\(\int \frac {\sqrt {c+d x} \sqrt {e+f x} (A+B x+C x^2)}{(a+b x)^{9/2}} \, dx\) [88]

Optimal result

Integrand size = 38, antiderivative size = 1714 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{9/2}} \, dx =\text {Too large to display} \] Output:

-2/105*(24*a^4*C*d^2*f^2-a^3*b*d*f*(-4*B*d*f+61*C*c*f+43*C*d*e)+b^4*(8*A*d 
^2*e^2-c*d*e*(A*f+14*B*e)+c^2*(-4*A*f^2+7*B*e*f+35*C*e^2))+3*a*b^3*(d^2*e* 
(-5*A*f+2*B*e)-c^2*f*(B*f+28*C*e)-c*d*(-3*A*f^2-5*B*e*f+14*C*e^2))-3*a^2*b 
^2*(d*f*(-A*d*f+3*B*c*f+2*B*d*e)-C*(15*c^2*f^2+37*c*d*e*f+5*d^2*e^2)))*(d* 
x+c)^(1/2)*(f*x+e)^(1/2)/b^3/(-a*d+b*c)^2/(-a*f+b*e)^2/(b*x+a)^(3/2)+2/105 
*(48*a^5*C*d^3*f^3+8*a^4*b*d^2*f^2*(B*d*f-16*C*(c*f+d*e))-b^5*(8*A*d^3*e^3 
-c*d^2*e^2*(5*A*f+14*B*e)+c^2*d*e*(-5*A*f^2+14*B*e*f+35*C*e^2)+c^3*f*(8*A* 
f^2-14*B*e*f+35*C*e^2))-a*b^4*(d^3*e^2*(-19*A*f+6*B*e)-6*c^3*f^2*(-B*f+7*C 
*e)-c^2*d*f*(238*C*e^2-19*f*(-A*f+B*e))-c*d^2*e*(42*C*e^2-f*(20*A*f+19*B*e 
)))+a^3*b^2*d*f*(C*(103*c^2*f^2+344*c*d*e*f+103*d^2*e^2)+d*f*(6*A*d*f-19*B 
*(c*f+d*e)))-3*a^2*b^3*(C*(5*c^3*f^3+94*c^2*d*e*f^2+94*c*d^2*e^2*f+5*d^3*e 
^3)+d*f*(3*A*d*f*(c*f+d*e)-B*(3*c^2*f^2+16*c*d*e*f+3*d^2*e^2))))*(d*x+c)^( 
1/2)*(f*x+e)^(1/2)/b^3/(-a*d+b*c)^3/(-a*f+b*e)^3/(b*x+a)^(1/2)+2/35*(6*a^3 
*C*d*f+a*b^2*(-8*A*d*f+3*B*c*f+3*B*d*e+14*C*c*e)-b^3*(7*B*c*e-4*A*(c*f+d*e 
))+a^2*b*(B*d*f-10*C*(c*f+d*e)))*(d*x+c)^(3/2)*(f*x+e)^(1/2)/b^2/(-a*d+b*c 
)^2/(-a*f+b*e)/(b*x+a)^(5/2)-2/7*(A*b^2-a*(B*b-C*a))*(d*x+c)^(3/2)*(f*x+e) 
^(3/2)/b/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^(7/2)+2/105*d^(1/2)*(48*a^5*C*d^3*f 
^3+8*a^4*b*d^2*f^2*(B*d*f-16*C*(c*f+d*e))-b^5*(8*A*d^3*e^3-c*d^2*e^2*(5*A* 
f+14*B*e)+c^2*d*e*(-5*A*f^2+14*B*e*f+35*C*e^2)+c^3*f*(8*A*f^2-14*B*e*f+35* 
C*e^2))-a*b^4*(d^3*e^2*(-19*A*f+6*B*e)-6*c^3*f^2*(-B*f+7*C*e)-c^2*d*f*(...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 36.99 (sec) , antiderivative size = 15963, normalized size of antiderivative = 9.31 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{9/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x)^(9/2), 
x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 4.40 (sec) , antiderivative size = 1770, 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 = {2117, 27, 167, 27, 167, 27, 169, 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.

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

\(\Big \downarrow \) 2117

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 167

\(\displaystyle \frac {\frac {2 \int -\frac {\sqrt {e+f x} \left (6 C d f (d e+3 c f) a^3+b \left (B d f (d e+3 c f)-5 C \left (2 d^2 e^2+11 c d f e+3 c^2 f^2\right )\right ) a^2+b^2 \left (6 f (7 C e-B f) c^2+d \left (49 C e^2-8 B f e+11 A f^2\right ) c+d^2 e (3 B e-8 A f)\right ) a+b^3 \left (-\left (\left (35 C e^2-14 B f e+8 A f^2\right ) c^2\right )-d e (7 B e-A f) c+4 A d^2 e^2\right )+d \left (24 C d f^2 a^3+b f (4 B d f-5 C (11 d e+5 c f)) a^2+b^2 (7 C e (5 d e+8 c f)-f (8 B d e+3 B c f-3 A d f)) a-b^3 \left (35 c C e^2-A d f e-c f (7 B e-4 A f)\right )\right ) x\right )}{2 (a+b x)^{5/2} \sqrt {c+d x}}dx}{5 b (b e-a f)}+\frac {2 \sqrt {c+d x} (e+f x)^{3/2} \left (6 a^3 C d f+a^2 b (B d f-10 C (c f+d e))+a b^2 (-8 A d f+3 B c f+3 B d e+14 c C e)-b^3 (7 B c e-4 A (c f+d e))\right )}{5 b (a+b x)^{5/2} (b e-a f)}}{7 b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{7 b (a+b x)^{7/2} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \sqrt {c+d x} (e+f x)^{3/2} \left (6 a^3 C d f+a^2 b (B d f-10 C (c f+d e))+a b^2 (-8 A d f+3 B c f+3 B d e+14 c C e)-b^3 (7 B c e-4 A (c f+d e))\right )}{5 b (a+b x)^{5/2} (b e-a f)}-\frac {\int \frac {\sqrt {e+f x} \left (6 C d f (d e+3 c f) a^3+b \left (B d f (d e+3 c f)-5 C \left (2 d^2 e^2+11 c d f e+3 c^2 f^2\right )\right ) a^2+b^2 \left (6 f (7 C e-B f) c^2+d \left (49 C e^2-8 B f e+11 A f^2\right ) c+d^2 e (3 B e-8 A f)\right ) a+b^3 \left (-\left (\left (35 C e^2-14 B f e+8 A f^2\right ) c^2\right )-d e (7 B e-A f) c+4 A d^2 e^2\right )+d \left (24 C d f^2 a^3+b f (4 B d f-5 C (11 d e+5 c f)) a^2+b^2 (7 C e (5 d e+8 c f)-f (8 B d e+3 B c f-3 A d f)) a-b^3 \left (35 c C e^2-A d f e-c f (7 B e-4 A f)\right )\right ) x\right )}{(a+b x)^{5/2} \sqrt {c+d x}}dx}{5 b (b e-a f)}}{7 b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{7 b (a+b x)^{7/2} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {\frac {2 \left (6 C d f a^3+b (B d f-10 C (d e+c f)) a^2+b^2 (14 c C e+3 B d e+3 B c f-8 A d f) a-b^3 (7 B c e-4 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{5 b (b e-a f) (a+b x)^{5/2}}-\frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (24 C d^2 f^2 a^4-b d f (61 C d e+43 c C f-4 B d f) a^3-3 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d f e+5 c^2 f^2\right )\right ) a^2-3 b^3 \left (2 f (7 C e-B f) c^2+d \left (28 C e^2-5 B f e+5 A f^2\right ) c+d^2 e (B e-3 A f)\right ) a-b^4 \left (-\left (\left (35 C e^2-14 B f e+8 A f^2\right ) c^2\right )-d e (7 B e-A f) c+4 A d^2 e^2\right )\right )}{3 b (b c-a d) (a+b x)^{3/2}}+\frac {2 \int -\frac {24 C d^2 f^2 (d e+c f) a^4+b d f \left (4 B d f (d e+c f)-C \left (43 d^2 e^2+122 c d f e+43 c^2 f^2\right )\right ) a^3+3 b^2 \left (C \left (5 d^3 e^3+52 c d^2 f e^2+52 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (A d f (d e+c f)-2 B \left (d^2 e^2+3 c d f e+c^2 f^2\right )\right )\right ) a^2+3 b^3 \left (-2 f^2 (7 C e-B f) c^3-d f \left (56 C e^2-4 B f e+5 A f^2\right ) c^2-2 d^2 e \left (7 C e^2-2 B f e-3 A f^2\right ) c+d^3 e^2 (2 B e-5 A f)\right ) a+b^4 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )+d f \left (48 C d^2 f^2 a^4+8 b d f (B d f-13 C (d e+c f)) a^3+3 b^2 \left (C \left (20 d^2 e^2+74 c d f e+20 c^2 f^2\right )+d f (2 A d f-5 B (d e+c f))\right ) a^2+3 b^3 \left (-f (42 C e-B f) c^2-2 d \left (21 C e^2-f (5 B e-A f)\right ) c+d^2 e (B e-2 A f)\right ) a+b^4 \left (\left (70 C e^2-f (7 B e-4 A f)\right ) c^2-d e (7 B e+2 A f) c+4 A d^2 e^2\right )\right ) x}{2 (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 b (b c-a d)}}{5 b (b e-a f)}}{7 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \left (6 C d f a^3+b (B d f-10 C (d e+c f)) a^2+b^2 (14 c C e+3 B d e+3 B c f-8 A d f) a-b^3 (7 B c e-4 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{5 b (b e-a f) (a+b x)^{5/2}}-\frac {\frac {2 \left (24 C d^2 f^2 a^4-b d f (61 C d e+43 c C f-4 B d f) a^3-3 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d f e+5 c^2 f^2\right )\right ) a^2-3 b^3 \left (2 f (7 C e-B f) c^2+d \left (28 C e^2-5 B f e+5 A f^2\right ) c+d^2 e (B e-3 A f)\right ) a-b^4 \left (-\left (\left (35 C e^2-14 B f e+8 A f^2\right ) c^2\right )-d e (7 B e-A f) c+4 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^{3/2}}-\frac {\int \frac {24 C d^2 f^2 (d e+c f) a^4+b d f \left (4 B d f (d e+c f)-C \left (43 d^2 e^2+122 c d f e+43 c^2 f^2\right )\right ) a^3+3 b^2 \left (C \left (5 d^3 e^3+52 c d^2 f e^2+52 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (A d f (d e+c f)-2 B \left (d^2 e^2+3 c d f e+c^2 f^2\right )\right )\right ) a^2+3 b^3 \left (-2 f^2 (7 C e-B f) c^3-d f \left (56 C e^2-f (4 B e-5 A f)\right ) c^2-2 d^2 e \left (7 C e^2-f (2 B e+3 A f)\right ) c+d^3 e^2 (2 B e-5 A f)\right ) a+b^4 \left (f \left (35 C e^2-2 f (7 B e-4 A f)\right ) c^3+d e \left (35 C e^2+f (14 B e-5 A f)\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )+d f \left (48 C d^2 f^2 a^4+8 b d f (B d f-13 C (d e+c f)) a^3+3 b^2 \left (C \left (20 d^2 e^2+74 c d f e+20 c^2 f^2\right )+d f (2 A d f-5 B (d e+c f))\right ) a^2+3 b^3 \left (-f (42 C e-B f) c^2-2 d \left (21 C e^2-f (5 B e-A f)\right ) c+d^2 e (B e-2 A f)\right ) a+b^4 \left (\left (70 C e^2-f (7 B e-4 A f)\right ) c^2-d e (7 B e+2 A f) c+4 A d^2 e^2\right )\right ) x}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 b (b c-a d)}}{5 b (b e-a f)}}{7 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\frac {2 \left (6 C d f a^3+b (B d f-10 C (d e+c f)) a^2+b^2 (14 c C e+3 B d e+3 B c f-8 A d f) a-b^3 (7 B c e-4 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{5 b (b e-a f) (a+b x)^{5/2}}-\frac {\frac {2 \left (24 C d^2 f^2 a^4-b d f (61 C d e+43 c C f-4 B d f) a^3-3 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d f e+5 c^2 f^2\right )\right ) a^2-3 b^3 \left (2 f (7 C e-B f) c^2+d \left (28 C e^2-5 B f e+5 A f^2\right ) c+d^2 e (B e-3 A f)\right ) a-b^4 \left (-\left (\left (35 C e^2-14 B f e+8 A f^2\right ) c^2\right )-d e (7 B e-A f) c+4 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^{3/2}}-\frac {\frac {2 \left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {2 \int \frac {d f \left (24 C d^2 f^2 (d e+c f) a^5+b d f \left (4 B d f (d e+c f)-C \left (61 d^2 e^2+134 c d f e+61 c^2 f^2\right )\right ) a^4+b^2 \left (5 C \left (9 d^3 e^3+46 c d^2 f e^2+46 c^2 d f^2 e+9 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (9 d^2 e^2+20 c d f e+9 c^2 f^2\right )\right )\right ) a^3-3 b^3 \left (f^2 (48 C e+B f) c^3+3 d f \left (34 C e^2-4 B f e-A f^2\right ) c^2+12 d^2 e \left (4 C e^2-B f e+A f^2\right ) c+d^3 e^2 (B e-3 A f)\right ) a^2-b^4 \left (-f \left (161 C e^2+4 B f e-4 A f^2\right ) c^3-d e \left (161 C e^2-58 B f e+13 A f^2\right ) c^2-d^2 e^2 (4 B e+13 A f) c+4 A d^3 e^3\right ) a-b^5 c e \left (\left (70 C e^2-7 B f e+4 A f^2\right ) c^2-d e (7 B e+2 A f) c+4 A d^2 e^2\right )+\left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\right ) x\right )}{2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{3 b (b c-a d)}}{5 b (b e-a f)}}{7 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \left (6 C d f a^3+b (B d f-10 C (d e+c f)) a^2+b^2 (14 c C e+3 B d e+3 B c f-8 A d f) a-b^3 (7 B c e-4 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{5 b (b e-a f) (a+b x)^{5/2}}-\frac {\frac {2 \left (24 C d^2 f^2 a^4-b d f (61 C d e+43 c C f-4 B d f) a^3-3 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d f e+5 c^2 f^2\right )\right ) a^2-3 b^3 \left (2 f (7 C e-B f) c^2+d \left (28 C e^2-5 B f e+5 A f^2\right ) c+d^2 e (B e-3 A f)\right ) a-b^4 \left (-\left (\left (35 C e^2-14 B f e+8 A f^2\right ) c^2\right )-d e (7 B e-A f) c+4 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^{3/2}}-\frac {\frac {2 \left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d f \int \frac {24 C d^2 f^2 (d e+c f) a^5+b d f \left (4 B d f (d e+c f)-C \left (61 d^2 e^2+134 c d f e+61 c^2 f^2\right )\right ) a^4+b^2 \left (5 C \left (9 d^3 e^3+46 c d^2 f e^2+46 c^2 d f^2 e+9 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (9 d^2 e^2+20 c d f e+9 c^2 f^2\right )\right )\right ) a^3-3 b^3 \left (f^2 (48 C e+B f) c^3+3 d f \left (34 C e^2-4 B f e-A f^2\right ) c^2+12 d^2 e \left (4 C e^2-B f e+A f^2\right ) c+d^3 e^2 (B e-3 A f)\right ) a^2-b^4 \left (-f \left (161 C e^2+4 B f e-4 A f^2\right ) c^3-d e \left (161 C e^2-58 B f e+13 A f^2\right ) c^2-d^2 e^2 (4 B e+13 A f) c+4 A d^3 e^3\right ) a-b^5 c e \left (\left (70 C e^2-7 B f e+4 A f^2\right ) c^2-d e (7 B e+2 A f) c+4 A d^2 e^2\right )+\left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{3 b (b c-a d)}}{5 b (b e-a f)}}{7 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {\frac {2 \left (6 C d f a^3+b (B d f-10 C (d e+c f)) a^2+b^2 (14 c C e+3 B d e+3 B c f-8 A d f) a-b^3 (7 B c e-4 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{5 b (b e-a f) (a+b x)^{5/2}}-\frac {\frac {2 \left (24 C d^2 f^2 a^4-b d f (61 C d e+43 c C f-4 B d f) a^3-3 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d f e+5 c^2 f^2\right )\right ) a^2-3 b^3 \left (2 f (7 C e-B f) c^2+d \left (28 C e^2-5 B f e+5 A f^2\right ) c+d^2 e (B e-3 A f)\right ) a-b^4 \left (-\left (\left (35 C e^2-14 B f e+8 A f^2\right ) c^2\right )-d e (7 B e-A f) c+4 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^{3/2}}-\frac {\frac {2 \left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d f \left (\frac {(b e-a f) (d e-c f) \left (24 C d^2 f^2 a^4-b d f (43 C d e+61 c C f-4 B d f) a^3-3 b^2 \left (d f (2 B d e+3 B c f-A d f)-C \left (5 d^2 e^2+37 c d f e+15 c^2 f^2\right )\right ) a^2+3 b^3 \left (-f (28 C e+B f) c^2-d \left (14 C e^2-5 B f e-3 A f^2\right ) c+d^2 e (2 B e-5 A f)\right ) a+b^4 \left (\left (35 C e^2+7 B f e-4 A f^2\right ) c^2-d e (14 B e+A f) c+8 A d^2 e^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {\left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}}dx}{f}\right )}{(b c-a d) (b e-a f)}}{3 b (b c-a d)}}{5 b (b e-a f)}}{7 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {\frac {2 \left (6 C d f a^3+b (B d f-10 C (d e+c f)) a^2+b^2 (14 c C e+3 B d e+3 B c f-8 A d f) a-b^3 (7 B c e-4 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{5 b (b e-a f) (a+b x)^{5/2}}-\frac {\frac {2 \left (24 C d^2 f^2 a^4-b d f (61 C d e+43 c C f-4 B d f) a^3-3 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d f e+5 c^2 f^2\right )\right ) a^2-3 b^3 \left (2 f (7 C e-B f) c^2+d \left (28 C e^2-5 B f e+5 A f^2\right ) c+d^2 e (B e-3 A f)\right ) a-b^4 \left (-\left (\left (35 C e^2-14 B f e+8 A f^2\right ) c^2\right )-d e (7 B e-A f) c+4 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^{3/2}}-\frac {\frac {2 \left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d f \left (\frac {(b e-a f) (d e-c f) \left (24 C d^2 f^2 a^4-b d f (43 C d e+61 c C f-4 B d f) a^3-3 b^2 \left (d f (2 B d e+3 B c f-A d f)-C \left (5 d^2 e^2+37 c d f e+15 c^2 f^2\right )\right ) a^2+3 b^3 \left (-f (28 C e+B f) c^2-d \left (14 C e^2-5 B f e-3 A f^2\right ) c+d^2 e (2 B e-5 A f)\right ) a+b^4 \left (\left (35 C e^2+7 B f e-4 A f^2\right ) c^2-d e (14 B e+A f) c+8 A d^2 e^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {\left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\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}}}\right )}{(b c-a d) (b e-a f)}}{3 b (b c-a d)}}{5 b (b e-a f)}}{7 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {\frac {2 \left (6 C d f a^3+b (B d f-10 C (d e+c f)) a^2+b^2 (14 c C e+3 B d e+3 B c f-8 A d f) a-b^3 (7 B c e-4 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{5 b (b e-a f) (a+b x)^{5/2}}-\frac {\frac {2 \left (24 C d^2 f^2 a^4-b d f (61 C d e+43 c C f-4 B d f) a^3-3 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d f e+5 c^2 f^2\right )\right ) a^2-3 b^3 \left (2 f (7 C e-B f) c^2+d \left (28 C e^2-5 B f e+5 A f^2\right ) c+d^2 e (B e-3 A f)\right ) a-b^4 \left (-\left (\left (35 C e^2-14 B f e+8 A f^2\right ) c^2\right )-d e (7 B e-A f) c+4 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^{3/2}}-\frac {\frac {2 \left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d f \left (\frac {2 \sqrt {a d-b c} \left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\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 (24 C d^2 f^2 a^4-b d f (43 C d e+61 c C f-4 B d f) a^3-3 b^2 \left (d f (2 B d e+3 B c f-A d f)-C \left (5 d^2 e^2+37 c d f e+15 c^2 f^2\right )\right ) a^2+3 b^3 \left (-f (28 C e+B f) c^2-d \left (14 C e^2-5 B f e-3 A f^2\right ) c+d^2 e (2 B e-5 A f)\right ) a+b^4 \left (\left (35 C e^2+7 B f e-4 A f^2\right ) c^2-d e (14 B e+A f) c+8 A d^2 e^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}\right )}{(b c-a d) (b e-a f)}}{3 b (b c-a d)}}{5 b (b e-a f)}}{7 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {\frac {2 \left (6 C d f a^3+b (B d f-10 C (d e+c f)) a^2+b^2 (14 c C e+3 B d e+3 B c f-8 A d f) a-b^3 (7 B c e-4 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{5 b (b e-a f) (a+b x)^{5/2}}-\frac {\frac {2 \left (24 C d^2 f^2 a^4-b d f (61 C d e+43 c C f-4 B d f) a^3-3 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d f e+5 c^2 f^2\right )\right ) a^2-3 b^3 \left (2 f (7 C e-B f) c^2+d \left (28 C e^2-5 B f e+5 A f^2\right ) c+d^2 e (B e-3 A f)\right ) a-b^4 \left (-\left (\left (35 C e^2-14 B f e+8 A f^2\right ) c^2\right )-d e (7 B e-A f) c+4 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^{3/2}}-\frac {\frac {2 \left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d f \left (\frac {2 \sqrt {a d-b c} \left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\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 (24 C d^2 f^2 a^4-b d f (43 C d e+61 c C f-4 B d f) a^3-3 b^2 \left (d f (2 B d e+3 B c f-A d f)-C \left (5 d^2 e^2+37 c d f e+15 c^2 f^2\right )\right ) a^2+3 b^3 \left (-f (28 C e+B f) c^2-d \left (14 C e^2-5 B f e-3 A f^2\right ) c+d^2 e (2 B e-5 A f)\right ) a+b^4 \left (\left (35 C e^2+7 B f e-4 A f^2\right ) c^2-d e (14 B e+A f) c+8 A d^2 e^2\right )\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}}\right )}{(b c-a d) (b e-a f)}}{3 b (b c-a d)}}{5 b (b e-a f)}}{7 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {\frac {2 \left (6 C d f a^3+b (B d f-10 C (d e+c f)) a^2+b^2 (14 c C e+3 B d e+3 B c f-8 A d f) a-b^3 (7 B c e-4 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{5 b (b e-a f) (a+b x)^{5/2}}-\frac {\frac {2 \left (24 C d^2 f^2 a^4-b d f (61 C d e+43 c C f-4 B d f) a^3-3 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d f e+5 c^2 f^2\right )\right ) a^2-3 b^3 \left (2 f (7 C e-B f) c^2+d \left (28 C e^2-5 B f e+5 A f^2\right ) c+d^2 e (B e-3 A f)\right ) a-b^4 \left (-\left (\left (35 C e^2-14 B f e+8 A f^2\right ) c^2\right )-d e (7 B e-A f) c+4 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^{3/2}}-\frac {\frac {2 \left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d f \left (\frac {2 \sqrt {a d-b c} \left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\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 (24 C d^2 f^2 a^4-b d f (43 C d e+61 c C f-4 B d f) a^3-3 b^2 \left (d f (2 B d e+3 B c f-A d f)-C \left (5 d^2 e^2+37 c d f e+15 c^2 f^2\right )\right ) a^2+3 b^3 \left (-f (28 C e+B f) c^2-d \left (14 C e^2-5 B f e-3 A f^2\right ) c+d^2 e (2 B e-5 A f)\right ) a+b^4 \left (\left (35 C e^2+7 B f e-4 A f^2\right ) c^2-d e (14 B e+A f) c+8 A d^2 e^2\right )\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}}\right )}{(b c-a d) (b e-a f)}}{3 b (b c-a d)}}{5 b (b e-a f)}}{7 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}\)

\(\Big \downarrow \) 130

\(\displaystyle \frac {\frac {2 \left (6 C d f a^3+b (B d f-10 C (d e+c f)) a^2+b^2 (14 c C e+3 B d e+3 B c f-8 A d f) a-b^3 (7 B c e-4 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{5 b (b e-a f) (a+b x)^{5/2}}-\frac {\frac {2 \left (24 C d^2 f^2 a^4-b d f (61 C d e+43 c C f-4 B d f) a^3-3 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d f e+5 c^2 f^2\right )\right ) a^2-3 b^3 \left (2 f (7 C e-B f) c^2+d \left (28 C e^2-5 B f e+5 A f^2\right ) c+d^2 e (B e-3 A f)\right ) a-b^4 \left (-\left (\left (35 C e^2-14 B f e+8 A f^2\right ) c^2\right )-d e (7 B e-A f) c+4 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^{3/2}}-\frac {\frac {2 \left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {d f \left (\frac {2 \sqrt {a d-b c} \left (48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left (C \left (103 d^2 e^2+344 c d f e+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right ) a^3-3 b^3 \left (C \left (5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^4 \left (-6 f^2 (7 C e-B f) c^3-d f \left (238 C e^2-19 f (B e-A f)\right ) c^2-d^2 e \left (42 C e^2-f (19 B e+20 A f)\right ) c+d^3 e^2 (6 B e-19 A f)\right ) a-b^5 \left (f \left (35 C e^2-14 B f e+8 A f^2\right ) c^3+d e \left (35 C e^2+14 B f e-5 A f^2\right ) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right )\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 (24 C d^2 f^2 a^4-b d f (43 C d e+61 c C f-4 B d f) a^3-3 b^2 \left (d f (2 B d e+3 B c f-A d f)-C \left (5 d^2 e^2+37 c d f e+15 c^2 f^2\right )\right ) a^2+3 b^3 \left (-f (28 C e+B f) c^2-d \left (14 C e^2-5 B f e-3 A f^2\right ) c+d^2 e (2 B e-5 A f)\right ) a+b^4 \left (\left (35 C e^2+7 B f e-4 A f^2\right ) c^2-d e (14 B e+A f) c+8 A d^2 e^2\right )\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}}\right )}{(b c-a d) (b e-a f)}}{3 b (b c-a d)}}{5 b (b e-a f)}}{7 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}\)

Input:

Int[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x)^(9/2),x]
 

Output:

(-2*(A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(7*b*(b*c - a 
*d)*(b*e - a*f)*(a + b*x)^(7/2)) + ((2*(6*a^3*C*d*f + a*b^2*(14*c*C*e + 3* 
B*d*e + 3*B*c*f - 8*A*d*f) - b^3*(7*B*c*e - 4*A*(d*e + c*f)) + a^2*b*(B*d* 
f - 10*C*(d*e + c*f)))*Sqrt[c + d*x]*(e + f*x)^(3/2))/(5*b*(b*e - a*f)*(a 
+ b*x)^(5/2)) - ((2*(24*a^4*C*d^2*f^2 - a^3*b*d*f*(61*C*d*e + 43*c*C*f - 4 
*B*d*f) - 3*a*b^3*(d^2*e*(B*e - 3*A*f) + 2*c^2*f*(7*C*e - B*f) + c*d*(28*C 
*e^2 - 5*B*e*f + 5*A*f^2)) - b^4*(4*A*d^2*e^2 - c*d*e*(7*B*e - A*f) - c^2* 
(35*C*e^2 - 14*B*e*f + 8*A*f^2)) - 3*a^2*b^2*(d*f*(3*B*d*e + 2*B*c*f - A*d 
*f) - C*(15*d^2*e^2 + 37*c*d*e*f + 5*c^2*f^2)))*Sqrt[c + d*x]*Sqrt[e + f*x 
])/(3*b*(b*c - a*d)*(a + b*x)^(3/2)) - ((2*(48*a^5*C*d^3*f^3 + 8*a^4*b*d^2 
*f^2*(B*d*f - 16*C*(d*e + c*f)) - b^5*(8*A*d^3*e^3 - c*d^2*e^2*(14*B*e + 5 
*A*f) + c^2*d*e*(35*C*e^2 + 14*B*e*f - 5*A*f^2) + c^3*f*(35*C*e^2 - 14*B*e 
*f + 8*A*f^2)) - a*b^4*(d^3*e^2*(6*B*e - 19*A*f) - 6*c^3*f^2*(7*C*e - B*f) 
 - c^2*d*f*(238*C*e^2 - 19*f*(B*e - A*f)) - c*d^2*e*(42*C*e^2 - f*(19*B*e 
+ 20*A*f))) + a^3*b^2*d*f*(C*(103*d^2*e^2 + 344*c*d*e*f + 103*c^2*f^2) + d 
*f*(6*A*d*f - 19*B*(d*e + c*f))) - 3*a^2*b^3*(C*(5*d^3*e^3 + 94*c*d^2*e^2* 
f + 94*c^2*d*e*f^2 + 5*c^3*f^3) + d*f*(3*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 
+ 16*c*d*e*f + 3*c^2*f^2))))*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b* 
e - a*f)*Sqrt[a + b*x]) - (d*f*((2*Sqrt[-(b*c) + a*d]*(48*a^5*C*d^3*f^3 + 
8*a^4*b*d^2*f^2*(B*d*f - 16*C*(d*e + c*f)) - b^5*(8*A*d^3*e^3 - c*d^2*e...
 

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 167

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

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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 2117

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_ 
.)*(x_))^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[Px, a + b*x, x], 
 R = PolynomialRemainder[Px, a + b*x, x]}, Simp[b*R*(a + b*x)^(m + 1)*(c + 
d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Si 
mp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n* 
(e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f*R*(m + 1 
) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x] 
, x], x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && PolyQ[Px, x] && LtQ[m, - 
1] && IntegersQ[2*m, 2*n, 2*p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3899\) vs. \(2(1644)=3288\).

Time = 8.70 (sec) , antiderivative size = 3900, normalized size of antiderivative = 2.28

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(3900\)
default	\(\text {Expression too large to display}\)	\(70236\)

Input:

int((d*x+c)^(1/2)*(f*x+e)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(9/2),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)* 
(-2/7*(A*b^2-B*a*b+C*a^2)/b^7*(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)/(x+a/b)^4+2/35*(2*A*a*b^2*d*f-A*b^3*c*f- 
A*b^3*d*e-9*B*a^2*b*d*f+8*B*a*b^2*c*f+8*B*a*b^2*d*e-7*B*b^3*c*e+16*C*a^3*d 
*f-15*C*a^2*b*c*f-15*C*a^2*b*d*e+14*C*a*b^2*c*e)/(a^2*d*f-a*b*c*f-a*b*d*e+ 
b^2*c*e)/b^6*(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)/(x+a/b)^3+2/105*(6*A*a^2*b^2*d^2*f^2-6*A*a*b^3*c*d*f^2-6* 
A*a*b^3*d^2*e*f+4*A*b^4*c^2*f^2-2*A*b^4*c*d*e*f+4*A*b^4*d^2*e^2+8*B*a^3*b* 
d^2*f^2-15*B*a^2*b^2*c*d*f^2-15*B*a^2*b^2*d^2*e*f+3*B*a*b^3*c^2*f^2+30*B*a 
*b^3*c*d*e*f+3*B*a*b^3*d^2*e^2-7*B*b^4*c^2*e*f-7*B*b^4*c*d*e^2-57*C*a^4*d^ 
2*f^2+106*C*a^3*b*c*d*f^2+106*C*a^3*b*d^2*e*f-45*C*a^2*b^2*c^2*f^2-198*C*a 
^2*b^2*c*d*e*f-45*C*a^2*b^2*d^2*e^2+84*C*a*b^3*c^2*e*f+84*C*a*b^3*c*d*e^2- 
35*C*b^4*c^2*e^2)/b^5/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)^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)/(x+a/b)^2+2/ 
105*(b*d*f*x^2+b*c*f*x+b*d*e*x+b*c*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)^3/ 
b^4*(6*A*a^3*b^2*d^3*f^3-9*A*a^2*b^3*c*d^2*f^3-9*A*a^2*b^3*d^3*e*f^2+19*A* 
a*b^4*c^2*d*f^3-20*A*a*b^4*c*d^2*e*f^2+19*A*a*b^4*d^3*e^2*f-8*A*b^5*c^3*f^ 
3+5*A*b^5*c^2*d*e*f^2+5*A*b^5*c*d^2*e^2*f-8*A*b^5*d^3*e^3+8*B*a^4*b*d^3*f^ 
3-19*B*a^3*b^2*c*d^2*f^3-19*B*a^3*b^2*d^3*e*f^2+9*B*a^2*b^3*c^2*d*f^3+48*B 
*a^2*b^3*c*d^2*e*f^2+9*B*a^2*b^3*d^3*e^2*f-6*B*a*b^4*c^3*f^3-19*B*a*b^4...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 9150 vs. \(2 (1643) = 3286\).

Time = 2.80 (sec) , antiderivative size = 9150, normalized size of antiderivative = 5.34 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{9/2}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(1/2)*(f*x+e)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(9/2),x, algor 
ithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(1/2)*(f*x+e)**(1/2)*(C*x**2+B*x+A)/(b*x+a)**(9/2),x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{9/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c} \sqrt {f x + e}}{{\left (b x + a\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((d*x+c)^(1/2)*(f*x+e)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(9/2),x, algor 
ithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)*sqrt(f*x + e)/(b*x + a)^(9/2), x 
)
 

Giac [F]

\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{9/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c} \sqrt {f x + e}}{{\left (b x + a\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((d*x+c)^(1/2)*(f*x+e)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(9/2),x, algor 
ithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)*sqrt(f*x + e)/(b*x + a)^(9/2), x 
)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{9/2}} \, dx=\int \frac {\sqrt {e+f\,x}\,\sqrt {c+d\,x}\,\left (C\,x^2+B\,x+A\right )}{{\left (a+b\,x\right )}^{9/2}} \,d x \] Input:

int(((e + f*x)^(1/2)*(c + d*x)^(1/2)*(A + B*x + C*x^2))/(a + b*x)^(9/2),x)
 

Output:

int(((e + f*x)^(1/2)*(c + d*x)^(1/2)*(A + B*x + C*x^2))/(a + b*x)^(9/2), x 
)
 

Reduce [F]

\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{9/2}} \, dx=\int \frac {\sqrt {d x +c}\, \sqrt {f x +e}\, \left (C \,x^{2}+B x +A \right )}{\left (b x +a \right )^{\frac {9}{2}}}d x \] Input:

int((d*x+c)^(1/2)*(f*x+e)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(9/2),x)
 

Output:

int((d*x+c)^(1/2)*(f*x+e)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(9/2),x)