\(\int \frac {(a+b x)^{3/2} (d e+c f+2 d f x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [11]

Optimal result

Integrand size = 49, antiderivative size = 913 \[ \int \frac {(a+b x)^{3/2} (d e+c f+2 d f x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {b (5 a d f h-b (3 d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h^2 \sqrt {a+b x}}+\frac {b \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{h}-\frac {\sqrt {b e-a f} (d g-c h) (5 a d f h-b (3 d f g+d e h+c f h)) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right )|\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{2 d f \sqrt {d e-c f} h^2 \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}-\frac {(b c-a d) \sqrt {b e-a f} (3 a d f h-b (d f g+d e h+c f h)) \sqrt {e+f x} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{2 b d f \sqrt {d e-c f} h \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}-\frac {\sqrt {d e-c f} \left (6 a b d^2 f^2 g h-3 a^2 d^2 f^2 h^2+b^2 \left (2 c d e f h^2-c^2 f^2 h^2-d^2 \left (3 f^2 g^2+e^2 h^2\right )\right )\right ) (a+b x) \sqrt {-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}} \sqrt {-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}} \operatorname {EllipticPi}\left (\frac {b (d e-c f)}{d (b e-a f)},\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {d e-c f} \sqrt {a+b x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{2 b d^2 f \sqrt {b e-a f} h^2 \sqrt {e+f x} \sqrt {g+h x}} \] Output:

1/2*b*(5*a*d*f*h-b*(c*f*h+d*e*h+3*d*f*g))*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x 
+g)^(1/2)/d/f/h^2/(b*x+a)^(1/2)+b*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2 
)*(h*x+g)^(1/2)/h-1/2*(-a*f+b*e)^(1/2)*(-c*h+d*g)*(5*a*d*f*h-b*(c*f*h+d*e* 
h+3*d*f*g))*(f*x+e)^(1/2)*(-(-a*d+b*c)*(h*x+g)/(-c*h+d*g)/(b*x+a))^(1/2)*E 
llipticE((-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))/d/f/(-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)-1/2*(-a*d+b* 
c)*(-a*f+b*e)^(1/2)*(3*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)*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/d/f/(-c*f+d*e)^(1/2)/h/(-(-a*d+b*c)*(f*x+e)/(-c*f+d*e) 
/(b*x+a))^(1/2)/(h*x+g)^(1/2)-1/2*(-c*f+d*e)^(1/2)*(6*a*b*d^2*f^2*g*h-3*a^ 
2*d^2*f^2*h^2+b^2*(2*c*d*e*f*h^2-c^2*f^2*h^2-d^2*(e^2*h^2+3*f^2*g^2)))*(b* 
x+a)*(-(-a*d+b*c)*(f*x+e)/(-c*f+d*e)/(b*x+a))^(1/2)*(-(-a*d+b*c)*(h*x+g)/( 
-c*h+d*g)/(b*x+a))^(1/2)*EllipticPi((-a*f+b*e)^(1/2)*(d*x+c)^(1/2)/(-c*f+d 
*e)^(1/2)/(b*x+a)^(1/2),b*(-c*f+d*e)/d/(-a*f+b*e),((-c*f+d*e)*(-a*h+b*g)/( 
-a*f+b*e)/(-c*h+d*g))^(1/2))/b/d^2/f/(-a*f+b*e)^(1/2)/h^2/(f*x+e)^(1/2)/(h 
*x+g)^(1/2)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(15131\) vs. \(2(913)=1826\).

Time = 35.42 (sec) , antiderivative size = 15131, normalized size of antiderivative = 16.57 \[ \int \frac {(a+b x)^{3/2} (d e+c f+2 d f x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \] Input:

Integrate[((a + b*x)^(3/2)*(d*e + c*f + 2*d*f*x))/(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 = 2.48 (sec) , antiderivative size = 892, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.224, Rules used = {2100, 27, 2105, 27, 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.

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

\(\Big \downarrow \) 2100

\(\displaystyle \frac {\int \frac {2 \left (b d f (5 a d f h-b (3 d f g+d e h+c f h)) x^2+2 d f \left (2 d f h a^2-b (d f g-d e h-c f h) a-b^2 (d e g+c f g+c e h)\right ) x+d f \left (2 a^2 (d e+c f) h-b (b c e g+a (d e g+c f g+c e h))\right )\right )}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{4 d f h}+\frac {b \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{h}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2105

\(\displaystyle \frac {\frac {\int \frac {b d f \left (d e g (3 d f g+d e h-c f h) b^2-a f h \left (-f h c^2-d (f g-e h) c+7 d^2 e g\right ) b+a^2 d f (4 d e-c f) h^2-\left (\left (-\left (\left (3 f^2 g^2+e^2 h^2\right ) d^2\right )+2 c e f h^2 d-c^2 f^2 h^2\right ) b^2+6 a d^2 f^2 g h b-3 a^2 d^2 f^2 h^2\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 b d f h}+\frac {(d e-c f) (d g-c h) (5 a d f h-b (c f h+d e h+3 d f g)) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 h}+\frac {d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (5 a d f h-b (c f h+d e h+3 d f g))}{h \sqrt {c+d x}}}{2 d f h}+\frac {b \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{h}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {d e g (3 d f g+d e h-c f h) b^2-a f h \left (-f h c^2-d (f g-e h) c+7 d^2 e g\right ) b+a^2 d f (4 d e-c f) h^2-\left (\left (-\left (\left (3 f^2 g^2+e^2 h^2\right ) d^2\right )+2 c e f h^2 d-c^2 f^2 h^2\right ) b^2+6 a d^2 f^2 g h b-3 a^2 d^2 f^2 h^2\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 h}+\frac {(d e-c f) (d g-c h) (5 a d f h-b (c f h+d e h+3 d f g)) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 h}+\frac {d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (5 a d f h-b (c f h+d e h+3 d f g))}{h \sqrt {c+d x}}}{2 d f h}+\frac {b \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{h}\)

\(\Big \downarrow \) 194

\(\displaystyle \frac {\frac {\int \frac {d e g (3 d f g+d e h-c f h) b^2-a f h \left (-f h c^2-d (f g-e h) c+7 d^2 e g\right ) b+a^2 d f (4 d e-c f) h^2-\left (\left (-\left (\left (3 f^2 g^2+e^2 h^2\right ) d^2\right )+2 c e f h^2 d-c^2 f^2 h^2\right ) b^2+6 a d^2 f^2 g h b-3 a^2 d^2 f^2 h^2\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 h}-\frac {\sqrt {a+b x} (d g-c h) \sqrt {-\frac {(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} (5 a d f h-b (c f h+d e h+3 d f g)) \int \frac {\sqrt {1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}}}{\sqrt {1-\frac {(d g-c h) (e+f x)}{(f g-e h) (c+d x)}}}d\frac {\sqrt {e+f x}}{\sqrt {c+d x}}}{h \sqrt {g+h x} \sqrt {\frac {(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}}+\frac {d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (5 a d f h-b (c f h+d e h+3 d f g))}{h \sqrt {c+d x}}}{2 d f h}+\frac {b \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{h}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {\int \frac {d e g (3 d f g+d e h-c f h) b^2-a f h \left (-f h c^2-d (f g-e h) c+7 d^2 e g\right ) b+a^2 d f (4 d e-c f) h^2-\left (\left (-\left (\left (3 f^2 g^2+e^2 h^2\right ) d^2\right )+2 c e f h^2 d-c^2 f^2 h^2\right ) b^2+6 a d^2 f^2 g h b-3 a^2 d^2 f^2 h^2\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 h}-\frac {\sqrt {a+b x} \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} (5 a d f h-b (c f h+d e h+3 d f g)) E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{h \sqrt {g+h x} \sqrt {\frac {(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}}+\frac {d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (5 a d f h-b (c f h+d e h+3 d f g))}{h \sqrt {c+d x}}}{2 d f h}+\frac {b \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{h}\)

\(\Big \downarrow \) 2101

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

\(\Big \downarrow \) 183

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

\(\Big \downarrow \) 188

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} b}{h}+\frac {-\frac {\sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right ) (5 a d f h-b (3 d f g+d e h+c f h))}{h \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (5 a d f h-b (3 d f g+d e h+c f h))}{h \sqrt {c+d x}}+\frac {-\frac {2 d (b e-a f) (b g-a h) (3 a d f h-b (3 d f g+d e h-c f h)) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \int \frac {1}{\sqrt {\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}+1} \sqrt {1-\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}}d\frac {\sqrt {e+f x}}{\sqrt {a+b x}}}{b (f g-e h) \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac {2 \left (\left (-\left (\left (3 f^2 g^2+e^2 h^2\right ) d^2\right )+2 c e f h^2 d-c^2 f^2 h^2\right ) b^2+6 a d^2 f^2 g h b-3 a^2 d^2 f^2 h^2\right ) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \int \frac {1}{\left (h-\frac {b (g+h x)}{a+b x}\right ) \sqrt {\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}+1} \sqrt {\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}+1}}d\frac {\sqrt {g+h x}}{\sqrt {a+b x}}}{b \sqrt {c+d x} \sqrt {e+f x}}}{2 h}}{2 d f h}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} b}{h}+\frac {-\frac {\sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right ) (5 a d f h-b (3 d f g+d e h+c f h))}{h \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (5 a d f h-b (3 d f g+d e h+c f h))}{h \sqrt {c+d x}}+\frac {-\frac {2 d (b e-a f) \sqrt {b g-a h} (3 a d f h-b (3 d f g+d e h-c f h)) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{b \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac {2 \left (\left (-\left (\left (3 f^2 g^2+e^2 h^2\right ) d^2\right )+2 c e f h^2 d-c^2 f^2 h^2\right ) b^2+6 a d^2 f^2 g h b-3 a^2 d^2 f^2 h^2\right ) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \int \frac {1}{\left (h-\frac {b (g+h x)}{a+b x}\right ) \sqrt {\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}+1} \sqrt {\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}+1}}d\frac {\sqrt {g+h x}}{\sqrt {a+b x}}}{b \sqrt {c+d x} \sqrt {e+f x}}}{2 h}}{2 d f h}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} b}{h}+\frac {-\frac {\sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right ) (5 a d f h-b (3 d f g+d e h+c f h))}{h \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (5 a d f h-b (3 d f g+d e h+c f h))}{h \sqrt {c+d x}}+\frac {-\frac {2 d (b e-a f) \sqrt {b g-a h} (3 a d f h-b (3 d f g+d e h-c f h)) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{b \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac {2 \sqrt {c h-d g} \left (\left (-\left (\left (3 f^2 g^2+e^2 h^2\right ) d^2\right )+2 c e f h^2 d-c^2 f^2 h^2\right ) b^2+6 a d^2 f^2 g h b-3 a^2 d^2 f^2 h^2\right ) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {c h-d g} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{b \sqrt {b c-a d} h \sqrt {c+d x} \sqrt {e+f x}}}{2 h}}{2 d f h}\)

Input:

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

Output:

(b*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/h + ((d*(5*a*d 
*f*h - b*(3*d*f*g + d*e*h + c*f*h))*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h 
*x])/(h*Sqrt[c + d*x]) - (Sqrt[d*g - c*h]*Sqrt[f*g - e*h]*(5*a*d*f*h - b*( 
3*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])/(S 
qrt[f*g - e*h]*Sqrt[c + d*x])], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f)*(d* 
g - c*h))])/(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*d*f*h - b*(3*d*f*g + d 
*e*h - c*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)))]) 
/(b*Sqrt[f*g - e*h]*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e 
*h)*(a + b*x)))]) - (2*Sqrt[-(d*g) + c*h]*(6*a*b*d^2*f^2*g*h - 3*a^2*d^2*f 
^2*h^2 + b^2*(2*c*d*e*f*h^2 - c^2*f^2*h^2 - d^2*(3*f^2*g^2 + e^2*h^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))]*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))])/(b* 
Sqrt[b*c - a*d]*h*Sqrt[c + d*x]*Sqrt[e + f*x]))/(2*h))/(2*d*f*h)
 

Defintions of rubi rules used

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 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 2100

Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (B_.)*(x_)))/(Sqrt[(c_.) + (d_.)*(x 
_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[2 
*b*B*(a + b*x)^(m - 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/(d*f*h*(2 
*m + 1))), x] + Simp[1/(d*f*h*(2*m + 1))   Int[((a + b*x)^(m - 2)/(Sqrt[c + 
 d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[(-b)*B*(a*(d*e*g + c*f*g + c*e*h) 
+ 2*b*c*e*g*(m - 1)) + a^2*A*d*f*h*(2*m + 1) + (2*a*A*b*d*f*h*(2*m + 1) - B 
*(2*a*b*(d*f*g + d*e*h + c*f*h) + b^2*(d*e*g + c*f*g + c*e*h)*(2*m - 1) - a 
^2*d*f*h*(2*m + 1)))*x + b*(A*b*d*f*h*(2*m + 1) - B*(2*b*(d*f*g + d*e*h + c 
*f*h)*m - a*d*f*h*(4*m - 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
, h, A, B}, x] && IntegerQ[2*m] && GtQ[m, 1]
 

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 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. \(1808\) vs. \(2(832)=1664\).

Time = 19.38 (sec) , antiderivative size = 1809, normalized size of antiderivative = 1.98

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1809\)
default	\(\text {Expression too large to display}\)	\(37880\)

Input:

int((b*x+a)^(3/2)*(2*d*f*x+c*f+d*e)/(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)*(b/h*(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* 
c*f+a^2*d*e-b/h*(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)*Ell 
ipticF(((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^2*d*f+2*a*b*c*f+2*a*b*d*e-b/h*(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)/(-c/d+g/h)/( 
h*d*b*f*(x+g/h)*(x+c/d)*(x+a/b)*(x+e/f))^(1/2)*(-c/d*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))+(c/d-g/h)*EllipticPi(((c/d-e/f)*(x+g/h)/(-e/f+g/h)/(x+c/d))^(1/ 
2),(-e/f+g/h)/(c/d-e/f),((-c/d+a/b)*(e/f-g/h)/(a/b-g/h)/(-c/d+e/f))^(1/2)) 
)+(4*d*f*a*b+b^2*c*f+b^2*d*e-b/h*(3/2*a*d*f*h+3/2*b*c*f*h+3/2*b*d*e*h+3/2* 
b*g*d*f))*((x+g/h)*(x+a/b)*(x+e/f)+(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)...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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