Integrand size = 42, antiderivative size = 738 \[ \int \frac {A+C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {\left (A b^2+a^2 C\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac {\left (A b+\frac {a^2 C}{b}\right ) \sqrt {f} \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {\sqrt {-d e+c f} \left (a^2 C d f-2 a b C (d e+c f)+b^2 (2 c C e-A d f)\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 d (b c-a d) \sqrt {f} (b e-a f) \sqrt {e+f x} \sqrt {g+h x}}-\frac {\sqrt {-d e+c f} \left (a^4 C d f h-A b^4 (d e g+c f g+c e h)-2 a^3 b C (d f g+d e h+c f h)-2 a b^3 (2 c C e g-A d f g-A d e h-A c f h)-3 a^2 b^2 (A d f h-C (d e g+c f g+c e h))\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 (b c-a d)^2 \sqrt {f} (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {g+h x}} \]
-(A*b^2+C*a^2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)/(-a*f+ b*e)/(-a*h+b*g)/(b*x+a)+(A*b+a^2*C/b)*EllipticE(f^(1/2)*(d*x+c)^(1/2)/(c*f -d*e)^(1/2),((-c*f+d*e)*h/f/(-c*h+d*g))^(1/2))*f^(1/2)*(c*f-d*e)^(1/2)*(d* (f*x+e)/(-c*f+d*e))^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)/(-a*f+b*e)/(-a*h+b*g)/( f*x+e)^(1/2)/(d*(h*x+g)/(-c*h+d*g))^(1/2)+(a^2*C*d*f-2*a*b*C*(c*f+d*e)+b^2 *(-A*d*f+2*C*c*e))*EllipticF(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),((-c*f+ d*e)*h/f/(-c*h+d*g))^(1/2))*(c*f-d*e)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*( d*(h*x+g)/(-c*h+d*g))^(1/2)/b^2/d/(-a*d+b*c)/(-a*f+b*e)/f^(1/2)/(f*x+e)^(1 /2)/(h*x+g)^(1/2)-(a^4*C*d*f*h-A*b^4*(c*e*h+c*f*g+d*e*g)-2*a^3*b*C*(c*f*h+ d*e*h+d*f*g)-2*a*b^3*(-A*c*f*h-A*d*e*h-A*d*f*g+2*C*c*e*g)-3*a^2*b^2*(A*d*f *h-C*(c*e*h+c*f*g+d*e*g)))*EllipticPi(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2 ),-b*(-c*f+d*e)/(-a*d+b*c)/f,((-c*f+d*e)*h/f/(-c*h+d*g))^(1/2))*(c*f-d*e)^ (1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*(d*(h*x+g)/(-c*h+d*g))^(1/2)/b^2/(-a*d+ b*c)^2/(-a*f+b*e)/(-a*h+b*g)/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)
Result contains complex when optimal does not.
Time = 35.65 (sec) , antiderivative size = 3935, normalized size of antiderivative = 5.33 \[ \int \frac {A+C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \]
((-(A*b^2) - a^2*C)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d )*(b*e - a*f)*(b*g - a*h)*(a + b*x)) - ((c + d*x)^(3/2)*(A*b^4*c*Sqrt[-c + (d*e)/f]*f*h + a^2*b^2*c*C*Sqrt[-c + (d*e)/f]*f*h - a*A*b^3*d*Sqrt[-c + ( d*e)/f]*f*h - a^3*b*C*d*Sqrt[-c + (d*e)/f]*f*h + (A*b^4*c*d^2*e*Sqrt[-c + (d*e)/f]*g)/(c + d*x)^2 + (a^2*b^2*c*C*d^2*e*Sqrt[-c + (d*e)/f]*g)/(c + d* x)^2 - (a*A*b^3*d^3*e*Sqrt[-c + (d*e)/f]*g)/(c + d*x)^2 - (a^3*b*C*d^3*e*S qrt[-c + (d*e)/f]*g)/(c + d*x)^2 - (A*b^4*c^2*d*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x)^2 - (a^2*b^2*c^2*C*d*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x)^2 + (a*A*b^ 3*c*d^2*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x)^2 + (a^3*b*c*C*d^2*Sqrt[-c + (d* e)/f]*f*g)/(c + d*x)^2 - (A*b^4*c^2*d*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x)^2 - (a^2*b^2*c^2*C*d*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x)^2 + (a*A*b^3*c*d^2*e* Sqrt[-c + (d*e)/f]*h)/(c + d*x)^2 + (a^3*b*c*C*d^2*e*Sqrt[-c + (d*e)/f]*h) /(c + d*x)^2 + (A*b^4*c^3*Sqrt[-c + (d*e)/f]*f*h)/(c + d*x)^2 + (a^2*b^2*c ^3*C*Sqrt[-c + (d*e)/f]*f*h)/(c + d*x)^2 - (a*A*b^3*c^2*d*Sqrt[-c + (d*e)/ f]*f*h)/(c + d*x)^2 - (a^3*b*c^2*C*d*Sqrt[-c + (d*e)/f]*f*h)/(c + d*x)^2 + (A*b^4*c*d*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x) + (a^2*b^2*c*C*d*Sqrt[-c + ( d*e)/f]*f*g)/(c + d*x) - (a*A*b^3*d^2*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x) - (a^3*b*C*d^2*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x) + (A*b^4*c*d*e*Sqrt[-c + (d *e)/f]*h)/(c + d*x) + (a^2*b^2*c*C*d*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x) - ( a*A*b^3*d^2*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x) - (a^3*b*C*d^2*e*Sqrt[-c ...
Time = 2.08 (sec) , antiderivative size = 745, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2108, 25, 2110, 176, 124, 123, 131, 131, 130, 187, 413, 413, 412}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {A+C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) |
\(\Big \downarrow \) 2108 |
\(\displaystyle \frac {\int -\frac {(2 A d f h-C (d e g+c f g+c e h)) a^2+2 b (c C e g-A d f g-A d e h-A c f h) a-\left (C a^2+A b^2\right ) d f h x^2+A b^2 (d e g+c f g+c e h)-2 \left (C (d f g+d e h+c f h) a^2+b (A d f h-C (d e g+c f g+c e h)) a+b^2 c C e g\right ) x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {(2 A d f h-C (d e g+c f g+c e h)) a^2+2 b (c C e g-A d f g-A d e h-A c f h) a-\left (C a^2+A b^2\right ) d f h x^2+A b^2 (d e g+c f g+c e h)-2 \left (C (d f g+d e h+c f h) a^2+b (A d f h-C (d e g+c f g+c e h)) a+b^2 c C e g\right ) x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 2110 |
\(\displaystyle -\frac {\int \frac {\frac {C d f h a^3}{b^2}-\frac {2 C d f g a^2}{b}-\frac {2 C d e h a^2}{b}-\frac {2 c C f h a^2}{b}+2 C d e g a+2 c C f g a+2 c C e h a-A d f h a-2 b c C e g+\left (-\frac {C d f h a^2}{b}-A b d f h\right ) x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx-\frac {\left (a^4 C d f h-2 a^3 b C (c f h+d e h+d f g)-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b^2}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 176 |
\(\displaystyle -\frac {-\frac {(b g-a h) \left (a^2 C d f-2 a b C (c f+d e)+b^2 (2 c C e-A d f)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b^2}-d f \left (\frac {a^2 C}{b}+A b\right ) \int \frac {\sqrt {g+h x}}{\sqrt {c+d x} \sqrt {e+f x}}dx-\frac {\left (a^4 C d f h-2 a^3 b C (c f h+d e h+d f g)-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b^2}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 124 |
\(\displaystyle -\frac {-\frac {(b g-a h) \left (a^2 C d f-2 a b C (c f+d e)+b^2 (2 c C e-A d f)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b^2}-\frac {d f \sqrt {g+h x} \left (\frac {a^2 C}{b}+A b\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {\sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {\left (a^4 C d f h-2 a^3 b C (c f h+d e h+d f g)-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b^2}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 123 |
\(\displaystyle -\frac {-\frac {(b g-a h) \left (a^2 C d f-2 a b C (c f+d e)+b^2 (2 c C e-A d f)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b^2}-\frac {\left (a^4 C d f h-2 a^3 b C (c f h+d e h+d f g)-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b^2}-\frac {2 \sqrt {f} \sqrt {g+h x} \left (\frac {a^2 C}{b}+A b\right ) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 131 |
\(\displaystyle -\frac {-\frac {(b g-a h) \sqrt {\frac {d (e+f x)}{d e-c f}} \left (a^2 C d f-2 a b C (c f+d e)+b^2 (2 c C e-A d f)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {g+h x}}dx}{b^2 \sqrt {e+f x}}-\frac {\left (a^4 C d f h-2 a^3 b C (c f h+d e h+d f g)-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b^2}-\frac {2 \sqrt {f} \sqrt {g+h x} \left (\frac {a^2 C}{b}+A b\right ) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 131 |
\(\displaystyle -\frac {-\frac {(b g-a h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (a^2 C d f-2 a b C (c f+d e)+b^2 (2 c C e-A d f)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}dx}{b^2 \sqrt {e+f x} \sqrt {g+h x}}-\frac {\left (a^4 C d f h-2 a^3 b C (c f h+d e h+d f g)-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b^2}-\frac {2 \sqrt {f} \sqrt {g+h x} \left (\frac {a^2 C}{b}+A b\right ) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 130 |
\(\displaystyle -\frac {-\frac {\left (a^4 C d f h-2 a^3 b C (c f h+d e h+d f g)-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b^2}-\frac {2 (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (a^2 C d f-2 a b C (c f+d e)+b^2 (2 c C e-A d f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \sqrt {f} \sqrt {g+h x} \left (\frac {a^2 C}{b}+A b\right ) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 187 |
\(\displaystyle -\frac {\frac {2 \left (a^4 C d f h-2 a^3 b C (c f h+d e h+d f g)-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {e-\frac {c f}{d}+\frac {f (c+d x)}{d}} \sqrt {g-\frac {c h}{d}+\frac {h (c+d x)}{d}}}d\sqrt {c+d x}}{b^2}-\frac {2 (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (a^2 C d f-2 a b C (c f+d e)+b^2 (2 c C e-A d f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \sqrt {f} \sqrt {g+h x} \left (\frac {a^2 C}{b}+A b\right ) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 413 |
\(\displaystyle -\frac {\frac {2 \sqrt {\frac {f (c+d x)}{d e-c f}+1} \left (a^4 C d f h-2 a^3 b C (c f h+d e h+d f g)-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {g-\frac {c h}{d}+\frac {h (c+d x)}{d}}}d\sqrt {c+d x}}{b^2 \sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e}}-\frac {2 (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (a^2 C d f-2 a b C (c f+d e)+b^2 (2 c C e-A d f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \sqrt {f} \sqrt {g+h x} \left (\frac {a^2 C}{b}+A b\right ) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 413 |
\(\displaystyle -\frac {\frac {2 \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1} \left (a^4 C d f h-2 a^3 b C (c f h+d e h+d f g)-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1}}d\sqrt {c+d x}}{b^2 \sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e} \sqrt {\frac {h (c+d x)}{d}-\frac {c h}{d}+g}}-\frac {2 (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (a^2 C d f-2 a b C (c f+d e)+b^2 (2 c C e-A d f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \sqrt {f} \sqrt {g+h x} \left (\frac {a^2 C}{b}+A b\right ) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle -\frac {\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}-\frac {-\frac {2 (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (a^2 C d f-2 a b C (c f+d e)+b^2 (2 c C e-A d f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \sqrt {f} \sqrt {g+h x} \left (\frac {a^2 C}{b}+A b\right ) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {2 \sqrt {c f-d e} \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1} \left (a^4 C d f h-2 a^3 b C (c f h+d e h+d f g)-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt {f} (b c-a d) \sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e} \sqrt {\frac {h (c+d x)}{d}-\frac {c h}{d}+g}}}{2 (b c-a d) (b e-a f) (b g-a h)}\) |
-(((A*b^2 + a^2*C)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d) *(b*e - a*f)*(b*g - a*h)*(a + b*x))) - ((-2*(A*b + (a^2*C)/b)*Sqrt[f]*Sqrt [-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[Arc Sin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) - (2*Sqrt[-(d*e) + c*f]*(a^2*C*d*f - 2*a*b*C*(d*e + c*f) + b^2*(2*c*C*e - A*d*f))*(b*g - a* h)*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*Ellipti cF[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f* (d*g - c*h))])/(b^2*d*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h*x]) + (2*Sqrt[-(d*e ) + c*f]*(a^4*C*d*f*h - A*b^4*(d*e*g + c*f*g + c*e*h) - 2*a^3*b*C*(d*f*g + d*e*h + c*f*h) - 2*a*b^3*(2*c*C*e*g - A*d*f*g - A*d*e*h - A*c*f*h) - 3*a^ 2*b^2*(A*d*f*h - C*(d*e*g + c*f*g + c*e*h)))*Sqrt[1 + (f*(c + d*x))/(d*e - c*f)]*Sqrt[1 + (h*(c + d*x))/(d*g - c*h)]*EllipticPi[-((b*(d*e - c*f))/(( b*c - a*d)*f)), ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(b^2*(b*c - a*d)*Sqrt[f]*Sqrt[e - (c*f)/d + (f *(c + d*x))/d]*Sqrt[g - (c*h)/d + (h*(c + d*x))/d]))/(2*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))
3.1.30.3.1 Defintions of rubi rules used
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d *x]*Sqrt[b*((e + f*x)/(b*e - a*f))])) Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x /(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && Gt Q[b/(b*e - a*f), 0]) && !LtQ[-(b*c - a*d)/d, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ [b/(b*e - a*f), 0] && SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f *x] && (PosQ[-(b*c - a*d)/d] || NegQ[-(b*e - a*f)/f])
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[Sqrt[b*((c + d*x)/(b*c - a*d))]/Sqrt[c + d*x] Int[1/(Sq rt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e + f*x]), x ], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[(b*c - a*d)/b, 0] && Simpler Q[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x]
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && !SimplerQ[e + f*x, c + d*x] && !SimplerQ[g + h*x, c + d*x]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (C_.)*(x_)^2))/(Sqrt[(c_.) + (d_.)* (x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp [(A*b^2 + a^2*C)*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h* x]/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))), x] - Simp[1/(2*(m + 1)*( b*c - a*d)*(b*e - a*f)*(b*g - a*h)) Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x] *Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*d*f*h*(m + 1) - 2*a*b*(m + 1)* (d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) + a*C*(a*( d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*(A*b*(a*d*f*h*(m + 1) - b*( m + 2)*(d*f*g + d*e*h + c*f*h)) - C*(a^2*(d*f*g + d*e*h + c*f*h) - b^2*c*e* g*(m + 1) + a*b*(m + 1)*(d*e*g + c*f*g + c*e*h)))*x + d*f*h*(2*m + 5)*(A*b^ 2 + a^2*C)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, C}, x] && I ntegerQ[2*m] && LtQ[m, -1]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.)*((g_.) + (h_.)*(x_))^(q_.), x_Symbol] :> Simp[PolynomialRem ainder[Px, a + b*x, x] Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^ q, x], x] + Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d *x)^n*(e + f*x)^p*(g + h*x)^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p , q}, x] && PolyQ[Px, x] && EqQ[m, -1]
Time = 3.94 (sec) , antiderivative size = 1269, normalized size of antiderivative = 1.72
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1269\) |
default | \(\text {Expression too large to display}\) | \(17416\) |
((d*x+c)*(f*x+e)*(h*x+g))^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)* (1/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+ a*b^2*d*e*g-b^3*c*e*g)*(A*b^2+C*a^2)*(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g* x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)/(b*x+a)+2*(C/b^2-1/2*a/b^2*d*f*h* (A*b^2+C*a^2)/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a *b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g))*(g/h-e/f)*((x+g/h)/(g/h-e/f))^(1/2)*((x +c/d)/(-g/h+c/d))^(1/2)*((x+e/f)/(-g/h+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+d* e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)*EllipticF(((x+g/h)/ (g/h-e/f))^(1/2),((-g/h+e/f)/(-g/h+c/d))^(1/2))-d*f*h*(A*b^2+C*a^2)/(a^3*d *f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e *g-b^3*c*e*g)/b*(g/h-e/f)*((x+g/h)/(g/h-e/f))^(1/2)*((x+c/d)/(-g/h+c/d))^( 1/2)*((x+e/f)/(-g/h+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c *e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)*((-g/h+c/d)*EllipticE(((x+g/h)/(g/h-e/ f))^(1/2),((-g/h+e/f)/(-g/h+c/d))^(1/2))-c/d*EllipticF(((x+g/h)/(g/h-e/f)) ^(1/2),((-g/h+e/f)/(-g/h+c/d))^(1/2)))+(3*A*a^2*b^2*d*f*h-2*A*a*b^3*c*f*h- 2*A*a*b^3*d*e*h-2*A*a*b^3*d*f*g+A*b^4*c*e*h+A*b^4*c*f*g+A*b^4*d*e*g-C*a^4* d*f*h+2*C*a^3*b*c*f*h+2*C*a^3*b*d*e*h+2*C*a^3*b*d*f*g-3*C*a^2*b^2*c*e*h-3* C*a^2*b^2*c*f*g-3*C*a^2*b^2*d*e*g+4*C*a*b^3*c*e*g)/(a^3*d*f*h-a^2*b*c*f*h- a^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)/b^3 *(g/h-e/f)*((x+g/h)/(g/h-e/f))^(1/2)*((x+c/d)/(-g/h+c/d))^(1/2)*((x+e/f...
Timed out. \[ \int \frac {A+C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
integrate((C*x^2+A)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="fricas")
Timed out. \[ \int \frac {A+C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
\[ \int \frac {A+C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C x^{2} + A}{{\left (b x + a\right )}^{2} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
integrate((C*x^2+A)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="maxima")
\[ \int \frac {A+C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C x^{2} + A}{{\left (b x + a\right )}^{2} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
Timed out. \[ \int \frac {A+C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {C\,x^2+A}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^2\,\sqrt {c+d\,x}} \,d x \]