\(\int \frac {A+B x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [10]

Optimal result

Integrand size = 42, antiderivative size = 737 \[ \int \frac {A+B x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 b (A b-a B) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {2 (d g-c h) \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right )|\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^{3/2} \sqrt {d e-c f} (b g-a h)^2 \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}-\frac {2 \left (B \left (3 b^2 c e g h+3 a^2 d f g h-a b (c h (2 f g+e h)+d g (f g+2 e h))\right )-A \left (3 a^2 d f h^2-3 a b (d e+c f) h^2-b^2 (d g (f g-e h)-c h (f g+2 e h))\right )\right ) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d) (b e-a f)^{3/2} \sqrt {d e-c f} (b g-a h)^2 \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}} \] Output:

-2/3*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)^(3/2)-2/3*(-c*h+d*g)*(3*a^3*B*d*f*h+b^3*(3*B*c*e 
*g-2*A*(c*e*h+c*f*g+d*e*g))-a*b^2*(B*(c*e*h+c*f*g+d*e*g)-4*A*(c*f*h+d*e*h+ 
d*f*g))-a^2*b*(6*A*d*f*h+B*(c*f*h+d*e*h+d*f*g)))*(f*x+e)^(1/2)*(-(-a*d+b*c 
)*(h*x+g)/(-c*h+d*g)/(b*x+a))^(1/2)*EllipticE((-a*f+b*e)^(1/2)*(d*x+c)^(1/ 
2)/(-c*f+d*e)^(1/2)/(b*x+a)^(1/2),((-c*f+d*e)*(-a*h+b*g)/(-a*f+b*e)/(-c*h+ 
d*g))^(1/2))/(-a*d+b*c)^2/(-a*f+b*e)^(3/2)/(-c*f+d*e)^(1/2)/(-a*h+b*g)^2/( 
-(-a*d+b*c)*(f*x+e)/(-c*f+d*e)/(b*x+a))^(1/2)/(h*x+g)^(1/2)-2/3*(B*(3*b^2* 
c*e*g*h+3*a^2*d*f*g*h-a*b*(c*h*(e*h+2*f*g)+d*g*(2*e*h+f*g)))-A*(3*a^2*d*f* 
h^2-3*a*b*(c*f+d*e)*h^2-b^2*(d*g*(-e*h+f*g)-c*h*(2*e*h+f*g))))*(f*x+e)^(1/ 
2)*(-(-a*d+b*c)*(h*x+g)/(-c*h+d*g)/(b*x+a))^(1/2)*EllipticF((-a*f+b*e)^(1/ 
2)*(d*x+c)^(1/2)/(-c*f+d*e)^(1/2)/(b*x+a)^(1/2),((-c*f+d*e)*(-a*h+b*g)/(-a 
*f+b*e)/(-c*h+d*g))^(1/2))/(-a*d+b*c)/(-a*f+b*e)^(3/2)/(-c*f+d*e)^(1/2)/(- 
a*h+b*g)^2/(-(-a*d+b*c)*(f*x+e)/(-c*f+d*e)/(b*x+a))^(1/2)/(h*x+g)^(1/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(10828\) vs. \(2(737)=1474\).

Time = 39.66 (sec) , antiderivative size = 10828, normalized size of antiderivative = 14.69 \[ \int \frac {A+B x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \] Input:

Integrate[(A + B*x)/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + 
h*x]),x]
 

Output:

Result too large to show
 

Rubi [A] (warning: unable to verify)

Time = 3.98 (sec) , antiderivative size = 1068, normalized size of antiderivative = 1.45, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2102, 25, 2102, 25, 2105, 27, 188, 194, 321, 327}

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.

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

\(\Big \downarrow \) 2102

\(\displaystyle \frac {\int -\frac {3 A d f h a^2+b (B (d e g+c f g+c e h)-3 A (d f g+d e h+c f h)) a-b^2 (3 B c e g-2 A (d e g+c f g+c e h))-(A b-a B) (3 a d f h-b (d f g+d e h+c f h)) x}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{3 (b c-a d) (b e-a f) (b g-a h)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{3 (a+b x)^{3/2} (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {3 A d f h a^2+b (B (d e g+c f g+c e h)-3 A (d f g+d e h+c f h)) a-b^2 (3 B c e g-2 A (d e g+c f g+c e h))-(A b-a B) (3 a d f h-b (d f g+d e h+c f h)) x}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{3 (b c-a d) (b e-a f) (b g-a h)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{3 (a+b x)^{3/2} (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 2102

\(\displaystyle -\frac {\frac {\int -\frac {2 b d f h \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) x^2+(a d f h+b (d f g+d e h+c f h)) \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) x+b (A b-a B) (b c e g-a (d e g+c f g+c e h)) (3 a d f h-b (d f g+d e h+c f h))+a (a d f h-b (d f g+d e h+c f h)) \left (3 A d f h a^2+b (B (d e g+c f g+c e h)-3 A (d f g+d e h+c f h)) a-b^2 (3 B c e g-2 A (d e g+c f g+c e h))\right )}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{(b c-a d) (b e-a f) (b g-a h)}+\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (3 a^3 B d f h-a^2 b (6 A d f h+B (c f h+d e h+d f g))-a b^2 (B (c e h+c f g+d e g)-4 A (c f h+d e h+d f g))+b^3 (3 B c e g-2 A (c e h+c f g+d e g))\right )}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{3 (a+b x)^{3/2} (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (3 a^3 B d f h-a^2 b (6 A d f h+B (c f h+d e h+d f g))-a b^2 (B (c e h+c f g+d e g)-4 A (c f h+d e h+d f g))+b^3 (3 B c e g-2 A (c e h+c f g+d e g))\right )}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}-\frac {\int \frac {2 b d f h \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) x^2+(a d f h+b (d f g+d e h+c f h)) \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) x+b (A b-a B) (b c e g-a (d e g+c f g+c e h)) (3 a d f h-b (d f g+d e h+c f h))+a (a d f h-b (d f g+d e h+c f h)) \left (3 A d f h a^2+b (B (d e g+c f g+c e h)-3 A (d f g+d e h+c f h)) a-b^2 (3 B c e g-2 A (d e g+c f g+c e h))\right )}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{(b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{3 (a+b x)^{3/2} (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 2105

\(\displaystyle -\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {\frac {2 b \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {(d e-c f) (d g-c h) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right )+\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right )}{\sqrt {c+d x}}+\frac {\int -\frac {2 b d f (b e-a f) h (b g-a h) \left (3 d (B c-A d) f h a^2+b \left (3 A d^2 (f g+e h)-B \left (f h c^2+2 d (f g+e h) c+d^2 e g\right )\right ) a+b^2 \left (A f h c^2+3 B d e g c-A d (f g+e h) c-2 A d^2 e g\right )\right )}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 b d f h}}{(b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (3 a^3 B d f h-a^2 b (6 A d f h+B (c f h+d e h+d f g))-a b^2 (B (c e h+c f g+d e g)-4 A (c f h+d e h+d f g))+b^3 (3 B c e g-2 A (c e h+c f g+d e g))\right )}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}-\frac {-(b e-a f) (b g-a h) \left (3 a^2 d f h (B c-A d)+a b \left (3 A d^2 (e h+f g)-B \left (c^2 f h+2 c d (e h+f g)+d^2 e g\right )\right )+b^2 \left (A c^2 f h-A c d (e h+f g)-2 A d^2 e g+3 B c d e g\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+(d e-c f) (d g-c h) \left (3 a^3 B d f h-a^2 b (6 A d f h+B (c f h+d e h+d f g))-a b^2 (B (c e h+c f g+d e g)-4 A (c f h+d e h+d f g))+b^3 (3 B c e g-2 A (c e h+c f g+d e g))\right ) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx+\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (3 a^3 B d f h-a^2 b (6 A d f h+B (c f h+d e h+d f g))-a b^2 (B (c e h+c f g+d e g)-4 A (c f h+d e h+d f g))+b^3 (3 B c e g-2 A (c e h+c f g+d e g))\right )}{\sqrt {c+d x}}}{(b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{3 (a+b x)^{3/2} (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 188

\(\displaystyle -\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {\frac {2 b \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {(d e-c f) (d g-c h) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right )+\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right )}{\sqrt {c+d x}}-\frac {2 (b e-a f) (b g-a h) \left (3 d (B c-A d) f h a^2+b \left (3 A d^2 (f g+e h)-B \left (f h c^2+2 d (f g+e h) c+d^2 e g\right )\right ) a+b^2 \left (A f h c^2+3 B d e g c-A d (f g+e h) c-2 A d^2 e g\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \int \frac {1}{\sqrt {\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}+1} \sqrt {1-\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}}d\frac {\sqrt {e+f x}}{\sqrt {a+b x}}}{(f g-e h) \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}}{(b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 194

\(\displaystyle -\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {\frac {2 b \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {-\frac {2 (d g-c h) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} \int \frac {\sqrt {1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}}}{\sqrt {1-\frac {(d g-c h) (e+f x)}{(f g-e h) (c+d x)}}}d\frac {\sqrt {e+f x}}{\sqrt {c+d x}} \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right )}{\sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right )}{\sqrt {c+d x}}-\frac {2 (b e-a f) (b g-a h) \left (3 d (B c-A d) f h a^2+b \left (3 A d^2 (f g+e h)-B \left (f h c^2+2 d (f g+e h) c+d^2 e g\right )\right ) a+b^2 \left (A f h c^2+3 B d e g c-A d (f g+e h) c-2 A d^2 e g\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \int \frac {1}{\sqrt {\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}+1} \sqrt {1-\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}}d\frac {\sqrt {e+f x}}{\sqrt {a+b x}}}{(f g-e h) \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}}{(b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {\frac {2 b \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {-\frac {2 (d g-c h) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} \int \frac {\sqrt {1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}}}{\sqrt {1-\frac {(d g-c h) (e+f x)}{(f g-e h) (c+d x)}}}d\frac {\sqrt {e+f x}}{\sqrt {c+d x}} \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right )}{\sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right )}{\sqrt {c+d x}}-\frac {2 (b e-a f) \sqrt {b g-a h} \left (3 d (B c-A d) f h a^2+b \left (3 A d^2 (f g+e h)-B \left (f h c^2+2 d (f g+e h) c+d^2 e g\right )\right ) a+b^2 \left (A f h c^2+3 B d e g c-A d (f g+e h) c-2 A d^2 e g\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}}{(b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {\frac {2 b \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {-\frac {2 \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right ) \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right )}{\sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right )}{\sqrt {c+d x}}-\frac {2 (b e-a f) \sqrt {b g-a h} \left (3 d (B c-A d) f h a^2+b \left (3 A d^2 (f g+e h)-B \left (f h c^2+2 d (f g+e h) c+d^2 e g\right )\right ) a+b^2 \left (A f h c^2+3 B d e g c-A d (f g+e h) c-2 A d^2 e g\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}}{(b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}\)

Input:

Int[(A + B*x)/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), 
x]
 

Output:

(-2*b*(A*b - a*B)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c - a*d 
)*(b*e - a*f)*(b*g - a*h)*(a + b*x)^(3/2)) - ((2*b*(3*a^3*B*d*f*h + b^3*(3 
*B*c*e*g - 2*A*(d*e*g + c*f*g + c*e*h)) - a*b^2*(B*(d*e*g + c*f*g + c*e*h) 
 - 4*A*(d*f*g + d*e*h + c*f*h)) - a^2*b*(6*A*d*f*h + B*(d*f*g + d*e*h + c* 
f*h)))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)*(b*e - a*f) 
*(b*g - a*h)*Sqrt[a + b*x]) - ((2*d*(3*a^3*B*d*f*h + b^3*(3*B*c*e*g - 2*A* 
(d*e*g + c*f*g + c*e*h)) - a*b^2*(B*(d*e*g + c*f*g + c*e*h) - 4*A*(d*f*g + 
 d*e*h + c*f*h)) - a^2*b*(6*A*d*f*h + B*(d*f*g + d*e*h + c*f*h)))*Sqrt[a + 
 b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/Sqrt[c + d*x] - (2*Sqrt[d*g - c*h]*Sqrt 
[f*g - e*h]*(3*a^3*B*d*f*h + b^3*(3*B*c*e*g - 2*A*(d*e*g + c*f*g + c*e*h)) 
 - a*b^2*(B*(d*e*g + c*f*g + c*e*h) - 4*A*(d*f*g + d*e*h + c*f*h)) - a^2*b 
*(6*A*d*f*h + B*(d*f*g + d*e*h + c*f*h)))*Sqrt[a + b*x]*Sqrt[-(((d*e - c*f 
)*(g + h*x))/((f*g - e*h)*(c + d*x)))]*EllipticE[ArcSin[(Sqrt[d*g - c*h]*S 
qrt[e + f*x])/(Sqrt[f*g - e*h]*Sqrt[c + d*x])], ((b*c - a*d)*(f*g - e*h))/ 
((b*e - a*f)*(d*g - c*h))])/(Sqrt[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c 
+ d*x))]*Sqrt[g + h*x]) - (2*(b*e - a*f)*Sqrt[b*g - a*h]*(3*a^2*d*(B*c - A 
*d)*f*h + b^2*(3*B*c*d*e*g - 2*A*d^2*e*g + A*c^2*f*h - A*c*d*(f*g + e*h)) 
+ a*b*(3*A*d^2*(f*g + e*h) - B*(d^2*e*g + c^2*f*h + 2*c*d*(f*g + e*h))))*S 
qrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]*Ellipti 
cF[ArcSin[(Sqrt[b*g - a*h]*Sqrt[e + f*x])/(Sqrt[f*g - e*h]*Sqrt[a + b*x...
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 188

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.) 
*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[2*Sqrt[g + h*x]*(Sqrt[(b*e - 
a*f)*((c + d*x)/((d*e - c*f)*(a + b*x)))]/((f*g - e*h)*Sqrt[c + d*x]*Sqrt[( 
-(b*e - a*f))*((g + h*x)/((f*g - e*h)*(a + b*x)))]))   Subst[Int[1/(Sqrt[1 
+ (b*c - a*d)*(x^2/(d*e - c*f))]*Sqrt[1 - (b*g - a*h)*(x^2/(f*g - e*h))]), 
x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 
x]
 

rule 194

Int[Sqrt[(c_.) + (d_.)*(x_)]/(((a_.) + (b_.)*(x_))^(3/2)*Sqrt[(e_.) + (f_.) 
*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2*Sqrt[c + d*x]*(Sqrt[(-(b*e 
 - a*f))*((g + h*x)/((f*g - e*h)*(a + b*x)))]/((b*e - a*f)*Sqrt[g + h*x]*Sq 
rt[(b*e - a*f)*((c + d*x)/((d*e - c*f)*(a + b*x)))]))   Subst[Int[Sqrt[1 + 
(b*c - a*d)*(x^2/(d*e - c*f))]/Sqrt[1 - (b*g - a*h)*(x^2/(f*g - e*h))], x], 
 x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 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 2105

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_. 
) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbo 
l] :> Simp[C*Sqrt[a + b*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/(b*f*h*Sqrt[c + d*x 
])), x] + (Simp[1/(2*b*d*f*h)   Int[(1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e 
+ f*x]*Sqrt[g + h*x]))*Simp[2*A*b*d*f*h - C*(b*d*e*g + a*c*f*h) + (2*b*B*d* 
f*h - C*(a*d*f*h + b*(d*f*g + d*e*h + c*f*h)))*x, x], x], x] + Simp[C*(d*e 
- c*f)*((d*g - c*h)/(2*b*d*f*h))   Int[Sqrt[a + b*x]/((c + d*x)^(3/2)*Sqrt[ 
e + f*x]*Sqrt[g + h*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C} 
, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3388\) vs. \(2(685)=1370\).

Time = 37.81 (sec) , antiderivative size = 3389, normalized size of antiderivative = 4.60

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(3389\)
default	\(\text {Expression too large to display}\)	\(112872\)

Input:

int((B*x+A)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x,meth 
od=_RETURNVERBOSE)
 

Output:

((h*x+g)*(d*x+c)*(b*x+a)*(f*x+e))^(1/2)/(h*x+g)^(1/2)/(d*x+c)^(1/2)/(b*x+a 
)^(1/2)/(f*x+e)^(1/2)*(2/3/(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*(A*b-B*a)*(b*d*f*h*x^4+a* 
d*f*h*x^3+b*c*f*h*x^3+b*d*e*h*x^3+b*d*f*g*x^3+a*c*f*h*x^2+a*d*e*h*x^2+a*d* 
f*g*x^2+b*c*e*h*x^2+b*c*f*g*x^2+b*d*e*g*x^2+a*c*e*h*x+a*c*f*g*x+a*d*e*g*x+ 
b*c*e*g*x+a*c*e*g)^(1/2)/(x+a/b)^2+2/3*(b*d*f*h*x^3+b*c*f*h*x^2+b*d*e*h*x^ 
2+b*d*f*g*x^2+b*c*e*h*x+b*c*f*g*x+b*d*e*g*x+b*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)^2 
*(6*A*a^2*b*d*f*h-4*A*a*b^2*c*f*h-4*A*a*b^2*d*e*h-4*A*a*b^2*d*f*g+2*A*b^3* 
c*e*h+2*A*b^3*c*f*g+2*A*b^3*d*e*g-3*B*a^3*d*f*h+B*a^2*b*c*f*h+B*a^2*b*d*e* 
h+B*a^2*b*d*f*g+B*a*b^2*c*e*h+B*a*b^2*c*f*g+B*a*b^2*d*e*g-3*B*b^3*c*e*g)/( 
(x+a/b)*(b*d*f*h*x^3+b*c*f*h*x^2+b*d*e*h*x^2+b*d*f*g*x^2+b*c*e*h*x+b*c*f*g 
*x+b*d*e*g*x+b*c*e*g))^(1/2)+2*(-1/3*(3*A*a*b*d*f*h-A*b^2*c*f*h-A*b^2*d*e* 
h-A*b^2*d*f*g-3*B*a^2*d*f*h+B*a*b*c*f*h+B*a*b*d*e*h+B*a*b*d*f*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+1/3/b*(a^2*d*f*h-a*b*c*f*h-a*b*d*e*h-a*b*d*f*g+b^2*c*e*h+b^2* 
c*f*g+b^2*d*e*g)*(6*A*a^2*b*d*f*h-4*A*a*b^2*c*f*h-4*A*a*b^2*d*e*h-4*A*a*b^ 
2*d*f*g+2*A*b^3*c*e*h+2*A*b^3*c*f*g+2*A*b^3*d*e*g-3*B*a^3*d*f*h+B*a^2*b*c* 
f*h+B*a^2*b*d*e*h+B*a^2*b*d*f*g+B*a*b^2*c*e*h+B*a*b^2*c*f*g+B*a*b^2*d*e*g- 
3*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...
 

Fricas [F]

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

integrate((B*x+A)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2), 
x, algorithm="fricas")
 

Output:

integral((B*x + A)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g) 
/(b^3*d*f*h*x^6 + a^3*c*e*g + (b^3*d*f*g + (b^3*d*e + (b^3*c + 3*a*b^2*d)* 
f)*h)*x^5 + ((b^3*d*e + (b^3*c + 3*a*b^2*d)*f)*g + ((b^3*c + 3*a*b^2*d)*e 
+ 3*(a*b^2*c + a^2*b*d)*f)*h)*x^4 + (((b^3*c + 3*a*b^2*d)*e + 3*(a*b^2*c + 
 a^2*b*d)*f)*g + (3*(a*b^2*c + a^2*b*d)*e + (3*a^2*b*c + a^3*d)*f)*h)*x^3 
+ ((3*(a*b^2*c + a^2*b*d)*e + (3*a^2*b*c + a^3*d)*f)*g + (a^3*c*f + (3*a^2 
*b*c + a^3*d)*e)*h)*x^2 + (a^3*c*e*h + (a^3*c*f + (3*a^2*b*c + a^3*d)*e)*g 
)*x), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)/(b*x+a)**(5/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1 
/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((B*x+A)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2), 
x, algorithm="maxima")
 

Output:

integrate((B*x + A)/((b*x + a)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x 
+ g)), x)
 

Giac [F]

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

integrate((B*x+A)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2), 
x, algorithm="giac")
 

Output:

integrate((B*x + A)/((b*x + a)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x 
+ g)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {A+B\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \] Input:

int((A + B*x)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)^(5/2)*(c + d*x)^( 
1/2)),x)
 

Output:

int((A + B*x)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)^(5/2)*(c + d*x)^( 
1/2)), x)
 

Reduce [F]

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

int((B*x+A)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)
 

Output:

int((B*x+A)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)