\(\int \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (A+B x+C x^2) \, dx\) [83]

Optimal result

Integrand size = 38, antiderivative size = 1183 \[ \int \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx =\text {Too large to display} \] Output:

2/315*(24*a^2*b*B*d^3*f^3-16*a^3*C*d^3*f^3-3*a*b^2*d*f*(d*f*(14*A*d*f+B*c* 
f+B*d*e)-C*(c^2*f^2+d^2*e^2))+b^3*(C*(8*c^3*f^3-3*c^2*d*e*f^2-3*c*d^2*e^2* 
f+8*d^3*e^3)+3*d*f*(7*A*d*f*(c*f+d*e)-B*(4*c^2*f^2-2*c*d*e*f+4*d^2*e^2)))) 
*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^3/d^3/f^3-2/105*(7*d*f*(-3*A* 
b*d*f+C*a*c*f+C*a*d*e+C*b*c*e)-(-4*a*d*f-4*b*c*f+b*d*e)*(3*b*B*d*f-2*a*C*d 
*f-2*b*C*(c*f+d*e))/b)*(b*x+a)^(3/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^2/d^2/f 
^2+2/21*(3*B*d*f-2*C*(a*d*f+b*c*f+b*d*e)/b)*(b*x+a)^(3/2)*(d*x+c)^(3/2)*(f 
*x+e)^(1/2)/b/d^2/f+2/9*C*(b*x+a)^(3/2)*(d*x+c)^(3/2)*(f*x+e)^(3/2)/b/d/f- 
2/315*(a*d-b*c)^(1/2)*(16*a^4*C*d^4*f^4-8*a^3*b*d^3*f^3*(3*B*d*f+C*c*f+C*d 
*e)+3*a^2*b^2*d^2*f^2*(d*f*(14*A*d*f+5*B*c*f+5*B*d*e)-2*C*(c^2*f^2-c*d*e*f 
+d^2*e^2))-a*b^3*d*f*(C*(8*c^3*f^3-6*c^2*d*e*f^2-6*c*d^2*e^2*f+8*d^3*e^3)+ 
3*d*f*(14*A*d*f*(c*f+d*e)-B*(5*c^2*f^2-6*c*d*e*f+5*d^2*e^2)))+b^4*(2*C*(8* 
c^4*f^4-4*c^3*d*e*f^3-3*c^2*d^2*e^2*f^2-4*c*d^3*e^3*f+8*d^4*e^4)+3*d*f*(14 
*A*d*f*(c^2*f^2-c*d*e*f+d^2*e^2)-B*(8*c^3*f^3-5*c^2*d*e*f^2-5*c*d^2*e^2*f+ 
8*d^3*e^3))))*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(f*x+e)^(1/2)*EllipticE(d^(1/2) 
*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/(-a*f+b*e))^(1/2))/b^4/d^(7 
/2)/f^4/(d*x+c)^(1/2)/(b*(f*x+e)/(-a*f+b*e))^(1/2)-2/315*(a*d-b*c)^(1/2)*( 
-a*f+b*e)*(-c*f+d*e)*(8*a^3*C*d^3*f^3+3*a^2*b*d^2*f^2*(-4*B*d*f-C*c*f+C*d* 
e)-3*a*b^2*d*f^2*((-7*A*d^2+C*c^2)*f+B*d*(-2*c*f+d*e))-b^3*(C*(-8*c^3*f^3- 
3*c^2*d*e*f^2+16*d^3*e^3)+3*d*f*(7*A*d*f*(-c*f+2*d*e)-B*(-4*c^2*f^2-c*d...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 33.72 (sec) , antiderivative size = 1422, normalized size of antiderivative = 1.20 \[ \int \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(2*(-(b^2*Sqrt[-a + (b*c)/d]*(16*a^4*C*d^4*f^4 - 8*a^3*b*d^3*f^3*(C*d*e + 
c*C*f + 3*B*d*f) + 3*a^2*b^2*d^2*f^2*(d*f*(5*B*d*e + 5*B*c*f + 14*A*d*f) - 
 2*C*(d^2*e^2 - c*d*e*f + c^2*f^2)) + a*b^3*d*f*(C*(-8*d^3*e^3 + 6*c*d^2*e 
^2*f + 6*c^2*d*e*f^2 - 8*c^3*f^3) - 3*d*f*(14*A*d*f*(d*e + c*f) + B*(-5*d^ 
2*e^2 + 6*c*d*e*f - 5*c^2*f^2))) + b^4*(2*C*(8*d^4*e^4 - 4*c*d^3*e^3*f - 3 
*c^2*d^2*e^2*f^2 - 4*c^3*d*e*f^3 + 8*c^4*f^4) + 3*d*f*(14*A*d*f*(d^2*e^2 - 
 c*d*e*f + c^2*f^2) + B*(-8*d^3*e^3 + 5*c*d^2*e^2*f + 5*c^2*d*e*f^2 - 8*c^ 
3*f^3))))*(c + d*x)*(e + f*x)) + b^2*Sqrt[-a + (b*c)/d]*d*f*(a + b*x)*(c + 
 d*x)*(e + f*x)*(8*a^3*C*d^3*f^3 - 3*a^2*b*d^2*f^2*(c*C*f + 4*B*d*f + C*d* 
(e + 2*f*x)) + a*b^2*d*f*(3*d*f*(7*A*d*f + B*(2*d*e + 2*c*f + 3*d*f*x)) + 
C*(-3*c^2*f^2 + 2*c*d*f*(e + f*x) + d^2*(-3*e^2 + 2*e*f*x + 5*f^2*x^2))) + 
 b^3*(C*(8*c^3*f^3 - 3*c^2*d*f^2*(e + 2*f*x) + c*d^2*f*(-3*e^2 + 2*e*f*x + 
 5*f^2*x^2) + d^3*(8*e^3 - 6*e^2*f*x + 5*e*f^2*x^2 + 35*f^3*x^3)) + 3*d*f* 
(7*A*d*f*(c*f + d*(e + 3*f*x)) + B*(-4*c^2*f^2 + c*d*f*(2*e + 3*f*x) + d^2 
*(-4*e^2 + 3*e*f*x + 15*f^2*x^2))))) - I*(b*c - a*d)*f*(16*a^4*C*d^4*f^4 - 
 8*a^3*b*d^3*f^3*(C*d*e + c*C*f + 3*B*d*f) + 3*a^2*b^2*d^2*f^2*(d*f*(5*B*d 
*e + 5*B*c*f + 14*A*d*f) - 2*C*(d^2*e^2 - c*d*e*f + c^2*f^2)) + a*b^3*d*f* 
(C*(-8*d^3*e^3 + 6*c*d^2*e^2*f + 6*c^2*d*e*f^2 - 8*c^3*f^3) - 3*d*f*(14*A* 
d*f*(d*e + c*f) + B*(-5*d^2*e^2 + 6*c*d*e*f - 5*c^2*f^2))) + b^4*(2*C*(8*d 
^4*e^4 - 4*c*d^3*e^3*f - 3*c^2*d^2*e^2*f^2 - 4*c^3*d*e*f^3 + 8*c^4*f^4)...
 

Rubi [A] (verified)

Time = 2.83 (sec) , antiderivative size = 1213, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2118, 27, 171, 27, 171, 27, 171, 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.

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

\(\Big \downarrow \) 2118

\(\displaystyle \frac {2 \int -\frac {3}{2} b \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (b c C e+a C d e+a c C f-3 A b d f-(3 b B d f-2 a C d f-2 b C (d e+c f)) x)dx}{9 b^2 d f}+\frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {\int \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (b c C e+a C d e+a c C f-3 A b d f-(3 b B d f-2 a C d f-2 b C (d e+c f)) x)dx}{3 b d f}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {\frac {2 \int \frac {\sqrt {c+d x} \sqrt {e+f x} (7 a d f (b c C e+a C d e+a c C f-3 A b d f)+(b c e+3 a (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))+(7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) x)}{2 \sqrt {a+b x}}dx}{7 d f}-\frac {2 \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2} (-2 a C d f+3 b B d f-2 b C (c f+d e))}{7 d f}}{3 b d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {\frac {\int \frac {\sqrt {c+d x} \sqrt {e+f x} (7 a d f (b c C e+a C d e+a c C f-3 A b d f)+(b c e+3 a (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))+(7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) x)}{\sqrt {a+b x}}dx}{7 d f}-\frac {2 \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2} (-2 a C d f+3 b B d f-2 b C (c f+d e))}{7 d f}}{3 b d f}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {\frac {\frac {2 \int \frac {\sqrt {e+f x} \left (5 b c f (7 a d f (b c C e+a C d e+a c C f-3 A b d f)+(b c e+3 a (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f)))-(b c e+a d e+3 a c f) (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f)))-\left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) x\right )}{2 \sqrt {a+b x} \sqrt {c+d x}}dx}{5 b f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} (7 b d f (a c C f+a C d e-3 A b d f+b c C e)-(a d f-4 b (c f+d e)) (-2 a C d f+3 b B d f-2 b C (c f+d e)))}{5 b f}}{7 d f}-\frac {2 \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2} (-2 a C d f+3 b B d f-2 b C (c f+d e))}{7 d f}}{3 b d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {\frac {\frac {\int \frac {\sqrt {e+f x} \left (5 b c f (7 a d f (b c C e+a C d e+a c C f-3 A b d f)+(b c e+3 a (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f)))-(b c e+a d e+3 a c f) (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f)))-\left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x}}dx}{5 b f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} (7 b d f (a c C f+a C d e-3 A b d f+b c C e)-(a d f-4 b (c f+d e)) (-2 a C d f+3 b B d f-2 b C (c f+d e)))}{5 b f}}{7 d f}-\frac {2 \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2} (-2 a C d f+3 b B d f-2 b C (c f+d e))}{7 d f}}{3 b d f}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {\frac {\frac {2 (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{5 b f}+\frac {\frac {2 \int \frac {3 b d e (5 b c f (7 a d f (b c C e+a C d e+a c C f-3 A b d f)+(b c e+3 a (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f)))-(b c e+a d e+3 a c f) (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))))+(b c e+a d e+a c f) \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right )+\left (\left (2 C \left (8 d^4 e^4-4 c d^3 f e^3-3 c^2 d^2 f^2 e^2-4 c^3 d f^3 e+8 c^4 f^4\right )+3 d f \left (14 A d f \left (d^2 e^2-c d f e+c^2 f^2\right )-B \left (8 d^3 e^3-5 c d^2 f e^2-5 c^2 d f^2 e+8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (8 d^3 e^3-6 c d^2 f e^2-6 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (14 A d f (d e+c f)-B \left (5 d^2 e^2-6 c d f e+5 c^2 f^2\right )\right )\right ) b^3+3 a^2 d^2 f^2 \left (d f (5 B d e+5 B c f+14 A d f)-2 C \left (d^2 e^2-c d f e+c^2 f^2\right )\right ) b^2-8 a^3 d^3 f^3 (C d e+c C f+3 B d f) b+16 a^4 C d^4 f^4\right ) x}{2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 b d}-\frac {2 \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{3 b d}}{5 b f}}{7 d f}-\frac {2 (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 d f}}{3 b d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {\frac {\frac {2 (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{5 b f}+\frac {\frac {\int \frac {3 b d e (5 b c f (7 a d f (b c C e+a C d e+a c C f-3 A b d f)+(b c e+3 a (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f)))-(b c e+a d e+3 a c f) (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))))+(b c e+a d e+a c f) \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right )+\left (\left (2 C \left (8 d^4 e^4-4 c d^3 f e^3-3 c^2 d^2 f^2 e^2-4 c^3 d f^3 e+8 c^4 f^4\right )+3 d f \left (14 A d f \left (d^2 e^2-c d f e+c^2 f^2\right )-B \left (8 d^3 e^3-5 c d^2 f e^2-5 c^2 d f^2 e+8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (8 d^3 e^3-6 c d^2 f e^2-6 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (14 A d f (d e+c f)-B \left (5 d^2 e^2-6 c d f e+5 c^2 f^2\right )\right )\right ) b^3+3 a^2 d^2 f^2 \left (d f (5 B d e+5 B c f+14 A d f)-2 C \left (d^2 e^2-c d f e+c^2 f^2\right )\right ) b^2-8 a^3 d^3 f^3 (C d e+c C f+3 B d f) b+16 a^4 C d^4 f^4\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 b d}-\frac {2 \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{3 b d}}{5 b f}}{7 d f}-\frac {2 (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 d f}}{3 b d f}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {\frac {\frac {2 (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{5 b f}+\frac {\frac {\frac {(b e-a f) (d e-c f) \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {\left (\left (2 C \left (8 d^4 e^4-4 c d^3 f e^3-3 c^2 d^2 f^2 e^2-4 c^3 d f^3 e+8 c^4 f^4\right )+3 d f \left (14 A d f \left (d^2 e^2-c d f e+c^2 f^2\right )-B \left (8 d^3 e^3-5 c d^2 f e^2-5 c^2 d f^2 e+8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (8 d^3 e^3-6 c d^2 f e^2-6 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (14 A d f (d e+c f)-B \left (5 d^2 e^2-6 c d f e+5 c^2 f^2\right )\right )\right ) b^3+3 a^2 d^2 f^2 \left (d f (5 B d e+5 B c f+14 A d f)-2 C \left (d^2 e^2-c d f e+c^2 f^2\right )\right ) b^2-8 a^3 d^3 f^3 (C d e+c C f+3 B d f) b+16 a^4 C d^4 f^4\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}}dx}{f}}{3 b d}-\frac {2 \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{3 b d}}{5 b f}}{7 d f}-\frac {2 (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 d f}}{3 b d f}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {\frac {\frac {2 (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{5 b f}+\frac {\frac {\frac {(b e-a f) (d e-c f) \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {\left (\left (2 C \left (8 d^4 e^4-4 c d^3 f e^3-3 c^2 d^2 f^2 e^2-4 c^3 d f^3 e+8 c^4 f^4\right )+3 d f \left (14 A d f \left (d^2 e^2-c d f e+c^2 f^2\right )-B \left (8 d^3 e^3-5 c d^2 f e^2-5 c^2 d f^2 e+8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (8 d^3 e^3-6 c d^2 f e^2-6 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (14 A d f (d e+c f)-B \left (5 d^2 e^2-6 c d f e+5 c^2 f^2\right )\right )\right ) b^3+3 a^2 d^2 f^2 \left (d f (5 B d e+5 B c f+14 A d f)-2 C \left (d^2 e^2-c d f e+c^2 f^2\right )\right ) b^2-8 a^3 d^3 f^3 (C d e+c C f+3 B d f) b+16 a^4 C d^4 f^4\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}dx}{f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 b d}-\frac {2 \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{3 b d}}{5 b f}}{7 d f}-\frac {2 (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 d f}}{3 b d f}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {\frac {\frac {2 (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{5 b f}+\frac {\frac {\frac {2 \sqrt {a d-b c} \left (\left (2 C \left (8 d^4 e^4-4 c d^3 f e^3-3 c^2 d^2 f^2 e^2-4 c^3 d f^3 e+8 c^4 f^4\right )+3 d f \left (14 A d f \left (d^2 e^2-c d f e+c^2 f^2\right )-B \left (8 d^3 e^3-5 c d^2 f e^2-5 c^2 d f^2 e+8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (8 d^3 e^3-6 c d^2 f e^2-6 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (14 A d f (d e+c f)-B \left (5 d^2 e^2-6 c d f e+5 c^2 f^2\right )\right )\right ) b^3+3 a^2 d^2 f^2 \left (d f (5 B d e+5 B c f+14 A d f)-2 C \left (d^2 e^2-c d f e+c^2 f^2\right )\right ) b^2-8 a^3 d^3 f^3 (C d e+c C f+3 B d f) b+16 a^4 C d^4 f^4\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {(b e-a f) (d e-c f) \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}}{3 b d}-\frac {2 \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{3 b d}}{5 b f}}{7 d f}-\frac {2 (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 d f}}{3 b d f}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {\frac {\frac {2 (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{5 b f}+\frac {\frac {\frac {2 \sqrt {a d-b c} \left (\left (2 C \left (8 d^4 e^4-4 c d^3 f e^3-3 c^2 d^2 f^2 e^2-4 c^3 d f^3 e+8 c^4 f^4\right )+3 d f \left (14 A d f \left (d^2 e^2-c d f e+c^2 f^2\right )-B \left (8 d^3 e^3-5 c d^2 f e^2-5 c^2 d f^2 e+8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (8 d^3 e^3-6 c d^2 f e^2-6 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (14 A d f (d e+c f)-B \left (5 d^2 e^2-6 c d f e+5 c^2 f^2\right )\right )\right ) b^3+3 a^2 d^2 f^2 \left (d f (5 B d e+5 B c f+14 A d f)-2 C \left (d^2 e^2-c d f e+c^2 f^2\right )\right ) b^2-8 a^3 d^3 f^3 (C d e+c C f+3 B d f) b+16 a^4 C d^4 f^4\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {(b e-a f) (d e-c f) \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}}dx}{f \sqrt {c+d x}}}{3 b d}-\frac {2 \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{3 b d}}{5 b f}}{7 d f}-\frac {2 (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 d f}}{3 b d f}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {\frac {\frac {2 (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{5 b f}+\frac {\frac {\frac {2 \sqrt {a d-b c} \left (\left (2 C \left (8 d^4 e^4-4 c d^3 f e^3-3 c^2 d^2 f^2 e^2-4 c^3 d f^3 e+8 c^4 f^4\right )+3 d f \left (14 A d f \left (d^2 e^2-c d f e+c^2 f^2\right )-B \left (8 d^3 e^3-5 c d^2 f e^2-5 c^2 d f^2 e+8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (8 d^3 e^3-6 c d^2 f e^2-6 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (14 A d f (d e+c f)-B \left (5 d^2 e^2-6 c d f e+5 c^2 f^2\right )\right )\right ) b^3+3 a^2 d^2 f^2 \left (d f (5 B d e+5 B c f+14 A d f)-2 C \left (d^2 e^2-c d f e+c^2 f^2\right )\right ) b^2-8 a^3 d^3 f^3 (C d e+c C f+3 B d f) b+16 a^4 C d^4 f^4\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {(b e-a f) (d e-c f) \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}dx}{f \sqrt {c+d x} \sqrt {e+f x}}}{3 b d}-\frac {2 \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{3 b d}}{5 b f}}{7 d f}-\frac {2 (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 d f}}{3 b d f}\)

\(\Big \downarrow \) 130

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac {\frac {\frac {2 (7 b d f (b c C e+a C d e+a c C f-3 A b d f)-(a d f-4 b (d e+c f)) (3 b B d f-2 a C d f-2 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{5 b f}+\frac {\frac {\frac {2 \sqrt {a d-b c} \left (\left (2 C \left (8 d^4 e^4-4 c d^3 f e^3-3 c^2 d^2 f^2 e^2-4 c^3 d f^3 e+8 c^4 f^4\right )+3 d f \left (14 A d f \left (d^2 e^2-c d f e+c^2 f^2\right )-B \left (8 d^3 e^3-5 c d^2 f e^2-5 c^2 d f^2 e+8 c^3 f^3\right )\right )\right ) b^4-a d f \left (C \left (8 d^3 e^3-6 c d^2 f e^2-6 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (14 A d f (d e+c f)-B \left (5 d^2 e^2-6 c d f e+5 c^2 f^2\right )\right )\right ) b^3+3 a^2 d^2 f^2 \left (d f (5 B d e+5 B c f+14 A d f)-2 C \left (d^2 e^2-c d f e+c^2 f^2\right )\right ) b^2-8 a^3 d^3 f^3 (C d e+c C f+3 B d f) b+16 a^4 C d^4 f^4\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \sqrt {a d-b c} (b e-a f) (d e-c f) \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {e+f x}}}{3 b d}-\frac {2 \left (-\left (\left (C \left (16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right )+3 d f \left (7 A d f (2 d e-c f)-B \left (8 d^2 e^2-c d f e-4 c^2 f^2\right )\right )\right ) b^3\right )-3 a d f^2 \left (\left (c^2 C-7 A d^2\right ) f+B d (d e-2 c f)\right ) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{3 b d}}{5 b f}}{7 d f}-\frac {2 (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 d f}}{3 b d f}\)

Input:

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

Output:

(2*C*(a + b*x)^(3/2)*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(9*b*d*f) - ((-2*(3* 
b*B*d*f - 2*a*C*d*f - 2*b*C*(d*e + c*f))*Sqrt[a + b*x]*(c + d*x)^(3/2)*(e 
+ f*x)^(3/2))/(7*d*f) + ((2*(7*b*d*f*(b*c*C*e + a*C*d*e + a*c*C*f - 3*A*b* 
d*f) - (a*d*f - 4*b*(d*e + c*f))*(3*b*B*d*f - 2*a*C*d*f - 2*b*C*(d*e + c*f 
)))*Sqrt[a + b*x]*Sqrt[c + d*x]*(e + f*x)^(3/2))/(5*b*f) + ((-2*(8*a^3*C*d 
^3*f^3 + 3*a^2*b*d^2*f^2*(C*d*e - c*C*f - 4*B*d*f) - 3*a*b^2*d*f^2*((c^2*C 
 - 7*A*d^2)*f + B*d*(d*e - 2*c*f)) - b^3*(C*(16*d^3*e^3 - 3*c^2*d*e*f^2 - 
8*c^3*f^3) + 3*d*f*(7*A*d*f*(2*d*e - c*f) - B*(8*d^2*e^2 - c*d*e*f - 4*c^2 
*f^2))))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b*d) + ((2*Sqrt[-(b 
*c) + a*d]*(16*a^4*C*d^4*f^4 - 8*a^3*b*d^3*f^3*(C*d*e + c*C*f + 3*B*d*f) + 
 3*a^2*b^2*d^2*f^2*(d*f*(5*B*d*e + 5*B*c*f + 14*A*d*f) - 2*C*(d^2*e^2 - c* 
d*e*f + c^2*f^2)) - a*b^3*d*f*(C*(8*d^3*e^3 - 6*c*d^2*e^2*f - 6*c^2*d*e*f^ 
2 + 8*c^3*f^3) + 3*d*f*(14*A*d*f*(d*e + c*f) - B*(5*d^2*e^2 - 6*c*d*e*f + 
5*c^2*f^2))) + b^4*(2*C*(8*d^4*e^4 - 4*c*d^3*e^3*f - 3*c^2*d^2*e^2*f^2 - 4 
*c^3*d*e*f^3 + 8*c^4*f^4) + 3*d*f*(14*A*d*f*(d^2*e^2 - c*d*e*f + c^2*f^2) 
- B*(8*d^3*e^3 - 5*c*d^2*e^2*f - 5*c^2*d*e*f^2 + 8*c^3*f^3))))*Sqrt[(b*(c 
+ d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x] 
)/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b*Sqrt[d]*f*Sqrt 
[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) + (2*Sqrt[-(b*c) + a*d]*(b*e - 
a*f)*(d*e - c*f)*(8*a^3*C*d^3*f^3 + 3*a^2*b*d^2*f^2*(C*d*e - c*C*f - 4*...
 

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 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegersQ[2*m, 2*n, 2*p]
 

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 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 = 3.13 (sec) , antiderivative size = 2077, normalized size of antiderivative = 1.76

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2077\)
default	\(\text {Expression too large to display}\)	\(14974\)

Input:

int((b*x+a)^(1/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(C*x^2+B*x+A),x,method=_RETU 
RNVERBOSE)
 

Output:

((f*x+e)*(b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)* 
(2/9*C*x^3*(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e* 
x+a*c*e)^(1/2)+2/7*(b*B*d*f+a*C*d*f+C*b*f*c+C*b*d*e-2/9*(4*a*d*f+4*b*c*f+4 
*b*d*e)*C)/b/d/f*x^2*(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d* 
e*x+b*c*e*x+a*c*e)^(1/2)+2/5*(A*b*d*f+a*B*d*f+B*b*c*f+b*B*d*e+C*a*c*f+C*a* 
d*e+C*b*c*e-2/9*C*(7/2*a*c*f+7/2*a*d*e+7/2*b*c*e)-2/7*(b*B*d*f+a*C*d*f+C*b 
*f*c+C*b*d*e-2/9*(4*a*d*f+4*b*c*f+4*b*d*e)*C)/b/d/f*(3*a*d*f+3*b*c*f+3*b*d 
*e))/b/d/f*x*(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c* 
e*x+a*c*e)^(1/2)+2/3*(A*a*d*f+A*b*c*f+A*b*d*e+B*a*c*f+B*a*d*e+B*b*c*e+1/3* 
C*a*c*e-2/7*(b*B*d*f+a*C*d*f+C*b*f*c+C*b*d*e-2/9*(4*a*d*f+4*b*c*f+4*b*d*e) 
*C)/b/d/f*(5/2*a*c*f+5/2*a*d*e+5/2*b*c*e)-2/5*(A*b*d*f+a*B*d*f+B*b*c*f+b*B 
*d*e+C*a*c*f+C*a*d*e+C*b*c*e-2/9*C*(7/2*a*c*f+7/2*a*d*e+7/2*b*c*e)-2/7*(b* 
B*d*f+a*C*d*f+C*b*f*c+C*b*d*e-2/9*(4*a*d*f+4*b*c*f+4*b*d*e)*C)/b/d/f*(3*a* 
d*f+3*b*c*f+3*b*d*e))/b/d/f*(2*a*d*f+2*b*c*f+2*b*d*e))/b/d/f*(b*d*f*x^3+a* 
d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)^(1/2)+2*(A*a*c* 
e-2/5*(A*b*d*f+a*B*d*f+B*b*c*f+b*B*d*e+C*a*c*f+C*a*d*e+C*b*c*e-2/9*C*(7/2* 
a*c*f+7/2*a*d*e+7/2*b*c*e)-2/7*(b*B*d*f+a*C*d*f+C*b*f*c+C*b*d*e-2/9*(4*a*d 
*f+4*b*c*f+4*b*d*e)*C)/b/d/f*(3*a*d*f+3*b*c*f+3*b*d*e))/b/d/f*a*c*e-2/3*(A 
*a*d*f+A*b*c*f+A*b*d*e+B*a*c*f+B*a*d*e+B*b*c*e+1/3*C*a*c*e-2/7*(b*B*d*f+a* 
C*d*f+C*b*f*c+C*b*d*e-2/9*(4*a*d*f+4*b*c*f+4*b*d*e)*C)/b/d/f*(5/2*a*c*f...
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1916, normalized size of antiderivative = 1.62 \[ \int \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(1/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(C*x^2+B*x+A),x, algor 
ithm="fricas")
 

Output:

2/945*(3*(35*C*b^5*d^5*f^5*x^3 + 8*C*b^5*d^5*e^3*f^2 - 3*(C*b^5*c*d^4 + (C 
*a*b^4 + 4*B*b^5)*d^5)*e^2*f^3 - (3*C*b^5*c^2*d^3 - 2*(C*a*b^4 + 3*B*b^5)* 
c*d^4 + 3*(C*a^2*b^3 - 2*B*a*b^4 - 7*A*b^5)*d^5)*e*f^4 + (8*C*b^5*c^3*d^2 
- 3*(C*a*b^4 + 4*B*b^5)*c^2*d^3 - 3*(C*a^2*b^3 - 2*B*a*b^4 - 7*A*b^5)*c*d^ 
4 + (8*C*a^3*b^2 - 12*B*a^2*b^3 + 21*A*a*b^4)*d^5)*f^5 + 5*(C*b^5*d^5*e*f^ 
4 + (C*b^5*c*d^4 + (C*a*b^4 + 9*B*b^5)*d^5)*f^5)*x^2 - (6*C*b^5*d^5*e^2*f^ 
3 - (2*C*b^5*c*d^4 + (2*C*a*b^4 + 9*B*b^5)*d^5)*e*f^4 + (6*C*b^5*c^2*d^3 - 
 (2*C*a*b^4 + 9*B*b^5)*c*d^4 + 3*(2*C*a^2*b^3 - 3*B*a*b^4 - 21*A*b^5)*d^5) 
*f^5)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e) + (16*C*b^5*d^5*e^5 - 8 
*(2*C*b^5*c*d^4 + (2*C*a*b^4 + 3*B*b^5)*d^5)*e^4*f - (5*C*b^5*c^2*d^3 - (2 
0*C*a*b^4 + 27*B*b^5)*c*d^4 + (5*C*a^2*b^3 - 27*B*a*b^4 - 42*A*b^5)*d^5)*e 
^3*f^2 - (5*C*b^5*c^3*d^2 - 6*(C*a*b^4 + 2*B*b^5)*c^2*d^3 - 3*(2*C*a^2*b^3 
 - 14*B*a*b^4 - 21*A*b^5)*c*d^4 + (5*C*a^3*b^2 - 12*B*a^2*b^3 + 63*A*a*b^4 
)*d^5)*e^2*f^3 - (16*C*b^5*c^4*d - (20*C*a*b^4 + 27*B*b^5)*c^3*d^2 - 3*(2* 
C*a^2*b^3 - 14*B*a*b^4 - 21*A*b^5)*c^2*d^3 - 2*(10*C*a^3*b^2 - 21*B*a^2*b^ 
3 + 126*A*a*b^4)*c*d^4 + (16*C*a^4*b - 27*B*a^3*b^2 + 63*A*a^2*b^3)*d^5)*e 
*f^4 + (16*C*b^5*c^5 - 8*(2*C*a*b^4 + 3*B*b^5)*c^4*d - (5*C*a^2*b^3 - 27*B 
*a*b^4 - 42*A*b^5)*c^3*d^2 - (5*C*a^3*b^2 - 12*B*a^2*b^3 + 63*A*a*b^4)*c^2 
*d^3 - (16*C*a^4*b - 27*B*a^3*b^2 + 63*A*a^2*b^3)*c*d^4 + 2*(8*C*a^5 - 12* 
B*a^4*b + 21*A*a^3*b^2)*d^5)*f^5)*sqrt(b*d*f)*weierstrassPInverse(4/3*(...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((b*x+a)^(1/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(C*x^2+B*x+A),x, algor 
ithm="maxima")
 

Output:

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

Giac [F]

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

integrate((b*x+a)^(1/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(C*x^2+B*x+A),x, algor 
ithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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