Integrand size = 28, antiderivative size = 1241 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac {\left (3 a e^2 g+c d (2 e f+d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 \left (c d^2+a e^2\right ) (e f-d g)^2 (d+e x)}+\frac {\sqrt {-a} \sqrt {c} \left (3 a e^2 g+c d (2 e f+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 \left (c d^2+a e^2\right ) (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {\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^2 (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\sqrt {-a} \sqrt {c} f \left (3 a e^2 g+c d (2 e f+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 \left (c d^2+a e^2\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\sqrt {-a} \sqrt {c} d g \left (3 a e^2 g+c d (2 e f+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)^2 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {c (e f+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^2 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \left (3 a e^2 g+c d (2 e f+d g)\right ) \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^2 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \]
-1/2*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/(-d*g+e*f)/(e*x+d)^2+1/4*(3*a*e^2*g+c*d *(d*g+2*e*f))*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)/(-d*g+e*f)^2/(e* x+d)+1/4*(3*a*e^2*g+c*d*(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/(a*e^2+c*d^2)/(-d*g+e*f)^2/(c*x^2+a )^(1/2)/((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)+1/2*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^2/(-d*g+e*f)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)-1/4*f *(3*a*e^2*g+c*d*(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/(a*e^2+c *d^2)/(-d*g+e*f)^2/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)+1/4*d*g*(3*a*e^2*g+c*d*(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) ^2/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)-c*(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^2/(-d*g+e*f)/(e+d*c^(1/2)/(-a)^(1/2))/(g*...
Result contains complex when optimal does not.
Time = 28.72 (sec) , antiderivative size = 2197, normalized size of antiderivative = 1.77 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\text {Result too large to show} \]
(c^2*d^2*f^3 - 3*a*c*e^2*f^3 - (2*c^2*d*e*f^4)/g + (c^2*d^3*f^2*g)/e + a*c *d*e*f^2*g + a*c*d^2*f*g^2 - 3*a^2*e^2*f*g^2 + (a*c*d^3*g^3)/e + 3*a^2*d*e *g^3 - 2*c^2*d^2*f^2*(f + g*x) + 6*a*c*e^2*f^2*(f + g*x) + (4*c^2*d*e*f^3* (f + g*x))/g - (2*c^2*d^3*f*g*(f + g*x))/e - 6*a*c*d*e*f*g*(f + g*x) + c^2 *d^2*f*(f + g*x)^2 - 3*a*c*e^2*f*(f + g*x)^2 - (2*c^2*d*e*f^2*(f + g*x)^2) /g + (c^2*d^3*g*(f + g*x)^2)/e + 3*a*c*d*e*g*(f + g*x)^2 - ((e*f - d*g)*(f + g*x)*(a + c*x^2)*(a*e^2*(2*e*f - 5*d*g - 3*e*g*x) - c*d*(3*d^2*g + 2*e^ 2*f*x + d*e*g*x)))/(d + e*x)^2 + (Sqrt[c]*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(-( e*f) + d*g)*(3*a*e^2*g + c*d*(2*e*f + d*g))*Sqrt[(g*((I*Sqrt[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)/Sqrt[c]]/Sqrt[f + g*x]] , (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/(e*g*Sqrt[-f - (I* Sqrt[a]*g)/Sqrt[c]]) + ((I*Sqrt[c]*d + Sqrt[a]*e)*(Sqrt[c]*f + I*Sqrt[a]*g )*(-3*a*e^2*g - (6*I)*Sqrt[a]*Sqrt[c]*e*(e*f - d*g) + c*d*(-4*e*f + d*g))* Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c ] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt [a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*S qrt[a]*g)])/(e*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]) + ((4*I)*a*c*e^2*f^2*Sqrt [(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(S...
Leaf count is larger than twice the leaf count of optimal. \(2774\) vs. \(2(1241)=2482\).
Time = 5.31 (sec) , antiderivative size = 2774, normalized size of antiderivative = 2.24, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.679, Rules used = {725, 25, 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.
| \(\displaystyle \int \frac {\sqrt {a+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx\) |
| \(\Big \downarrow \) 725 |
| \(\displaystyle \frac {\int -\frac {c g x^2-2 c f x+3 a g}{(d+e x)^2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{4 (e f-d g)}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {c g x^2-2 c f x+3 a g}{(d+e x)^2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{4 (e f-d g)}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 2349 |
| \(\displaystyle -\frac {\left (3 a g+\frac {c d (d g+2 e f)}{e^2}\right ) \int \frac {1}{(d+e x)^2 \sqrt {f+g x} \sqrt {c x^2+a}}dx+\int \frac {-\frac {2 c f}{e}-\frac {c d g}{e^2}+\frac {c g x}{e}}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{4 (e f-d g)}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 734 |
| \(\displaystyle -\frac {\left (3 a g+\frac {c d (d g+2 e f)}{e^2}\right ) \left (-\frac {\int \frac {-c g x^2 e^2+a g e^2-2 c d g x e-2 c d (e f-d g)}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )+\int \frac {-\frac {2 c f}{e}-\frac {c d g}{e^2}+\frac {c g x}{e}}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{4 (e f-d g)}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 2349 |
| \(\displaystyle -\frac {\left (3 a g+\frac {c d (d g+2 e f)}{e^2}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\int \frac {-c d g-c e x g}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-\frac {2 c (d g+e f) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}+\int \frac {c g}{e^2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{4 (e f-d g)}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\left (3 a g+\frac {c d (d g+2 e f)}{e^2}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\int \frac {-c d g-c e x g}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-\frac {2 c (d g+e f) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}+\frac {c g \int \frac {1}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}}{4 (e f-d g)}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 510 |
| \(\displaystyle -\frac {\left (3 a g+\frac {c d (d g+2 e f)}{e^2}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\int \frac {-c d g-c e x g}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-\frac {2 c (d g+e f) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}+\frac {2 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}}{4 (e f-d g)}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle -\frac {\left (3 a g+\frac {c d (d g+2 e f)}{e^2}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx-\frac {2 \int -\frac {c g (e f-d g-e (f+g x))}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-\frac {2 c (d g+e f) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}+\frac {2 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}}{4 (e f-d g)}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\left (3 a g+\frac {c d (d g+2 e f)}{e^2}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\frac {2 \int \frac {c g (e f-d g-e (f+g x))}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-\frac {2 c (d g+e f) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}+\frac {2 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}}{4 (e f-d g)}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\left (3 a g+\frac {c d (d g+2 e f)}{e^2}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\frac {2 c \int \frac {e f-d g-e (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-\frac {2 c (d g+e f) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}+\frac {2 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}}{4 (e f-d g)}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 729 |
| \(\displaystyle -\frac {\left (3 a g+\frac {c d (d g+2 e f)}{e^2}\right ) \left (-\frac {2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \int -\frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}+\frac {2 c \int \frac {e f-d g-e (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-\frac {4 c (d g+e f) \int -\frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}+\frac {2 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}}{4 (e f-d g)}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\left (3 a g+\frac {c d (d g+2 e f)}{e^2}\right ) \left (-\frac {\frac {2 c \int \frac {e f-d g-e (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g}-2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )+\frac {4 c (d g+e f) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}+\frac {2 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}}{4 (e f-d g)}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {\left (3 a g+\frac {c d (d g+2 e f)}{e^2}\right ) \left (-\frac {\frac {2 c \int \frac {e f-d g-e (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g}-2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )+\frac {4 c (d g+e f) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}+\frac {c^{3/4} \sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{e^2 \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}}{4 (e f-d g)}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle -\frac {\left (3 a g+\frac {c d (d g+2 e f)}{e^2}\right ) \left (-\frac {\frac {2 c \left (\frac {e \sqrt {a g^2+c f^2} \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}-\left (d g-e \left (f-\frac {\sqrt {a g^2+c f^2}}{\sqrt {c}}\right )\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}\right )}{g}-2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )+\frac {4 c (d g+e f) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e^2}+\frac {c^{3/4} \sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{e^2 \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}}{4 (e f-d g)}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 (e f-d g)}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {\frac {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+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 (3 a g+\frac {c d (2 e f+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 f-d g)}-\frac {\sqrt {f+g x} \sqrt {c x^2+a}}{2 (e f-d g) (d+e x)^2}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle -\frac {\frac {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+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 (3 a g+\frac {c d (2 e f+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 f-d g)}-\frac {\sqrt {f+g x} \sqrt {c x^2+a}}{2 (e f-d g) (d+e x)^2}\) |
| \(\Big \downarrow \) 1540 |
| \(\displaystyle -\frac {\frac {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+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 (3 a g+\frac {c d (2 e f+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 f-d g)}-\frac {\sqrt {f+g x} \sqrt {c x^2+a}}{2 (e f-d g) (d+e x)^2}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {\frac {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+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 (3 a g+\frac {c d (2 e f+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 f-d g)}-\frac {\sqrt {f+g x} \sqrt {c x^2+a}}{2 (e f-d g) (d+e x)^2}\) |
| \(\Big \downarrow \) 2222 |
| \(\displaystyle -\frac {\frac {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+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 (3 a g+\frac {c d (2 e f+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 f-d g)}-\frac {\sqrt {f+g x} \sqrt {c x^2+a}}{2 (e f-d g) (d+e x)^2}\) |
-1/2*(Sqrt[f + g*x]*Sqrt[a + c*x^2])/((e*f - d*g)*(d + e*x)^2) - ((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*Arc Tan[(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 + 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)]*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])/(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[(Sqr t[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 + (Sq...
3.7.35.3.1 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[(a_) + (c_.)*(x_)^2])/Sqrt[(f_.) + (g_ .)*(x_)], x_Symbol] :> Simp[(d + e*x)^(m + 1)*Sqrt[f + g*x]*(Sqrt[a + 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 + c*x^2]))*Simp[a*g*(2*m + 3) + 2*(c*f)*x + c*g*(2*m + 5)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && Integer Q[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 = 2.72 (sec) , antiderivative size = 1196, normalized size of antiderivative = 0.96
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1196\) |
| default | \(\text {Expression too large to display}\) | \(19170\) |
((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(1/2/(d*g-e*f)*(c* g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)/(e*x+d)^2+1/4*(3*a*e^2*g+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)/(d*g-e*f)*(c*g*x^3+c*f*x^2+a*g*x+a* f)^(1/2)/(e*x+d)-1/4*c*g*(a*d*e^2*g+2*a*e^3*f-c*d^3*g+4*c*d^2*e*f)/(a*d*e^ 2*g-a*e^3*f+c*d^3*g-c*d^2*e*f)/(d*g-e*f)/e^2*(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*g*(3*a*e^2*g+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)/(d*g-e*f)/e*(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*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*(3*a^2*e^4*g^2+6*a*c*d^2*e^2*g^2 -4*a*c*d*e^3*f*g+4*a*c*e^4*f^2-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^3/(d*g-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*...
Timed out. \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + a}}{{\left (e x + d\right )}^{3} \sqrt {g x + f}} \,d x } \]
\[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + a}}{{\left (e x + d\right )}^{3} \sqrt {g x + f}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\int \frac {\sqrt {c\,x^2+a}}{\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^3} \,d x \]