]/Sqrt[c + d*x]], x] /; Free Q[{a, b, c, d, e, f}, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]*Sqrt[(d_) + (f_.)*(x_)^2]), x _Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[b + r + 2*c*x]*(Sqrt[2 *a + (b + r)*x]/Sqrt[a + b*x + c*x^2]) Int[1/(Sqrt[b + r + 2*c*x]*Sqrt[2* a + (b + r)*x]*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, f}, x] && Ne Q[b^2 - 4*a*c, 0]
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 = 5.29 (sec) , antiderivative size = 721, normalized size of antiderivative = 0.67
method | result | size |
default | \(\frac {16 \left (b c f \,x^{2}-2 c^{2} x^{2} \sqrt {-d f}-c f \,x^{2} \sqrt {-4 a c +b^{2}}+4 a c f x -2 b c x \sqrt {-d f}-2 c x \sqrt {-d f}\, \sqrt {-4 a c +b^{2}}+b a f +2 a c \sqrt {-d f}+a f \sqrt {-4 a c +b^{2}}-b^{2} \sqrt {-d f}-b \sqrt {-d f}\, \sqrt {-4 a c +b^{2}}\right ) F\left (\sqrt {-\frac {\left (2 \sqrt {-d f}\, c -f \sqrt {-4 a c +b^{2}}-b f \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right )}{\left (f \sqrt {-4 a c +b^{2}}+2 \sqrt {-d f}\, c -b f \right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}}, \sqrt {\frac {\left (f \sqrt {-4 a c +b^{2}}+2 \sqrt {-d f}\, c +b f \right ) \left (f \sqrt {-4 a c +b^{2}}+2 \sqrt {-d f}\, c -b f \right )}{\left (2 \sqrt {-d f}\, c -f \sqrt {-4 a c +b^{2}}+b f \right ) \left (2 \sqrt {-d f}\, c -f \sqrt {-4 a c +b^{2}}-b f \right )}}\right ) \sqrt {\frac {\sqrt {-4 a c +b^{2}}\, \left (f x +\sqrt {-d f}\right ) c}{\left (f \sqrt {-4 a c +b^{2}}+2 \sqrt {-d f}\, c -b f \right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}}\, \sqrt {\frac {\sqrt {-4 a c +b^{2}}\, \left (-f x +\sqrt {-d f}\right ) c}{\left (2 \sqrt {-d f}\, c -f \sqrt {-4 a c +b^{2}}+b f \right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}}\, \sqrt {-\frac {\left (2 \sqrt {-d f}\, c -f \sqrt {-4 a c +b^{2}}-b f \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right )}{\left (f \sqrt {-4 a c +b^{2}}+2 \sqrt {-d f}\, c -b f \right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}}\, \sqrt {f \,x^{2}+d}\, \sqrt {c \,x^{2}+b x +a}}{\sqrt {\frac {\left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right ) \left (-f x +\sqrt {-d f}\right ) \left (f x +\sqrt {-d f}\right )}{c f}}\, \sqrt {-4 a c +b^{2}}\, \left (f \sqrt {-4 a c +b^{2}}-2 \sqrt {-d f}\, c +b f \right ) \sqrt {\left (c \,x^{2}+b x +a \right ) \left (f \,x^{2}+d \right )}}\) | \(721\) |
elliptic | \(\frac {2 \sqrt {\left (c \,x^{2}+b x +a \right ) \left (f \,x^{2}+d \right )}\, \left (\frac {\sqrt {-d f}}{f}+\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {\left (-\frac {\sqrt {-d f}}{f}+\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 (-\frac {\sqrt {-d f}}{f}-\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 {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 {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x -\frac {\sqrt {-d f}}{f}\right )}{\left (\frac {\sqrt {-d f}}{f}-\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 {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x +\frac {\sqrt {-d f}}{f}\right )}{\left (-\frac {\sqrt {-d f}}{f}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}\, F\left (\sqrt {\frac {\left (-\frac {\sqrt {-d f}}{f}+\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 (-\frac {\sqrt {-d f}}{f}-\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 {\sqrt {-d f}}{f}\right ) \left (\frac {\sqrt {-d f}}{f}+\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}{\left (-\frac {\sqrt {-d f}}{f}+\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (\frac {\sqrt {-d f}}{f}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}\right )}{\sqrt {c \,x^{2}+b x +a}\, \sqrt {f \,x^{2}+d}\, \left (-\frac {\sqrt {-d f}}{f}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {c f \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 {\sqrt {-d f}}{f}\right ) \left (x +\frac {\sqrt {-d f}}{f}\right )}}\) | \(787\) |
16*(b*c*f*x^2-2*c^2*x^2*(-d*f)^(1/2)-c*f*x^2*(-4*a*c+b^2)^(1/2)+4*a*c*f*x- 2*b*c*x*(-d*f)^(1/2)-2*c*x*(-d*f)^(1/2)*(-4*a*c+b^2)^(1/2)+b*a*f+2*a*c*(-d *f)^(1/2)+a*f*(-4*a*c+b^2)^(1/2)-b^2*(-d*f)^(1/2)-b*(-d*f)^(1/2)*(-4*a*c+b ^2)^(1/2))*EllipticF((-(2*(-d*f)^(1/2)*c-f*(-4*a*c+b^2)^(1/2)-b*f)*(-b-2*c *x+(-4*a*c+b^2)^(1/2))/(f*(-4*a*c+b^2)^(1/2)+2*(-d*f)^(1/2)*c-b*f)/(b+2*c* x+(-4*a*c+b^2)^(1/2)))^(1/2),((f*(-4*a*c+b^2)^(1/2)+2*(-d*f)^(1/2)*c+b*f)* (f*(-4*a*c+b^2)^(1/2)+2*(-d*f)^(1/2)*c-b*f)/(2*(-d*f)^(1/2)*c-f*(-4*a*c+b^ 2)^(1/2)+b*f)/(2*(-d*f)^(1/2)*c-f*(-4*a*c+b^2)^(1/2)-b*f))^(1/2))*((-4*a*c +b^2)^(1/2)*(f*x+(-d*f)^(1/2))*c/(f*(-4*a*c+b^2)^(1/2)+2*(-d*f)^(1/2)*c-b* f)/(b+2*c*x+(-4*a*c+b^2)^(1/2)))^(1/2)*((-4*a*c+b^2)^(1/2)*(-f*x+(-d*f)^(1 /2))*c/(2*(-d*f)^(1/2)*c-f*(-4*a*c+b^2)^(1/2)+b*f)/(b+2*c*x+(-4*a*c+b^2)^( 1/2)))^(1/2)*(-(2*(-d*f)^(1/2)*c-f*(-4*a*c+b^2)^(1/2)-b*f)*(-b-2*c*x+(-4*a *c+b^2)^(1/2))/(f*(-4*a*c+b^2)^(1/2)+2*(-d*f)^(1/2)*c-b*f)/(b+2*c*x+(-4*a* c+b^2)^(1/2)))^(1/2)*(f*x^2+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/(1/c/f*(-b-2*c*x+ (-4*a*c+b^2)^(1/2))*(b+2*c*x+(-4*a*c+b^2)^(1/2))*(-f*x+(-d*f)^(1/2))*(f*x+ (-d*f)^(1/2)))^(1/2)/(-4*a*c+b^2)^(1/2)/(f*(-4*a*c+b^2)^(1/2)-2*(-d*f)^(1/ 2)*c+b*f)/((c*x^2+b*x+a)*(f*x^2+d))^(1/2)
\[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {f x^{2} + d}} \,d x } \]
integral(sqrt(c*x^2 + b*x + a)*sqrt(f*x^2 + d)/(c*f*x^4 + b*f*x^3 + b*d*x + (c*d + a*f)*x^2 + a*d), x)
\[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx=\int \frac {1}{\sqrt {d + f x^{2}} \sqrt {a + b x + c x^{2}}}\, dx \]
\[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {f x^{2} + d}} \,d x } \]
\[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {f x^{2} + d}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx=\int \frac {1}{\sqrt {f\,x^2+d}\,\sqrt {c\,x^2+b\,x+a}} \,d x \]