\(\int \frac {(a+b x)^2 (A+C x^2)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [27]

Optimal result

Integrand size = 42, antiderivative size = 1084 \[ \int \frac {(a+b x)^2 \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 \left (8 a^2 C d f h-38 a b C (d f g+d e h+c f h)+b^2 \left (35 A d f h+C \left (\frac {24 c^2 f h}{d}+23 c (f g+e h)+d \left (23 e g+\frac {24 f g^2}{h}+\frac {24 e^2 h}{f}\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{105 d^2 f^2 h^2}+\frac {4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{35 d^2 f^2 h^2}+\frac {2 C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{7 d f h}-\frac {4 \sqrt {-d e+c f} \left (35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)-7 a b d f h \left (15 A d^2 f^2 h^2+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 )+b^2 \left (35 A d^2 f^2 h^2 (d f g+d e h+c f h)+2 C \left (12 c^3 f^3 h^3+10 c^2 d f^2 h^2 (f g+e h)+c d^2 f h \left (10 f^2 g^2+9 e f g h+10 e^2 h^2\right )+2 d^3 \left (6 f^3 g^3+5 e f^2 g^2 h+5 e^2 f g h^2+6 e^3 h^3\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 )}{105 d^4 f^{7/2} h^4 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {2 \sqrt {-d e+c f} \left (35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-14 a b d f h \left (15 A d^2 f^2 g h^2+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 )+b^2 \left (35 A d^2 f^2 h^2 (c h (f g-e h)+d g (2 f g+e h))+C \left (24 c^3 f^2 h^3 (f g-e h)+c^2 d f h^2 \left (17 f^2 g^2+6 e f g h-23 e^2 h^2\right )+2 c d^2 h \left (8 f^3 g^3+e f^2 g^2 h+3 e^2 f g h^2-12 e^3 h^3\right )+d^3 g \left (48 f^3 g^3+16 e f^2 g^2 h+17 e^2 f g h^2+24 e^3 h^3\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 )}{105 d^4 f^{7/2} h^4 \sqrt {e+f x} \sqrt {g+h x}} \] Output:

2/105*(8*a^2*C*d*f*h-38*a*b*C*(c*f*h+d*e*h+d*f*g)+b^2*(35*A*d*f*h+C*(24*c^ 
2*f*h/d+23*c*(e*h+f*g)+d*(23*e*g+24*f*g^2/h+24*e^2*h/f))))*(d*x+c)^(1/2)*( 
f*x+e)^(1/2)*(h*x+g)^(1/2)/d^2/f^2/h^2+4/35*C*(2*a*d*f*h-3*b*(c*f*h+d*e*h+ 
d*f*g))*(b*x+a)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/d^2/f^2/h^2+2/7* 
C*(b*x+a)^2*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/d/f/h-4/105*(c*f-d*e 
)^(1/2)*(35*a^2*C*d^2*f^2*h^2*(c*f*h+d*e*h+d*f*g)-7*a*b*d*f*h*(15*A*d^2*f^ 
2*h^2+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)))+b^2*(35*A*d^2*f^2*h^2*(c*f*h+d*e*h+d*f*g)+2*C*(12*c^3*f^3*h^3+10*c^ 
2*d*f^2*h^2*(e*h+f*g)+c*d^2*f*h*(10*e^2*h^2+9*e*f*g*h+10*f^2*g^2)+2*d^3*(6 
*e^3*h^3+5*e^2*f*g*h^2+5*e*f^2*g^2*h+6*f^3*g^3))))*(d*(f*x+e)/(-c*f+d*e))^ 
(1/2)*(h*x+g)^(1/2)*EllipticE(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))/d^4/f^(7/2)/h^4/(f*x+e)^(1/2)/(d*(h*x+g)/(-c* 
h+d*g))^(1/2)+2/105*(c*f-d*e)^(1/2)*(35*a^2*d^2*f^2*h^2*(3*A*d*f*h^2+c*C*h 
*(-e*h+f*g)+C*d*g*(e*h+2*f*g))-14*a*b*d*f*h*(15*A*d^2*f^2*g*h^2+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)))+b^2*(35*A*d^2*f^2*h^2*(c*h*(-e*h+f*g)+d*g*(e*h+2*f*g))+ 
C*(24*c^3*f^2*h^3*(-e*h+f*g)+c^2*d*f*h^2*(-23*e^2*h^2+6*e*f*g*h+17*f^2*g^2 
)+2*c*d^2*h*(-12*e^3*h^3+3*e^2*f*g*h^2+e*f^2*g^2*h+8*f^3*g^3)+d^3*g*(24*e^ 
3*h^3+17*e^2*f*g*h^2+16*e*f^2*g^2*h+48*f^3*g^3))))*(d*(f*x+e)/(-c*f+d*e))^ 
(1/2)*(d*(h*x+g)/(-c*h+d*g))^(1/2)*EllipticF(f^(1/2)*(d*x+c)^(1/2)/(c*f...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 32.97 (sec) , antiderivative size = 1291, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b x)^2 \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*(-2*d^2*Sqrt[-c + (d*e)/f]*(35*a^2*C*d^2*f^2*h^2*(d*f*g + d*e*h + c*f*h 
) - 7*a*b*d*f*h*(15*A*d^2*f^2*h^2 + 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))) + b^2*(35*A*d^2*f^2*h^2*(d* 
f*g + d*e*h + c*f*h) + 2*C*(12*c^3*f^3*h^3 + 10*c^2*d*f^2*h^2*(f*g + e*h) 
+ c*d^2*f*h*(10*f^2*g^2 + 9*e*f*g*h + 10*e^2*h^2) + 2*d^3*(6*f^3*g^3 + 5*e 
*f^2*g^2*h + 5*e^2*f*g*h^2 + 6*e^3*h^3))))*(e + f*x)*(g + h*x) + d^2*Sqrt[ 
-c + (d*e)/f]*f*h*(c + d*x)*(e + f*x)*(g + h*x)*(35*a^2*C*d^2*f^2*h^2 - 14 
*a*b*C*d*f*h*(4*c*f*h + d*(4*f*g + 4*e*h - 3*f*h*x)) + b^2*(35*A*d^2*f^2*h 
^2 + C*(24*c^2*f^2*h^2 + c*d*f*h*(23*f*g + 23*e*h - 18*f*h*x) + d^2*(24*e^ 
2*h^2 + e*f*h*(23*g - 18*h*x) + 3*f^2*(8*g^2 - 6*g*h*x + 5*h^2*x^2))))) - 
(2*I)*(d*e - c*f)*h*(35*a^2*C*d^2*f^2*h^2*(d*f*g + d*e*h + c*f*h) - 7*a*b* 
d*f*h*(15*A*d^2*f^2*h^2 + 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))) + b^2*(35*A*d^2*f^2*h^2*(d*f*g + d*e* 
h + c*f*h) + 2*C*(12*c^3*f^3*h^3 + 10*c^2*d*f^2*h^2*(f*g + e*h) + c*d^2*f* 
h*(10*f^2*g^2 + 9*e*f*g*h + 10*e^2*h^2) + 2*d^3*(6*f^3*g^3 + 5*e*f^2*g^2*h 
 + 5*e^2*f*g*h^2 + 6*e^3*h^3))))*(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*(35*a^2*d 
^2*f^2*h^2*(3*A*d*f^2*h + c*C*f*(-(f*g) + e*h) + C*d*e*(f*g + 2*e*h)) - 14 
*a*b*d*f*h*(15*A*d^2*e*f^2*h^2 + C*(4*c^2*f^2*h*(-(f*g) + e*h) + c*d*f*...
 

Rubi [A] (verified)

Time = 3.23 (sec) , antiderivative size = 1112, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.262, Rules used = {2104, 25, 2103, 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.

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

\(\Big \downarrow \) 2104

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2103

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

\(\Big \downarrow \) 2118

\(\displaystyle \frac {2 C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{7 d f h}-\frac {\frac {\frac {2 \int -\frac {d \left (-\left (\left (35 A d^2 f^2 (d e g+c f g+c e h) h^2+C \left (24 f^2 h^2 (f g+e h) c^3+d f h \left (23 f^2 g^2+34 e f h g+23 e^2 h^2\right ) c^2+2 d^2 \left (12 f^3 g^3+17 e f^2 h g^2+17 e^2 f h^2 g+12 e^3 h^3\right ) c+d^3 e g \left (24 f^2 g^2+23 e f h g+24 e^2 h^2\right )\right )\right ) b^2\right )+28 a C d f h \left (2 f h (f g+e h) c^2+d \left (2 f^2 g^2+3 e f h g+2 e^2 h^2\right ) c+2 d^2 e g (f g+e h)\right ) b+35 a^2 d^2 f^2 h^2 (3 A d f h-C (d e g+c f g+c e h))-2 \left (\left (35 A d^2 f^2 (d f g+d e h+c f h) h^2+2 C \left (2 \left (6 f^3 g^3+5 e f^2 h g^2+5 e^2 f h^2 g+6 e^3 h^3\right ) d^3+c f h \left (10 f^2 g^2+9 e f h g+10 e^2 h^2\right ) d^2+10 c^2 f^2 h^2 (f g+e h) d+12 c^3 f^3 h^3\right )\right ) b^2-7 a d f h \left (15 A d^2 f^2 h^2+C \left (\left (8 f^2 g^2+7 e f h g+8 e^2 h^2\right ) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right )\right ) b+35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)\right ) x\right )}{2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{3 d^2 f h}-\frac {2}{3} \left (8 C d f h a^2-38 b C (d f g+d e h+c f h) a+\frac {24 b^2 C (d f g+d e h+c f h)^2}{d f h}+35 A b^2 d f h-25 b^2 C (d e g+c f g+c e h)\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}-\frac {4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}}{7 d f h}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{7 d f h}-\frac {\frac {-\frac {2}{3} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (8 C d f h a^2-38 b C (d f g+d e h+c f h) a+\frac {24 b^2 C (d f g+d e h+c f h)^2}{d f h}+35 A b^2 d f h-25 b^2 C (d e g+c f g+c e h)\right )-\frac {\int \frac {-\left (\left (35 A d^2 f^2 (d e g+c f g+c e h) h^2+C \left (24 f^2 h^2 (f g+e h) c^3+d f h \left (23 f^2 g^2+34 e f h g+23 e^2 h^2\right ) c^2+2 d^2 \left (12 f^3 g^3+17 e f^2 h g^2+17 e^2 f h^2 g+12 e^3 h^3\right ) c+d^3 e g \left (24 f^2 g^2+23 e f h g+24 e^2 h^2\right )\right )\right ) b^2\right )+28 a C d f h \left (2 f h (f g+e h) c^2+d \left (2 f^2 g^2+3 e f h g+2 e^2 h^2\right ) c+2 d^2 e g (f g+e h)\right ) b+35 a^2 d^2 f^2 h^2 (3 A d f h-C (d e g+c f g+c e h))-2 \left (\left (35 A d^2 f^2 (d f g+d e h+c f h) h^2+2 C \left (2 \left (6 f^3 g^3+5 e f^2 h g^2+5 e^2 f h^2 g+6 e^3 h^3\right ) d^3+c f h \left (10 f^2 g^2+9 e f h g+10 e^2 h^2\right ) d^2+10 c^2 f^2 h^2 (f g+e h) d+12 c^3 f^3 h^3\right )\right ) b^2-7 a d f h \left (15 A d^2 f^2 h^2+C \left (\left (8 f^2 g^2+7 e f h g+8 e^2 h^2\right ) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right )\right ) b+35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)\right ) x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{3 d f h}}{5 d f h}-\frac {4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}}{7 d f h}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {2 C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{7 d f h}-\frac {\frac {-\frac {2}{3} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (8 C d f h a^2-38 b C (d f g+d e h+c f h) a+\frac {24 b^2 C (d f g+d e h+c f h)^2}{d f h}+35 A b^2 d f h-25 b^2 C (d e g+c f g+c e h)\right )-\frac {\frac {\left (\left (35 A d^2 f^2 (c h (f g-e h)+d g (2 f g+e h)) h^2+C \left (g \left (48 f^3 g^3+16 e f^2 h g^2+17 e^2 f h^2 g+24 e^3 h^3\right ) d^3+2 c h \left (8 f^3 g^3+e f^2 h g^2+3 e^2 f h^2 g-12 e^3 h^3\right ) d^2+c^2 f h^2 \left (17 f^2 g^2+6 e f h g-23 e^2 h^2\right ) d+24 c^3 f^2 h^3 (f g-e h)\right )\right ) b^2-14 a d f h \left (15 A d^2 f^2 g h^2+C \left (g \left (8 f^2 g^2+3 e f h g+4 e^2 h^2\right ) d^2+c h \left (3 f^2 g^2+e f h g-4 e^2 h^2\right ) d+4 c^2 f h^2 (f g-e h)\right )\right ) b+35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C (f g-e h) h+C d g (2 f g+e h)\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{h}-\frac {2 \left (\left (35 A d^2 f^2 (d f g+d e h+c f h) h^2+2 C \left (2 \left (6 f^3 g^3+5 e f^2 h g^2+5 e^2 f h^2 g+6 e^3 h^3\right ) d^3+c f h \left (10 f^2 g^2+9 e f h g+10 e^2 h^2\right ) d^2+10 c^2 f^2 h^2 (f g+e h) d+12 c^3 f^3 h^3\right )\right ) b^2-7 a d f h \left (15 A d^2 f^2 h^2+C \left (\left (8 f^2 g^2+7 e f h g+8 e^2 h^2\right ) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right )\right ) b+35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)\right ) \int \frac {\sqrt {g+h x}}{\sqrt {c+d x} \sqrt {e+f x}}dx}{h}}{3 d f h}}{5 d f h}-\frac {4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}}{7 d f h}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {2 C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{7 d f h}-\frac {\frac {-\frac {2}{3} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (8 C d f h a^2-38 b C (d f g+d e h+c f h) a+\frac {24 b^2 C (d f g+d e h+c f h)^2}{d f h}+35 A b^2 d f h-25 b^2 C (d e g+c f g+c e h)\right )-\frac {\frac {\left (\left (35 A d^2 f^2 (c h (f g-e h)+d g (2 f g+e h)) h^2+C \left (g \left (48 f^3 g^3+16 e f^2 h g^2+17 e^2 f h^2 g+24 e^3 h^3\right ) d^3+2 c h \left (8 f^3 g^3+e f^2 h g^2+3 e^2 f h^2 g-12 e^3 h^3\right ) d^2+c^2 f h^2 \left (17 f^2 g^2+6 e f h g-23 e^2 h^2\right ) d+24 c^3 f^2 h^3 (f g-e h)\right )\right ) b^2-14 a d f h \left (15 A d^2 f^2 g h^2+C \left (g \left (8 f^2 g^2+3 e f h g+4 e^2 h^2\right ) d^2+c h \left (3 f^2 g^2+e f h g-4 e^2 h^2\right ) d+4 c^2 f h^2 (f g-e h)\right )\right ) b+35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C (f g-e h) h+C d g (2 f g+e h)\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{h}-\frac {2 \left (\left (35 A d^2 f^2 (d f g+d e h+c f h) h^2+2 C \left (2 \left (6 f^3 g^3+5 e f^2 h g^2+5 e^2 f h^2 g+6 e^3 h^3\right ) d^3+c f h \left (10 f^2 g^2+9 e f h g+10 e^2 h^2\right ) d^2+10 c^2 f^2 h^2 (f g+e h) d+12 c^3 f^3 h^3\right )\right ) b^2-7 a d f h \left (15 A d^2 f^2 h^2+C \left (\left (8 f^2 g^2+7 e f h g+8 e^2 h^2\right ) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right )\right ) b+35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} \int \frac {\sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{3 d f h}}{5 d f h}-\frac {4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}}{7 d f h}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {2 C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{7 d f h}-\frac {\frac {-\frac {2}{3} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (8 C d f h a^2-38 b C (d f g+d e h+c f h) a+\frac {24 b^2 C (d f g+d e h+c f h)^2}{d f h}+35 A b^2 d f h-25 b^2 C (d e g+c f g+c e h)\right )-\frac {\frac {\left (\left (35 A d^2 f^2 (c h (f g-e h)+d g (2 f g+e h)) h^2+C \left (g \left (48 f^3 g^3+16 e f^2 h g^2+17 e^2 f h^2 g+24 e^3 h^3\right ) d^3+2 c h \left (8 f^3 g^3+e f^2 h g^2+3 e^2 f h^2 g-12 e^3 h^3\right ) d^2+c^2 f h^2 \left (17 f^2 g^2+6 e f h g-23 e^2 h^2\right ) d+24 c^3 f^2 h^3 (f g-e h)\right )\right ) b^2-14 a d f h \left (15 A d^2 f^2 g h^2+C \left (g \left (8 f^2 g^2+3 e f h g+4 e^2 h^2\right ) d^2+c h \left (3 f^2 g^2+e f h g-4 e^2 h^2\right ) d+4 c^2 f h^2 (f g-e h)\right )\right ) b+35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C (f g-e h) h+C d g (2 f g+e h)\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{h}-\frac {4 \sqrt {c f-d e} \left (\left (35 A d^2 f^2 (d f g+d e h+c f h) h^2+2 C \left (2 \left (6 f^3 g^3+5 e f^2 h g^2+5 e^2 f h^2 g+6 e^3 h^3\right ) d^3+c f h \left (10 f^2 g^2+9 e f h g+10 e^2 h^2\right ) d^2+10 c^2 f^2 h^2 (f g+e h) d+12 c^3 f^3 h^3\right )\right ) b^2-7 a d f h \left (15 A d^2 f^2 h^2+C \left (\left (8 f^2 g^2+7 e f h g+8 e^2 h^2\right ) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right )\right ) b+35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{3 d f h}}{5 d f h}-\frac {4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}}{7 d f h}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {2 C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{7 d f h}-\frac {\frac {-\frac {2}{3} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (8 C d f h a^2-38 b C (d f g+d e h+c f h) a+\frac {24 b^2 C (d f g+d e h+c f h)^2}{d f h}+35 A b^2 d f h-25 b^2 C (d e g+c f g+c e h)\right )-\frac {\frac {\left (\left (35 A d^2 f^2 (c h (f g-e h)+d g (2 f g+e h)) h^2+C \left (g \left (48 f^3 g^3+16 e f^2 h g^2+17 e^2 f h^2 g+24 e^3 h^3\right ) d^3+2 c h \left (8 f^3 g^3+e f^2 h g^2+3 e^2 f h^2 g-12 e^3 h^3\right ) d^2+c^2 f h^2 \left (17 f^2 g^2+6 e f h g-23 e^2 h^2\right ) d+24 c^3 f^2 h^3 (f g-e h)\right )\right ) b^2-14 a d f h \left (15 A d^2 f^2 g h^2+C \left (g \left (8 f^2 g^2+3 e f h g+4 e^2 h^2\right ) d^2+c h \left (3 f^2 g^2+e f h g-4 e^2 h^2\right ) d+4 c^2 f h^2 (f g-e h)\right )\right ) b+35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C (f g-e h) h+C d g (2 f g+e h)\right )\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {g+h x}}dx}{h \sqrt {e+f x}}-\frac {4 \sqrt {c f-d e} \left (\left (35 A d^2 f^2 (d f g+d e h+c f h) h^2+2 C \left (2 \left (6 f^3 g^3+5 e f^2 h g^2+5 e^2 f h^2 g+6 e^3 h^3\right ) d^3+c f h \left (10 f^2 g^2+9 e f h g+10 e^2 h^2\right ) d^2+10 c^2 f^2 h^2 (f g+e h) d+12 c^3 f^3 h^3\right )\right ) b^2-7 a d f h \left (15 A d^2 f^2 h^2+C \left (\left (8 f^2 g^2+7 e f h g+8 e^2 h^2\right ) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right )\right ) b+35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{3 d f h}}{5 d f h}-\frac {4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}}{7 d f h}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {2 C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{7 d f h}-\frac {\frac {-\frac {2}{3} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (8 C d f h a^2-38 b C (d f g+d e h+c f h) a+\frac {24 b^2 C (d f g+d e h+c f h)^2}{d f h}+35 A b^2 d f h-25 b^2 C (d e g+c f g+c e h)\right )-\frac {\frac {\left (\left (35 A d^2 f^2 (c h (f g-e h)+d g (2 f g+e h)) h^2+C \left (g \left (48 f^3 g^3+16 e f^2 h g^2+17 e^2 f h^2 g+24 e^3 h^3\right ) d^3+2 c h \left (8 f^3 g^3+e f^2 h g^2+3 e^2 f h^2 g-12 e^3 h^3\right ) d^2+c^2 f h^2 \left (17 f^2 g^2+6 e f h g-23 e^2 h^2\right ) d+24 c^3 f^2 h^3 (f g-e h)\right )\right ) b^2-14 a d f h \left (15 A d^2 f^2 g h^2+C \left (g \left (8 f^2 g^2+3 e f h g+4 e^2 h^2\right ) d^2+c h \left (3 f^2 g^2+e f h g-4 e^2 h^2\right ) d+4 c^2 f h^2 (f g-e h)\right )\right ) b+35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C (f g-e h) h+C d g (2 f g+e h)\right )\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}dx}{h \sqrt {e+f x} \sqrt {g+h x}}-\frac {4 \sqrt {c f-d e} \left (\left (35 A d^2 f^2 (d f g+d e h+c f h) h^2+2 C \left (2 \left (6 f^3 g^3+5 e f^2 h g^2+5 e^2 f h^2 g+6 e^3 h^3\right ) d^3+c f h \left (10 f^2 g^2+9 e f h g+10 e^2 h^2\right ) d^2+10 c^2 f^2 h^2 (f g+e h) d+12 c^3 f^3 h^3\right )\right ) b^2-7 a d f h \left (15 A d^2 f^2 h^2+C \left (\left (8 f^2 g^2+7 e f h g+8 e^2 h^2\right ) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right )\right ) b+35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{3 d f h}}{5 d f h}-\frac {4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}}{7 d f h}\)

\(\Big \downarrow \) 130

\(\displaystyle \frac {2 C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{7 d f h}-\frac {\frac {-\frac {2}{3} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (8 C d f h a^2-38 b C (d f g+d e h+c f h) a+\frac {24 b^2 C (d f g+d e h+c f h)^2}{d f h}+35 A b^2 d f h-25 b^2 C (d e g+c f g+c e h)\right )-\frac {\frac {2 \sqrt {c f-d e} \left (\left (35 A d^2 f^2 (c h (f g-e h)+d g (2 f g+e h)) h^2+C \left (g \left (48 f^3 g^3+16 e f^2 h g^2+17 e^2 f h^2 g+24 e^3 h^3\right ) d^3+2 c h \left (8 f^3 g^3+e f^2 h g^2+3 e^2 f h^2 g-12 e^3 h^3\right ) d^2+c^2 f h^2 \left (17 f^2 g^2+6 e f h g-23 e^2 h^2\right ) d+24 c^3 f^2 h^3 (f g-e h)\right )\right ) b^2-14 a d f h \left (15 A d^2 f^2 g h^2+C \left (g \left (8 f^2 g^2+3 e f h g+4 e^2 h^2\right ) d^2+c h \left (3 f^2 g^2+e f h g-4 e^2 h^2\right ) d+4 c^2 f h^2 (f g-e h)\right )\right ) b+35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C (f g-e h) h+C d g (2 f g+e h)\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 {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}}-\frac {4 \sqrt {c f-d e} \left (\left (35 A d^2 f^2 (d f g+d e h+c f h) h^2+2 C \left (2 \left (6 f^3 g^3+5 e f^2 h g^2+5 e^2 f h^2 g+6 e^3 h^3\right ) d^3+c f h \left (10 f^2 g^2+9 e f h g+10 e^2 h^2\right ) d^2+10 c^2 f^2 h^2 (f g+e h) d+12 c^3 f^3 h^3\right )\right ) b^2-7 a d f h \left (15 A d^2 f^2 h^2+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+35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)\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 {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{3 d f h}}{5 d f h}-\frac {4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{5 d f h}}{7 d f h}\)

Input:

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

Output:

(2*C*(a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(7*d*f*h) - (( 
-4*C*(2*a*d*f*h - 3*b*(d*f*g + d*e*h + c*f*h))*(a + b*x)*Sqrt[c + d*x]*Sqr 
t[e + f*x]*Sqrt[g + h*x])/(5*d*f*h) + ((-2*(35*A*b^2*d*f*h + 8*a^2*C*d*f*h 
 - 25*b^2*C*(d*e*g + c*f*g + c*e*h) - 38*a*b*C*(d*f*g + d*e*h + c*f*h) + ( 
24*b^2*C*(d*f*g + d*e*h + c*f*h)^2)/(d*f*h))*Sqrt[c + d*x]*Sqrt[e + f*x]*S 
qrt[g + h*x])/3 - ((-4*Sqrt[-(d*e) + c*f]*(35*a^2*C*d^2*f^2*h^2*(d*f*g + d 
*e*h + c*f*h) - 7*a*b*d*f*h*(15*A*d^2*f^2*h^2 + 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))) + b^2*(35*A*d^2 
*f^2*h^2*(d*f*g + d*e*h + c*f*h) + 2*C*(12*c^3*f^3*h^3 + 10*c^2*d*f^2*h^2* 
(f*g + e*h) + c*d^2*f*h*(10*f^2*g^2 + 9*e*f*g*h + 10*e^2*h^2) + 2*d^3*(6*f 
^3*g^3 + 5*e*f^2*g^2*h + 5*e^2*f*g*h^2 + 6*e^3*h^3))))*Sqrt[(d*(e + f*x))/ 
(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[ArcSin[(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[(d*(g + h*x))/(d*g - c*h)]) + (2*Sqrt[-(d*e) + c*f]*(35*a^2*d^2*f^2 
*h^2*(3*A*d*f*h^2 + c*C*h*(f*g - e*h) + C*d*g*(2*f*g + e*h)) - 14*a*b*d*f* 
h*(15*A*d^2*f^2*g*h^2 + 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))) + b^2*(35 
*A*d^2*f^2*h^2*(c*h*(f*g - e*h) + d*g*(2*f*g + e*h)) + C*(24*c^3*f^2*h^3*( 
f*g - e*h) + c^2*d*f*h^2*(17*f^2*g^2 + 6*e*f*g*h - 23*e^2*h^2) + 2*c*d^2*h 
*(8*f^3*g^3 + e*f^2*g^2*h + 3*e^2*f*g*h^2 - 12*e^3*h^3) + d^3*g*(48*f^3...
 

Defintions of rubi rules used

rule 25

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

rule 27

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

rule 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 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 2104

Int[(((a_.) + (b_.)*(x_))^(m_.)*((A_.) + (C_.)*(x_)^2))/(Sqrt[(c_.) + (d_.) 
*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Sim 
p[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*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 + 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, C}, x] && IntegerQ[2*m] && GtQ[m, 0]
 

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]
 

Maple [A] (verified)

Time = 8.29 (sec) , antiderivative size = 1238, normalized size of antiderivative = 1.14

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1238\)
default	\(\text {Expression too large to display}\)	\(12279\)

Input:

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

Output:

((h*x+g)*(f*x+e)*(d*x+c))^(1/2)/(h*x+g)^(1/2)/(f*x+e)^(1/2)/(d*x+c)^(1/2)* 
(2/7*C*b^2/d/f/h*x^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)+2/5*(2*C*a*b-2/7*C*b^2/d/f/h*(3*c*f*h+3*d*e*h+3*d 
*f*g))/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^2*A+a^2*C-2/7*C*b^2/d/f/h*(5/2*c*e*h+5/2*c*f*g+5 
/2*d*e*g)-2/5*(2*C*a*b-2/7*C*b^2/d/f/h*(3*c*f*h+3*d*e*h+3*d*f*g))/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*A-2/5*(2*C*a*b-2/7*C*b^2/d/f/h*( 
3*c*f*h+3*d*e*h+3*d*f*g))/d/f/h*c*e*g-2/3*(b^2*A+a^2*C-2/7*C*b^2/d/f/h*(5/ 
2*c*e*h+5/2*c*f*g+5/2*d*e*g)-2/5*(2*C*a*b-2/7*C*b^2/d/f/h*(3*c*f*h+3*d*e*h 
+3*d*f*g))/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))*(c/d-e/f)*((x+c/d)/(c/d-e/f))^(1/2)*((x+g/h)/(-c/d+g/h))^(1/2)*(( 
x+e/f)/(-c/d+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+c/d)/(c/d-e/f))^(1/2),((-c/d+e/ 
f)/(-c/d+g/h))^(1/2))+2*(2*a*b*A-4/7*C*b^2/d/f/h*c*e*g-2/5*(2*C*a*b-2/7*C* 
b^2/d/f/h*(3*c*f*h+3*d*e*h+3*d*f*g))/d/f/h*(3/2*c*e*h+3/2*c*f*g+3/2*d*e*g) 
-2/3*(b^2*A+a^2*C-2/7*C*b^2/d/f/h*(5/2*c*e*h+5/2*c*f*g+5/2*d*e*g)-2/5*(2*C 
*a*b-2/7*C*b^2/d/f/h*(3*c*f*h+3*d*e*h+3*d*f*g))/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))*(c/d-e/f)*((x+c/d)/(c/d-e/f))^(1/2)*((x+ 
g/h)/(-c/d+g/h))^(1/2)*((x+e/f)/(-c/d+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+...
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 1665, normalized size of antiderivative = 1.54 \[ \int \frac {(a+b x)^2 \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Too large to display} \] Input:

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

Output:

2/315*(3*(15*C*b^2*d^4*f^4*h^4*x^2 + 24*C*b^2*d^4*f^4*g^2*h^2 + (23*C*b^2* 
d^4*e*f^3 + (23*C*b^2*c*d^3 - 56*C*a*b*d^4)*f^4)*g*h^3 + (24*C*b^2*d^4*e^2 
*f^2 + (23*C*b^2*c*d^3 - 56*C*a*b*d^4)*e*f^3 + (24*C*b^2*c^2*d^2 - 56*C*a* 
b*c*d^3 + 35*(C*a^2 + A*b^2)*d^4)*f^4)*h^4 - 6*(3*C*b^2*d^4*f^4*g*h^3 + (3 
*C*b^2*d^4*e*f^3 + (3*C*b^2*c*d^3 - 7*C*a*b*d^4)*f^4)*h^4)*x)*sqrt(d*x + c 
)*sqrt(f*x + e)*sqrt(h*x + g) + (48*C*b^2*d^4*f^4*g^4 + 16*(C*b^2*d^4*e*f^ 
3 + (C*b^2*c*d^3 - 7*C*a*b*d^4)*f^4)*g^3*h + (11*C*b^2*d^4*e^2*f^2 + 14*(C 
*b^2*c*d^3 - 3*C*a*b*d^4)*e*f^3 + (11*C*b^2*c^2*d^2 - 42*C*a*b*c*d^3 + 70* 
(C*a^2 + A*b^2)*d^4)*f^4)*g^2*h^2 + (16*C*b^2*d^4*e^3*f + 14*(C*b^2*c*d^3 
- 3*C*a*b*d^4)*e^2*f^2 + 7*(2*C*b^2*c^2*d^2 - 6*C*a*b*c*d^3 + 5*(C*a^2 + A 
*b^2)*d^4)*e*f^3 + (16*C*b^2*c^3*d - 42*C*a*b*c^2*d^2 - 210*A*a*b*d^4 + 35 
*(C*a^2 + A*b^2)*c*d^3)*f^4)*g*h^3 + (48*C*b^2*d^4*e^4 + 16*(C*b^2*c*d^3 - 
 7*C*a*b*d^4)*e^3*f + (11*C*b^2*c^2*d^2 - 42*C*a*b*c*d^3 + 70*(C*a^2 + A*b 
^2)*d^4)*e^2*f^2 + (16*C*b^2*c^3*d - 42*C*a*b*c^2*d^2 - 210*A*a*b*d^4 + 35 
*(C*a^2 + A*b^2)*c*d^3)*e*f^3 + (48*C*b^2*c^4 - 112*C*a*b*c^3*d - 210*A*a* 
b*c*d^3 + 315*A*a^2*d^4 + 70*(C*a^2 + A*b^2)*c^2*d^2)*f^4)*h^4)*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...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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