\(\int \frac {(a+b x)^{3/2} (A+B x+C x^2)}{\sqrt {c+d x} \sqrt {e+f x}} \, dx\) [95]

Optimal result

Integrand size = 38, antiderivative size = 825 \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {2 \left (5 d f (5 b c C e+a C d e+a c C f-7 A b d f)-\frac {(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))}{b}\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{105 d^3 f^3}+\frac {2 \left (7 B d f-\frac {2 a C d f}{b}-6 C (d e+c f)\right ) (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{35 d^2 f^2}+\frac {2 C (a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}}{7 b d f}-\frac {2 \sqrt {-b c+a d} \left (3 d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))+(a d f-2 b (d e+c f)) \left (5 d f (5 b c C e+a C d e+a c C f-7 A b d f)-\frac {(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))}{b}\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{105 b d^{7/2} f^4 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {-b c+a d} (b e-a f) \left (3 a^2 C d^2 f^2 (d e-c f)-3 a b d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d e f+11 c^2 f^2\right )\right )-b^2 \left (C \left (48 d^3 e^3+16 c d^2 e^2 f+17 c^2 d e f^2+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d e f+4 c^2 f^2\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt {c+d x} \sqrt {e+f x}} \] Output:

-2/105*(5*d*f*(-7*A*b*d*f+C*a*c*f+C*a*d*e+5*C*b*c*e)-(3*a*d*f-4*b*(c*f+d*e 
))*(7*b*B*d*f-2*a*C*d*f-6*b*C*(c*f+d*e))/b)*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(f 
*x+e)^(1/2)/d^3/f^3+2/35*(7*B*d*f-2*a*C*d*f/b-6*C*(c*f+d*e))*(b*x+a)^(3/2) 
*(d*x+c)^(1/2)*(f*x+e)^(1/2)/d^2/f^2+2/7*C*(b*x+a)^(5/2)*(d*x+c)^(1/2)*(f* 
x+e)^(1/2)/b/d/f-2/105*(a*d-b*c)^(1/2)*(3*d*f*(5*a*d*f*(-7*A*b*d*f+C*a*c*f 
+C*a*d*e+5*C*b*c*e)+(a*c*f+a*d*e+3*b*c*e)*(7*b*B*d*f-2*a*C*d*f-6*b*C*(c*f+ 
d*e)))+(a*d*f-2*b*(c*f+d*e))*(5*d*f*(-7*A*b*d*f+C*a*c*f+C*a*d*e+5*C*b*c*e) 
-(3*a*d*f-4*b*(c*f+d*e))*(7*b*B*d*f-2*a*C*d*f-6*b*C*(c*f+d*e))/b))*(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/d^(7/2)/f^4/(d*x+c)^(1/2)/( 
b*(f*x+e)/(-a*f+b*e))^(1/2)-2/105*(a*d-b*c)^(1/2)*(-a*f+b*e)*(3*a^2*C*d^2* 
f^2*(-c*f+d*e)-3*a*b*d*f*(7*d*f*(-5*A*d*f+2*B*c*f+3*B*d*e)-C*(11*c^2*f^2+8 
*c*d*e*f+16*d^2*e^2))-b^2*(C*(24*c^3*f^3+17*c^2*d*e*f^2+16*c*d^2*e^2*f+48* 
d^3*e^3)+7*d*f*(5*A*d*f*(c*f+2*d*e)-B*(4*c^2*f^2+3*c*d*e*f+8*d^2*e^2))))*( 
b*(d*x+c)/(-a*d+b*c))^(1/2)*(b*(f*x+e)/(-a*f+b*e))^(1/2)*EllipticF(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^2/d^(7 
/2)/f^4/(d*x+c)^(1/2)/(f*x+e)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 29.49 (sec) , antiderivative size = 1000, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {2 \left (-b^2 \sqrt {-a+\frac {b c}{d}} \left (6 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (-7 B d f+4 C (d e+c f))-a b^2 d f \left (C \left (72 d^2 e^2+62 c d e f+72 c^2 f^2\right )+7 d f (20 A d f-13 B (d e+c f))\right )+b^3 \left (8 C \left (6 d^3 e^3+5 c d^2 e^2 f+5 c^2 d e f^2+6 c^3 f^3\right )+7 d f \left (10 A d f (d e+c f)-B \left (8 d^2 e^2+7 c d e f+8 c^2 f^2\right )\right )\right )\right ) (c+d x) (e+f x)+b^2 \sqrt {-a+\frac {b c}{d}} d f (a+b x) (c+d x) (e+f x) \left (3 a^2 C d^2 f^2+3 a b d f (14 B d f+C (-11 d e-11 c f+8 d f x))+b^2 \left (7 d f (5 A d f+B (-4 d e-4 c f+3 d f x))+C \left (24 c^2 f^2+c d f (23 e-18 f x)+3 d^2 \left (8 e^2-6 e f x+5 f^2 x^2\right )\right )\right )\right )-i (b c-a d) f \left (6 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (-7 B d f+4 C (d e+c f))-a b^2 d f \left (C \left (72 d^2 e^2+62 c d e f+72 c^2 f^2\right )+7 d f (20 A d f-13 B (d e+c f))\right )+b^3 \left (8 C \left (6 d^3 e^3+5 c d^2 e^2 f+5 c^2 d e f^2+6 c^3 f^3\right )+7 d f \left (10 A d f (d e+c f)-B \left (8 d^2 e^2+7 c d e f+8 c^2 f^2\right )\right )\right )\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )+i b (b c-a d) f \left (3 a^2 C d^2 f^2 (d e-c f)-3 a b d f \left (7 d f (-2 B d e-3 B c f+5 A d f)+C \left (11 d^2 e^2+8 c d e f+16 c^2 f^2\right )\right )+b^2 \left (C \left (24 d^3 e^3+17 c d^2 e^2 f+16 c^2 d e f^2+48 c^3 f^3\right )+7 d f \left (5 A d f (d e+2 c f)-B \left (4 d^2 e^2+3 c d e f+8 c^2 f^2\right )\right )\right )\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )\right )}{105 b^3 \sqrt {-a+\frac {b c}{d}} d^4 f^4 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \] Input:

Integrate[((a + b*x)^(3/2)*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x] 
),x]
 

Output:

(2*(-(b^2*Sqrt[-a + (b*c)/d]*(6*a^3*C*d^3*f^3 + 3*a^2*b*d^2*f^2*(-7*B*d*f 
+ 4*C*(d*e + c*f)) - a*b^2*d*f*(C*(72*d^2*e^2 + 62*c*d*e*f + 72*c^2*f^2) + 
 7*d*f*(20*A*d*f - 13*B*(d*e + c*f))) + b^3*(8*C*(6*d^3*e^3 + 5*c*d^2*e^2* 
f + 5*c^2*d*e*f^2 + 6*c^3*f^3) + 7*d*f*(10*A*d*f*(d*e + c*f) - B*(8*d^2*e^ 
2 + 7*c*d*e*f + 8*c^2*f^2))))*(c + d*x)*(e + f*x)) + b^2*Sqrt[-a + (b*c)/d 
]*d*f*(a + b*x)*(c + d*x)*(e + f*x)*(3*a^2*C*d^2*f^2 + 3*a*b*d*f*(14*B*d*f 
 + C*(-11*d*e - 11*c*f + 8*d*f*x)) + b^2*(7*d*f*(5*A*d*f + B*(-4*d*e - 4*c 
*f + 3*d*f*x)) + C*(24*c^2*f^2 + c*d*f*(23*e - 18*f*x) + 3*d^2*(8*e^2 - 6* 
e*f*x + 5*f^2*x^2)))) - I*(b*c - a*d)*f*(6*a^3*C*d^3*f^3 + 3*a^2*b*d^2*f^2 
*(-7*B*d*f + 4*C*(d*e + c*f)) - a*b^2*d*f*(C*(72*d^2*e^2 + 62*c*d*e*f + 72 
*c^2*f^2) + 7*d*f*(20*A*d*f - 13*B*(d*e + c*f))) + b^3*(8*C*(6*d^3*e^3 + 5 
*c*d^2*e^2*f + 5*c^2*d*e*f^2 + 6*c^3*f^3) + 7*d*f*(10*A*d*f*(d*e + c*f) - 
B*(8*d^2*e^2 + 7*c*d*e*f + 8*c^2*f^2))))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x) 
)/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqr 
t[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)] + I*b*(b* 
c - a*d)*f*(3*a^2*C*d^2*f^2*(d*e - c*f) - 3*a*b*d*f*(7*d*f*(-2*B*d*e - 3*B 
*c*f + 5*A*d*f) + C*(11*d^2*e^2 + 8*c*d*e*f + 16*c^2*f^2)) + b^2*(C*(24*d^ 
3*e^3 + 17*c*d^2*e^2*f + 16*c^2*d*e*f^2 + 48*c^3*f^3) + 7*d*f*(5*A*d*f*(d* 
e + 2*c*f) - B*(4*d^2*e^2 + 3*c*d*e*f + 8*c^2*f^2))))*(a + b*x)^(3/2)*Sqrt 
[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*Ellipti...
 

Rubi [A] (verified)

Time = 1.70 (sec) , antiderivative size = 857, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2118, 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.

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

\(\Big \downarrow \) 2118

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 176

\(\displaystyle \frac {2 C (a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}}{7 b d f}-\frac {\frac {\frac {\frac {(b e-a f) \left (3 a^2 C d^2 f^2 (d e-c f)-3 a b d f \left (7 d f (-5 A d f+2 B c f+3 B d e)-C \left (11 c^2 f^2+8 c d e f+16 d^2 e^2\right )\right )-\left (b^2 \left (7 d f \left (5 A d f (c f+2 d e)-B \left (4 c^2 f^2+3 c d e f+8 d^2 e^2\right )\right )+C \left (24 c^3 f^3+17 c^2 d e f^2+16 c d^2 e^2 f+48 d^3 e^3\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {\left (3 b d f (5 a d f (a c C f+a C d e-7 A b d f+5 b c C e)+(a c f+a d e+3 b c e) (-2 a C d f+7 b B d f-6 b C (c f+d e)))+2 \left (\frac {a d f}{2}-b (c f+d e)\right ) (5 b d f (a c C f+a C d e-7 A b d f+5 b c C e)-(3 a d f-4 b (c f+d e)) (-2 a C d f+7 b B d f-6 b C (c f+d e)))\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}}dx}{f}}{3 d f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (a c C f+a C d e-7 A b d f+5 b c C e)-(3 a d f-4 b (c f+d e)) (-2 a C d f+7 b B d f-6 b C (c f+d e)))}{3 d f}}{5 d f}-\frac {2 (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+7 b B d f-6 b C (c f+d e))}{5 d f}}{7 b d f}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {2 C (a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}}{7 b d f}-\frac {\frac {\frac {\frac {(b e-a f) \left (3 a^2 C d^2 f^2 (d e-c f)-3 a b d f \left (7 d f (-5 A d f+2 B c f+3 B d e)-C \left (11 c^2 f^2+8 c d e f+16 d^2 e^2\right )\right )-\left (b^2 \left (7 d f \left (5 A d f (c f+2 d e)-B \left (4 c^2 f^2+3 c d e f+8 d^2 e^2\right )\right )+C \left (24 c^3 f^3+17 c^2 d e f^2+16 c d^2 e^2 f+48 d^3 e^3\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {\sqrt {e+f x} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (3 b d f (5 a d f (a c C f+a C d e-7 A b d f+5 b c C e)+(a c f+a d e+3 b c e) (-2 a C d f+7 b B d f-6 b C (c f+d e)))+2 \left (\frac {a d f}{2}-b (c f+d e)\right ) (5 b d f (a c C f+a C d e-7 A b d f+5 b c C e)-(3 a d f-4 b (c f+d e)) (-2 a C d f+7 b B d f-6 b C (c f+d e)))\right ) \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}dx}{f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 d f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (a c C f+a C d e-7 A b d f+5 b c C e)-(3 a d f-4 b (c f+d e)) (-2 a C d f+7 b B d f-6 b C (c f+d e)))}{3 d f}}{5 d f}-\frac {2 (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+7 b B d f-6 b C (c f+d e))}{5 d f}}{7 b d f}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {2 C (a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}}{7 b d f}-\frac {\frac {\frac {\frac {(b e-a f) \left (3 a^2 C d^2 f^2 (d e-c f)-3 a b d f \left (7 d f (-5 A d f+2 B c f+3 B d e)-C \left (11 c^2 f^2+8 c d e f+16 d^2 e^2\right )\right )-\left (b^2 \left (7 d f \left (5 A d f (c f+2 d e)-B \left (4 c^2 f^2+3 c d e f+8 d^2 e^2\right )\right )+C \left (24 c^3 f^3+17 c^2 d e f^2+16 c d^2 e^2 f+48 d^3 e^3\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (3 b d f (5 a d f (a c C f+a C d e-7 A b d f+5 b c C e)+(a c f+a d e+3 b c e) (-2 a C d f+7 b B d f-6 b C (c f+d e)))+2 \left (\frac {a d f}{2}-b (c f+d e)\right ) (5 b d f (a c C f+a C d e-7 A b d f+5 b c C e)-(3 a d f-4 b (c f+d e)) (-2 a C d f+7 b B d f-6 b C (c f+d e)))\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 d f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (a c C f+a C d e-7 A b d f+5 b c C e)-(3 a d f-4 b (c f+d e)) (-2 a C d f+7 b B d f-6 b C (c f+d e)))}{3 d f}}{5 d f}-\frac {2 (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+7 b B d f-6 b C (c f+d e))}{5 d f}}{7 b d f}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {2 C (a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}}{7 b d f}-\frac {\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))}{3 d f}+\frac {\frac {2 \sqrt {a d-b c} \left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))+2 \left (\frac {a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {(b e-a f) \left (-\left (\left (C \left (48 d^3 e^3+16 c d^2 f e^2+17 c^2 d f^2 e+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d f e+4 c^2 f^2\right )\right )\right ) b^2\right )-3 a d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d f e+11 c^2 f^2\right )\right ) b+3 a^2 C d^2 f^2 (d e-c f)\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}}dx}{f \sqrt {c+d x}}}{3 d f}}{5 d f}-\frac {2 (7 b B d f-2 a C d f-6 b C (d e+c f)) (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{5 d f}}{7 b d f}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {2 C (a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}}{7 b d f}-\frac {\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))}{3 d f}+\frac {\frac {2 \sqrt {a d-b c} \left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))+2 \left (\frac {a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {(b e-a f) \left (-\left (\left (C \left (48 d^3 e^3+16 c d^2 f e^2+17 c^2 d f^2 e+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d f e+4 c^2 f^2\right )\right )\right ) b^2\right )-3 a d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d f e+11 c^2 f^2\right )\right ) b+3 a^2 C d^2 f^2 (d e-c f)\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}dx}{f \sqrt {c+d x} \sqrt {e+f x}}}{3 d f}}{5 d f}-\frac {2 (7 b B d f-2 a C d f-6 b C (d e+c f)) (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{5 d f}}{7 b d f}\)

\(\Big \downarrow \) 130

\(\displaystyle \frac {2 C (a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}}{7 b d f}-\frac {\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))}{3 d f}+\frac {\frac {2 \sqrt {a d-b c} \left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))+2 \left (\frac {a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \sqrt {a d-b c} (b e-a f) \left (-\left (\left (C \left (48 d^3 e^3+16 c d^2 f e^2+17 c^2 d f^2 e+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d f e+4 c^2 f^2\right )\right )\right ) b^2\right )-3 a d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d f e+11 c^2 f^2\right )\right ) b+3 a^2 C d^2 f^2 (d e-c f)\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {e+f x}}}{3 d f}}{5 d f}-\frac {2 (7 b B d f-2 a C d f-6 b C (d e+c f)) (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{5 d f}}{7 b d f}\)

Input:

Int[((a + b*x)^(3/2)*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]),x]
 

Output:

(2*C*(a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x])/(7*b*d*f) - ((-2*(7*b*B* 
d*f - 2*a*C*d*f - 6*b*C*(d*e + c*f))*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e 
+ f*x])/(5*d*f) + ((2*(5*b*d*f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A*b*d*f) 
 - (3*a*d*f - 4*b*(d*e + c*f))*(7*b*B*d*f - 2*a*C*d*f - 6*b*C*(d*e + c*f)) 
)*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*d*f) + ((2*Sqrt[-(b*c) + a 
*d]*(3*b*d*f*(5*a*d*f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A*b*d*f) + (3*b*c 
*e + a*d*e + a*c*f)*(7*b*B*d*f - 2*a*C*d*f - 6*b*C*(d*e + c*f))) + 2*((a*d 
*f)/2 - b*(d*e + c*f))*(5*b*d*f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A*b*d*f 
) - (3*a*d*f - 4*b*(d*e + c*f))*(7*b*B*d*f - 2*a*C*d*f - 6*b*C*(d*e + c*f) 
)))*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)*(3*a^2*C*d^2*f^2*(d*e - c*f) - 3*a*b*d*f*(7*d*f*(3*B*d 
*e + 2*B*c*f - 5*A*d*f) - C*(16*d^2*e^2 + 8*c*d*e*f + 11*c^2*f^2)) - b^2*( 
C*(48*d^3*e^3 + 16*c*d^2*e^2*f + 17*c^2*d*e*f^2 + 24*c^3*f^3) + 7*d*f*(5*A 
*d*f*(2*d*e + c*f) - B*(8*d^2*e^2 + 3*c*d*e*f + 4*c^2*f^2))))*Sqrt[(b*(c + 
 d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[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[e + f*x]))/(3*d*f))/(5*d*f))/(7*b*d*f)
 

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 = 6.52 (sec) , antiderivative size = 1233, normalized size of antiderivative = 1.49

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1233\)
default	\(\text {Expression too large to display}\)	\(10454\)

Input:

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

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 1388, normalized size of antiderivative = 1.68 \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algor 
ithm="fricas")
 

Output:

2/315*(3*(15*C*b^4*d^4*f^4*x^2 + 24*C*b^4*d^4*e^2*f^2 + (23*C*b^4*c*d^3 - 
(33*C*a*b^3 + 28*B*b^4)*d^4)*e*f^3 + (24*C*b^4*c^2*d^2 - (33*C*a*b^3 + 28* 
B*b^4)*c*d^3 + (3*C*a^2*b^2 + 42*B*a*b^3 + 35*A*b^4)*d^4)*f^4 - 3*(6*C*b^4 
*d^4*e*f^3 + (6*C*b^4*c*d^3 - (8*C*a*b^3 + 7*B*b^4)*d^4)*f^4)*x)*sqrt(b*x 
+ a)*sqrt(d*x + c)*sqrt(f*x + e) + (48*C*b^4*d^4*e^4 + 8*(2*C*b^4*c*d^3 - 
(12*C*a*b^3 + 7*B*b^4)*d^4)*e^3*f + (11*C*b^4*c^2*d^2 - 7*(4*C*a*b^3 + 3*B 
*b^4)*c*d^3 + (39*C*a^2*b^2 + 119*B*a*b^3 + 70*A*b^4)*d^4)*e^2*f^2 + (16*C 
*b^4*c^3*d - 7*(4*C*a*b^3 + 3*B*b^4)*c^2*d^2 + 7*(C*a^2*b^2 + 7*B*a*b^3 + 
5*A*b^4)*c*d^3 + (9*C*a^3*b - 56*B*a^2*b^2 - 175*A*a*b^3)*d^4)*e*f^3 + (48 
*C*b^4*c^4 - 8*(12*C*a*b^3 + 7*B*b^4)*c^3*d + (39*C*a^2*b^2 + 119*B*a*b^3 
+ 70*A*b^4)*c^2*d^2 + (9*C*a^3*b - 56*B*a^2*b^2 - 175*A*a*b^3)*c*d^3 + (6* 
C*a^4 - 21*B*a^3*b + 175*A*a^2*b^2)*d^4)*f^4)*sqrt(b*d*f)*weierstrassPInve 
rse(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b*c*d + a^2* 
d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b^2*d^3)* 
e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - 3*a 
*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f^3), 1/3*(3*b*d*f*x 
 + b*d*e + (b*c + a*d)*f)/(b*d*f)) + 3*(48*C*b^4*d^4*e^3*f + 8*(5*C*b^4*c* 
d^3 - (9*C*a*b^3 + 7*B*b^4)*d^4)*e^2*f^2 + (40*C*b^4*c^2*d^2 - (62*C*a*b^3 
 + 49*B*b^4)*c*d^3 + (12*C*a^2*b^2 + 91*B*a*b^3 + 70*A*b^4)*d^4)*e*f^3 + ( 
48*C*b^4*c^3*d - 8*(9*C*a*b^3 + 7*B*b^4)*c^2*d^2 + (12*C*a^2*b^2 + 91*B...
 

Sympy [F]

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

integrate((b*x+a)**(3/2)*(C*x**2+B*x+A)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)
 

Output:

Integral((a + b*x)**(3/2)*(A + B*x + C*x**2)/(sqrt(c + d*x)*sqrt(e + f*x)) 
, x)
 

Maxima [F]

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

integrate((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algor 
ithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)*(b*x + a)^(3/2)/(sqrt(d*x + c)*sqrt(f*x + e)), 
 x)
 

Giac [F]

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

integrate((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algor 
ithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)*(b*x + a)^(3/2)/(sqrt(d*x + c)*sqrt(f*x + e)), 
 x)
 

Mupad [F(-1)]

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

int(((a + b*x)^(3/2)*(A + B*x + C*x^2))/((e + f*x)^(1/2)*(c + d*x)^(1/2)), 
x)
                                                                                    
                                                                                    
 

Output:

int(((a + b*x)^(3/2)*(A + B*x + C*x^2))/((e + f*x)^(1/2)*(c + d*x)^(1/2)), 
 x)
 

Reduce [F]

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

int((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x)
 

Output:

int((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x)