Integrand size = 28, antiderivative size = 983 \[ \int \frac {\sqrt {f+g x}}{(d+e x)^3 \sqrt {a+c x^2}} \, dx=-\frac {e \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {e \left (a e^2 g+c d (6 e f-5 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g) (d+e x)}-\frac {\sqrt [4]{c} \left (\sqrt {c} f-\sqrt {-a} g\right ) \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (a e^2 g+c d (6 e f-5 d g)\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} E\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right )|\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{4 \left (c d^2+a e^2\right )^2 g (e f-d g) \sqrt {a+c x^2}}+\frac {\sqrt [4]{c} \sqrt {\sqrt {c} f+\sqrt {-a} g} \left ((-a)^{3/2} e^3 g-\sqrt {-a} c d e (6 e f-5 d g)-a \sqrt {c} e^2 (2 e f-3 d g)+c^{3/2} d^2 (4 e f-3 d g)\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{4 e \left (c d^2+a e^2\right )^2 (e f-d g) \sqrt {a+c x^2}}+\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \left (a^2 e^4 g^2-c^2 d^2 \left (8 e^2 f^2-12 d e f g+3 d^2 g^2\right )+2 a c e^2 \left (2 e^2 f^2-6 d e f g+5 d^2 g^2\right )\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticPi}\left (\frac {e \left (f+\frac {\sqrt {-a} g}{\sqrt {c}}\right )}{e f-d g},\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{4 \sqrt [4]{c} e \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt {a+c x^2}} \] Output:
-1/2*e*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)/(e*x+d)^2-1/4*e*(a*e^2* g+c*d*(-5*d*g+6*e*f))*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^2/(-d*g+ e*f)/(e*x+d)-1/4*c^(1/4)*(c^(1/2)*f-(-a)^(1/2)*g)*(c^(1/2)*f+(-a)^(1/2)*g) ^(1/2)*(a*e^2*g+c*d*(-5*d*g+6*e*f))*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f-(-a)^(1/ 2)*g))^(1/2)*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f+(-a)^(1/2)*g))^(1/2)*EllipticE( c^(1/4)*(g*x+f)^(1/2)/(c^(1/2)*f+(-a)^(1/2)*g)^(1/2),((c^(1/2)*f+(-a)^(1/2 )*g)/(c^(1/2)*f-(-a)^(1/2)*g))^(1/2))/(a*e^2+c*d^2)^2/g/(-d*g+e*f)/(c*x^2+ a)^(1/2)+1/4*c^(1/4)*(c^(1/2)*f+(-a)^(1/2)*g)^(1/2)*((-a)^(3/2)*e^3*g-(-a) ^(1/2)*c*d*e*(-5*d*g+6*e*f)-a*c^(1/2)*e^2*(-3*d*g+2*e*f)+c^(3/2)*d^2*(-3*d *g+4*e*f))*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f-(-a)^(1/2)*g))^(1/2)*(1-c^(1/2)*( g*x+f)/(c^(1/2)*f+(-a)^(1/2)*g))^(1/2)*EllipticF(c^(1/4)*(g*x+f)^(1/2)/(c^ (1/2)*f+(-a)^(1/2)*g)^(1/2),((c^(1/2)*f+(-a)^(1/2)*g)/(c^(1/2)*f-(-a)^(1/2 )*g))^(1/2))/e/(a*e^2+c*d^2)^2/(-d*g+e*f)/(c*x^2+a)^(1/2)+1/4*(c^(1/2)*f+( -a)^(1/2)*g)^(1/2)*(a^2*e^4*g^2-c^2*d^2*(3*d^2*g^2-12*d*e*f*g+8*e^2*f^2)+2 *a*c*e^2*(5*d^2*g^2-6*d*e*f*g+2*e^2*f^2))*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f-(- a)^(1/2)*g))^(1/2)*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f+(-a)^(1/2)*g))^(1/2)*Elli pticPi(c^(1/4)*(g*x+f)^(1/2)/(c^(1/2)*f+(-a)^(1/2)*g)^(1/2),e*(f+(-a)^(1/2 )*g/c^(1/2))/(-d*g+e*f),((c^(1/2)*f+(-a)^(1/2)*g)/(c^(1/2)*f-(-a)^(1/2)*g) )^(1/2))/c^(1/4)/e/(a*e^2+c*d^2)^2/(-d*g+e*f)^2/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 32.81 (sec) , antiderivative size = 2937, normalized size of antiderivative = 2.99 \[ \int \frac {\sqrt {f+g x}}{(d+e x)^3 \sqrt {a+c x^2}} \, dx=\text {Result too large to show} \] Input:
Integrate[Sqrt[f + g*x]/((d + e*x)^3*Sqrt[a + c*x^2]),x]
Output:
Sqrt[f + g*x]*Sqrt[a + c*x^2]*(-1/2*e/((c*d^2 + a*e^2)*(d + e*x)^2) - (e*( 6*c*d*e*f - 5*c*d^2*g + a*e^2*g))/(4*(c*d^2 + a*e^2)^2*(e*f - d*g)*(d + e* x))) - ((f + g*x)^(3/2)*(6*c^2*d*e^3*f^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 11*c^2*d^2*e^2*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + a*c*e^4*f*g*Sqrt[- f - (I*Sqrt[a]*g)/Sqrt[c]] + 5*c^2*d^3*e*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[ c]] - a*c*d*e^3*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + (6*c^2*d*e^3*f^4*Sq rt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (11*c^2*d^2*e^2*f^3*g*Sqrt[- f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 + (a*c*e^4*f^3*g*Sqrt[-f - (I*Sqrt [a]*g)/Sqrt[c]])/(f + g*x)^2 + (5*c^2*d^3*e*f^2*g^2*Sqrt[-f - (I*Sqrt[a]*g )/Sqrt[c]])/(f + g*x)^2 + (5*a*c*d*e^3*f^2*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqr t[c]])/(f + g*x)^2 - (11*a*c*d^2*e^2*f*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c] ])/(f + g*x)^2 + (a^2*e^4*f*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x )^2 + (5*a*c*d^3*e*g^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (a^ 2*d*e^3*g^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (12*c^2*d*e^3* f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x) + (22*c^2*d^2*e^2*f^2*g*Sq rt[-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) - (10*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) + (Sqrt[c]*e*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(e*f - d*g)*(a*e^2 *g + c*d*(6*e*f - 5*d*g))*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]...
Leaf count is larger than twice the leaf count of optimal. \(2782\) vs. \(2(983)=1966\).
Time = 9.16 (sec) , antiderivative size = 2782, normalized size of antiderivative = 2.83, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {737, 25, 2349, 734, 2349, 25, 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.
| \(\displaystyle \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2} (d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 737 |
| \(\displaystyle -\frac {\int -\frac {-c e g x^2-2 c (e f-2 d g) x+4 c d f+a e g}{(d+e x)^2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{4 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-c e g x^2-2 c (e f-2 d g) x+4 c d f+a e g}{(d+e x)^2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{4 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 2349 |
| \(\displaystyle \frac {\left (a e g+\frac {c d (6 e f-5 d g)}{e}\right ) \int \frac {1}{(d+e x)^2 \sqrt {f+g x} \sqrt {c x^2+a}}dx+\int \frac {-2 c f+\frac {5 c d g}{e}-c g x}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{4 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 734 |
| \(\displaystyle \frac {\left (a e g+\frac {c d (6 e f-5 d g)}{e}\right ) \left (-\frac {\int \frac {-c g x^2 e^2+a g e^2-2 c d g x e-2 c d (e f-d g)}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )+\int \frac {-2 c f+\frac {5 c d g}{e}-c g x}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{4 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 2349 |
| \(\displaystyle \frac {\left (a e g+\frac {c d (6 e f-5 d g)}{e}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\int \frac {-c d g-c e x g}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-2 c \left (f-\frac {3 d g}{e}\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\int -\frac {c g}{e \sqrt {f+g x} \sqrt {c x^2+a}}dx}{4 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a e g+\frac {c d (6 e f-5 d g)}{e}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\int \frac {-c d g-c e x g}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-2 c \left (f-\frac {3 d g}{e}\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx-\int \frac {c g}{e \sqrt {f+g x} \sqrt {c x^2+a}}dx}{4 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a e g+\frac {c d (6 e f-5 d g)}{e}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\int \frac {-c d g-c e x g}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-2 c \left (f-\frac {3 d g}{e}\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx-\frac {c g \int \frac {1}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{e}}{4 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 510 |
| \(\displaystyle \frac {\left (a e g+\frac {c d (6 e f-5 d g)}{e}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\int \frac {-c d g-c e x g}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-2 c \left (f-\frac {3 d g}{e}\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx-\frac {2 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e}}{4 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {\left (a e g+\frac {c d (6 e f-5 d g)}{e}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx-\frac {2 \int -\frac {c g (e f-d g-e (f+g x))}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-2 c \left (f-\frac {3 d g}{e}\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx-\frac {2 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e}}{4 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a e g+\frac {c d (6 e f-5 d g)}{e}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\frac {2 \int \frac {c g (e f-d g-e (f+g x))}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-2 c \left (f-\frac {3 d g}{e}\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx-\frac {2 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e}}{4 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a e g+\frac {c d (6 e f-5 d g)}{e}\right ) \left (-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\frac {2 c \int \frac {e f-d g-e (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-2 c \left (f-\frac {3 d g}{e}\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx-\frac {2 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e}}{4 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 729 |
| \(\displaystyle \frac {\left (a e g+\frac {c d (6 e f-5 d g)}{e}\right ) \left (-\frac {2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \int -\frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}+\frac {2 c \int \frac {e f-d g-e (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )-4 c \left (f-\frac {3 d g}{e}\right ) \int -\frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-\frac {2 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e}}{4 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a e g+\frac {c d (6 e f-5 d g)}{e}\right ) \left (-\frac {\frac {2 c \int \frac {e f-d g-e (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g}-2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )+4 c \left (f-\frac {3 d g}{e}\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-\frac {2 c \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e}}{4 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\left (a e g+\frac {c d (6 e f-5 d g)}{e}\right ) \left (-\frac {\frac {2 c \int \frac {e f-d g-e (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g}-2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )+4 c \left (f-\frac {3 d g}{e}\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-\frac {c^{3/4} \sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{e \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}}{4 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\left (a e g+\frac {c d (6 e f-5 d g)}{e}\right ) \left (-\frac {\frac {2 c \left (\frac {e \sqrt {a g^2+c f^2} \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}-\left (d g-e \left (f-\frac {\sqrt {a g^2+c f^2}}{\sqrt {c}}\right )\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}\right )}{g}-2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{2 \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)}\right )+4 c \left (f-\frac {3 d g}{e}\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-\frac {c^{3/4} \sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{e \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}}{4 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 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 \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}}+4 c \left (f-\frac {3 d g}{e}\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}+\left (a e g+\frac {c d (6 e f-5 d g)}{e}\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 \left (c d^2+a e^2\right )}-\frac {e \sqrt {f+g x} \sqrt {c x^2+a}}{2 \left (c d^2+a e^2\right ) (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 \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}}+4 c \left (f-\frac {3 d g}{e}\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}+\left (a e g+\frac {c d (6 e f-5 d g)}{e}\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 \left (c d^2+a e^2\right )}-\frac {e \sqrt {f+g x} \sqrt {c x^2+a}}{2 \left (c d^2+a e^2\right ) (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 \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}}-4 c \left (f-\frac {3 d g}{e}\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 )+\left (a e g+\frac {c d (6 e f-5 d g)}{e}\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 \left (c d^2+a e^2\right )}-\frac {e \sqrt {f+g x} \sqrt {c x^2+a}}{2 \left (c d^2+a e^2\right ) (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 \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}}-4 c \left (f-\frac {3 d g}{e}\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 )+\left (a e g+\frac {c d (6 e f-5 d g)}{e}\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 \left (c d^2+a e^2\right )}-\frac {e \sqrt {f+g x} \sqrt {c x^2+a}}{2 \left (c d^2+a e^2\right ) (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 \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}}-4 c \left (f-\frac {3 d g}{e}\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 )+\left (a e g+\frac {c d (6 e f-5 d g)}{e}\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 \left (c d^2+a e^2\right )}-\frac {e \sqrt {f+g x} \sqrt {c x^2+a}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}\) |
Input:
Int[Sqrt[f + g*x]/((d + e*x)^3*Sqrt[a + c*x^2]),x]
Output:
-1/2*(e*Sqrt[f + g*x]*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*(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)]*Ellipti cF[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*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^ 2 + (c*(f + g*x)^2)/g^2])) - 4*c*(f - (3*d*g)/e)*(-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])*Ar cTanh[(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...
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[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[(((d_.) + (e_.)*(x_))^(m_)*Sqrt[(f_.) + (g_.)*(x_)])/Sqrt[(a_) + (c_.)* (x_)^2], x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*Sqrt[f + g*x]*(Sqrt[a + c*x^ 2]/((m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/(2*(m + 1)*(c*d^2 + a*e^2)) In t[((d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqrt[a + c*x^2]))*Simp[2*c*d*f*(m + 1) - e*(a*g) + 2*c*(d*g*(m + 1) - e*f*(m + 2))*x - c*e*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 = 7.60 (sec) , antiderivative size = 1224, normalized size of antiderivative = 1.25
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1224\) |
| default | \(\text {Expression too large to display}\) | \(20359\) |
Input:
int((g*x+f)^(1/2)/(e*x+d)^3/(c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(-1/2*e/(a*e^2+c*d ^2)*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)/(e*x+d)^2+1/4*e*(a*e^2*g-5*c*d^2*g+6 *c*d*e*f)/(a*d*e^2*g-a*e^3*f+c*d^3*g-c*d^2*e*f)/(a*e^2+c*d^2)*(c*g*x^3+c*f *x^2+a*g*x+a*f)^(1/2)/(e*x+d)-1/4*c*g*(3*a*d*e^2*g-2*a*e^3*f-3*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)/(a*e^2+c*d^2)/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)*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*(a*e^2*g -5*c*d^2*g+6*c*d*e*f)/(a*d*e^2*g-a*e^3*f+c*d^3*g-c*d^2*e*f)/(a*e^2+c*d^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)*((-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*(a^2*e^4*g^ 2+10*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+12*c^2*d ^3*e*f*g-8*c^2*d^2*e^2*f^2)/(a*d*e^2*g-a*e^3*f+c*d^3*g-c*d^2*e*f)/(a*e^2+c *d^2)/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+(-...
Timed out. \[ \int \frac {\sqrt {f+g x}}{(d+e x)^3 \sqrt {a+c x^2}} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)^(1/2)/(e*x+d)^3/(c*x^2+a)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {f+g x}}{(d+e x)^3 \sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {f + g x}}{\sqrt {a + c x^{2}} \left (d + e x\right )^{3}}\, dx \] Input:
integrate((g*x+f)**(1/2)/(e*x+d)**3/(c*x**2+a)**(1/2),x)
Output:
Integral(sqrt(f + g*x)/(sqrt(a + c*x**2)*(d + e*x)**3), x)
\[ \int \frac {\sqrt {f+g x}}{(d+e x)^3 \sqrt {a+c x^2}} \, dx=\int { \frac {\sqrt {g x + f}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{3}} \,d x } \] Input:
integrate((g*x+f)^(1/2)/(e*x+d)^3/(c*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(g*x + f)/(sqrt(c*x^2 + a)*(e*x + d)^3), x)
\[ \int \frac {\sqrt {f+g x}}{(d+e x)^3 \sqrt {a+c x^2}} \, dx=\int { \frac {\sqrt {g x + f}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{3}} \,d x } \] Input:
integrate((g*x+f)^(1/2)/(e*x+d)^3/(c*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(g*x + f)/(sqrt(c*x^2 + a)*(e*x + d)^3), x)
Timed out. \[ \int \frac {\sqrt {f+g x}}{(d+e x)^3 \sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {f+g\,x}}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^3} \,d x \] Input:
int((f + g*x)^(1/2)/((a + c*x^2)^(1/2)*(d + e*x)^3),x)
Output:
int((f + g*x)^(1/2)/((a + c*x^2)^(1/2)*(d + e*x)^3), x)
\[ \int \frac {\sqrt {f+g x}}{(d+e x)^3 \sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c \,e^{3} x^{5}+3 c d \,e^{2} x^{4}+a \,e^{3} x^{3}+3 c \,d^{2} e \,x^{3}+3 a d \,e^{2} x^{2}+c \,d^{3} x^{2}+3 a \,d^{2} e x +a \,d^{3}}d x \] Input:
int((g*x+f)^(1/2)/(e*x+d)^3/(c*x^2+a)^(1/2),x)
Output:
int((sqrt(f + g*x)*sqrt(a + c*x**2))/(a*d**3 + 3*a*d**2*e*x + 3*a*d*e**2*x **2 + a*e**3*x**3 + c*d**3*x**2 + 3*c*d**2*e*x**3 + 3*c*d*e**2*x**4 + c*e* *3*x**5),x)