3.2.7 \(\int \frac {(a+b x)^{3/2}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [107]

3.2.7.1 Optimal result

Integrand size = 37, antiderivative size = 968 \[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {b \sqrt {a+b x} \sqrt {c+d x} \sqrt {g+h x}}{d h \sqrt {e+f x}}-\frac {b \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {\frac {(d e-c f) (g+h x)}{(d g-c h) (e+f x)}} E\left (\arcsin \left (\frac {\sqrt {f g-e h} \sqrt {c+d x}}{\sqrt {d g-c h} \sqrt {e+f x}}\right )|\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{d f h \sqrt {-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}} \sqrt {g+h x}}+\frac {b (d e-c f) (b f g+b e h-2 a f h) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{d f^2 h \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {b \sqrt {b g-a h} (a d f h-b (d f g+d e h-c f h)) \sqrt {\frac {(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} \sqrt {\frac {(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} (e+f x) \operatorname {EllipticPi}\left (\frac {f (b g-a h)}{(b e-a f) h},\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {g+h x}}{\sqrt {b g-a h} \sqrt {e+f x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{d f^2 \sqrt {b e-a f} h^2 \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 \sqrt {b c-a d} \sqrt {-d g+c h} (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {-d g+c h} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{d h \sqrt {c+d x} \sqrt {e+f x}} \]

output

b*(a*d*f*h-b*(-c*f*h+d*e*h+d*f*g))*(f*x+e)*EllipticPi((-a*f+b*e)^(1/2)*(h* 
x+g)^(1/2)/(-a*h+b*g)^(1/2)/(f*x+e)^(1/2),f*(-a*h+b*g)/(-a*f+b*e)/h,((-c*f 
+d*e)*(-a*h+b*g)/(-a*f+b*e)/(-c*h+d*g))^(1/2))*(-a*h+b*g)^(1/2)*((-e*h+f*g 
)*(b*x+a)/(-a*h+b*g)/(f*x+e))^(1/2)*((-e*h+f*g)*(d*x+c)/(-c*h+d*g)/(f*x+e) 
)^(1/2)/d/f^2/h^2/(-a*f+b*e)^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)-2*(b*x+a)*E 
llipticPi((-a*d+b*c)^(1/2)*(h*x+g)^(1/2)/(c*h-d*g)^(1/2)/(b*x+a)^(1/2),-b* 
(-c*h+d*g)/(-a*d+b*c)/h,((-a*f+b*e)*(-c*h+d*g)/(-a*d+b*c)/(-e*h+f*g))^(1/2 
))*(-a*d+b*c)^(1/2)*(c*h-d*g)^(1/2)*((-a*h+b*g)*(d*x+c)/(-c*h+d*g)/(b*x+a) 
)^(1/2)*((-a*h+b*g)*(f*x+e)/(-e*h+f*g)/(b*x+a))^(1/2)/d/h/(d*x+c)^(1/2)/(f 
*x+e)^(1/2)+b*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(h*x+g)^(1/2)/d/h/(f*x+e)^(1/2)+ 
b*(-c*f+d*e)*(-2*a*f*h+b*e*h+b*f*g)*EllipticF((-a*h+b*g)^(1/2)*(f*x+e)^(1/ 
2)/(-e*h+f*g)^(1/2)/(b*x+a)^(1/2),(-(-a*d+b*c)*(-e*h+f*g)/(-c*f+d*e)/(-a*h 
+b*g))^(1/2))*((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^(1/2)*(h*x+g)^(1/2)/ 
d/f^2/h/(-a*h+b*g)^(1/2)/(-e*h+f*g)^(1/2)/(d*x+c)^(1/2)/(-(-a*f+b*e)*(h*x+ 
g)/(-e*h+f*g)/(b*x+a))^(1/2)-b*EllipticE((-e*h+f*g)^(1/2)*(d*x+c)^(1/2)/(- 
c*h+d*g)^(1/2)/(f*x+e)^(1/2),((-a*f+b*e)*(-c*h+d*g)/(-a*d+b*c)/(-e*h+f*g)) 
^(1/2))*(-c*h+d*g)^(1/2)*(-e*h+f*g)^(1/2)*(b*x+a)^(1/2)*((-c*f+d*e)*(h*x+g 
)/(-c*h+d*g)/(f*x+e))^(1/2)/d/f/h/(-(-c*f+d*e)*(b*x+a)/(-a*d+b*c)/(f*x+e)) 
^(1/2)/(h*x+g)^(1/2)
 

3.2.7.2 Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(7319\) vs. \(2(968)=1936\).

Time = 30.31 (sec) , antiderivative size = 7319, normalized size of antiderivative = 7.56 \[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \]

input

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

output

Result too large to show
 

3.2.7.3 Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 958, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {184, 183, 191, 183, 188, 194, 321, 327, 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.

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

\(\Big \downarrow \) 184

\(\displaystyle \frac {b \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{\sqrt {e+f x} \sqrt {g+h x}}dx}{d}-\frac {(b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{d}\)

\(\Big \downarrow \) 183

\(\displaystyle \frac {b \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{\sqrt {e+f x} \sqrt {g+h x}}dx}{d}-\frac {2 (a+b x) (b c-a d) \sqrt {\frac {(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt {\frac {(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \int \frac {1}{\left (h-\frac {b (g+h x)}{a+b x}\right ) \sqrt {\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}+1} \sqrt {\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}+1}}d\frac {\sqrt {g+h x}}{\sqrt {a+b x}}}{d \sqrt {c+d x} \sqrt {e+f x}}\)

\(\Big \downarrow \) 191

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

\(\Big \downarrow \) 183

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

\(\Big \downarrow \) 188

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

\(\Big \downarrow \) 194

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 412

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

input

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

output

(b*((Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[g + h*x])/(h*Sqrt[e + f*x]) - (Sqrt[ 
d*g - c*h]*Sqrt[f*g - e*h]*Sqrt[a + b*x]*Sqrt[((d*e - c*f)*(g + h*x))/((d* 
g - c*h)*(e + f*x))]*EllipticE[ArcSin[(Sqrt[f*g - e*h]*Sqrt[c + d*x])/(Sqr 
t[d*g - c*h]*Sqrt[e + f*x])], ((b*e - a*f)*(d*g - c*h))/((b*c - a*d)*(f*g 
- e*h))])/(f*h*Sqrt[-(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x)))]*Sq 
rt[g + h*x]) + ((d*e - c*f)*(b*f*g + b*e*h - 2*a*f*h)*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]*Sqrt[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)))])/(f^2*h*Sqrt[b*g - a*h]*Sqrt[f*g - 
e*h]*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))) 
]) + (Sqrt[b*g - a*h]*(a*d*f*h - b*(d*f*g + d*e*h - c*f*h))*Sqrt[((f*g - e 
*h)*(a + b*x))/((b*g - a*h)*(e + f*x))]*Sqrt[((f*g - e*h)*(c + d*x))/((d*g 
 - c*h)*(e + f*x))]*(e + f*x)*EllipticPi[(f*(b*g - a*h))/((b*e - a*f)*h), 
ArcSin[(Sqrt[b*e - a*f]*Sqrt[g + h*x])/(Sqrt[b*g - a*h]*Sqrt[e + f*x])], ( 
(d*e - c*f)*(b*g - a*h))/((b*e - a*f)*(d*g - c*h))])/(f^2*Sqrt[b*e - a*f]* 
h^2*Sqrt[a + b*x]*Sqrt[c + d*x])))/d - (2*Sqrt[b*c - a*d]*Sqrt[-(d*g) + c* 
h]*(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))]*EllipticPi[-((b*(d*g - c*h) 
)/((b*c - a*d)*h)), ArcSin[(Sqrt[b*c - a*d]*Sqrt[g + h*x])/(Sqrt[-(d*g) + 
c*h]*Sqrt[a + b*x])], ((b*e - a*f)*(d*g - c*h))/((b*c - a*d)*(f*g - e*h...
 

3.2.7.3.1 Defintions of rubi rules used

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 184

Int[((a_.) + (b_.)*(x_))^(3/2)/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.) 
*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[b/d   Int[Sqrt[a + b*x]*(Sqrt 
[c + d*x]/(Sqrt[e + f*x]*Sqrt[g + h*x])), x], x] - Simp[(b*c - a*d)/d   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}, 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 191

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

3.2.7.4 Maple [A] (verified)

Time = 2.60 (sec) , antiderivative size = 1541, normalized size of antiderivative = 1.59

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1541\)
default	\(\text {Expression too large to display}\)	\(17031\)

input

int((b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x,method=_RETU 
RNVERBOSE)
 

output

((b*x+a)*(d*x+c)*(f*x+e)*(h*x+g))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e 
)^(1/2)/(h*x+g)^(1/2)*(2*a^2*(g/h-a/b)*((-g/h+c/d)*(x+a/b)/(-g/h+a/b)/(x+c 
/d))^(1/2)*(x+c/d)^2*((-c/d+a/b)*(x+e/f)/(-e/f+a/b)/(x+c/d))^(1/2)*((-c/d+ 
a/b)*(x+g/h)/(-g/h+a/b)/(x+c/d))^(1/2)/(-g/h+c/d)/(-c/d+a/b)/(b*d*f*h*(x+a 
/b)*(x+c/d)*(x+e/f)*(x+g/h))^(1/2)*EllipticF(((-g/h+c/d)*(x+a/b)/(-g/h+a/b 
)/(x+c/d))^(1/2),((e/f-c/d)*(g/h-a/b)/(-a/b+e/f)/(-c/d+g/h))^(1/2))+4*a*b* 
(g/h-a/b)*((-g/h+c/d)*(x+a/b)/(-g/h+a/b)/(x+c/d))^(1/2)*(x+c/d)^2*((-c/d+a 
/b)*(x+e/f)/(-e/f+a/b)/(x+c/d))^(1/2)*((-c/d+a/b)*(x+g/h)/(-g/h+a/b)/(x+c/ 
d))^(1/2)/(-g/h+c/d)/(-c/d+a/b)/(b*d*f*h*(x+a/b)*(x+c/d)*(x+e/f)*(x+g/h))^ 
(1/2)*(-c/d*EllipticF(((-g/h+c/d)*(x+a/b)/(-g/h+a/b)/(x+c/d))^(1/2),((e/f- 
c/d)*(g/h-a/b)/(-a/b+e/f)/(-c/d+g/h))^(1/2))+(c/d-a/b)*EllipticPi(((-g/h+c 
/d)*(x+a/b)/(-g/h+a/b)/(x+c/d))^(1/2),(-g/h+a/b)/(-g/h+c/d),((e/f-c/d)*(g/ 
h-a/b)/(-a/b+e/f)/(-c/d+g/h))^(1/2)))+b^2*((x+a/b)*(x+e/f)*(x+g/h)+(g/h-a/ 
b)*((-g/h+c/d)*(x+a/b)/(-g/h+a/b)/(x+c/d))^(1/2)*(x+c/d)^2*((-c/d+a/b)*(x+ 
e/f)/(-e/f+a/b)/(x+c/d))^(1/2)*((-c/d+a/b)*(x+g/h)/(-g/h+a/b)/(x+c/d))^(1/ 
2)*((a*c/b/d-g/h*a/b+g/h*c/d+c^2/d^2)/(-g/h+c/d)/(-c/d+a/b)*EllipticF(((-g 
/h+c/d)*(x+a/b)/(-g/h+a/b)/(x+c/d))^(1/2),((e/f-c/d)*(g/h-a/b)/(-a/b+e/f)/ 
(-c/d+g/h))^(1/2))+(-a/b+e/f)*EllipticE(((-g/h+c/d)*(x+a/b)/(-g/h+a/b)/(x+ 
c/d))^(1/2),((e/f-c/d)*(g/h-a/b)/(-a/b+e/f)/(-c/d+g/h))^(1/2))/(-c/d+a/b)+ 
(a*d*f*h+b*c*f*h+b*d*e*h+b*d*f*g)/b/d/f/h/(-g/h+c/d)*EllipticPi(((-g/h+...
 

3.2.7.5 Fricas [F(-1)]

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

input

integrate((b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algor 
ithm="fricas")
 

output

Timed out
 

3.2.7.6 Sympy [F]

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

input

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

output

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

3.2.7.7 Maxima [F]

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

input

integrate((b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algor 
ithm="maxima")
 

output

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

3.2.7.8 Giac [F]

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

input

integrate((b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algor 
ithm="giac")
 

output

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

3.2.7.9 Mupad [F(-1)]

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

input

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

output

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