3.1.16 \(\int \frac {(a+b x) (a b B-a^2 C+b^2 B x+b^2 C x^2)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [16]

3.1.16.1 Optimal result

Integrand size = 58, antiderivative size = 721 \[ \int \frac {(a+b x) \left (a b B-a^2 C+b^2 B x+b^2 C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 b^2 (5 b B d f h+2 C (a d f h-2 b (d f g+d e h+c f h))) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{15 d^2 f^2 h^2}+\frac {2 b^2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}-\frac {2 b \sqrt {-d e+c f} \left (15 a^2 C d^2 f^2 h^2-10 a b d f h (3 B d f h-C (d f g+d e h+c f h))+b^2 \left (10 B d f h (d f g+d e h+c f h)-C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 \sqrt {-d e+c f} \left (15 a^3 C d^2 f^2 h^3-15 a^2 b d^2 f^2 h^2 (C g+B h)+5 a b^2 d f h (6 B d f g h-C (c h (f g-e h)+d g (2 f g+e h)))-b^3 \left (5 B d f h (c h (f g-e h)+d g (2 f g+e h))-C \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{15 d^3 f^{5/2} h^3 \sqrt {e+f x} \sqrt {g+h x}} \]

output

2/15*b^2*(5*b*B*d*f*h+2*C*(a*d*f*h-2*b*(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+2/5*b^2*C*(b*x+a)*(d*x+c)^(1/2)*( 
f*x+e)^(1/2)*(h*x+g)^(1/2)/d/f/h-2/15*b*(15*a^2*C*d^2*f^2*h^2-10*a*b*d*f*h 
*(3*B*d*f*h-C*(c*f*h+d*e*h+d*f*g))+b^2*(10*B*d*f*h*(c*f*h+d*e*h+d*f*g)-C*( 
8*c^2*f^2*h^2+7*c*d*f*h*(e*h+f*g)+d^2*(8*e^2*h^2+7*e*f*g*h+8*f^2*g^2))))*E 
llipticE(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),((-c*f+d*e)*h/f/(-c*h+d*g)) 
^(1/2))*(c*f-d*e)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*(h*x+g)^(1/2)/d^3/f^( 
5/2)/h^3/(f*x+e)^(1/2)/(d*(h*x+g)/(-c*h+d*g))^(1/2)-2/15*(15*a^3*C*d^2*f^2 
*h^3-15*a^2*b*d^2*f^2*h^2*(B*h+C*g)+5*a*b^2*d*f*h*(6*B*d*f*g*h-C*(c*h*(-e* 
h+f*g)+d*g*(e*h+2*f*g)))-b^3*(5*B*d*f*h*(c*h*(-e*h+f*g)+d*g*(e*h+2*f*g))-C 
*(4*c^2*f*h^2*(-e*h+f*g)+c*d*h*(-4*e^2*h^2+e*f*g*h+3*f^2*g^2)+d^2*g*(4*e^2 
*h^2+3*e*f*g*h+8*f^2*g^2))))*EllipticF(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/ 
2),((-c*f+d*e)*h/f/(-c*h+d*g))^(1/2))*(c*f-d*e)^(1/2)*(d*(f*x+e)/(-c*f+d*e 
))^(1/2)*(d*(h*x+g)/(-c*h+d*g))^(1/2)/d^3/f^(5/2)/h^3/(f*x+e)^(1/2)/(h*x+g 
)^(1/2)
 

3.1.16.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 28.44 (sec) , antiderivative size = 825, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b x) \left (a b B-a^2 C+b^2 B x+b^2 C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \left (b d^2 \sqrt {-c+\frac {d e}{f}} \left (15 a^2 C d^2 f^2 h^2+10 a b d f h (-3 B d f h+C (d f g+d e h+c f h))-b^2 \left (-10 B d f h (d f g+d e h+c f h)+C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )\right ) (e+f x) (g+h x)+b^2 d^2 \sqrt {-c+\frac {d e}{f}} f h (c+d x) (e+f x) (g+h x) (-5 b B d f h-5 a C d f h+b C (4 c f h+d (4 f g+4 e h-3 f h x)))+i b (d e-c f) h \left (15 a^2 C d^2 f^2 h^2+10 a b d f h (-3 B d f h+C (d f g+d e h+c f h))-b^2 \left (-10 B d f h (d f g+d e h+c f h)+C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )+i d h \left (15 a^3 C d^2 f^3 h^2-15 a^2 b d^2 f^2 (C e+B f) h^2-5 a b^2 d f h (-6 B d e f h+c C f (-f g+e h)+C d e (f g+2 e h))+b^3 \left (-5 B d f h (c f (-f g+e h)+d e (f g+2 e h))+C \left (4 c^2 f^2 h (-f g+e h)+c d f \left (-4 f^2 g^2+e f g h+3 e^2 h^2\right )+d^2 e \left (4 f^2 g^2+3 e f g h+8 e^2 h^2\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )\right )}{15 d^4 \sqrt {-c+\frac {d e}{f}} f^3 h^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \]

input

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

output

(-2*(b*d^2*Sqrt[-c + (d*e)/f]*(15*a^2*C*d^2*f^2*h^2 + 10*a*b*d*f*h*(-3*B*d 
*f*h + C*(d*f*g + d*e*h + c*f*h)) - b^2*(-10*B*d*f*h*(d*f*g + d*e*h + c*f* 
h) + C*(8*c^2*f^2*h^2 + 7*c*d*f*h*(f*g + e*h) + d^2*(8*f^2*g^2 + 7*e*f*g*h 
 + 8*e^2*h^2))))*(e + f*x)*(g + h*x) + b^2*d^2*Sqrt[-c + (d*e)/f]*f*h*(c + 
 d*x)*(e + f*x)*(g + h*x)*(-5*b*B*d*f*h - 5*a*C*d*f*h + b*C*(4*c*f*h + d*( 
4*f*g + 4*e*h - 3*f*h*x))) + I*b*(d*e - c*f)*h*(15*a^2*C*d^2*f^2*h^2 + 10* 
a*b*d*f*h*(-3*B*d*f*h + C*(d*f*g + d*e*h + c*f*h)) - b^2*(-10*B*d*f*h*(d*f 
*g + d*e*h + c*f*h) + C*(8*c^2*f^2*h^2 + 7*c*d*f*h*(f*g + e*h) + d^2*(8*f^ 
2*g^2 + 7*e*f*g*h + 8*e^2*h^2))))*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c 
 + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + 
(d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)] + I*d*h*(15*a^3* 
C*d^2*f^3*h^2 - 15*a^2*b*d^2*f^2*(C*e + B*f)*h^2 - 5*a*b^2*d*f*h*(-6*B*d*e 
*f*h + c*C*f*(-(f*g) + e*h) + C*d*e*(f*g + 2*e*h)) + b^3*(-5*B*d*f*h*(c*f* 
(-(f*g) + e*h) + d*e*(f*g + 2*e*h)) + C*(4*c^2*f^2*h*(-(f*g) + e*h) + c*d* 
f*(-4*f^2*g^2 + e*f*g*h + 3*e^2*h^2) + d^2*e*(4*f^2*g^2 + 3*e*f*g*h + 8*e^ 
2*h^2))))*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h 
*x))/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], 
 (d*f*g - c*f*h)/(d*e*h - c*f*h)]))/(15*d^4*Sqrt[-c + (d*e)/f]*f^3*h^3*Sqr 
t[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])
 

3.1.16.3 Rubi [A] (verified)

Time = 1.83 (sec) , antiderivative size = 744, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2004, 2100, 2118, 27, 176, 124, 123, 131, 131, 130}

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.

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

\(\Big \downarrow \) 2004

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

\(\Big \downarrow \) 2100

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

\(\Big \downarrow \) 2118

\(\displaystyle \frac {\frac {2 \int \frac {d \left (-15 C d^2 f^2 h^2 a^3+15 b B d^2 f^2 h^2 a^2-5 b^2 C d f h (d e g+c f g+c e h) a-b^3 \left (5 B d f h (d e g+c f g+c e h)-C \left (4 f h (f g+e h) c^2+2 d \left (2 f^2 g^2+3 e f h g+2 e^2 h^2\right ) c+4 d^2 e g (f g+e h)\right )\right )-b \left (\left (10 B d f h (d f g+d e h+c f h)-C \left (\left (8 f^2 g^2+7 e f h g+8 e^2 h^2\right ) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right )\right ) b^2-10 a d f h (3 B d f h-C (d f g+d e h+c f h)) b+15 a^2 C d^2 f^2 h^2\right ) x\right )}{2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{3 d^2 f h}+\frac {2 b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (2 a C d f h+5 b B d f h-4 b C (c f h+d e h+d f g))}{3 d f h}}{5 d f h}+\frac {2 b^2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {-15 C d^2 f^2 h^2 a^3+15 b B d^2 f^2 h^2 a^2-5 b^2 C d f h (d e g+c f g+c e h) a-b^3 \left (5 B d f h (d e g+c f g+c e h)-C \left (4 f h (f g+e h) c^2+2 d \left (2 f^2 g^2+3 e f h g+2 e^2 h^2\right ) c+4 d^2 e g (f g+e h)\right )\right )-b \left (\left (10 B d f h (d f g+d e h+c f h)-C \left (\left (8 f^2 g^2+7 e f h g+8 e^2 h^2\right ) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right )\right ) b^2-10 a d f h (3 B d f h-C (d f g+d e h+c f h)) b+15 a^2 C d^2 f^2 h^2\right ) x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{3 d f h}+\frac {2 b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (2 a C d f h+5 b B d f h-4 b C (c f h+d e h+d f g))}{3 d f h}}{5 d f h}+\frac {2 b^2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {\frac {-\frac {b \left (15 a^2 C d^2 f^2 h^2-10 a b d f h (3 B d f h-C (c f h+d e h+d f g))+b^2 \left (10 B d f h (c f h+d e h+d f g)-C \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )\right ) \int \frac {\sqrt {g+h x}}{\sqrt {c+d x} \sqrt {e+f x}}dx}{h}-\frac {\left (15 a^3 C d^2 f^2 h^3-15 a^2 b d^2 f^2 h^2 (B h+C g)+5 a b^2 d f h (6 B d f g h-c C h (f g-e h)-C d g (e h+2 f g))-\left (b^3 \left (5 B d f h (c h (f g-e h)+d g (e h+2 f g))-C \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{h}}{3 d f h}+\frac {2 b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (2 a C d f h+5 b B d f h-4 b C (c f h+d e h+d f g))}{3 d f h}}{5 d f h}+\frac {2 b^2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {\frac {-\frac {b \sqrt {g+h x} \sqrt {\frac {d (e+f x)}{d e-c f}} \left (15 a^2 C d^2 f^2 h^2-10 a b d f h (3 B d f h-C (c f h+d e h+d f g))+b^2 \left (10 B d f h (c f h+d e h+d f g)-C \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )\right ) \int \frac {\sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {\left (15 a^3 C d^2 f^2 h^3-15 a^2 b d^2 f^2 h^2 (B h+C g)+5 a b^2 d f h (6 B d f g h-c C h (f g-e h)-C d g (e h+2 f g))-\left (b^3 \left (5 B d f h (c h (f g-e h)+d g (e h+2 f g))-C \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{h}}{3 d f h}+\frac {2 b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (2 a C d f h+5 b B d f h-4 b C (c f h+d e h+d f g))}{3 d f h}}{5 d f h}+\frac {2 b^2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {\frac {-\frac {\left (15 a^3 C d^2 f^2 h^3-15 a^2 b d^2 f^2 h^2 (B h+C g)+5 a b^2 d f h (6 B d f g h-c C h (f g-e h)-C d g (e h+2 f g))-\left (b^3 \left (5 B d f h (c h (f g-e h)+d g (e h+2 f g))-C \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{h}-\frac {2 b \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right ) \left (15 a^2 C d^2 f^2 h^2-10 a b d f h (3 B d f h-C (c f h+d e h+d f g))+b^2 \left (10 B d f h (c f h+d e h+d f g)-C \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{3 d f h}+\frac {2 b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (2 a C d f h+5 b B d f h-4 b C (c f h+d e h+d f g))}{3 d f h}}{5 d f h}+\frac {2 b^2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {\frac {-\frac {\sqrt {\frac {d (e+f x)}{d e-c f}} \left (15 a^3 C d^2 f^2 h^3-15 a^2 b d^2 f^2 h^2 (B h+C g)+5 a b^2 d f h (6 B d f g h-c C h (f g-e h)-C d g (e h+2 f g))-\left (b^3 \left (5 B d f h (c h (f g-e h)+d g (e h+2 f g))-C \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {g+h x}}dx}{h \sqrt {e+f x}}-\frac {2 b \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right ) \left (15 a^2 C d^2 f^2 h^2-10 a b d f h (3 B d f h-C (c f h+d e h+d f g))+b^2 \left (10 B d f h (c f h+d e h+d f g)-C \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{3 d f h}+\frac {2 b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (2 a C d f h+5 b B d f h-4 b C (c f h+d e h+d f g))}{3 d f h}}{5 d f h}+\frac {2 b^2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {\frac {-\frac {\sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (15 a^3 C d^2 f^2 h^3-15 a^2 b d^2 f^2 h^2 (B h+C g)+5 a b^2 d f h (6 B d f g h-c C h (f g-e h)-C d g (e h+2 f g))-\left (b^3 \left (5 B d f h (c h (f g-e h)+d g (e h+2 f g))-C \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}dx}{h \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 b \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right ) \left (15 a^2 C d^2 f^2 h^2-10 a b d f h (3 B d f h-C (c f h+d e h+d f g))+b^2 \left (10 B d f h (c f h+d e h+d f g)-C \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{3 d f h}+\frac {2 b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (2 a C d f h+5 b B d f h-4 b C (c f h+d e h+d f g))}{3 d f h}}{5 d f h}+\frac {2 b^2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}\)

\(\Big \downarrow \) 130

\(\displaystyle \frac {\frac {-\frac {2 b \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right ) \left (15 a^2 C d^2 f^2 h^2-10 a b d f h (3 B d f h-C (c f h+d e h+d f g))+b^2 \left (10 B d f h (c f h+d e h+d f g)-C \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right ) \left (15 a^3 C d^2 f^2 h^3-15 a^2 b d^2 f^2 h^2 (B h+C g)+5 a b^2 d f h (6 B d f g h-c C h (f g-e h)-C d g (e h+2 f g))-\left (b^3 \left (5 B d f h (c h (f g-e h)+d g (e h+2 f g))-C \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right )\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}}}{3 d f h}+\frac {2 b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (2 a C d f h+5 b B d f h-4 b C (c f h+d e h+d f g))}{3 d f h}}{5 d f h}+\frac {2 b^2 C (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}\)

input

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

output

(2*b^2*C*(a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(5*d*f*h) + 
((2*b^2*(5*b*B*d*f*h + 2*a*C*d*f*h - 4*b*C*(d*f*g + d*e*h + c*f*h))*Sqrt[c 
 + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*d*f*h) + ((-2*b*Sqrt[-(d*e) + c*f] 
*(15*a^2*C*d^2*f^2*h^2 - 10*a*b*d*f*h*(3*B*d*f*h - C*(d*f*g + d*e*h + c*f* 
h)) + b^2*(10*B*d*f*h*(d*f*g + d*e*h + c*f*h) - C*(8*c^2*f^2*h^2 + 7*c*d*f 
*h*(f*g + e*h) + d^2*(8*f^2*g^2 + 7*e*f*g*h + 8*e^2*h^2))))*Sqrt[(d*(e + f 
*x))/(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/S 
qrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(d*Sqrt[f]*h*Sqrt[e 
+ f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) - (2*Sqrt[-(d*e) + c*f]*(15*a^3*C* 
d^2*f^2*h^3 - 15*a^2*b*d^2*f^2*h^2*(C*g + B*h) + 5*a*b^2*d*f*h*(6*B*d*f*g* 
h - c*C*h*(f*g - e*h) - C*d*g*(2*f*g + e*h)) - b^3*(5*B*d*f*h*(c*h*(f*g - 
e*h) + d*g*(2*f*g + e*h)) - C*(4*c^2*f*h^2*(f*g - e*h) + c*d*h*(3*f^2*g^2 
+ e*f*g*h - 4*e^2*h^2) + d^2*g*(8*f^2*g^2 + 3*e*f*g*h + 4*e^2*h^2))))*Sqrt 
[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticF[ArcS 
in[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - 
c*h))])/(d*Sqrt[f]*h*Sqrt[e + f*x]*Sqrt[g + h*x]))/(3*d*f*h))/(5*d*f*h)
 

3.1.16.3.1 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 123

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] 
/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, 
b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !L 
tQ[-(b*c - a*d)/d, 0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d 
), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])
 

rule 124

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d 
*x]*Sqrt[b*((e + f*x)/(b*e - a*f))]))   Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x 
/(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] 
), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && Gt 
Q[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]
 

rule 130

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x 
_)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ 
Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - 
a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ 
[b/(b*e - a*f), 0] && SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f 
*x] && (PosQ[-(b*c - a*d)/d] || NegQ[-(b*e - a*f)/f])
 

rule 131

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x 
_)]), x_] :> Simp[Sqrt[b*((c + d*x)/(b*c - a*d))]/Sqrt[c + d*x]   Int[1/(Sq 
rt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e + f*x]), x 
], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && Simpler 
Q[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x]
 

rule 176

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

rule 2004

Int[(u_)*((d_) + (e_.)*(x_))^(q_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.) 
, x_Symbol] :> Int[u*(d + e*x)^(p + q)*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, b 
, c, d, e, q}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
 

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 2118

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f 
_.)*(x_))^(p_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expo 
n[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 
1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Simp[1/(d*f*b^q*(m + n + p + 
q + 1))   Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + 
n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q 
- 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + 
 c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m 
 + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] && PolyQ[Px, x]
 

3.1.16.4 Maple [A] (verified)

Time = 2.50 (sec) , antiderivative size = 880, normalized size of antiderivative = 1.22

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {2 C \,b^{3} x \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}{5 d f h}+\frac {2 \left (B \,b^{3}+C \,b^{2} a -\frac {2 C \,b^{3} \left (2 c f h +2 d e h +2 d f g \right )}{5 d f h}\right ) \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}{3 d f h}+\frac {2 \left (a^{2} b B -C \,a^{3}-\frac {2 C \,b^{3} c e g}{5 d f h}-\frac {2 \left (B \,b^{3}+C \,b^{2} a -\frac {2 C \,b^{3} \left (2 c f h +2 d e h +2 d f g \right )}{5 d f h}\right ) \left (\frac {1}{2} c e h +\frac {1}{2} c f g +\frac {1}{2} d e g \right )}{3 d f h}\right ) \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, F\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}+\frac {2 \left (2 a \,b^{2} B -C \,a^{2} b -\frac {2 C \,b^{3} \left (\frac {3}{2} c e h +\frac {3}{2} c f g +\frac {3}{2} d e g \right )}{5 d f h}-\frac {2 \left (B \,b^{3}+C \,b^{2} a -\frac {2 C \,b^{3} \left (2 c f h +2 d e h +2 d f g \right )}{5 d f h}\right ) \left (c f h +d e h +d f g \right )}{3 d f h}\right ) \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \left (\left (-\frac {g}{h}+\frac {c}{d}\right ) E\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )-\frac {c F\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{d}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}\right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\)	\(880\)
default	\(\text {Expression too large to display}\)	\(8421\)

input

int((b*x+a)*(C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h 
*x+g)^(1/2),x,method=_RETURNVERBOSE)
 

output

((d*x+c)*(f*x+e)*(h*x+g))^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)* 
(2/5*C*b^3/d/f/h*x*(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g* 
x+d*e*g*x+c*e*g)^(1/2)+2/3*(B*b^3+C*b^2*a-2/5*C*b^3/d/f/h*(2*c*f*h+2*d*e*h 
+2*d*f*g))/d/f/h*(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+ 
d*e*g*x+c*e*g)^(1/2)+2*(a^2*b*B-C*a^3-2/5*C*b^3/d/f/h*c*e*g-2/3*(B*b^3+C*b 
^2*a-2/5*C*b^3/d/f/h*(2*c*f*h+2*d*e*h+2*d*f*g))/d/f/h*(1/2*c*e*h+1/2*c*f*g 
+1/2*d*e*g))*(g/h-e/f)*((x+g/h)/(g/h-e/f))^(1/2)*((x+c/d)/(-g/h+c/d))^(1/2 
)*((x+e/f)/(-g/h+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e* 
h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)*EllipticF(((x+g/h)/(g/h-e/f))^(1/2),((-g/ 
h+e/f)/(-g/h+c/d))^(1/2))+2*(2*a*b^2*B-C*a^2*b-2/5*C*b^3/d/f/h*(3/2*c*e*h+ 
3/2*c*f*g+3/2*d*e*g)-2/3*(B*b^3+C*b^2*a-2/5*C*b^3/d/f/h*(2*c*f*h+2*d*e*h+2 
*d*f*g))/d/f/h*(c*f*h+d*e*h+d*f*g))*(g/h-e/f)*((x+g/h)/(g/h-e/f))^(1/2)*(( 
x+c/d)/(-g/h+c/d))^(1/2)*((x+e/f)/(-g/h+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+d 
*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)*((-g/h+c/d)*Ellipt 
icE(((x+g/h)/(g/h-e/f))^(1/2),((-g/h+e/f)/(-g/h+c/d))^(1/2))-c/d*EllipticF 
(((x+g/h)/(g/h-e/f))^(1/2),((-g/h+e/f)/(-g/h+c/d))^(1/2))))
 

3.1.16.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.16 (sec) , antiderivative size = 1267, normalized size of antiderivative = 1.76 \[ \int \frac {(a+b x) \left (a b B-a^2 C+b^2 B x+b^2 C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Too large to display} \]

input

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

output

2/45*(3*(3*C*b^3*d^3*f^3*h^3*x - 4*C*b^3*d^3*f^3*g*h^2 - (4*C*b^3*d^3*e*f^ 
2 + (4*C*b^3*c*d^2 - 5*(C*a*b^2 + B*b^3)*d^3)*f^3)*h^3)*sqrt(d*x + c)*sqrt 
(f*x + e)*sqrt(h*x + g) - (8*C*b^3*d^3*f^3*g^3 + (3*C*b^3*d^3*e*f^2 + (3*C 
*b^3*c*d^2 - 10*(C*a*b^2 + B*b^3)*d^3)*f^3)*g^2*h + (3*C*b^3*d^3*e^2*f + ( 
3*C*b^3*c*d^2 - 5*(C*a*b^2 + B*b^3)*d^3)*e*f^2 + (3*C*b^3*c^2*d - 5*(C*a*b 
^2 + B*b^3)*c*d^2 - 15*(C*a^2*b - 2*B*a*b^2)*d^3)*f^3)*g*h^2 + (8*C*b^3*d^ 
3*e^3 + (3*C*b^3*c*d^2 - 10*(C*a*b^2 + B*b^3)*d^3)*e^2*f + (3*C*b^3*c^2*d 
- 5*(C*a*b^2 + B*b^3)*c*d^2 - 15*(C*a^2*b - 2*B*a*b^2)*d^3)*e*f^2 + (8*C*b 
^3*c^3 - 10*(C*a*b^2 + B*b^3)*c^2*d - 15*(C*a^2*b - 2*B*a*b^2)*c*d^2 + 45* 
(C*a^3 - B*a^2*b)*d^3)*f^3)*h^3)*sqrt(d*f*h)*weierstrassPInverse(4/3*(d^2* 
f^2*g^2 - (d^2*e*f + c*d*f^2)*g*h + (d^2*e^2 - c*d*e*f + c^2*f^2)*h^2)/(d^ 
2*f^2*h^2), -4/27*(2*d^3*f^3*g^3 - 3*(d^3*e*f^2 + c*d^2*f^3)*g^2*h - 3*(d^ 
3*e^2*f - 4*c*d^2*e*f^2 + c^2*d*f^3)*g*h^2 + (2*d^3*e^3 - 3*c*d^2*e^2*f - 
3*c^2*d*e*f^2 + 2*c^3*f^3)*h^3)/(d^3*f^3*h^3), 1/3*(3*d*f*h*x + d*f*g + (d 
*e + c*f)*h)/(d*f*h)) - 3*(8*C*b^3*d^3*f^3*g^2*h + (7*C*b^3*d^3*e*f^2 + (7 
*C*b^3*c*d^2 - 10*(C*a*b^2 + B*b^3)*d^3)*f^3)*g*h^2 + (8*C*b^3*d^3*e^2*f + 
 (7*C*b^3*c*d^2 - 10*(C*a*b^2 + B*b^3)*d^3)*e*f^2 + (8*C*b^3*c^2*d - 10*(C 
*a*b^2 + B*b^3)*c*d^2 - 15*(C*a^2*b - 2*B*a*b^2)*d^3)*f^3)*h^3)*sqrt(d*f*h 
)*weierstrassZeta(4/3*(d^2*f^2*g^2 - (d^2*e*f + c*d*f^2)*g*h + (d^2*e^2 - 
c*d*e*f + c^2*f^2)*h^2)/(d^2*f^2*h^2), -4/27*(2*d^3*f^3*g^3 - 3*(d^3*e*...
 

3.1.16.6 Sympy [F]

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

input

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

output

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

3.1.16.7 Maxima [F]

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

input

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

output

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

3.1.16.8 Giac [F]

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

input

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

output

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

3.1.16.9 Mupad [F(-1)]

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

input

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

output

\text{Hanged}