\(\int \frac {\sqrt {c+d x} \sqrt {e+f x} (A+B x+C x^2)}{\sqrt {a+b x}} \, dx\) [84]

Optimal result

Integrand size = 38, antiderivative size = 762 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{\sqrt {a+b x}} \, dx=\frac {2 \left (\frac {(b d e-2 b c f-4 a d f) (7 b B d f-6 a C d f-4 b C (d e+c f))}{b}-5 d f (3 a C (d e+c f)+b (c C e-7 A d f))\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{105 b^2 d^2 f^2}+\frac {2 (7 b B d f-6 a C d f-4 b C (d e+c f)) \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x}}{35 b^2 d^2 f}+\frac {2 C \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f}-\frac {2 \sqrt {-b c+a d} \left (3 d f ((b c e+3 a d e+a c f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b d e (3 a C (d e+c f)+b (c C e-7 A d f)))-\frac {(b c f-2 d (b e+a f)) ((b d e-2 b c f-4 a d f) (7 b B d f-6 a C d f-4 b C (d e+c f))-5 b d f (3 a C (d e+c f)+b (c C e-7 A d f)))}{b}\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{105 b^3 d^{5/2} f^3 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {-b c+a d} (b e-a f) (d e-c f) \left (24 a^2 C d^2 f^2+a b d f (13 C d e-5 c C f-28 B d f)-b^2 \left (7 d f (2 B d e-B c f-5 A d f)-C \left (8 d^2 e^2-c d e f-4 c^2 f^2\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{105 b^4 d^{5/2} f^3 \sqrt {c+d x} \sqrt {e+f x}} \] Output:

2/105*((-4*a*d*f-2*b*c*f+b*d*e)*(7*b*B*d*f-6*a*C*d*f-4*b*C*(c*f+d*e))/b-5* 
d*f*(3*a*C*(c*f+d*e)+b*(-7*A*d*f+C*c*e)))*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(f*x 
+e)^(1/2)/b^2/d^2/f^2+2/35*(7*b*B*d*f-6*a*C*d*f-4*b*C*(c*f+d*e))*(b*x+a)^( 
1/2)*(d*x+c)^(3/2)*(f*x+e)^(1/2)/b^2/d^2/f+2/7*C*(b*x+a)^(1/2)*(d*x+c)^(3/ 
2)*(f*x+e)^(3/2)/b/d/f-2/105*(a*d-b*c)^(1/2)*(3*d*f*((a*c*f+3*a*d*e+b*c*e) 
*(7*b*B*d*f-6*a*C*d*f-4*b*C*(c*f+d*e))+5*b*d*e*(3*a*C*(c*f+d*e)+b*(-7*A*d* 
f+C*c*e)))-(b*c*f-2*d*(a*f+b*e))*((-4*a*d*f-2*b*c*f+b*d*e)*(7*b*B*d*f-6*a* 
C*d*f-4*b*C*(c*f+d*e))-5*b*d*f*(3*a*C*(c*f+d*e)+b*(-7*A*d*f+C*c*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^3/d^(5/2)/f^3/(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)*(-c*f+ 
d*e)*(24*a^2*C*d^2*f^2+a*b*d*f*(-28*B*d*f-5*C*c*f+13*C*d*e)-b^2*(7*d*f*(-5 
*A*d*f-B*c*f+2*B*d*e)-C*(-4*c^2*f^2-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^4/d^(5/2)/f^3/(d*x+c)^ 
(1/2)/(f*x+e)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 29.02 (sec) , antiderivative size = 917, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{\sqrt {a+b x}} \, dx=-\frac {2 \left (b^2 \sqrt {-a+\frac {b c}{d}} \left (48 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (7 B d f+2 C (d e+c f))+a b^2 d f \left (7 d f (3 B d e+3 B c f+10 A d f)+C \left (-9 d^2 e^2+8 c d e f-9 c^2 f^2\right )\right )+b^3 \left (C \left (-8 d^3 e^3+5 c d^2 e^2 f+5 c^2 d e f^2-8 c^3 f^3\right )-7 d f \left (5 A d f (d e+c f)-2 B \left (d^2 e^2-c d e f+c^2 f^2\right )\right )\right )\right ) (c+d x) (e+f x)+b^2 \sqrt {-a+\frac {b c}{d}} d f (a+b x) (c+d x) (e+f x) \left (-24 a^2 C d^2 f^2+a b d f (28 B d f+C (5 d e+5 c f+18 d f x))+b^2 \left (-7 d f (B c f+5 A d f+B d (e+3 f x))+C \left (4 c^2 f^2-c d f (2 e+3 f x)+d^2 \left (4 e^2-3 e f x-15 f^2 x^2\right )\right )\right )\right )+i (b c-a d) f \left (48 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (7 B d f+2 C (d e+c f))+a b^2 d f \left (7 d f (3 B d e+3 B c f+10 A d f)+C \left (-9 d^2 e^2+8 c d e f-9 c^2 f^2\right )\right )+b^3 \left (C \left (-8 d^3 e^3+5 c d^2 e^2 f+5 c^2 d e f^2-8 c^3 f^3\right )-7 d f \left (5 A d f (d e+c f)-2 B \left (d^2 e^2-c d e f+c^2 f^2\right )\right )\right )\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )-i b (b c-a d) f (d e-c f) \left (24 a^2 C d^2 f^2+a b d f (-5 C d e+13 c C f-28 B d f)+b^2 \left (7 d f (B d e-2 B c f+5 A d f)-C \left (4 d^2 e^2+c d e f-8 c^2 f^2\right )\right )\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )\right )}{105 b^5 \sqrt {-a+\frac {b c}{d}} d^3 f^3 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \] Input:

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

Output:

(-2*(b^2*Sqrt[-a + (b*c)/d]*(48*a^3*C*d^3*f^3 - 8*a^2*b*d^2*f^2*(7*B*d*f + 
 2*C*(d*e + c*f)) + a*b^2*d*f*(7*d*f*(3*B*d*e + 3*B*c*f + 10*A*d*f) + C*(- 
9*d^2*e^2 + 8*c*d*e*f - 9*c^2*f^2)) + b^3*(C*(-8*d^3*e^3 + 5*c*d^2*e^2*f + 
 5*c^2*d*e*f^2 - 8*c^3*f^3) - 7*d*f*(5*A*d*f*(d*e + c*f) - 2*B*(d^2*e^2 - 
c*d*e*f + 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)*(-24*a^2*C*d^2*f^2 + a*b*d*f*(28*B*d*f + C*(5*d 
*e + 5*c*f + 18*d*f*x)) + b^2*(-7*d*f*(B*c*f + 5*A*d*f + B*d*(e + 3*f*x)) 
+ C*(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*(48*a^3*C*d^3*f^3 - 8*a^2*b*d^2*f^2*(7*B*d*f + 2*C*(d 
*e + c*f)) + a*b^2*d*f*(7*d*f*(3*B*d*e + 3*B*c*f + 10*A*d*f) + C*(-9*d^2*e 
^2 + 8*c*d*e*f - 9*c^2*f^2)) + b^3*(C*(-8*d^3*e^3 + 5*c*d^2*e^2*f + 5*c^2* 
d*e*f^2 - 8*c^3*f^3) - 7*d*f*(5*A*d*f*(d*e + c*f) - 2*B*(d^2*e^2 - c*d*e*f 
 + 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[Sqrt[-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*(d*e - c*f)*(24 
*a^2*C*d^2*f^2 + a*b*d*f*(-5*C*d*e + 13*c*C*f - 28*B*d*f) + b^2*(7*d*f*(B* 
d*e - 2*B*c*f + 5*A*d*f) - C*(4*d^2*e^2 + 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))] 
*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b 
*c*f - a*d*f)]))/(105*b^5*Sqrt[-a + (b*c)/d]*d^3*f^3*Sqrt[a + b*x]*Sqrt...
 

Rubi [A] (verified)

Time = 1.79 (sec) , antiderivative size = 789, 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.

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

\(\Big \downarrow \) 2118

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 176

\(\displaystyle \frac {2 C \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f}-\frac {\frac {\frac {\frac {(b e-a f) (d e-c f) \left (24 a^2 C d^2 f^2+a b d f (-28 B d f-5 c C f+13 C d e)-\left (b^2 \left (7 d f (-5 A d f-B c f+2 B d e)-C \left (-4 c^2 f^2-c d e f+8 d^2 e^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {\left (3 b d f (5 b c f (3 a C (c f+d e)+b (c C e-7 A d f))+(3 a c f+a d e+b c e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))+2 \left (\frac {b d e}{2}-f (a d+b c)\right ) (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))+(4 a d f-b c f+2 b d e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}}dx}{f}}{3 b d}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))+(4 a d f-b c f+2 b d e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))}{3 b d}}{5 b f}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} (-6 a C d f+7 b B d f-4 b C (c f+d e))}{5 b f}}{7 b d f}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {2 C \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f}-\frac {\frac {\frac {\frac {(b e-a f) (d e-c f) \left (24 a^2 C d^2 f^2+a b d f (-28 B d f-5 c C f+13 C d e)-\left (b^2 \left (7 d f (-5 A d f-B c f+2 B d e)-C \left (-4 c^2 f^2-c d e f+8 d^2 e^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {\sqrt {e+f x} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (3 b d f (5 b c f (3 a C (c f+d e)+b (c C e-7 A d f))+(3 a c f+a d e+b c e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))+2 \left (\frac {b d e}{2}-f (a d+b c)\right ) (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))+(4 a d f-b c f+2 b d e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))\right ) \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}dx}{f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 b d}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))+(4 a d f-b c f+2 b d e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))}{3 b d}}{5 b f}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} (-6 a C d f+7 b B d f-4 b C (c f+d e))}{5 b f}}{7 b d f}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {2 C \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f}-\frac {\frac {\frac {\frac {(b e-a f) (d e-c f) \left (24 a^2 C d^2 f^2+a b d f (-28 B d f-5 c C f+13 C d e)-\left (b^2 \left (7 d f (-5 A d f-B c f+2 B d e)-C \left (-4 c^2 f^2-c d e f+8 d^2 e^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (3 b d f (5 b c f (3 a C (c f+d e)+b (c C e-7 A d f))+(3 a c f+a d e+b c e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))+2 \left (\frac {b d e}{2}-f (a d+b c)\right ) (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))+(4 a d f-b c f+2 b d e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 b d}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))+(4 a d f-b c f+2 b d e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))}{3 b d}}{5 b f}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} (-6 a C d f+7 b B d f-4 b C (c f+d e))}{5 b f}}{7 b d f}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {2 C \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f}-\frac {\frac {\frac {\frac {(b e-a f) (d e-c f) \sqrt {\frac {b (c+d x)}{b c-a d}} \left (24 a^2 C d^2 f^2+a b d f (-28 B d f-5 c C f+13 C d e)-\left (b^2 \left (7 d f (-5 A d f-B c f+2 B d e)-C \left (-4 c^2 f^2-c d e f+8 d^2 e^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}}dx}{f \sqrt {c+d x}}+\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (3 b d f (5 b c f (3 a C (c f+d e)+b (c C e-7 A d f))+(3 a c f+a d e+b c e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))+2 \left (\frac {b d e}{2}-f (a d+b c)\right ) (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))+(4 a d f-b c f+2 b d e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 b d}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))+(4 a d f-b c f+2 b d e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))}{3 b d}}{5 b f}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} (-6 a C d f+7 b B d f-4 b C (c f+d e))}{5 b f}}{7 b d f}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {2 C \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f}-\frac {\frac {\frac {\frac {(b e-a f) (d e-c f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \left (24 a^2 C d^2 f^2+a b d f (-28 B d f-5 c C f+13 C d e)-\left (b^2 \left (7 d f (-5 A d f-B c f+2 B d e)-C \left (-4 c^2 f^2-c d e f+8 d^2 e^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}dx}{f \sqrt {c+d x} \sqrt {e+f x}}+\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (3 b d f (5 b c f (3 a C (c f+d e)+b (c C e-7 A d f))+(3 a c f+a d e+b c e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))+2 \left (\frac {b d e}{2}-f (a d+b c)\right ) (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))+(4 a d f-b c f+2 b d e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 b d}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))+(4 a d f-b c f+2 b d e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))}{3 b d}}{5 b f}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} (-6 a C d f+7 b B d f-4 b C (c f+d e))}{5 b f}}{7 b d f}\)

\(\Big \downarrow \) 130

\(\displaystyle \frac {2 C \sqrt {a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f}-\frac {\frac {\frac {\frac {2 \sqrt {a d-b c} (b e-a f) (d e-c f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \left (24 a^2 C d^2 f^2+a b d f (-28 B d f-5 c C f+13 C d e)-\left (b^2 \left (7 d f (-5 A d f-B c f+2 B d e)-C \left (-4 c^2 f^2-c d e f+8 d^2 e^2\right )\right )\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {e+f x}}+\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (3 b d f (5 b c f (3 a C (c f+d e)+b (c C e-7 A d f))+(3 a c f+a d e+b c e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))+2 \left (\frac {b d e}{2}-f (a d+b c)\right ) (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))+(4 a d f-b c f+2 b d e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 b d}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))+(4 a d f-b c f+2 b d e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))}{3 b d}}{5 b f}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} (-6 a C d f+7 b B d f-4 b C (c f+d e))}{5 b f}}{7 b d f}\)

Input:

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

Output:

(2*C*Sqrt[a + b*x]*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(7*b*d*f) - ((-2*(7*b* 
B*d*f - 6*a*C*d*f - 4*b*C*(d*e + c*f))*Sqrt[a + b*x]*Sqrt[c + d*x]*(e + f* 
x)^(3/2))/(5*b*f) + ((2*((2*b*d*e - b*c*f + 4*a*d*f)*(7*b*B*d*f - 6*a*C*d* 
f - 4*b*C*(d*e + c*f)) + 5*b*d*f*(3*a*C*(d*e + c*f) + b*(c*C*e - 7*A*d*f)) 
)*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b*d) + ((2*Sqrt[-(b*c) + a 
*d]*(3*b*d*f*((b*c*e + a*d*e + 3*a*c*f)*(7*b*B*d*f - 6*a*C*d*f - 4*b*C*(d* 
e + c*f)) + 5*b*c*f*(3*a*C*(d*e + c*f) + b*(c*C*e - 7*A*d*f))) + 2*((b*d*e 
)/2 - (b*c + a*d)*f)*((2*b*d*e - b*c*f + 4*a*d*f)*(7*b*B*d*f - 6*a*C*d*f - 
 4*b*C*(d*e + c*f)) + 5*b*d*f*(3*a*C*(d*e + c*f) + b*(c*C*e - 7*A*d*f))))* 
Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sq 
rt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b*Sqr 
t[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)*(24*a^2*C*d^2*f^2 + a*b*d*f*(13*C*d*e - 5*c*C* 
f - 28*B*d*f) - b^2*(7*d*f*(2*B*d*e - B*c*f - 5*A*d*f) - C*(8*d^2*e^2 - 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*b*d))/(5*b*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 = 4.14 (sec) , antiderivative size = 1205, normalized size of antiderivative = 1.58

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1205\)
default	\(\text {Expression too large to display}\)	\(10175\)

Input:

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

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 1393, normalized size of antiderivative = 1.83 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{\sqrt {a+b x}} \, dx=\text {Too large to display} \] Input:

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

Output:

2/315*(3*(15*C*b^4*d^4*f^4*x^2 - 4*C*b^4*d^4*e^2*f^2 + (2*C*b^4*c*d^3 - (5 
*C*a*b^3 - 7*B*b^4)*d^4)*e*f^3 - (4*C*b^4*c^2*d^2 + (5*C*a*b^3 - 7*B*b^4)* 
c*d^3 - (24*C*a^2*b^2 - 28*B*a*b^3 + 35*A*b^4)*d^4)*f^4 + 3*(C*b^4*d^4*e*f 
^3 + (C*b^4*c*d^3 - (6*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) - (8*C*b^4*d^4*e^4 - (9*C*b^4*c*d^3 - (5*C*a*b^3 - 
14*B*b^4)*d^4)*e^3*f - (4*C*b^4*c^2*d^2 + 7*(C*a*b^3 - 3*B*b^4)*c*d^3 - (1 
0*C*a^2*b^2 - 14*B*a*b^3 + 35*A*b^4)*d^4)*e^2*f^2 - (9*C*b^4*c^3*d + 7*(C* 
a*b^3 - 3*B*b^4)*c^2*d^2 + 14*(3*C*a^2*b^2 - 4*B*a*b^3 + 10*A*b^4)*c*d^3 - 
 (40*C*a^3*b - 49*B*a^2*b^2 + 70*A*a*b^3)*d^4)*e*f^3 + (8*C*b^4*c^4 + (5*C 
*a*b^3 - 14*B*b^4)*c^3*d + (10*C*a^2*b^2 - 14*B*a*b^3 + 35*A*b^4)*c^2*d^2 
+ (40*C*a^3*b - 49*B*a^2*b^2 + 70*A*a*b^3)*c*d^3 - 2*(24*C*a^4 - 28*B*a^3* 
b + 35*A*a^2*b^2)*d^4)*f^4)*sqrt(b*d*f)*weierstrassPInverse(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*(8*C*b^4*d^4*e^3*f - (5*C*b^4*c*d^3 - (9*C*a*b^3 - 14 
*B*b^4)*d^4)*e^2*f^2 - (5*C*b^4*c^2*d^2 + 2*(4*C*a*b^3 - 7*B*b^4)*c*d^3 - 
(16*C*a^2*b^2 - 21*B*a*b^3 + 35*A*b^4)*d^4)*e*f^3 + (8*C*b^4*c^3*d + (9*C* 
a*b^3 - 14*B*b^4)*c^2*d^2 + (16*C*a^2*b^2 - 21*B*a*b^3 + 35*A*b^4)*c*d^...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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