\(\int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [5]

Optimal result

Integrand size = 40, antiderivative size = 678 \[ \int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {b (A b-a B) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac {(A b-a B) \sqrt {f} \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {(A b-a B) \sqrt {f} \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b (b c-a d) (b e-a f) \sqrt {e+f x} \sqrt {g+h x}}+\frac {\sqrt {-d e+c f} \left (3 a^2 A b d f h-a^3 B d f h-b^3 (2 B c e g-A (d e g+c f g+c e h))+a b^2 (B (d e g+c f g+c e h)-2 A (d f g+d e h+c f h))\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b (b c-a d)^2 \sqrt {f} (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {g+h x}} \] Output:

-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)*f^(1/2)*(c*f-d*e)^(1/2)*(d*(f*x+e)/(-c*f+d 
*e))^(1/2)*(h*x+g)^(1/2)*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))/(-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*b-B*a)*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)*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))/b/(-a*d+b*c 
)/(-a*f+b*e)/(f*x+e)^(1/2)/(h*x+g)^(1/2)+(c*f-d*e)^(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)))*(d*(f*x+e)/(-c*f+d*e))^(1/2)*(d*(h*x+g)/(-c 
*h+d*g))^(1/2)*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))/b/(-a*d+b*c)^2/f^(1/2) 
/(-a*f+b*e)/(-a*h+b*g)/(f*x+e)^(1/2)/(h*x+g)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 34.43 (sec) , antiderivative size = 3412, normalized size of antiderivative = 5.03 \[ \int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \] Input:

Integrate[(A + B*x)/((a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x] 
),x]
 

Output:

-((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...
 

Rubi [A] (verified)

Time = 1.76 (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

\(\displaystyle -\frac {-\frac {2 \sqrt {c f-d e} \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1} \left (a^3 (-B) d f h+3 a^2 A b d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right ) \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt {f} (b c-a d) \sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e} \sqrt {\frac {h (c+d x)}{d}-\frac {c h}{d}+g}}+\frac {2 \sqrt {f} (A b-a B) (b g-a h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \sqrt {f} \sqrt {g+h x} (A b-a B) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}\)

Input:

Int[(A + B*x)/((a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
 

Output:

-((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))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 123

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])
 

rule 124

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]
 

rule 130

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])
 

rule 131

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]
 

rule 176

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]
 

rule 187

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]
 

rule 412

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])
 

rule 413

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]
 

rule 2102

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]
 

rule 2110

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]
 

Maple [A] (verified)

Time = 20.61 (sec) , antiderivative size = 1208, normalized size of antiderivative = 1.78

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1208\)
default	\(\text {Expression too large to display}\)	\(13380\)

Input:

int((B*x+A)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x,method=_ 
RETURNVERBOSE)
 

Output:

((h*x+g)*(f*x+e)*(d*x+c))^(1/2)/(h*x+g)^(1/2)/(f*x+e)^(1/2)/(d*x+c)^(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*(c/d-e/f)*((x+c/d)/(c/d-e/f))^(1/2)*((x+g/h)/(-c/d+g/h))^(1/2)* 
((x+e/f)/(-c/d+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+c/d)/(c/d-e/f))^(1/2),((-c/d+ 
e/f)/(-c/d+g/h))^(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)*(c/d-e/f)*((x+ 
c/d)/(c/d-e/f))^(1/2)*((x+g/h)/(-c/d+g/h))^(1/2)*((x+e/f)/(-c/d+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)*((-c/d+g/h)*EllipticE(((x+c/d)/(c/d-e/f))^(1/2),((-c/d+e/f)/(-c/d+g/ 
h))^(1/2))-g/h*EllipticF(((x+c/d)/(c/d-e/f))^(1/2),((-c/d+e/f)/(-c/d+g/h)) 
^(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*(c/d-e/f)*((x+c/d)/ 
(c/d-e/f))^(1/2)*((x+g/h)/(-c/d+g/h))^(1/2)*((x+e/f)/(-c/d+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...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, a 
lgorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)/(b*x+a)**2/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2), 
x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {B x + A}{{\left (b x + a\right )}^{2} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \] Input:

integrate((B*x+A)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, a 
lgorithm="maxima")
 

Output:

integrate((B*x + A)/((b*x + a)^2*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g) 
), x)
 

Giac [F]

\[ \int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {B x + A}{{\left (b x + a\right )}^{2} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \] Input:

integrate((B*x+A)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, a 
lgorithm="giac")
 

Output:

integrate((B*x + A)/((b*x + a)^2*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g) 
), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {A+B\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^2\,\sqrt {c+d\,x}} \,d x \] Input:

int((A + B*x)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)^2*(c + d*x)^(1/2) 
),x)
 

Output:

int((A + B*x)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)^2*(c + d*x)^(1/2) 
), x)
 

Reduce [F]

\[ \int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {B x +A}{\left (b x +a \right )^{2} \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}d x \] Input:

int((B*x+A)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)
 

Output:

int((B*x+A)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)