Integrand size = 40, antiderivative size = 678 \[ \int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {b (A b-a B) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac {(A b-a B) \sqrt {f} \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {(A b-a B) \sqrt {f} \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b (b c-a d) (b e-a f) \sqrt {e+f x} \sqrt {g+h x}}+\frac {\sqrt {-d e+c f} \left (3 a^2 A b d f h-a^3 B d f h-b^3 (2 B c e g-A (d e g+c f g+c e h))+a b^2 (B (d e g+c f g+c e h)-2 A (d f g+d e h+c f h))\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b (b c-a d)^2 \sqrt {f} (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {g+h x}} \]
-b*(A*b-B*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)+(A*b-B*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)+(3*a^2*A*b*d*f*h-a^3*B*d*f*h-b^3*(2*B* c*e*g-A*(c*e*h+c*f*g+d*e*g))+a*b^2*(B*(c*e*h+c*f*g+d*e*g)-2*A*(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)/b/(-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)-(A*b-B*a)*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))*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)/b /(-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 = 34.13 (sec) , antiderivative size = 3412, normalized size of antiderivative = 5.03 \[ \int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \]
-((b*(A*b - a*B)*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^3*c*Sqrt[-c + ( d*e)/f]*f*h - a*b^2*B*c*Sqrt[-c + (d*e)/f]*f*h - a*A*b^2*d*Sqrt[-c + (d*e) /f]*f*h + a^2*b*B*d*Sqrt[-c + (d*e)/f]*f*h + (A*b^3*c*d^2*e*Sqrt[-c + (d*e )/f]*g)/(c + d*x)^2 - (a*b^2*B*c*d^2*e*Sqrt[-c + (d*e)/f]*g)/(c + d*x)^2 - (a*A*b^2*d^3*e*Sqrt[-c + (d*e)/f]*g)/(c + d*x)^2 + (a^2*b*B*d^3*e*Sqrt[-c + (d*e)/f]*g)/(c + d*x)^2 - (A*b^3*c^2*d*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x )^2 + (a*b^2*B*c^2*d*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x)^2 + (a*A*b^2*c*d^2* Sqrt[-c + (d*e)/f]*f*g)/(c + d*x)^2 - (a^2*b*B*c*d^2*Sqrt[-c + (d*e)/f]*f* g)/(c + d*x)^2 - (A*b^3*c^2*d*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x)^2 + (a*b^2 *B*c^2*d*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x)^2 + (a*A*b^2*c*d^2*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x)^2 - (a^2*b*B*c*d^2*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x) ^2 + (A*b^3*c^3*Sqrt[-c + (d*e)/f]*f*h)/(c + d*x)^2 - (a*b^2*B*c^3*Sqrt[-c + (d*e)/f]*f*h)/(c + d*x)^2 - (a*A*b^2*c^2*d*Sqrt[-c + (d*e)/f]*f*h)/(c + d*x)^2 + (a^2*b*B*c^2*d*Sqrt[-c + (d*e)/f]*f*h)/(c + d*x)^2 + (A*b^3*c*d* Sqrt[-c + (d*e)/f]*f*g)/(c + d*x) - (a*b^2*B*c*d*Sqrt[-c + (d*e)/f]*f*g)/( c + d*x) - (a*A*b^2*d^2*Sqrt[-c + (d*e)/f]*f*g)/(c + d*x) + (a^2*b*B*d^2*S qrt[-c + (d*e)/f]*f*g)/(c + d*x) + (A*b^3*c*d*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x) - (a*b^2*B*c*d*e*Sqrt[-c + (d*e)/f]*h)/(c + d*x) - (a*A*b^2*d^2*e*Sq rt[-c + (d*e)/f]*h)/(c + d*x) + (a^2*b*B*d^2*e*Sqrt[-c + (d*e)/f]*h)/(c...
Time = 1.70 (sec) , antiderivative size = 685, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {2102, 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.
\(\displaystyle \int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) |
\(\Big \downarrow \) 2102 |
\(\displaystyle \frac {\int -\frac {2 A d f h a^2+b (B (d e g+c f g+c e h)-2 A (d f g+d e h+c f h)) a-2 (A b-a B) d f h x a-b (A b-a B) d f h x^2-b^2 (2 B c e g-A (d e g+c f g+c e h))}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {2 A d f h a^2+b (B (d e g+c f g+c e h)-2 A (d f g+d e h+c f h)) a-2 (A b-a B) d f h x a-b (A b-a B) d f h x^2-b^2 (2 B c e g-A (d e g+c f g+c e h))}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 2110 |
\(\displaystyle -\frac {\int \frac {\frac {B d f h a^2}{b}-A d f h a+(a B d f h-A b d f h) x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+\frac {\left (a^3 (-B) d f h+3 a^2 A b d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 176 |
\(\displaystyle -\frac {\frac {\left (a^3 (-B) d f h+3 a^2 A b d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b}+\frac {d f (A b-a B) (b g-a h) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b}-d f (A b-a B) \int \frac {\sqrt {g+h x}}{\sqrt {c+d x} \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 124 |
\(\displaystyle -\frac {\frac {\left (a^3 (-B) d f h+3 a^2 A b d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b}+\frac {d f (A b-a B) (b g-a h) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b}-\frac {d f \sqrt {g+h x} (A b-a B) \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {\sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 123 |
\(\displaystyle -\frac {\frac {\left (a^3 (-B) d f h+3 a^2 A b d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b}+\frac {d f (A b-a B) (b g-a h) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b}-\frac {2 \sqrt {f} \sqrt {g+h x} (A b-a B) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 131 |
\(\displaystyle -\frac {\frac {\left (a^3 (-B) d f h+3 a^2 A b d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b}+\frac {d f (A b-a B) (b g-a h) \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {g+h x}}dx}{b \sqrt {e+f x}}-\frac {2 \sqrt {f} \sqrt {g+h x} (A b-a B) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 131 |
\(\displaystyle -\frac {\frac {\left (a^3 (-B) d f h+3 a^2 A b d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b}+\frac {d f (A b-a B) (b g-a h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}dx}{b \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \sqrt {f} \sqrt {g+h x} (A b-a B) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 130 |
\(\displaystyle -\frac {\frac {\left (a^3 (-B) d f h+3 a^2 A b d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b}+\frac {2 \sqrt {f} (A b-a B) (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \sqrt {f} \sqrt {g+h x} (A b-a B) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 187 |
\(\displaystyle -\frac {-\frac {2 \left (a^3 (-B) d f h+3 a^2 A b d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right ) \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {e-\frac {c f}{d}+\frac {f (c+d x)}{d}} \sqrt {g-\frac {c h}{d}+\frac {h (c+d x)}{d}}}d\sqrt {c+d x}}{b}+\frac {2 \sqrt {f} (A b-a B) (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \sqrt {f} \sqrt {g+h x} (A b-a B) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 413 |
\(\displaystyle -\frac {-\frac {2 \sqrt {\frac {f (c+d x)}{d e-c f}+1} \left (a^3 (-B) d f h+3 a^2 A b d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right ) \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {g-\frac {c h}{d}+\frac {h (c+d x)}{d}}}d\sqrt {c+d x}}{b \sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e}}+\frac {2 \sqrt {f} (A b-a B) (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \sqrt {f} \sqrt {g+h x} (A b-a B) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 413 |
\(\displaystyle -\frac {-\frac {2 \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1} \left (a^3 (-B) d f h+3 a^2 A b d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right ) \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1}}d\sqrt {c+d x}}{b \sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e} \sqrt {\frac {h (c+d x)}{d}-\frac {c h}{d}+g}}+\frac {2 \sqrt {f} (A b-a B) (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \sqrt {f} \sqrt {g+h x} (A b-a B) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle -\frac {-\frac {2 \sqrt {c f-d e} \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1} \left (a^3 (-B) d f h+3 a^2 A b d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right ) \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt {f} (b c-a d) \sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e} \sqrt {\frac {h (c+d x)}{d}-\frac {c h}{d}+g}}+\frac {2 \sqrt {f} (A b-a B) (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \sqrt {f} \sqrt {g+h x} (A b-a B) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\) |
-((b*(A*b - a*B)*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*B)*Sqrt[f]*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[ArcSin[(Sqr t[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*(A*b - a*B)*Sqrt[f]* Sqrt[-(d*e) + c*f]*(b*g - a*h)*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticF[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(b*Sqrt[e + f*x]*Sqrt[g + h*x]) - (2*Sqrt[-(d*e) + c*f]*(3*a^2*A*b*d*f*h - a^3*B*d*f*h - b^3*(2*B*c*e*g - A*(d*e*g + c*f*g + c*e*h)) + a*b^2*(B*(d*e*g + c*f*g + c*e*h) - 2*A*(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*(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))
3.1.5.3.1 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_.) + (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 = 3.96 (sec) , antiderivative size = 1208, normalized size of antiderivative = 1.78
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1208\) |
default | \(\text {Expression too large to display}\) | \(13344\) |
((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/(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-B*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*(A*b-B*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)/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)*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-B*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*A*a^2*b*d*f*h-2*A*a*b^2*c*f*h-2*A*a*b^2*d*e*h-2*A*a*b^2*d*f*g+ A*b^3*c*e*h+A*b^3*c*f*g+A*b^3*d*e*g-B*a^3*d*f*h+B*a*b^2*c*e*h+B*a*b^2*c*f* g+B*a*b^2*d*e*g-2*B*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^2*(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...
Timed out. \[ \int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {B x + A}{{\left (b x + a\right )}^{2} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
\[ \int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {B x + A}{{\left (b x + a\right )}^{2} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
Timed out. \[ \int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {A+B\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^2\,\sqrt {c+d\,x}} \,d x \]