Integrand size = 60, antiderivative size = 680 \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {b^2 (b B-2 a C) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac {b (b B-2 a C) \sqrt {f} \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {(b B-2 a C) \sqrt {f} \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) (b e-a f) \sqrt {e+f x} \sqrt {g+h x}}-\frac {\sqrt {-d e+c f} \left (4 a^3 C d f h+2 a b^2 B (d f g+d e h+c f h)-b^3 (B d e g-c (2 C e g-B f g-B e h))-a^2 b (3 B d f h+2 C (d f g+d e h+c f h))\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d)^2 \sqrt {f} (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {g+h x}} \]
-b^2*(B*b-2*C*a)*(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)+b*(B*b-2*C*a)*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)-(4*a^3*C*d*f*h+2*a*b^2*B*(c*f* h+d*e*h+d*f*g)-b^3*(B*d*e*g-c*(-B*e*h-B*f*g+2*C*e*g))-a^2*b*(3*B*d*f*h+2*C *(c*f*h+d*e*h+d*f*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)/(-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)-(B*b-2*C*a)*Ellip ticF(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)*(d*(h*x+g)/(-c*h+ d*g))^(1/2)/(-a*d+b*c)/(-a*f+b*e)/(f*x+e)^(1/2)/(h*x+g)^(1/2)
Result contains complex when optimal does not.
Time = 34.66 (sec) , antiderivative size = 3419, normalized size of antiderivative = 5.03 \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \]
Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^3*Sqrt[c + d*x] *Sqrt[e + f*x]*Sqrt[g + h*x]),x]
-((b^2*(b*B - 2*a*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)*(b^3*B*c*Sqrt[-c + (d*e)/f]*f*h - 2*a*b^2*c*C*Sqrt[-c + (d*e)/f]*f*h - a*b^2*B*d*Sqrt[-c + (d*e)/f]*f*h + 2*a^2*b*C*d*Sqrt[-c + (d*e)/f]*f*h + (b^3*B*c*d^2*e*Sqrt[- c + (d*e)/f]*g)/(c + d*x)^2 - (2*a*b^2*c*C*d^2*e*Sqrt[-c + (d*e)/f]*g)/(c + d*x)^2 - (a*b^2*B*d^3*e*Sqrt[-c + (d*e)/f]*g)/(c + d*x)^2 + (2*a^2*b*C*d ^3*e*Sqrt[-c + (d*e)/f]*g)/(c + d*x)^2 - (b^3*B*c^2*d*Sqrt[-c + (d*e)/f]*f *g)/(c + d*x)^2 + (2*a*b^2*c^2*C*d*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x)^2 + ( a*b^2*B*c*d^2*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x)^2 - (2*a^2*b*c*C*d^2*Sqrt[ -c + (d*e)/f]*f*g)/(c + d*x)^2 - (b^3*B*c^2*d*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x)^2 + (2*a*b^2*c^2*C*d*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x)^2 + (a*b^2*B* c*d^2*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x)^2 - (2*a^2*b*c*C*d^2*e*Sqrt[-c + ( d*e)/f]*h)/(c + d*x)^2 + (b^3*B*c^3*Sqrt[-c + (d*e)/f]*f*h)/(c + d*x)^2 - (2*a*b^2*c^3*C*Sqrt[-c + (d*e)/f]*f*h)/(c + d*x)^2 - (a*b^2*B*c^2*d*Sqrt[- c + (d*e)/f]*f*h)/(c + d*x)^2 + (2*a^2*b*c^2*C*d*Sqrt[-c + (d*e)/f]*f*h)/( c + d*x)^2 + (b^3*B*c*d*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x) - (2*a*b^2*c*C*d *Sqrt[-c + (d*e)/f]*f*g)/(c + d*x) - (a*b^2*B*d^2*Sqrt[-c + (d*e)/f]*f*g)/ (c + d*x) + (2*a^2*b*C*d^2*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x) + (b^3*B*c*d* e*Sqrt[-c + (d*e)/f]*h)/(c + d*x) - (2*a*b^2*c*C*d*e*Sqrt[-c + (d*e)/f]*h) /(c + d*x) - (a*b^2*B*d^2*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x) + (2*a^2*b*...
Time = 1.76 (sec) , antiderivative size = 686, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2004, 2102, 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.
| \(\displaystyle \int \frac {a^2 (-C)+a b B+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) |
| \(\Big \downarrow \) 2004 |
| \(\displaystyle \int \frac {\frac {a b B-a^2 C}{a}+b C x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx\) |
| \(\Big \downarrow \) 2102 |
| \(\displaystyle \frac {\int \frac {2 C d f h a^3-2 b (B d f h+C (d f g+d e h+c f h)) a^2+2 b^2 B (d f g+d e h+c f h) a+2 b (b B-2 a C) d f h x a+b^2 (b B-2 a C) d f h x^2-b^3 (B d e g-c (2 C e g-B f g-B e h))}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
| \(\Big \downarrow \) 2110 |
| \(\displaystyle \frac {\int \frac {-2 C d f h a^2+b B d f h a+\left (b^2 B d f h-2 a b C d f h\right ) x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+\left (4 a^3 C d f h-a^2 b (3 B d f h+2 C (c f h+d e h+d f g))+2 a b^2 B (c f h+d e h+d f g)-b^3 (B d e g-c (-B e h-B f g+2 C e g))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {\left (4 a^3 C d f h-a^2 b (3 B d f h+2 C (c f h+d e h+d f g))+2 a b^2 B (c f h+d e h+d f g)-b^3 (B d e g-c (-B e h-B f g+2 C e g))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx-\left (d f (b B-2 a C) (b g-a h) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx\right )+b d f (b B-2 a C) \int \frac {\sqrt {g+h x}}{\sqrt {c+d x} \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {\left (4 a^3 C d f h-a^2 b (3 B d f h+2 C (c f h+d e h+d f g))+2 a b^2 B (c f h+d e h+d f g)-b^3 (B d e g-c (-B e h-B f g+2 C e g))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx-\left (d f (b B-2 a C) (b g-a h) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx\right )+\frac {b d f \sqrt {g+h x} (b B-2 a C) \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {\sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {\left (4 a^3 C d f h-a^2 b (3 B d f h+2 C (c f h+d e h+d f g))+2 a b^2 B (c f h+d e h+d f g)-b^3 (B d e g-c (-B e h-B f g+2 C e g))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx-d f (b B-2 a C) (b g-a h) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+\frac {2 b \sqrt {f} \sqrt {g+h x} (b B-2 a C) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\left (4 a^3 C d f h-a^2 b (3 B d f h+2 C (c f h+d e h+d f g))+2 a b^2 B (c f h+d e h+d f g)-b^3 (B d e g-c (-B e h-B f g+2 C e g))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx-\frac {d f (b B-2 a C) (b g-a h) \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {g+h x}}dx}{\sqrt {e+f x}}+\frac {2 b \sqrt {f} \sqrt {g+h x} (b B-2 a C) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\left (4 a^3 C d f h-a^2 b (3 B d f h+2 C (c f h+d e h+d f g))+2 a b^2 B (c f h+d e h+d f g)-b^3 (B d e g-c (-B e h-B f g+2 C e g))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx-\frac {d f (b B-2 a C) (b g-a h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}dx}{\sqrt {e+f x} \sqrt {g+h x}}+\frac {2 b \sqrt {f} \sqrt {g+h x} (b B-2 a C) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
| \(\Big \downarrow \) 130 |
| \(\displaystyle \frac {\left (4 a^3 C d f h-a^2 b (3 B d f h+2 C (c f h+d e h+d f g))+2 a b^2 B (c f h+d e h+d f g)-b^3 (B d e g-c (-B e h-B f g+2 C e g))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx-\frac {2 \sqrt {f} (b B-2 a C) (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {g+h x}}+\frac {2 b \sqrt {f} \sqrt {g+h x} (b B-2 a C) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
| \(\Big \downarrow \) 187 |
| \(\displaystyle \frac {-2 \left (4 a^3 C d f h-a^2 b (3 B d f h+2 C (c f h+d e h+d f g))+2 a b^2 B (c f h+d e h+d f g)-b^3 (B d e g-c (-B e h-B f g+2 C e g))\right ) \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {e-\frac {c f}{d}+\frac {f (c+d x)}{d}} \sqrt {g-\frac {c h}{d}+\frac {h (c+d x)}{d}}}d\sqrt {c+d x}-\frac {2 \sqrt {f} (b B-2 a C) (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {g+h x}}+\frac {2 b \sqrt {f} \sqrt {g+h x} (b B-2 a C) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {-\frac {2 \sqrt {\frac {f (c+d x)}{d e-c f}+1} \left (4 a^3 C d f h-a^2 b (3 B d f h+2 C (c f h+d e h+d f g))+2 a b^2 B (c f h+d e h+d f g)-b^3 (B d e g-c (-B e h-B f g+2 C e g))\right ) \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {g-\frac {c h}{d}+\frac {h (c+d x)}{d}}}d\sqrt {c+d x}}{\sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e}}-\frac {2 \sqrt {f} (b B-2 a C) (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {g+h x}}+\frac {2 b \sqrt {f} \sqrt {g+h x} (b B-2 a C) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {-\frac {2 \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1} \left (4 a^3 C d f h-a^2 b (3 B d f h+2 C (c f h+d e h+d f g))+2 a b^2 B (c f h+d e h+d f g)-b^3 (B d e g-c (-B e h-B f g+2 C e g))\right ) \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1}}d\sqrt {c+d x}}{\sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e} \sqrt {\frac {h (c+d x)}{d}-\frac {c h}{d}+g}}-\frac {2 \sqrt {f} (b B-2 a C) (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {g+h x}}+\frac {2 b \sqrt {f} \sqrt {g+h x} (b B-2 a C) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {-\frac {2 \sqrt {c f-d e} \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1} \left (4 a^3 C d f h-a^2 b (3 B d f h+2 C (c f h+d e h+d f g))+2 a b^2 B (c f h+d e h+d f g)-b^3 (B d e g-c (-B e h-B f g+2 C e g))\right ) \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {f} (b c-a d) \sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e} \sqrt {\frac {h (c+d x)}{d}-\frac {c h}{d}+g}}-\frac {2 \sqrt {f} (b B-2 a C) (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {g+h x}}+\frac {2 b \sqrt {f} \sqrt {g+h x} (b B-2 a C) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
Int[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^3*Sqrt[c + d*x]*Sqrt[ e + f*x]*Sqrt[g + h*x]),x]
-((b^2*(b*B - 2*a*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*b*(b*B - 2*a*C)*Sqrt[f]*Sqrt[ -(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[ArcS in[(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*(b*B - 2*a*C) *Sqrt[f]*Sqrt[-(d*e) + c*f]*(b*g - a*h)*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sq rt[(d*(g + h*x))/(d*g - c*h)]*EllipticF[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqr t[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(Sqrt[e + f*x]*Sqrt[g + h*x]) - (2*Sqrt[-(d*e) + c*f]*(4*a^3*C*d*f*h + 2*a*b^2*B*(d*f*g + d*e*h + c*f*h) - b^3*(B*d*e*g - c*(2*C*e*g - B*f*g - B*e*h)) - a^2*b*(3*B*d*f*h + 2*C*(d*f*g + d*e*h + c*f*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*c - a*d)*Sqrt[f]*Sqrt[e - (c*f)/d + (f*(c + d*x))/d]*Sq rt[g - (c*h)/d + (h*(c + d*x))/d]))/(2*(b*c - a*d)*(b*e - a*f)*(b*g - a*h) )
3.1.20.3.1 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[(u_)*((d_) + (e_.)*(x_))^(q_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.) , x_Symbol] :> Int[u*(d + e*x)^(p + q)*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, b , c, d, e, q}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (B_.)*(x_)))/(Sqrt[(c_.) + (d_.)*(x _)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[( A*b^2 - a*b*B)*(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]*S qrt[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)) - b*B*(a*(d* e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*((A*b - a*B)*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h)))*x + d*f*h*(2*m + 5)*(A*b^2 - a*b*B)* x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && IntegerQ[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 = 3.93 (sec) , antiderivative size = 1211, normalized size of antiderivative = 1.78
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1211\) |
| default | \(\text {Expression too large to display}\) | \(13369\) |
int((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^3/(d*x+c)^(1/2)/(f*x+e)^(1/2)/ (h*x+g)^(1/2),x,method=_RETURNVERBOSE)
((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)* (b^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*b-2*C*a)*(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)-a*d*f*h*(B*b-2*C*a)/(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*b*(B*b-2*C*a)/(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)*((-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*B*a^2*b*d*f*h-2*B*a*b^2*c*f*h-2*B*a*b^2*d*e*h-2*B*a*b^ 2*d*f*g+B*b^3*c*e*h+B*b^3*c*f*g+B*b^3*d*e*g-4*C*a^3*d*f*h+2*C*a^2*b*c*f*h+ 2*C*a^2*b*d*e*h+2*C*a^2*b*d*f*g-2*C*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*(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...
Timed out. \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^3/(d*x+c)^(1/2)/(f*x+e)^ (1/2)/(h*x+g)^(1/2),x, algorithm="fricas")
Timed out. \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
integrate((C*b**2*x**2+B*b**2*x+B*a*b-C*a**2)/(b*x+a)**3/(d*x+c)**(1/2)/(f *x+e)**(1/2)/(h*x+g)**(1/2),x)
\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{3} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^3/(d*x+c)^(1/2)/(f*x+e)^ (1/2)/(h*x+g)^(1/2),x, algorithm="maxima")
integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/((b*x + a)^3*sqrt(d*x + c) *sqrt(f*x + e)*sqrt(h*x + g)), x)
\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{3} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^3/(d*x+c)^(1/2)/(f*x+e)^ (1/2)/(h*x+g)^(1/2),x, algorithm="giac")
integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/((b*x + a)^3*sqrt(d*x + c) *sqrt(f*x + e)*sqrt(h*x + g)), x)
Timed out. \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Hanged} \]