Integrand size = 31, antiderivative size = 1049 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 (d+e x)}-\frac {\sqrt {b^2-4 a c} (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{4 \sqrt {2} e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {\sqrt {b^2-4 a c} \left (e^2 (b f-a g)+c d (-2 e f+d g)\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} g}{-2 c f+\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{2 \sqrt {2} e^2 \left (c d^2+e (-b d+a e)\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+x (b+c x)}}-\frac {\sqrt {2 c f-b g+\sqrt {b^2-4 a c} g} \left (3 a^2 e^4 g^2+c^2 d^3 g (4 e f-d g)+b^2 e^3 f (-e f+4 d g)+2 a c e^2 \left (2 e^2 f^2-2 d e f g+3 d^2 g^2\right )-2 b e^2 g \left (3 c d^2 f+a e (e f+2 d g)\right )\right ) \sqrt {\frac {g \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}{2 c f+\left (-b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {\frac {g \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c f+\left (b+\sqrt {b^2-4 a c}\right ) g}} \operatorname {EllipticPi}\left (\frac {2 c e f-b e g+\sqrt {b^2-4 a c} e g}{2 c e f-2 c d g},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-b g+\sqrt {b^2-4 a c} g}}\right ),\frac {2 c f+\left (-b+\sqrt {b^2-4 a c}\right ) g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{4 \sqrt {2} \sqrt {c} e^2 \left (c d^2+e (-b d+a e)\right ) (e f-d g)^3 \sqrt {a+x (b+c x)}} \]
-1/2*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/(-d*g+e*f)/(e*x+d)^2+1/4*(c*d*(d*g+ 2*e*f)-e*(-3*a*e*g+2*b*d*g+b*e*f))*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/(a*e^ 2-b*d*e+c*d^2)/(-d*g+e*f)^2/(e*x+d)-1/8*(c*d*(d*g+2*e*f)-e*(-3*a*e*g+2*b*d *g+b*e*f))*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2)) ^(1/2)*2^(1/2),(-2*g*(-4*a*c+b^2)^(1/2)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^ (1/2))*(-4*a*c+b^2)^(1/2)*(g*x+f)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1 /2)/e/(a*e^2-b*d*e+c*d^2)/(-d*g+e*f)^2*2^(1/2)/(c*x^2+b*x+a)^(1/2)/(c*(g*x +f)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/2)-1/4*(e^2*(-a*g+b*f)+c*d*(d*g-2 *e*f))*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/ 2)*2^(1/2),2^(1/2)*(g*(-4*a*c+b^2)^(1/2)/(-2*c*f+g*(b+(-4*a*c+b^2)^(1/2))) )^(1/2))*(-4*a*c+b^2)^(1/2)*(c*(a+x*(c*x+b))/(4*a*c-b^2))^(1/2)*(c*(g*x+f) /(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/e^2/(c*d^2+e*(a*e-b*d))/(-d*g+e*f )*2^(1/2)/(g*x+f)^(1/2)/(a+x*(c*x+b))^(1/2)-1/8*(3*a^2*e^4*g^2+c^2*d^3*g*( -d*g+4*e*f)+b^2*e^3*f*(4*d*g-e*f)+2*a*c*e^2*(3*d^2*g^2-2*d*e*f*g+2*e^2*f^2 )-2*b*e^2*g*(3*c*d^2*f+a*e*(2*d*g+e*f)))*EllipticPi(2^(1/2)*c^(1/2)*(g*x+f )^(1/2)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2))^(1/2),(2*c*e*f-b*e*g+e*g*(-4*a*c+ b^2)^(1/2))/(-2*c*d*g+2*c*e*f),((2*c*f+g*(-b+(-4*a*c+b^2)^(1/2)))/(2*c*f-g *(b+(-4*a*c+b^2)^(1/2))))^(1/2))*(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2))^(1/2)*(g *(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*f+g*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*(g *(b+2*c*x+(-4*a*c+b^2)^(1/2))/(-2*c*f+g*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/...
Result contains complex when optimal does not.
Time = 36.62 (sec) , antiderivative size = 36617, normalized size of antiderivative = 34.91 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\text {Result too large to show} \]
Time = 3.55 (sec) , antiderivative size = 1613, normalized size of antiderivative = 1.54, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.581, Rules used = {1275, 2154, 1282, 25, 2154, 25, 27, 1172, 321, 1269, 1172, 321, 327, 1279, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx\) |
| \(\Big \downarrow \) 1275 |
| \(\displaystyle \frac {\int \frac {-c g x^2+2 (c f-b g) x+b f-3 a g}{(d+e x)^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 2154 |
| \(\displaystyle \frac {\int \frac {\frac {2 c f}{e}-\frac {2 b g}{e}+\frac {c d g}{e^2}-\frac {c g x}{e}}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx-\frac {(c d (d g+2 e f)-e (-3 a e g+2 b d g+b e f)) \int \frac {1}{(d+e x)^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e^2}}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 1282 |
| \(\displaystyle \frac {\int \frac {\frac {2 c f}{e}-\frac {2 b g}{e}+\frac {c d g}{e^2}-\frac {c g x}{e}}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx-\frac {(c d (d g+2 e f)-e (-3 a e g+2 b d g+b e f)) \left (-\frac {\int -\frac {c e^2 g x^2+2 c d e g x+2 c d (e f-d g)-e (b e f-2 b d g+a e g)}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{2 (e f-d g) \left (a e^2-b d e+c d^2\right )}-\frac {e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{(d+e x) (e f-d g) \left (a e^2-b d e+c d^2\right )}\right )}{e^2}}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\frac {2 c f}{e}-\frac {2 b g}{e}+\frac {c d g}{e^2}-\frac {c g x}{e}}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx-\frac {(c d (d g+2 e f)-e (-3 a e g+2 b d g+b e f)) \left (\frac {\int \frac {c e^2 g x^2+2 c d e g x+2 c d (e f-d g)-e (b e f-2 b d g+a e g)}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{2 (e f-d g) \left (a e^2-b d e+c d^2\right )}-\frac {e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{(d+e x) (e f-d g) \left (a e^2-b d e+c d^2\right )}\right )}{e^2}}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 2154 |
| \(\displaystyle \frac {-\frac {(c d (d g+2 e f)-e (-3 a e g+2 b d g+b e f)) \left (\frac {(c d (2 e f-3 d g)-e (a e g-2 b d g+b e f)) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx+\int \frac {c d g+c e x g}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{2 (e f-d g) \left (a e^2-b d e+c d^2\right )}-\frac {e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{(d+e x) (e f-d g) \left (a e^2-b d e+c d^2\right )}\right )}{e^2}+\frac {2 (-b e g+c d g+c e f) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e^2}+\int -\frac {c g}{e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {(c d (d g+2 e f)-e (-3 a e g+2 b d g+b e f)) \left (\frac {(c d (2 e f-3 d g)-e (a e g-2 b d g+b e f)) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx+\int \frac {c d g+c e x g}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{2 (e f-d g) \left (a e^2-b d e+c d^2\right )}-\frac {e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{(d+e x) (e f-d g) \left (a e^2-b d e+c d^2\right )}\right )}{e^2}+\frac {2 (-b e g+c d g+c e f) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e^2}-\int \frac {c g}{e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {(c d (d g+2 e f)-e (-3 a e g+2 b d g+b e f)) \left (\frac {(c d (2 e f-3 d g)-e (a e g-2 b d g+b e f)) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx+\int \frac {c d g+c e x g}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{2 (e f-d g) \left (a e^2-b d e+c d^2\right )}-\frac {e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{(d+e x) (e f-d g) \left (a e^2-b d e+c d^2\right )}\right )}{e^2}+\frac {2 (-b e g+c d g+c e f) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e^2}-\frac {c g \int \frac {1}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e^2}}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {-\frac {2 \sqrt {2} g \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {g \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {(c d (d g+2 e f)-e (-3 a e g+2 b d g+b e f)) \left (\frac {(c d (2 e f-3 d g)-e (a e g-2 b d g+b e f)) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx+\int \frac {c d g+c e x g}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{2 (e f-d g) \left (a e^2-b d e+c d^2\right )}-\frac {e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{(d+e x) (e f-d g) \left (a e^2-b d e+c d^2\right )}\right )}{e^2}+\frac {2 (-b e g+c d g+c e f) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e^2}}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {-\frac {(c d (d g+2 e f)-e (-3 a e g+2 b d g+b e f)) \left (\frac {(c d (2 e f-3 d g)-e (a e g-2 b d g+b e f)) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx+\int \frac {c d g+c e x g}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{2 (e f-d g) \left (a e^2-b d e+c d^2\right )}-\frac {e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{(d+e x) (e f-d g) \left (a e^2-b d e+c d^2\right )}\right )}{e^2}+\frac {2 (-b e g+c d g+c e f) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e^2}-\frac {2 \sqrt {2} g \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {-\frac {(c d (d g+2 e f)-e (-3 a e g+2 b d g+b e f)) \left (\frac {-c (e f-d g) \int \frac {1}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx+(c d (2 e f-3 d g)-e (a e g-2 b d g+b e f)) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx+c e \int \frac {\sqrt {f+g x}}{\sqrt {c x^2+b x+a}}dx}{2 (e f-d g) \left (a e^2-b d e+c d^2\right )}-\frac {e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{(d+e x) (e f-d g) \left (a e^2-b d e+c d^2\right )}\right )}{e^2}+\frac {2 (-b e g+c d g+c e f) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e^2}-\frac {2 \sqrt {2} g \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}+\frac {2 (c e f+c d g-b e g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e^2}-\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \left (\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {g \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} e \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {g \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{\sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}}{2 \left (c d^2-b e d+a e^2\right ) (e f-d g)}-\frac {e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) (e f-d g) (d+e x)}\right )}{e^2}}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {c x^2+b x+a}}{2 (e f-d g) (d+e x)^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}+\frac {2 (c e f+c d g-b e g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e^2}-\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \left (\frac {-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}+(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx+\frac {\sqrt {2} \sqrt {b^2-4 a c} e \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {g \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{\sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}}{2 \left (c d^2-b e d+a e^2\right ) (e f-d g)}-\frac {e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) (e f-d g) (d+e x)}\right )}{e^2}}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {c x^2+b x+a}}{2 (e f-d g) (d+e x)^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}+\frac {2 (c e f+c d g-b e g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e^2}-\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \left (\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} e \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}+(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{2 \left (c d^2-b e d+a e^2\right ) (e f-d g)}-\frac {e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) (e f-d g) (d+e x)}\right )}{e^2}}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {c x^2+b x+a}}{2 (e f-d g) (d+e x)^2}\) |
| \(\Big \downarrow \) 1279 |
| \(\displaystyle \frac {-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}+\frac {2 (c e f+c d g-b e g) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \int \frac {1}{\sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} (d+e x) \sqrt {f+g x}}dx}{e^2 \sqrt {c x^2+b x+a}}-\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \left (\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} e \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}+\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \int \frac {1}{\sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} (d+e x) \sqrt {f+g x}}dx}{\sqrt {c x^2+b x+a}}}{2 \left (c d^2-b e d+a e^2\right ) (e f-d g)}-\frac {e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) (e f-d g) (d+e x)}\right )}{e^2}}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {c x^2+b x+a}}{2 (e f-d g) (d+e x)^2}\) |
| \(\Big \downarrow \) 187 |
| \(\displaystyle \frac {-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}-\frac {4 (c e f+c d g-b e g) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {b+\frac {2 c (f+g x)}{g}-\sqrt {b^2-4 a c}-\frac {2 c f}{g}} \sqrt {b+\frac {2 c (f+g x)}{g}+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}}d\sqrt {f+g x}}{e^2 \sqrt {c x^2+b x+a}}-\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \left (\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} e \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}-\frac {2 (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {b+\frac {2 c (f+g x)}{g}-\sqrt {b^2-4 a c}-\frac {2 c f}{g}} \sqrt {b+\frac {2 c (f+g x)}{g}+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}}d\sqrt {f+g x}}{\sqrt {c x^2+b x+a}}}{2 \left (c d^2-b e d+a e^2\right ) (e f-d g)}-\frac {e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) (e f-d g) (d+e x)}\right )}{e^2}}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {c x^2+b x+a}}{2 (e f-d g) (d+e x)^2}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}-\frac {4 (c e f+c d g-b e g) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {b+\frac {2 c (f+g x)}{g}+\sqrt {b^2-4 a c}-\frac {2 c f}{g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}}d\sqrt {f+g x}}{e^2 \sqrt {c x^2+b x+a} \sqrt {b+\frac {2 c (f+g x)}{g}-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}}-\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \left (\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} e \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}-\frac {2 (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {b+\frac {2 c (f+g x)}{g}+\sqrt {b^2-4 a c}-\frac {2 c f}{g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}}d\sqrt {f+g x}}{\sqrt {c x^2+b x+a} \sqrt {b+\frac {2 c (f+g x)}{g}-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}}}{2 \left (c d^2-b e d+a e^2\right ) (e f-d g)}-\frac {e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) (e f-d g) (d+e x)}\right )}{e^2}}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {c x^2+b x+a}}{2 (e f-d g) (d+e x)^2}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}-\frac {4 (c e f+c d g-b e g) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}}}d\sqrt {f+g x}}{e^2 \sqrt {c x^2+b x+a} \sqrt {b+\frac {2 c (f+g x)}{g}-\sqrt {b^2-4 a c}-\frac {2 c f}{g}} \sqrt {b+\frac {2 c (f+g x)}{g}+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}}-\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \left (\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} e \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}-\frac {2 (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}}}d\sqrt {f+g x}}{\sqrt {c x^2+b x+a} \sqrt {b+\frac {2 c (f+g x)}{g}-\sqrt {b^2-4 a c}-\frac {2 c f}{g}} \sqrt {b+\frac {2 c (f+g x)}{g}+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}}}{2 \left (c d^2-b e d+a e^2\right ) (e f-d g)}-\frac {e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) (e f-d g) (d+e x)}\right )}{e^2}}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {c x^2+b x+a}}{2 (e f-d g) (d+e x)^2}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} (c e f+c d g-b e g) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \operatorname {EllipticPi}\left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right ),\frac {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {c} e^2 (e f-d g) \sqrt {c x^2+b x+a} \sqrt {b+\frac {2 c (f+g x)}{g}-\sqrt {b^2-4 a c}-\frac {2 c f}{g}} \sqrt {b+\frac {2 c (f+g x)}{g}+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}}-\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \left (\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} e \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (e f-d g) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}-\frac {\sqrt {2} \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \operatorname {EllipticPi}\left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right ),\frac {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {c} (e f-d g) \sqrt {c x^2+b x+a} \sqrt {b+\frac {2 c (f+g x)}{g}-\sqrt {b^2-4 a c}-\frac {2 c f}{g}} \sqrt {b+\frac {2 c (f+g x)}{g}+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}}}{2 \left (c d^2-b e d+a e^2\right ) (e f-d g)}-\frac {e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) (e f-d g) (d+e x)}\right )}{e^2}}{4 (e f-d g)}-\frac {\sqrt {f+g x} \sqrt {c x^2+b x+a}}{2 (e f-d g) (d+e x)^2}\) |
-1/2*(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/((e*f - d*g)*(d + e*x)^2) + ((- 2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*g*Sqrt[(c*(f + g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[ Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt [b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(e^2*Sqrt[f + g*x]* Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g ]*(c*e*f + c*d*g - b*e*g)*Sqrt[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Sqrt[b + Sqr t[b^2 - 4*a*c] + 2*c*x]*Sqrt[1 - (2*c*(f + g*x))/(2*c*f - (b - Sqrt[b^2 - 4*a*c])*g)]*Sqrt[1 - (2*c*(f + g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]* EllipticPi[(e*(2*c*f - b*g + Sqrt[b^2 - 4*a*c]*g))/(2*c*(e*f - d*g)), ArcS in[(Sqrt[2]*Sqrt[c]*Sqrt[f + g*x])/Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g] ], (2*c*f - (b - Sqrt[b^2 - 4*a*c])*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g) ])/(Sqrt[c]*e^2*(e*f - d*g)*Sqrt[a + b*x + c*x^2]*Sqrt[b - Sqrt[b^2 - 4*a* c] - (2*c*f)/g + (2*c*(f + g*x))/g]*Sqrt[b + Sqrt[b^2 - 4*a*c] - (2*c*f)/g + (2*c*(f + g*x))/g]) - ((c*d*(2*e*f + d*g) - e*(b*e*f + 2*b*d*g - 3*a*e* g))*(-((e^2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2)* (e*f - d*g)*(d + e*x))) + ((Sqrt[2]*Sqrt[b^2 - 4*a*c]*e*Sqrt[f + g*x]*Sqrt [-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b ^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g) /(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(Sqrt[(c*(f + g*x))/(2*c*f - (b ...
3.9.99.3.1 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/(((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/(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[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[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 )/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e *Rt[b^2 - 4*a*c, 2])))^m)) Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2])/Sq rt[(f_.) + (g_.)*(x_)], x_Symbol] :> Simp[(d + e*x)^(m + 1)*Sqrt[f + g*x]*( Sqrt[a + b*x + c*x^2]/((m + 1)*(e*f - d*g))), x] - Simp[1/(2*(m + 1)*(e*f - d*g)) Int[((d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp [b*f + a*g*(2*m + 3) + 2*(c*f + b*g*(m + 2))*x + c*g*(2*m + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[2*m] && LtQ[m, -1]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[b - q + 2*c*x]*(Sqrt[b + q + 2*c*x]/Sqrt[a + b*x + c*x^2]) Int[1/((d + e*x )*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[ {a, b, c, d, e, f, g}, x]
Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)* (x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[e^2*(d + e*x)^(m + 1)*Sqrt[f + g*x ]*(Sqrt[a + b*x + c*x^2]/((m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/(2*(m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2)) Int[((d + e*x)^ (m + 1)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[2*d*(c*e*f - c*d*g + b* e*g)*(m + 1) - e^2*(b*f + a*g)*(2*m + 3) + 2*e*(c*d*g*(m + 1) - e*(c*f + b* g)*(m + 2))*x - c*e^2*g*(2*m + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[2*m] && LeQ[m, -2]
Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b _.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, d + e*x, x]*(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x] + Simp[Polyn omialRemainder[Px, d + e*x, x] Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x ^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && PolynomialQ[Px, x ] && LtQ[m, 0] && !IntegerQ[n] && IntegersQ[2*m, 2*n, 2*p]
Time = 3.16 (sec) , antiderivative size = 1634, normalized size of antiderivative = 1.56
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1634\) |
| default | \(\text {Expression too large to display}\) | \(57841\) |
((g*x+f)*(c*x^2+b*x+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2)*(1/2/(d*g- e*f)*(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)^(1/2)/(e*x+d)^2+1/4*(3*a*e^ 2*g-2*b*d*e*g-b*e^2*f+c*d^2*g+2*c*d*e*f)/(d*g-e*f)/(a*d*e^2*g-a*e^3*f-b*d^ 2*e*g+b*d*e^2*f+c*d^3*g-c*d^2*e*f)*(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a* f)^(1/2)/(e*x+d)-1/4*c*g*(a*d*e^2*g+2*a*e^3*f-3*b*d*e^2*f-c*d^3*g+4*c*d^2* e*f)/(d*g-e*f)/(a*d*e^2*g-a*e^3*f-b*d^2*e*g+b*d*e^2*f+c*d^3*g-c*d^2*e*f)/e ^2*(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1 /2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-f/g-1/2/c*(-b+(-4*a*c+ b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g+1/2*(b+(-4*a*c +b^2)^(1/2))/c))^(1/2)/(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)^(1/2)*Ell ipticF(((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-f/g+1/2*(b+(- 4*a*c+b^2)^(1/2))/c)/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))-1/4*c*g* (3*a*e^2*g-2*b*d*e*g-b*e^2*f+c*d^2*g+2*c*d*e*f)/(d*g-e*f)/(a*d*e^2*g-a*e^3 *f-b*d^2*e*g+b*d*e^2*f+c*d^3*g-c*d^2*e*f)/e*(f/g-1/2*(b+(-4*a*c+b^2)^(1/2) )/c)*((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4* a*c+b^2)^(1/2)))/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(- 4*a*c+b^2)^(1/2))/c)/(-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*g*x^3+b *g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)^(1/2)*((-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2)) )*EllipticE(((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-f/g+1/2* (b+(-4*a*c+b^2)^(1/2))/c)/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+...
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\int \frac {\sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{3} \sqrt {f + g x}}\, dx \]
\[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{3} \sqrt {g x + f}} \,d x } \]
\[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{3} \sqrt {g x + f}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^3} \,d x \]