Integrand size = 28, antiderivative size = 1205 \[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(d+e x)^3} \, dx=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\sqrt {-a} \sqrt {c} \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{4 e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {3 \sqrt {-a} \sqrt {c} g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{2 e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\sqrt {-a} \sqrt {c} f \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{4 e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\sqrt {-a} \sqrt {c} d g \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{4 e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {c (e f-3 d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{e^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (a e^2 g-c d (2 e f-3 d g)\right )^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{4 e^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}} \]
-1/2*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/e/(e*x+d)^2-1/4*(a*e^2*g-c*d*(-3*d*g+2* e*f))*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/e/(a*e^2+c*d^2)/(-d*g+e*f)/(e*x+d)-1/4 *(a*e^2*g-c*d*(-3*d*g+2*e*f))*EllipticE(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2) *2^(1/2),(-2*a*g/(-a*g+f*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*c^(1/2)*(g *x+f)^(1/2)*(1+c*x^2/a)^(1/2)/e^2/(a*e^2+c*d^2)/(-d*g+e*f)/(c*x^2+a)^(1/2) /((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)-3/2*g*EllipticF(1/2*(1-x *c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*g/(-a*g+f*(-a)^(1/2)*c^(1/2)))^(1 /2))*(-a)^(1/2)*c^(1/2)*(1+c*x^2/a)^(1/2)*((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f *c^(1/2)))^(1/2)/e^3/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)+1/4*f*(a*e^2*g-c*d*(-3* d*g+2*e*f))*EllipticF(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*g/( -a*g+f*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*c^(1/2)*(1+c*x^2/a)^(1/2)*(( g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)/e^2/(a*e^2+c*d^2)/(-d*g+e*f )/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)-1/4*d*g*(a*e^2*g-c*d*(-3*d*g+2*e*f))*Ellip ticF(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*g/(-a*g+f*(-a)^(1/2) *c^(1/2)))^(1/2))*(-a)^(1/2)*c^(1/2)*(1+c*x^2/a)^(1/2)*((g*x+f)*c^(1/2)/(g *(-a)^(1/2)+f*c^(1/2)))^(1/2)/e^3/(a*e^2+c*d^2)/(-d*g+e*f)/(g*x+f)^(1/2)/( c*x^2+a)^(1/2)-c*(-3*d*g+e*f)*EllipticPi(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2 )*2^(1/2),2*e/(e+d*c^(1/2)/(-a)^(1/2)),2^(1/2)*(g*(-a)^(1/2)/(g*(-a)^(1/2) +f*c^(1/2)))^(1/2))*(1+c*x^2/a)^(1/2)*((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^( 1/2)))^(1/2)/e^3/(e+d*c^(1/2)/(-a)^(1/2))/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)...
Result contains complex when optimal does not.
Time = 29.36 (sec) , antiderivative size = 2526, normalized size of antiderivative = 2.10 \[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(d+e x)^3} \, dx=\text {Result too large to show} \]
(Sqrt[f + g*x]*Sqrt[a + c*x^2]*(-2 + ((a*e^2*g + c*d*(-2*e*f + 3*d*g))*(d + e*x))/((c*d^2 + a*e^2)*(-(e*f) + d*g))))/(4*e*(d + e*x)^2) + (-2*c^2*d*e ^3*f^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 5*c^2*d^2*e^2*f^3*g*Sqrt[-f - (I *Sqrt[a]*g)/Sqrt[c]] + a*c*e^4*f^3*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 3* c^2*d^3*e*f^2*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 3*a*c*d*e^3*f^2*g^2*S qrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 5*a*c*d^2*e^2*f*g^3*Sqrt[-f - (I*Sqrt[a] *g)/Sqrt[c]] + a^2*e^4*f*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 3*a*c*d^3* e*g^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - a^2*d*e^3*g^4*Sqrt[-f - (I*Sqrt[a ]*g)/Sqrt[c]] + 4*c^2*d*e^3*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - 10*c^2*d^2*e^2*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - 2*a*c *e^4*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) + 6*c^2*d^3*e*f*g^2* Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) + 2*a*c*d*e^3*f*g^2*Sqrt[-f - ( I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - 2*c^2*d*e^3*f^2*Sqrt[-f - (I*Sqrt[a]*g)/ Sqrt[c]]*(f + g*x)^2 + 5*c^2*d^2*e^2*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]* (f + g*x)^2 + a*c*e^4*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 - 3 *c^2*d^3*e*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 - a*c*d*e^3*g^ 2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + I*Sqrt[c]*e*(Sqrt[c]*f + I*Sqrt[a]*g)*(-(e*f) + d*g)*(a*e^2*g + c*d*(-2*e*f + 3*d*g))*Sqrt[(g*((I*S qrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqr...
Leaf count is larger than twice the leaf count of optimal. \(2763\) vs. \(2(1205)=2410\).
Time = 5.36 (sec) , antiderivative size = 2763, normalized size of antiderivative = 2.29, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {721, 2349, 734, 2349, 27, 510, 599, 25, 27, 729, 25, 1416, 1511, 1416, 1509, 1540, 1416, 2222}
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.
| \(\displaystyle \int \frac {\sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 721 |
| \(\displaystyle \frac {\int \frac {3 c g x^2+2 c f x+a g}{(d+e x)^2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{4 e}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 2349 |
| \(\displaystyle \frac {\left (a g-\frac {c d (2 e f-3 d g)}{e^2}\right ) \int \frac {1}{(d+e x)^2 \sqrt {f+g x} \sqrt {c x^2+a}}dx+\int \frac {\frac {2 c f}{e}-\frac {3 c d g}{e^2}+\frac {3 c g x}{e}}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{4 e}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 734 |
| \(\displaystyle \frac {\left (a g-\frac {c d (2 e f-3 d g)}{e^2}\right ) \left (-\frac {\int \frac {-c g x^2 e^2+a g e^2-2 c d g x e-2 c d (e f-d g)}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )+\int \frac {\frac {2 c f}{e}-\frac {3 c d g}{e^2}+\frac {3 c g x}{e}}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{4 e}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 2349 |
| \(\displaystyle \frac {\left (a g-\frac {c d (2 e f-3 d g)}{e^2}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\int \frac {-c d g-c e x g}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )+\frac {2 c (e f-3 d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}+\int \frac {3 c g}{e^2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{4 e}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a g-\frac {c d (2 e f-3 d g)}{e^2}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\int \frac {-c d g-c e x g}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )+\frac {2 c (e f-3 d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}+\frac {3 c g \int \frac {1}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}}{4 e}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 510 |
| \(\displaystyle \frac {\left (a g-\frac {c d (2 e f-3 d g)}{e^2}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\int \frac {-c d g-c e x g}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )+\frac {2 c (e f-3 d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}+\frac {6 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}}{4 e}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {\left (a g-\frac {c d (2 e f-3 d g)}{e^2}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx-\frac {2 \int -\frac {c g (e f-d g-e (f+g x))}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )+\frac {2 c (e f-3 d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}+\frac {6 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}}{4 e}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a g-\frac {c d (2 e f-3 d g)}{e^2}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\frac {2 \int \frac {c g (e f-d g-e (f+g x))}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )+\frac {2 c (e f-3 d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}+\frac {6 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}}{4 e}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a g-\frac {c d (2 e f-3 d g)}{e^2}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\frac {2 c \int \frac {e f-d g-e (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )+\frac {2 c (e f-3 d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}+\frac {6 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}}{4 e}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 729 |
| \(\displaystyle \frac {\left (a g-\frac {c d (2 e f-3 d g)}{e^2}\right ) \left (-\frac {2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \int -\frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}+\frac {2 c \int \frac {e f-d g-e (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )+\frac {4 c (e f-3 d g) \int -\frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}+\frac {6 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}}{4 e}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a g-\frac {c d (2 e f-3 d g)}{e^2}\right ) \left (-\frac {\frac {2 c \int \frac {e f-d g-e (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g}-2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-\frac {4 c (e f-3 d g) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}+\frac {6 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}}{4 e}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\left (a g-\frac {c d (2 e f-3 d g)}{e^2}\right ) \left (-\frac {\frac {2 c \int \frac {e f-d g-e (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g}-2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-\frac {4 c (e f-3 d g) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}+\frac {3 c^{3/4} \sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{e^2 \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}}{4 e}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\left (a g-\frac {c d (2 e f-3 d g)}{e^2}\right ) \left (-\frac {\frac {2 c \left (\frac {e \sqrt {a g^2+c f^2} \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}-\left (d g-e \left (f-\frac {\sqrt {a g^2+c f^2}}{\sqrt {c}}\right )\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}\right )}{g}-2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-\frac {4 c (e f-3 d g) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}+\frac {3 c^{3/4} \sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{e^2 \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}}{4 e}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {3 c^{3/4} \sqrt [4]{c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{e^2 \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}-\frac {4 c (e f-3 d g) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}+\left (a g-\frac {c d (2 e f-3 d g)}{e^2}\right ) \left (-\frac {\sqrt {f+g x} \sqrt {c x^2+a} e^2}{\left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\frac {2 c \left (\frac {e \sqrt {c f^2+a g^2} \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}-\frac {\sqrt [4]{c f^2+a g^2} \left (d g-e \left (f-\frac {\sqrt {c f^2+a g^2}}{\sqrt {c}}\right )\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right )}{g}-2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{2 \left (c d^2+a e^2\right ) (e f-d g)}\right )}{4 e}-\frac {\sqrt {f+g x} \sqrt {c x^2+a}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {\frac {3 c^{3/4} \sqrt [4]{c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{e^2 \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}-\frac {4 c (e f-3 d g) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}+\left (a g-\frac {c d (2 e f-3 d g)}{e^2}\right ) \left (-\frac {\sqrt {f+g x} \sqrt {c x^2+a} e^2}{\left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\frac {2 c \left (\frac {e \sqrt {c f^2+a g^2} \left (\frac {\sqrt [4]{c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}-\frac {\sqrt {f+g x} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )}\right )}{\sqrt {c}}-\frac {\sqrt [4]{c f^2+a g^2} \left (d g-e \left (f-\frac {\sqrt {c f^2+a g^2}}{\sqrt {c}}\right )\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right )}{g}-2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{2 \left (c d^2+a e^2\right ) (e f-d g)}\right )}{4 e}-\frac {\sqrt {f+g x} \sqrt {c x^2+a}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 1540 |
| \(\displaystyle \frac {\frac {3 c^{3/4} \sqrt [4]{c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{e^2 \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}+\frac {4 c (e f-3 d g) \left (\frac {e \sqrt {c f^2+a g^2} \left (\sqrt {c} (e f-d g)-e \sqrt {c f^2+a g^2}\right ) \int \frac {\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g \left (a g e^2+c d (2 e f-d g)\right )}-\frac {\sqrt {c} \left (c e f^2+a e g^2-\sqrt {c} (e f-d g) \sqrt {c f^2+a g^2}\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g \sqrt {c f^2+a g^2} \left (a g e^2+c d (2 e f-d g)\right )}\right )}{e^2}+\left (a g-\frac {c d (2 e f-3 d g)}{e^2}\right ) \left (-\frac {\sqrt {f+g x} \sqrt {c x^2+a} e^2}{\left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\frac {2 c \left (\frac {e \sqrt {c f^2+a g^2} \left (\frac {\sqrt [4]{c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}-\frac {\sqrt {f+g x} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )}\right )}{\sqrt {c}}-\frac {\sqrt [4]{c f^2+a g^2} \left (d g-e \left (f-\frac {\sqrt {c f^2+a g^2}}{\sqrt {c}}\right )\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right )}{g}+2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \left (\frac {e \sqrt {c f^2+a g^2} \left (\sqrt {c} (e f-d g)-e \sqrt {c f^2+a g^2}\right ) \int \frac {\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g \left (a g e^2+c d (2 e f-d g)\right )}-\frac {\sqrt {c} \left (c e f^2+a e g^2-\sqrt {c} (e f-d g) \sqrt {c f^2+a g^2}\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g \sqrt {c f^2+a g^2} \left (a g e^2+c d (2 e f-d g)\right )}\right )}{2 \left (c d^2+a e^2\right ) (e f-d g)}\right )}{4 e}-\frac {\sqrt {f+g x} \sqrt {c x^2+a}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {3 c^{3/4} \sqrt [4]{c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{e^2 \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}+\frac {4 c (e f-3 d g) \left (\frac {e \sqrt {c f^2+a g^2} \left (\sqrt {c} (e f-d g)-e \sqrt {c f^2+a g^2}\right ) \int \frac {\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g \left (a g e^2+c d (2 e f-d g)\right )}-\frac {\sqrt [4]{c} \left (c e f^2+a e g^2-\sqrt {c} (e f-d g) \sqrt {c f^2+a g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 g \sqrt [4]{c f^2+a g^2} \left (a g e^2+c d (2 e f-d g)\right ) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right )}{e^2}+\left (a g-\frac {c d (2 e f-3 d g)}{e^2}\right ) \left (-\frac {\sqrt {f+g x} \sqrt {c x^2+a} e^2}{\left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\frac {2 c \left (\frac {e \sqrt {c f^2+a g^2} \left (\frac {\sqrt [4]{c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}-\frac {\sqrt {f+g x} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )}\right )}{\sqrt {c}}-\frac {\sqrt [4]{c f^2+a g^2} \left (d g-e \left (f-\frac {\sqrt {c f^2+a g^2}}{\sqrt {c}}\right )\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right )}{g}+2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \left (\frac {e \sqrt {c f^2+a g^2} \left (\sqrt {c} (e f-d g)-e \sqrt {c f^2+a g^2}\right ) \int \frac {\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g \left (a g e^2+c d (2 e f-d g)\right )}-\frac {\sqrt [4]{c} \left (c e f^2+a e g^2-\sqrt {c} (e f-d g) \sqrt {c f^2+a g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 g \sqrt [4]{c f^2+a g^2} \left (a g e^2+c d (2 e f-d g)\right ) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right )}{2 \left (c d^2+a e^2\right ) (e f-d g)}\right )}{4 e}-\frac {\sqrt {f+g x} \sqrt {c x^2+a}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 2222 |
| \(\displaystyle \frac {\frac {3 c^{3/4} \sqrt [4]{c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{e^2 \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}+\frac {4 c (e f-3 d g) \left (\frac {e \sqrt {c f^2+a g^2} \left (\sqrt {c} (e f-d g)-e \sqrt {c f^2+a g^2}\right ) \left (\frac {\left (e+\frac {\sqrt {c} (e f-d g)}{\sqrt {c f^2+a g^2}}\right ) \text {arctanh}\left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {e} \sqrt {e f-d g} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right )}{2 \sqrt {e} \sqrt {c d^2+a e^2} \sqrt {e f-d g}}-\frac {\left (\frac {\sqrt {c}}{e}-\frac {\sqrt {c f^2+a g^2}}{e f-d g}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c f^2+a g^2} e+\sqrt {c} (e f-d g)\right )^2}{4 \sqrt {c} e (e f-d g) \sqrt {c f^2+a g^2}},2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{4 \sqrt [4]{c} \sqrt [4]{c f^2+a g^2} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right )}{g \left (a g e^2+c d (2 e f-d g)\right )}-\frac {\sqrt [4]{c} \left (c e f^2+a e g^2-\sqrt {c} (e f-d g) \sqrt {c f^2+a g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 g \sqrt [4]{c f^2+a g^2} \left (a g e^2+c d (2 e f-d g)\right ) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right )}{e^2}+\left (a g-\frac {c d (2 e f-3 d g)}{e^2}\right ) \left (-\frac {\sqrt {f+g x} \sqrt {c x^2+a} e^2}{\left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\frac {2 c \left (\frac {e \sqrt {c f^2+a g^2} \left (\frac {\sqrt [4]{c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}-\frac {\sqrt {f+g x} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )}\right )}{\sqrt {c}}-\frac {\sqrt [4]{c f^2+a g^2} \left (d g-e \left (f-\frac {\sqrt {c f^2+a g^2}}{\sqrt {c}}\right )\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right )}{g}+2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \left (\frac {e \sqrt {c f^2+a g^2} \left (\sqrt {c} (e f-d g)-e \sqrt {c f^2+a g^2}\right ) \left (\frac {\left (e+\frac {\sqrt {c} (e f-d g)}{\sqrt {c f^2+a g^2}}\right ) \text {arctanh}\left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {e} \sqrt {e f-d g} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right )}{2 \sqrt {e} \sqrt {c d^2+a e^2} \sqrt {e f-d g}}-\frac {\left (\frac {\sqrt {c}}{e}-\frac {\sqrt {c f^2+a g^2}}{e f-d g}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c f^2+a g^2} e+\sqrt {c} (e f-d g)\right )^2}{4 \sqrt {c} e (e f-d g) \sqrt {c f^2+a g^2}},2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{4 \sqrt [4]{c} \sqrt [4]{c f^2+a g^2} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right )}{g \left (a g e^2+c d (2 e f-d g)\right )}-\frac {\sqrt [4]{c} \left (c e f^2+a e g^2-\sqrt {c} (e f-d g) \sqrt {c f^2+a g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 g \sqrt [4]{c f^2+a g^2} \left (a g e^2+c d (2 e f-d g)\right ) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right )}{2 \left (c d^2+a e^2\right ) (e f-d g)}\right )}{4 e}-\frac {\sqrt {f+g x} \sqrt {c x^2+a}}{2 e (d+e x)^2}\) |
-1/2*(Sqrt[f + g*x]*Sqrt[a + c*x^2])/(e*(d + e*x)^2) + ((3*c^(3/4)*(c*f^2 + a*g^2)^(1/4)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])*Sqrt[(a + (c* f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2)/((a + (c*f^2)/g^2) *(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])^2)]*EllipticF[2*ArcTan[(c^( 1/4)*Sqrt[f + g*x])/(c*f^2 + a*g^2)^(1/4)], (1 + (Sqrt[c]*f)/Sqrt[c*f^2 + a*g^2])/2])/(e^2*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g* x)^2)/g^2]) + (4*c*(e*f - 3*d*g)*(-1/2*(c^(1/4)*(c*e*f^2 + a*e*g^2 - Sqrt[ c]*(e*f - d*g)*Sqrt[c*f^2 + a*g^2])*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])*Sqrt[(a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^ 2)/((a + (c*f^2)/g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])^2)]*El lipticF[2*ArcTan[(c^(1/4)*Sqrt[f + g*x])/(c*f^2 + a*g^2)^(1/4)], (1 + (Sqr t[c]*f)/Sqrt[c*f^2 + a*g^2])/2])/(g*(c*f^2 + a*g^2)^(1/4)*(a*e^2*g + c*d*( 2*e*f - d*g))*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^ 2)/g^2]) + (e*Sqrt[c*f^2 + a*g^2]*(Sqrt[c]*(e*f - d*g) - e*Sqrt[c*f^2 + a* g^2])*(((e + (Sqrt[c]*(e*f - d*g))/Sqrt[c*f^2 + a*g^2])*ArcTanh[(Sqrt[c*d^ 2 + a*e^2]*Sqrt[f + g*x])/(Sqrt[e]*Sqrt[e*f - d*g]*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2])])/(2*Sqrt[e]*Sqrt[c*d^2 + a* e^2]*Sqrt[e*f - d*g]) - ((Sqrt[c]/e - Sqrt[c*f^2 + a*g^2]/(e*f - d*g))*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])*Sqrt[(a + (c*f^2)/g^2 - (2*c*f* (f + g*x))/g^2 + (c*(f + g*x)^2)/g^2)/((a + (c*f^2)/g^2)*(1 + (Sqrt[c]*...
3.7.28.3.1 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[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[2/d Subst[Int[1/Sqrt[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2 )], x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && PosQ[b/a]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[-2/d^2 Subst[Int[(B*c - A*d - B*x^2)/Sqrt[(b*c^2 + a *d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)], x], x, Sqrt[c + d*x]], x] /; Fr eeQ[{a, b, c, d, A, B}, x] && PosQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*( x_)^2], x_Symbol] :> Simp[(d + e*x)^(m + 1)*Sqrt[f + g*x]*(Sqrt[a + c*x^2]/ (e*(m + 1))), x] - Simp[1/(2*e*(m + 1)) Int[((d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqrt[a + c*x^2]))*Simp[a*g + 2*c*f*x + 3*c*g*x^2, x], x], x] /; FreeQ[ {a, c, d, e, f, g}, x] && IntegerQ[2*m] && LtQ[m, -1]
Int[1/(Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))*Sqrt[(a_) + (b_.)*(x_) ^2]), x_Symbol] :> Simp[2 Subst[Int[1/((d*e - c*f + f*x^2)*Sqrt[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*( x_)^2]), x_Symbol] :> Simp[e^2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*(Sqrt[a + c* x^2]/((m + 1)*(e*f - d*g)*(c*d^2 + a*e^2))), x] + Simp[1/(2*(m + 1)*(e*f - d*g)*(c*d^2 + a*e^2)) Int[((d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqrt[a + c*x^ 2]))*Simp[2*c*d*(e*f - d*g)*(m + 1) - a*e^2*g*(2*m + 3) + 2*c*e*(d*g*(m + 1 ) - e*f*(m + 2))*x - c*e^2*g*(2*m + 5)*x^2, x], x], x] /; FreeQ[{a, c, d, e , f, g}, x] && IntegerQ[2*m] && LeQ[m, -2]
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]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S ymbol] :> With[{q = Rt[c/a, 2]}, Simp[(c*d + a*e*q)/(c*d^2 - a*e^2) Int[1 /Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[(a*e*(e + d*q))/(c*d^2 - a*e^2) I nt[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
Int[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. )*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(c + d *x)^(m + 1)*(e + f*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c + d*x, x] Int[(c + d*x)^m*(e + f*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && PolynomialQ[Px, x] && LtQ[m, 0] && !IntegerQ[n ] && IntegersQ[2*m, 2*n, 2*p]
Time = 0.89 (sec) , antiderivative size = 1161, normalized size of antiderivative = 0.96
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1161\) |
| default | \(\text {Expression too large to display}\) | \(19181\) |
((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(-1/2/e*(c*g*x^3+c *f*x^2+a*g*x+a*f)^(1/2)/(e*x+d)^2+1/4*(a*e^2*g+3*c*d^2*g-2*c*d*e*f)/e/(a*d *e^2*g-a*e^3*f+c*d^3*g-c*d^2*e*f)*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)/(e*x+d )+2*(c*g/e^3-1/8*c*g*(3*a*d*e^2*g-2*a*e^3*f+5*c*d^3*g-4*c*d^2*e*f)/e^3/(a* d*e^2*g-a*e^3*f+c*d^3*g-c*d^2*e*f))*(f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f/g-(-a *c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2)*((x+( -a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/2)/c))^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1 /2)*EllipticF(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),((-f/g+(-a*c)^(1/2)/c)/ (-f/g-(-a*c)^(1/2)/c))^(1/2))-1/4*c/e^2*g*(a*e^2*g+3*c*d^2*g-2*c*d*e*f)/(a *d*e^2*g-a*e^3*f+c*d^3*g-c*d^2*e*f)*(f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f/g-(-a *c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2)*((x+( -a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/2)/c))^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1 /2)*((-f/g-(-a*c)^(1/2)/c)*EllipticE(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2), ((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2))+(-a*c)^(1/2)/c*Ellipt icF(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a* c)^(1/2)/c))^(1/2)))+1/4*(a^2*e^4*g^2-6*a*c*d^2*e^2*g^2+12*a*c*d*e^3*f*g-4 *a*c*e^4*f^2-3*c^2*d^4*g^2+4*c^2*d^3*e*f*g)/(a*d*e^2*g-a*e^3*f+c*d^3*g-c*d ^2*e*f)/e^4*(f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2)*((x- (-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-f/g+(-a *c)^(1/2)/c))^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)/(-f/g+d/e)*Ellipt...
Timed out. \[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(d+e x)^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(d+e x)^3} \, dx=\int \frac {\sqrt {a + c x^{2}} \sqrt {f + g x}}{\left (d + e x\right )^{3}}\, dx \]
\[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(d+e x)^3} \, dx=\int { \frac {\sqrt {c x^{2} + a} \sqrt {g x + f}}{{\left (e x + d\right )}^{3}} \,d x } \]
\[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(d+e x)^3} \, dx=\int { \frac {\sqrt {c x^{2} + a} \sqrt {g x + f}}{{\left (e x + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(d+e x)^3} \, dx=\int \frac {\sqrt {f+g\,x}\,\sqrt {c\,x^2+a}}{{\left (d+e\,x\right )}^3} \,d x \]