\(\int \frac {(a+b x)^{3/2} (A+B x+C x^2)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [38]

Optimal result

Integrand size = 47, antiderivative size = 1524 \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx =\text {Too large to display} \] Output:

1/24*((a*d*f*h-3*b*(c*f*h+d*e*h+d*f*g))*(6*b*B*d*f*h+3*a*C*d*f*h-5*b*C*(c* 
f*h+d*e*h+d*f*g))/d/f/h+8*b*(3*(A*b+B*a)*d*f*h-C*(2*b*(c*e*h+c*f*g+d*e*g)+ 
a*(c*f*h+d*e*h+d*f*g))))*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/d^2/f^2 
/h^2/(b*x+a)^(1/2)+1/12*(6*b*B*d*f*h+3*a*C*d*f*h-5*b*C*(c*f*h+d*e*h+d*f*g) 
)*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/d^2/f^2/h^2+1/3* 
C*(b*x+a)^(3/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/d/f/h-1/24*(-a*f 
+b*e)^(1/2)*(-c*h+d*g)*(3*a^2*C*d^2*f^2*h^2+2*a*b*d*f*h*(15*B*d*f*h-11*C*( 
c*f*h+d*e*h+d*f*g))+b^2*(6*d*f*h*(4*A*d*f*h-3*B*(c*f*h+d*e*h+d*f*g))+C*(15 
*c^2*f^2*h^2+14*c*d*f*h*(e*h+f*g)+d^2*(15*e^2*h^2+14*e*f*g*h+15*f^2*g^2))) 
)*(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))/b/d^3/f^3/(-c*f+d*e)^(1/2)/h^3/(-( 
-a*d+b*c)*(f*x+e)/(-c*f+d*e)/(b*x+a))^(1/2)/(h*x+g)^(1/2)+1/24*(-a*d+b*c)* 
(-a*f+b*e)^(1/2)*(3*a^2*C*d^2*f^2*h^2-6*a*b*d*f*h*(3*B*d*f*h-C*(2*c*f*h+2* 
d*e*h+d*f*g))-b^2*(6*d*f*h*(4*A*d*f*h-B*(3*c*f*h+3*d*e*h+d*f*g))+C*(15*c^2 
*f^2*h^2+2*c*d*f*h*(7*e*h+2*f*g)+d^2*(15*e^2*h^2+4*e*f*g*h+5*f^2*g^2))))*( 
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))/b^2/d^3/f^3/(-c*f+d*e)^(1/2)/h^2/(-(- 
a*d+b*c)*(f*x+e)/(-c*f+d*e)/(b*x+a))^(1/2)/(h*x+g)^(1/2)-1/24*(-c*f+d*e...
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(55395\) vs. \(2(1524)=3048\).

Time = 43.08 (sec) , antiderivative size = 55395, normalized size of antiderivative = 36.35 \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \] Input:

Integrate[((a + b*x)^(3/2)*(A + B*x + C*x^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 = 5.78 (sec) , antiderivative size = 1488, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.234, Rules used = {2103, 25, 2103, 2105, 194, 327, 2101, 183, 188, 321, 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.

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

\(\Big \downarrow \) 2103

\(\displaystyle \frac {\int -\frac {\sqrt {a+b x} \left (-\left ((6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h)) x^2\right )-2 (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h))) x+3 b c C e g-6 a A d f h+a C (d e g+c f g+c e h)\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{6 d f h}+\frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {\int \frac {\sqrt {a+b x} \left (-\left ((6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h)) x^2\right )-2 (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h))) x+3 b c C e g-6 a A d f h+a C (d e g+c f g+c e h)\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{6 d f h}\)

\(\Big \downarrow \) 2103

\(\displaystyle \frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {\frac {\int \frac {-\left (((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) x^2\right )+2 \left ((b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+2 d f h \left (-2 (3 B d f h-C (d f g+d e h+c f h)) a^2-b (12 A d f h-5 C (d e g+c f g+c e h)) a+3 b^2 c C e g\right )\right ) x+4 a d f h (3 b c C e g-6 a A d f h+a C (d e g+c f g+c e h))+(b c e g+a (d e g+c f g+c e h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{4 d f h}-\frac {1}{2} \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (3 a C+6 b B-5 b C \left (\frac {c}{d}+\frac {e}{f}+\frac {g}{h}\right )\right )}{6 d f h}\)

\(\Big \downarrow \) 2105

\(\displaystyle \frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {\frac {-\frac {\sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (\frac {(a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))}{b f h}+8 d (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))\right )}{\sqrt {c+d x}}-\frac {(d e-c f) (d g-c h) ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 b d f h}+\frac {\int \frac {2 b d f h (4 a d f h (3 b c C e g-6 a A d f h+a C (d e g+c f g+c e h))+(b c e g+a (d e g+c f g+c e h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h)))+(b d e g+a c f h) ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h))))+\left ((a d f h+b (d f g+d e h+c f h)) ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h))))+4 b d f h \left ((b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+2 d f h \left (-2 (3 B d f h-C (d f g+d e h+c f h)) a^2-b (12 A d f h-5 C (d e g+c f g+c e h)) a+3 b^2 c C e g\right )\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 b d f h}}{4 d f h}-\frac {1}{2} \left (6 b B+3 a C-5 b C \left (\frac {c}{d}+\frac {e}{f}+\frac {g}{h}\right )\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{6 d f h}\)

\(\Big \downarrow \) 194

\(\displaystyle \frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {\frac {-\frac {\sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (\frac {(a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))}{b f h}+8 d (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))\right )}{\sqrt {c+d x}}+\frac {\int \frac {2 b d f h (4 a d f h (3 b c C e g-6 a A d f h+a C (d e g+c f g+c e h))+(b c e g+a (d e g+c f g+c e h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h)))+(b d e g+a c f h) ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h))))+\left ((a d f h+b (d f g+d e h+c f h)) ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h))))+4 b d f h \left ((b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+2 d f h \left (-2 (3 B d f h-C (d f g+d e h+c f h)) a^2-b (12 A d f h-5 C (d e g+c f g+c e h)) a+3 b^2 c C e g\right )\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 b d f h}+\frac {(d g-c h) ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} \int \frac {\sqrt {1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}}}{\sqrt {1-\frac {(d g-c h) (e+f x)}{(f g-e h) (c+d x)}}}d\frac {\sqrt {e+f x}}{\sqrt {c+d x}}}{b d f h \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}}{4 d f h}-\frac {1}{2} \left (6 b B+3 a C-5 b C \left (\frac {c}{d}+\frac {e}{f}+\frac {g}{h}\right )\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{6 d f h}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {\frac {-\frac {\sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (\frac {(a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))}{b f h}+8 d (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))\right )}{\sqrt {c+d x}}+\frac {\sqrt {d g-c h} \sqrt {f g-e h} ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{b d f h \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {\int \frac {2 b d f h (4 a d f h (3 b c C e g-6 a A d f h+a C (d e g+c f g+c e h))+(b c e g+a (d e g+c f g+c e h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h)))+(b d e g+a c f h) ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h))))+\left ((a d f h+b (d f g+d e h+c f h)) ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h))))+4 b d f h \left ((b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+2 d f h \left (-2 (3 B d f h-C (d f g+d e h+c f h)) a^2-b (12 A d f h-5 C (d e g+c f g+c e h)) a+3 b^2 c C e g\right )\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 b d f h}}{4 d f h}-\frac {1}{2} \left (6 b B+3 a C-5 b C \left (\frac {c}{d}+\frac {e}{f}+\frac {g}{h}\right )\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{6 d f h}\)

\(\Big \downarrow \) 2101

\(\displaystyle \frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {\frac {-\frac {\sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (\frac {(a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))}{b f h}+8 d (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))\right )}{\sqrt {c+d x}}+\frac {\sqrt {d g-c h} \sqrt {f g-e h} ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{b d f h \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {\frac {\left ((a d f h+b (d f g+d e h+c f h)) ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h))))+4 b d f h \left ((b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+2 d f h \left (-2 (3 B d f h-C (d f g+d e h+c f h)) a^2-b (12 A d f h-5 C (d e g+c f g+c e h)) a+3 b^2 c C e g\right )\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b}-\frac {d (b e-a f) (b g-a h) \left (-\left (\left (6 d f h (4 A d f h-B (3 d f g+3 d e h+c f h))+C \left (\left (15 f^2 g^2+14 e f h g+15 e^2 h^2\right ) d^2+4 c f h (f g+e h) d+5 c^2 f^2 h^2\right )\right ) b^2\right )+6 a d f h (c C f h-3 B d f h+2 C d (f g+e h)) b+3 a^2 C d^2 f^2 h^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b}}{2 b d f h}}{4 d f h}-\frac {1}{2} \left (6 b B+3 a C-5 b C \left (\frac {c}{d}+\frac {e}{f}+\frac {g}{h}\right )\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{6 d f h}\)

\(\Big \downarrow \) 183

\(\displaystyle \frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {\frac {-\frac {\sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (\frac {(a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))}{b f h}+8 d (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))\right )}{\sqrt {c+d x}}+\frac {\sqrt {d g-c h} \sqrt {f g-e h} ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f 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 )}{b d f h \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {\frac {2 \left ((a d f h+b (d f g+d e h+c f h)) ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h))))+4 b d f h \left ((b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+2 d f h \left (-2 (3 B d f h-C (d f g+d e h+c f h)) a^2-b (12 A d f h-5 C (d e g+c f g+c e h)) a+3 b^2 c C e g\right )\right )\right ) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \int \frac {1}{\left (h-\frac {b (g+h x)}{a+b x}\right ) \sqrt {\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}+1} \sqrt {\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}+1}}d\frac {\sqrt {g+h x}}{\sqrt {a+b x}}}{b \sqrt {c+d x} \sqrt {e+f x}}-\frac {d (b e-a f) (b g-a h) \left (-\left (\left (6 d f h (4 A d f h-B (3 d f g+3 d e h+c f h))+C \left (\left (15 f^2 g^2+14 e f h g+15 e^2 h^2\right ) d^2+4 c f h (f g+e h) d+5 c^2 f^2 h^2\right )\right ) b^2\right )+6 a d f h (c C f h-3 B d f h+2 C d (f g+e h)) b+3 a^2 C d^2 f^2 h^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{b}}{2 b d f h}}{4 d f h}-\frac {1}{2} \left (6 b B+3 a C-5 b C \left (\frac {c}{d}+\frac {e}{f}+\frac {g}{h}\right )\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{6 d f h}\)

\(\Big \downarrow \) 188

\(\displaystyle \frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {\frac {-\frac {\sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (\frac {(a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))}{b f h}+8 d (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))\right )}{\sqrt {c+d x}}+\frac {\sqrt {d g-c h} \sqrt {f g-e h} ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f 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 )}{b d f h \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {\frac {2 \left ((a d f h+b (d f g+d e h+c f h)) ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h))))+4 b d f h \left ((b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+2 d f h \left (-2 (3 B d f h-C (d f g+d e h+c f h)) a^2-b (12 A d f h-5 C (d e g+c f g+c e h)) a+3 b^2 c C e g\right )\right )\right ) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \int \frac {1}{\left (h-\frac {b (g+h x)}{a+b x}\right ) \sqrt {\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}+1} \sqrt {\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}+1}}d\frac {\sqrt {g+h x}}{\sqrt {a+b x}}}{b \sqrt {c+d x} \sqrt {e+f x}}-\frac {2 d (b e-a f) (b g-a h) \left (-\left (\left (6 d f h (4 A d f h-B (3 d f g+3 d e h+c f h))+C \left (\left (15 f^2 g^2+14 e f h g+15 e^2 h^2\right ) d^2+4 c f h (f g+e h) d+5 c^2 f^2 h^2\right )\right ) b^2\right )+6 a d f h (c C f h-3 B d f h+2 C d (f g+e h)) b+3 a^2 C d^2 f^2 h^2\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}}}{b (f g-e h) \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}}{2 b d f h}}{4 d f h}-\frac {1}{2} \left (6 b B+3 a C-5 b C \left (\frac {c}{d}+\frac {e}{f}+\frac {g}{h}\right )\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{6 d f h}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {\frac {-\frac {\sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (\frac {(a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))}{b f h}+8 d (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))\right )}{\sqrt {c+d x}}+\frac {\sqrt {d g-c h} \sqrt {f g-e h} ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f 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 )}{b d f h \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {\frac {2 \left ((a d f h+b (d f g+d e h+c f h)) ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h))))+4 b d f h \left ((b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+2 d f h \left (-2 (3 B d f h-C (d f g+d e h+c f h)) a^2-b (12 A d f h-5 C (d e g+c f g+c e h)) a+3 b^2 c C e g\right )\right )\right ) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \int \frac {1}{\left (h-\frac {b (g+h x)}{a+b x}\right ) \sqrt {\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}+1} \sqrt {\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}+1}}d\frac {\sqrt {g+h x}}{\sqrt {a+b x}}}{b \sqrt {c+d x} \sqrt {e+f x}}-\frac {2 d (b e-a f) \sqrt {b g-a h} \left (-\left (\left (6 d f h (4 A d f h-B (3 d f g+3 d e h+c f h))+C \left (\left (15 f^2 g^2+14 e f h g+15 e^2 h^2\right ) d^2+4 c f h (f g+e h) d+5 c^2 f^2 h^2\right )\right ) b^2\right )+6 a d f h (c C f h-3 B d f h+2 C d (f g+e h)) b+3 a^2 C d^2 f^2 h^2\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 )}{b \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)}}}}{2 b d f h}}{4 d f h}-\frac {1}{2} \left (6 b B+3 a C-5 b C \left (\frac {c}{d}+\frac {e}{f}+\frac {g}{h}\right )\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{6 d f h}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {C (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {\frac {-\frac {\sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (\frac {(a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))}{b f h}+8 d (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))\right )}{\sqrt {c+d x}}+\frac {\sqrt {d g-c h} \sqrt {f g-e h} ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f 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 )}{b d f h \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {\frac {2 \sqrt {c h-d g} \left ((a d f h+b (d f g+d e h+c f h)) ((a d f h-3 b (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+8 b d f h (3 (A b+a B) d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h))))+4 b d f h \left ((b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (6 b B d f h+3 a C d f h-5 b C (d f g+d e h+c f h))+2 d f h \left (-2 (3 B d f h-C (d f g+d e h+c f h)) a^2-b (12 A d f h-5 C (d e g+c f g+c e h)) a+3 b^2 c C e g\right )\right )\right ) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {c h-d g} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{b \sqrt {b c-a d} h \sqrt {c+d x} \sqrt {e+f x}}-\frac {2 d (b e-a f) \sqrt {b g-a h} \left (-\left (\left (6 d f h (4 A d f h-B (3 d f g+3 d e h+c f h))+C \left (\left (15 f^2 g^2+14 e f h g+15 e^2 h^2\right ) d^2+4 c f h (f g+e h) d+5 c^2 f^2 h^2\right )\right ) b^2\right )+6 a d f h (c C f h-3 B d f h+2 C d (f g+e h)) b+3 a^2 C d^2 f^2 h^2\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 )}{b \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)}}}}{2 b d f h}}{4 d f h}-\frac {1}{2} \left (6 b B+3 a C-5 b C \left (\frac {c}{d}+\frac {e}{f}+\frac {g}{h}\right )\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{6 d f h}\)

Input:

Int[((a + b*x)^(3/2)*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[ 
g + h*x]),x]
 

Output:

(C*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*d*f*h) - 
(-1/2*((6*b*B + 3*a*C - 5*b*C*(c/d + e/f + g/h))*Sqrt[a + b*x]*Sqrt[c + d* 
x]*Sqrt[e + f*x]*Sqrt[g + h*x]) + (-(((((a*d*f*h - 3*b*(d*f*g + d*e*h + c* 
f*h))*(6*b*B*d*f*h + 3*a*C*d*f*h - 5*b*C*(d*f*g + d*e*h + c*f*h)))/(b*f*h) 
 + 8*d*(3*(A*b + a*B)*d*f*h - C*(2*b*(d*e*g + c*f*g + c*e*h) + a*(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] 
) + (Sqrt[d*g - c*h]*Sqrt[f*g - e*h]*((a*d*f*h - 3*b*(d*f*g + d*e*h + c*f* 
h))*(6*b*B*d*f*h + 3*a*C*d*f*h - 5*b*C*(d*f*g + d*e*h + c*f*h)) + 8*b*d*f* 
h*(3*(A*b + a*B)*d*f*h - C*(2*b*(d*e*g + c*f*g + c*e*h) + a*(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]*Sqrt[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))])/( 
b*d*f*h*Sqrt[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x 
]) + ((-2*d*(b*e - a*f)*Sqrt[b*g - a*h]*(3*a^2*C*d^2*f^2*h^2 + 6*a*b*d*f*h 
*(c*C*f*h - 3*B*d*f*h + 2*C*d*(f*g + e*h)) - b^2*(6*d*f*h*(4*A*d*f*h - B*( 
3*d*f*g + 3*d*e*h + c*f*h)) + C*(5*c^2*f^2*h^2 + 4*c*d*f*h*(f*g + e*h) + d 
^2*(15*f^2*g^2 + 14*e*f*g*h + 15*e^2*h^2))))*Sqrt[((b*e - a*f)*(c + d*x))/ 
((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]*EllipticF[ArcSin[(Sqrt[b*g - a*h]*S 
qrt[e + f*x])/(Sqrt[f*g - e*h]*Sqrt[a + b*x])], -(((b*c - a*d)*(f*g - e*h) 
)/((d*e - c*f)*(b*g - a*h)))])/(b*Sqrt[f*g - e*h]*Sqrt[c + d*x]*Sqrt[-(...
 

Defintions of rubi rules used

rule 25

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

rule 183

Int[Sqrt[(a_.) + (b_.)*(x_)]/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*( 
x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[2*(a + b*x)*Sqrt[(b*g - a*h)*(( 
c + d*x)/((d*g - c*h)*(a + b*x)))]*(Sqrt[(b*g - a*h)*((e + f*x)/((f*g - e*h 
)*(a + b*x)))]/(Sqrt[c + d*x]*Sqrt[e + f*x]))   Subst[Int[1/((h - b*x^2)*Sq 
rt[1 + (b*c - a*d)*(x^2/(d*g - c*h))]*Sqrt[1 + (b*e - a*f)*(x^2/(f*g - e*h) 
)]), x], x, Sqrt[g + h*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, 
 h}, 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 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 2101

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

rule 2103

Int[(((a_.) + (b_.)*(x_))^(m_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2))/(Sqrt[ 
(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_S 
ymbol] :> Simp[2*C*(a + b*x)^m*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/( 
d*f*h*(2*m + 3))), x] + Simp[1/(d*f*h*(2*m + 3))   Int[((a + b*x)^(m - 1)/( 
Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[a*A*d*f*h*(2*m + 3) - C*(a 
*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*m) + ((A*b + a*B)*d*f*h*(2*m + 3) - C* 
(2*a*(d*f*g + d*e*h + c*f*h) + b*(2*m + 1)*(d*e*g + c*f*g + c*e*h)))*x + (b 
*B*d*f*h*(2*m + 3) + 2*C*(a*d*f*h*m - b*(m + 1)*(d*f*g + d*e*h + c*f*h)))*x 
^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x] && IntegerQ[2 
*m] && GtQ[m, 0]
 

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 [A] (verified)

Time = 21.75 (sec) , antiderivative size = 2258, normalized size of antiderivative = 1.48

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2258\)
default	\(\text {Expression too large to display}\)	\(125812\)

Input:

int((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2), 
x,method=_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)*(1/3*C*b/d/f/h*x*(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)+1/2*(B*b^2+2*C*a*b-1/3*C*b/d/f/h*(5/2*a*d*f*h+5/2*b*c*f*h+5/2*b*d*e*h 
+5/2*b*g*d*f))/h/d/b/f*(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)+2*(a^2*A-1/ 
3*C*b/d/f/h*a*c*e*g-1/2*(B*b^2+2*C*a*b-1/3*C*b/d/f/h*(5/2*a*d*f*h+5/2*b*c* 
f*h+5/2*b*d*e*h+5/2*b*g*d*f))/h/d/b/f*(1/2*a*c*e*h+1/2*a*c*f*g+1/2*a*d*e*g 
+1/2*b*c*e*g))*(e/f-g/h)*((c/d-e/f)*(x+g/h)/(-e/f+g/h)/(x+c/d))^(1/2)*(x+c 
/d)^2*((-c/d+g/h)*(x+a/b)/(-a/b+g/h)/(x+c/d))^(1/2)*((-c/d+g/h)*(x+e/f)/(- 
e/f+g/h)/(x+c/d))^(1/2)/(c/d-e/f)/(-c/d+g/h)/(h*d*b*f*(x+g/h)*(x+c/d)*(x+a 
/b)*(x+e/f))^(1/2)*EllipticF(((c/d-e/f)*(x+g/h)/(-e/f+g/h)/(x+c/d))^(1/2), 
((-c/d+a/b)*(e/f-g/h)/(a/b-g/h)/(-c/d+e/f))^(1/2))+2*(2*a*b*A+a^2*B-1/3*C* 
b/d/f/h*(3/2*a*c*e*h+3/2*a*c*f*g+3/2*a*d*e*g+3/2*b*c*e*g)-1/2*(B*b^2+2*C*a 
*b-1/3*C*b/d/f/h*(5/2*a*d*f*h+5/2*b*c*f*h+5/2*b*d*e*h+5/2*b*g*d*f))/h/d/b/ 
f*(a*c*f*h+a*d*e*h+a*d*f*g+b*c*e*h+b*c*f*g+b*d*e*g))*(e/f-g/h)*((c/d-e/f)* 
(x+g/h)/(-e/f+g/h)/(x+c/d))^(1/2)*(x+c/d)^2*((-c/d+g/h)*(x+a/b)/(-a/b+g/h) 
/(x+c/d))^(1/2)*((-c/d+g/h)*(x+e/f)/(-e/f+g/h)/(x+c/d))^(1/2)/(c/d-e/f)...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^ 
(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (A + B x + C x^{2}\right )}{\sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \] Input:

integrate((b*x+a)**(3/2)*(C*x**2+B*x+A)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x 
+g)**(1/2),x)
 

Output:

Integral((a + b*x)**(3/2)*(A + B*x + C*x**2)/(sqrt(c + d*x)*sqrt(e + f*x)* 
sqrt(g + h*x)), x)
 

Maxima [F]

\[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \] Input:

integrate((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^ 
(1/2),x, algorithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)*(b*x + a)^(3/2)/(sqrt(d*x + c)*sqrt(f*x + e)*s 
qrt(h*x + g)), x)
 

Giac [F]

\[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \] Input:

integrate((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^ 
(1/2),x, algorithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)*(b*x + a)^(3/2)/(sqrt(d*x + c)*sqrt(f*x + e)*s 
qrt(h*x + g)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}\,\left (C\,x^2+B\,x+A\right )}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \] Input:

int(((a + b*x)^(3/2)*(A + B*x + C*x^2))/((e + f*x)^(1/2)*(g + h*x)^(1/2)*( 
c + d*x)^(1/2)),x)
 

Output:

int(((a + b*x)^(3/2)*(A + B*x + C*x^2))/((e + f*x)^(1/2)*(g + h*x)^(1/2)*( 
c + d*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\left (b x +a \right )^{\frac {3}{2}} \left (C \,x^{2}+B x +A \right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}d x \] Input:

int((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2), 
x)
 

Output:

int((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2), 
x)