\(\int \frac {\sqrt {a+b x} \sqrt {c+d x} (A+B x+C x^2)}{\sqrt {e+f x}} \, dx\) [90]

Optimal result

Integrand size = 38, antiderivative size = 767 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=-\frac {2 \left (5 d f (3 b c C e+3 a C d e+a c C f-7 A b d f)-\frac {(4 b d e-b c f+2 a d f) (4 a C d f+b (6 C d e+4 c C f-7 B d f))}{b}\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{105 b d^2 f^3}-\frac {2 (4 a C d f+b (6 C d e+4 c C f-7 B d f)) (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{35 b^2 d f^2}+\frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x}}{7 b d f}-\frac {2 \sqrt {-b c+a d} \left (3 d f (5 b c f (3 b c C e+3 a C d e+a c C f-7 A b d f)-(3 b c e+a d e+a c f) (4 a C d f+b (6 C d e+4 c C f-7 B d f)))+(a d f-2 b (d e+c f)) \left (5 d f (3 b c C e+3 a C d e+a c C f-7 A b d f)-\frac {(4 b d e-b c f+2 a d f) (4 a C d f+b (6 C d e+4 c C f-7 B d f))}{b}\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{5/2} f^4 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \sqrt {-b c+a d} (b e-a f) (d e-c f) \left (4 a^2 C d^2 f^2+a b d f (8 C d e-2 c C f-7 B d f)-b^2 \left (7 d f (8 B d e+B c f-10 A d f)-4 C \left (12 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{105 b^3 d^{5/2} f^4 \sqrt {c+d x} \sqrt {e+f x}} \] Output:

-2/105*(5*d*f*(-7*A*b*d*f+C*a*c*f+3*C*a*d*e+3*C*b*c*e)-(2*a*d*f-b*c*f+4*b* 
d*e)*(4*a*C*d*f+b*(-7*B*d*f+4*C*c*f+6*C*d*e))/b)*(b*x+a)^(1/2)*(d*x+c)^(1/ 
2)*(f*x+e)^(1/2)/b/d^2/f^3-2/35*(4*a*C*d*f+b*(-7*B*d*f+4*C*c*f+6*C*d*e))*( 
b*x+a)^(3/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^2/d/f^2+2/7*C*(b*x+a)^(3/2)*(d* 
x+c)^(3/2)*(f*x+e)^(1/2)/b/d/f-2/105*(a*d-b*c)^(1/2)*(3*d*f*(5*b*c*f*(-7*A 
*b*d*f+C*a*c*f+3*C*a*d*e+3*C*b*c*e)-(a*c*f+a*d*e+3*b*c*e)*(4*a*C*d*f+b*(-7 
*B*d*f+4*C*c*f+6*C*d*e)))+(a*d*f-2*b*(c*f+d*e))*(5*d*f*(-7*A*b*d*f+C*a*c*f 
+3*C*a*d*e+3*C*b*c*e)-(2*a*d*f-b*c*f+4*b*d*e)*(4*a*C*d*f+b*(-7*B*d*f+4*C*c 
*f+6*C*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^2/d 
^(5/2)/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)*(-c*f+d*e)*(4*a^2*C*d^2*f^2+a*b*d*f*(-7*B*d*f-2*C*c*f+8*C*d*e 
)-b^2*(7*d*f*(-10*A*d*f+B*c*f+8*B*d*e)-4*C*(c^2*f^2+2*c*d*e*f+12*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^3/d 
^(5/2)/f^4/(d*x+c)^(1/2)/(f*x+e)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 29.06 (sec) , antiderivative size = 922, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=\frac {2 \left (b^2 \sqrt {-a+\frac {b c}{d}} \left (8 a^3 C d^3 f^3+a^2 b d^2 f^2 (9 C d e-5 c C f-14 B d f)+a b^2 d f \left (7 d f (-3 B d e+2 B c f+5 A d f)+C \left (16 d^2 e^2-8 c d e f-5 c^2 f^2\right )\right )+b^3 \left (C \left (-48 d^3 e^3+16 c d^2 e^2 f+9 c^2 d e f^2+8 c^3 f^3\right )+7 d f \left (5 A d f (-2 d e+c f)+B \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )\right )\right )\right ) (c+d x) (e+f x)+b^2 \sqrt {-a+\frac {b c}{d}} d f (a+b x) (c+d x) (e+f x) \left (-4 a^2 C d^2 f^2+a b d f (7 B d f+C (-5 d e+2 c f+3 d f x))+b^2 \left (7 d f (5 A d f+B (-4 d e+c f+3 d f x))+C \left (-4 c^2 f^2+c d f (-5 e+3 f x)+3 d^2 \left (8 e^2-6 e f x+5 f^2 x^2\right )\right )\right )\right )+i (b c-a d) f \left (8 a^3 C d^3 f^3+a^2 b d^2 f^2 (9 C d e-5 c C f-14 B d f)+a b^2 d f \left (7 d f (-3 B d e+2 B c f+5 A d f)+C \left (16 d^2 e^2-8 c d e f-5 c^2 f^2\right )\right )+b^3 \left (C \left (-48 d^3 e^3+16 c d^2 e^2 f+9 c^2 d e f^2+8 c^3 f^3\right )+7 d f \left (5 A d f (-2 d e+c f)+B \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )\right )\right )\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )-i b (b c-a d) f (d e-c f) \left (4 a^2 C d^2 f^2+a b d f (5 C d e+c C f-7 B d f)-b^2 \left (7 d f (-4 B d e-2 B c f+5 A d f)+C \left (24 d^2 e^2+13 c d e f+8 c^2 f^2\right )\right )\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )\right )}{105 b^4 \sqrt {-a+\frac {b c}{d}} d^3 f^4 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \] Input:

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

Output:

(2*(b^2*Sqrt[-a + (b*c)/d]*(8*a^3*C*d^3*f^3 + a^2*b*d^2*f^2*(9*C*d*e - 5*c 
*C*f - 14*B*d*f) + a*b^2*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 - 5*c^2*f^2)) + b^3*(C*(-48*d^3*e^3 + 16*c*d^2*e^2*f 
+ 9*c^2*d*e*f^2 + 8*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 - 2*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)*(-4*a^2*C*d^2*f^2 + a*b*d*f*(7*B*d*f + 
C*(-5*d*e + 2*c*f + 3*d*f*x)) + b^2*(7*d*f*(5*A*d*f + B*(-4*d*e + c*f + 3* 
d*f*x)) + C*(-4*c^2*f^2 + c*d*f*(-5*e + 3*f*x) + 3*d^2*(8*e^2 - 6*e*f*x + 
5*f^2*x^2)))) + I*(b*c - a*d)*f*(8*a^3*C*d^3*f^3 + a^2*b*d^2*f^2*(9*C*d*e 
- 5*c*C*f - 14*B*d*f) + a*b^2*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 - 5*c^2*f^2)) + b^3*(C*(-48*d^3*e^3 + 16*c*d^2*e 
^2*f + 9*c^2*d*e*f^2 + 8*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 - 2*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)*(4*a^2*C*d^2*f^2 + a*b*d*f*(5*C*d*e + c*C*f - 7*B*d*f) - 
b^2*(7*d*f*(-4*B*d*e - 2*B*c*f + 5*A*d*f) + C*(24*d^2*e^2 + 13*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^4*Sqrt[-a + (b*c)/d]*d^3*f...
 

Rubi [A] (verified)

Time = 1.68 (sec) , antiderivative size = 792, normalized size of antiderivative = 1.03, 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.

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

\(\Big \downarrow \) 2118

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 176

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x}}{7 b d f}-\frac {\frac {\frac {\frac {\left (3 b d f (5 a d f (a c C f+3 a C d e-7 A b d f+3 b c C e)-(a c f+3 a d e+b c e) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))+2 \left (\frac {b c f}{2}-d (a f+b e)\right ) (5 b d f (a c C f+3 a C d e-7 A b d f+3 b c C e)+(a d f-2 b (c f+2 d e)) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}}dx}{f}-\frac {(b e-a f) (d e-c f) \left (4 a^2 C d^2 f^2+a b d f (-7 B d f-2 c C f+8 C d e)-\left (b^2 \left (7 d f (-10 A d f+B c f+8 B d e)-4 C \left (c^2 f^2+2 c d e f+12 d^2 e^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}}{3 b f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (a c C f+3 a C d e-7 A b d f+3 b c C e)+(a d f-2 b (c f+2 d e)) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))}{3 b f}}{5 d f}+\frac {2 \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x} (4 a C d f+b (-7 B d f+4 c C f+6 C d e))}{5 d f}}{7 b d f}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x}}{7 b d f}-\frac {\frac {\frac {\frac {\sqrt {e+f x} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (3 b d f (5 a d f (a c C f+3 a C d e-7 A b d f+3 b c C e)-(a c f+3 a d e+b c e) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))+2 \left (\frac {b c f}{2}-d (a f+b e)\right ) (5 b d f (a c C f+3 a C d e-7 A b d f+3 b c C e)+(a d f-2 b (c f+2 d e)) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))\right ) \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}dx}{f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {(b e-a f) (d e-c f) \left (4 a^2 C d^2 f^2+a b d f (-7 B d f-2 c C f+8 C d e)-\left (b^2 \left (7 d f (-10 A d f+B c f+8 B d e)-4 C \left (c^2 f^2+2 c d e f+12 d^2 e^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}}{3 b f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (a c C f+3 a C d e-7 A b d f+3 b c C e)+(a d f-2 b (c f+2 d e)) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))}{3 b f}}{5 d f}+\frac {2 \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x} (4 a C d f+b (-7 B d f+4 c C f+6 C d e))}{5 d f}}{7 b d f}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x}}{7 b d f}-\frac {\frac {\frac {\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (3 b d f (5 a d f (a c C f+3 a C d e-7 A b d f+3 b c C e)-(a c f+3 a d e+b c e) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))+2 \left (\frac {b c f}{2}-d (a f+b e)\right ) (5 b d f (a c C f+3 a C d e-7 A b d f+3 b c C e)+(a d f-2 b (c f+2 d e)) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {(b e-a f) (d e-c f) \left (4 a^2 C d^2 f^2+a b d f (-7 B d f-2 c C f+8 C d e)-\left (b^2 \left (7 d f (-10 A d f+B c f+8 B d e)-4 C \left (c^2 f^2+2 c d e f+12 d^2 e^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}}{3 b f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (a c C f+3 a C d e-7 A b d f+3 b c C e)+(a d f-2 b (c f+2 d e)) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))}{3 b f}}{5 d f}+\frac {2 \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x} (4 a C d f+b (-7 B d f+4 c C f+6 C d e))}{5 d f}}{7 b d f}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x}}{7 b d f}-\frac {\frac {\frac {\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (3 b d f (5 a d f (a c C f+3 a C d e-7 A b d f+3 b c C e)-(a c f+3 a d e+b c e) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))+2 \left (\frac {b c f}{2}-d (a f+b e)\right ) (5 b d f (a c C f+3 a C d e-7 A b d f+3 b c C e)+(a d f-2 b (c f+2 d e)) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {(b e-a f) (d e-c f) \sqrt {\frac {b (c+d x)}{b c-a d}} \left (4 a^2 C d^2 f^2+a b d f (-7 B d f-2 c C f+8 C d e)-\left (b^2 \left (7 d f (-10 A d f+B c f+8 B d e)-4 C \left (c^2 f^2+2 c d e f+12 d^2 e^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}}dx}{f \sqrt {c+d x}}}{3 b f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (a c C f+3 a C d e-7 A b d f+3 b c C e)+(a d f-2 b (c f+2 d e)) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))}{3 b f}}{5 d f}+\frac {2 \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x} (4 a C d f+b (-7 B d f+4 c C f+6 C d e))}{5 d f}}{7 b d f}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x}}{7 b d f}-\frac {\frac {2 (4 a C d f+b (6 C d e+4 c C f-7 B d f)) \sqrt {a+b x} \sqrt {e+f x} (c+d x)^{3/2}}{5 d f}+\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (3 b c C e+3 a C d e+a c C f-7 A b d f)+(a d f-2 b (2 d e+c f)) (4 a C d f+b (6 C d e+4 c C f-7 B d f)))}{3 b f}+\frac {\frac {2 \sqrt {a d-b c} \left (3 b d f (5 a d f (3 b c C e+3 a C d e+a c C f-7 A b d f)-(b c e+3 a d e+a c f) (4 a C d f+b (6 C d e+4 c C f-7 B d f)))+2 \left (\frac {b c f}{2}-d (b e+a f)\right ) (5 b d f (3 b c C e+3 a C d e+a c C f-7 A b d f)+(a d f-2 b (2 d e+c f)) (4 a C d f+b (6 C d e+4 c C f-7 B d f)))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {(b e-a f) (d e-c f) \left (-\left (\left (7 d f (8 B d e+B c f-10 A d f)-4 C \left (12 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) b^2\right )+a d f (8 C d e-2 c C f-7 B d f) b+4 a^2 C d^2 f^2\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}dx}{f \sqrt {c+d x} \sqrt {e+f x}}}{3 b f}}{5 d f}}{7 b d f}\)

\(\Big \downarrow \) 130

\(\displaystyle \frac {2 C (a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x}}{7 b d f}-\frac {\frac {\frac {\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (3 b d f (5 a d f (a c C f+3 a C d e-7 A b d f+3 b c C e)-(a c f+3 a d e+b c e) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))+2 \left (\frac {b c f}{2}-d (a f+b e)\right ) (5 b d f (a c C f+3 a C d e-7 A b d f+3 b c C e)+(a d f-2 b (c f+2 d e)) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {a d-b c} (b e-a f) (d e-c f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \left (4 a^2 C d^2 f^2+a b d f (-7 B d f-2 c C f+8 C d e)-\left (b^2 \left (7 d f (-10 A d f+B c f+8 B d e)-4 C \left (c^2 f^2+2 c d e f+12 d^2 e^2\right )\right )\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {e+f x}}}{3 b f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (5 b d f (a c C f+3 a C d e-7 A b d f+3 b c C e)+(a d f-2 b (c f+2 d e)) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))}{3 b f}}{5 d f}+\frac {2 \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x} (4 a C d f+b (-7 B d f+4 c C f+6 C d e))}{5 d f}}{7 b d f}\)

Input:

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

Output:

(2*C*(a + b*x)^(3/2)*(c + d*x)^(3/2)*Sqrt[e + f*x])/(7*b*d*f) - ((2*(4*a*C 
*d*f + b*(6*C*d*e + 4*c*C*f - 7*B*d*f))*Sqrt[a + b*x]*(c + d*x)^(3/2)*Sqrt 
[e + f*x])/(5*d*f) + ((2*(5*b*d*f*(3*b*c*C*e + 3*a*C*d*e + a*c*C*f - 7*A*b 
*d*f) + (a*d*f - 2*b*(2*d*e + c*f))*(4*a*C*d*f + b*(6*C*d*e + 4*c*C*f - 7* 
B*d*f)))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b*f) + ((2*Sqrt[-(b 
*c) + a*d]*(3*b*d*f*(5*a*d*f*(3*b*c*C*e + 3*a*C*d*e + a*c*C*f - 7*A*b*d*f) 
 - (b*c*e + 3*a*d*e + a*c*f)*(4*a*C*d*f + b*(6*C*d*e + 4*c*C*f - 7*B*d*f)) 
) + 2*((b*c*f)/2 - d*(b*e + a*f))*(5*b*d*f*(3*b*c*C*e + 3*a*C*d*e + a*c*C* 
f - 7*A*b*d*f) + (a*d*f - 2*b*(2*d*e + c*f))*(4*a*C*d*f + b*(6*C*d*e + 4*c 
*C*f - 7*B*d*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)*(d*e - c*f)*(4*a^2*C*d^2*f^2 + a*b*d*f* 
(8*C*d*e - 2*c*C*f - 7*B*d*f) - b^2*(7*d*f*(8*B*d*e + B*c*f - 10*A*d*f) - 
4*C*(12*d^2*e^2 + 2*c*d*e*f + c^2*f^2)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*S 
qrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sq 
rt[-(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*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 = 3.99 (sec) , antiderivative size = 1205, normalized size of antiderivative = 1.57

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1205\)
default	\(\text {Expression too large to display}\)	\(9684\)

Input:

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

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1392, normalized size of antiderivative = 1.81 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(1/2)*(d*x+c)^(1/2)*(C*x^2+B*x+A)/(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 - (5*C*b^4*c*d^3 + ( 
5*C*a*b^3 + 28*B*b^4)*d^4)*e*f^3 - (4*C*b^4*c^2*d^2 - (2*C*a*b^3 + 7*B*b^4 
)*c*d^3 + (4*C*a^2*b^2 - 7*B*a*b^3 - 35*A*b^4)*d^4)*f^4 - 3*(6*C*b^4*d^4*e 
*f^3 - (C*b^4*c*d^3 + (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*(5*C*b^4*c*d^3 + (5*C*a*b^3 
 + 7*B*b^4)*d^4)*e^3*f - (10*C*b^4*c^2*d^2 - 7*(6*C*a*b^3 + 7*B*b^4)*c*d^3 
 + (10*C*a^2*b^2 - 49*B*a*b^3 - 70*A*b^4)*d^4)*e^2*f^2 - (5*C*b^4*c^3*d - 
7*(C*a*b^3 + 2*B*b^4)*c^2*d^2 - 7*(C*a^2*b^2 - 8*B*a*b^3 - 10*A*b^4)*c*d^3 
 + (5*C*a^3*b - 14*B*a^2*b^2 + 70*A*a*b^3)*d^4)*e*f^3 - (8*C*b^4*c^4 - (9* 
C*a*b^3 + 14*B*b^4)*c^3*d - (4*C*a^2*b^2 - 21*B*a*b^3 - 35*A*b^4)*c^2*d^2 
- (9*C*a^3*b - 21*B*a^2*b^2 + 140*A*a*b^3)*c*d^3 + (8*C*a^4 - 14*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*(48*C*b^4*d^4*e^3*f - 8*(2*C*b^4*c*d^3 + (2*C*a*b^3 + 7* 
B*b^4)*d^4)*e^2*f^2 - (9*C*b^4*c^2*d^2 - (8*C*a*b^3 + 21*B*b^4)*c*d^3 + (9 
*C*a^2*b^2 - 21*B*a*b^3 - 70*A*b^4)*d^4)*e*f^3 - (8*C*b^4*c^3*d - (5*C*a*b 
^3 + 14*B*b^4)*c^2*d^2 - (5*C*a^2*b^2 - 14*B*a*b^3 - 35*A*b^4)*c*d^3 + ...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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