\(\int \frac {\sqrt {c+d x} (A+B x+C x^2)}{(a+b x)^{3/2} \sqrt {e+f x}} \, dx\) [92]

Optimal result

Integrand size = 38, antiderivative size = 540 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2} \sqrt {e+f x}} \, dx=\frac {2 \left (4 a^2 C d f+b^2 (c C e+3 A d f)-a b (C d e+c C f+3 B d f)\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{3 b^2 (b c-a d) f (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{b (b c-a d) (b e-a f) \sqrt {a+b x}}+\frac {2 \sqrt {-b c+a d} \left (8 a^2 C d f^2-a b f (3 C d e+c C f+6 B d f)+b^2 (3 d f (B e+A f)-C e (2 d e-c f))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt {d} f^2 (b e-a f) \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \sqrt {-b c+a d} (d e-c f) (2 b C e-3 b B f+4 a C f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt {d} f^2 \sqrt {c+d x} \sqrt {e+f x}} \] Output:

2/3*(4*a^2*C*d*f+b^2*(3*A*d*f+C*c*e)-a*b*(3*B*d*f+C*c*f+C*d*e))*(b*x+a)^(1 
/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^2/(-a*d+b*c)/f/(-a*f+b*e)-2*(A*b^2-a*(B* 
b-C*a))*(d*x+c)^(3/2)*(f*x+e)^(1/2)/b/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^(1/2)+ 
2/3*(a*d-b*c)^(1/2)*(8*a^2*C*d*f^2-a*b*f*(6*B*d*f+C*c*f+3*C*d*e)+b^2*(3*d* 
f*(A*f+B*e)-C*e*(-c*f+2*d*e)))*(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^(1/2)/f^2/(-a*f+b*e)/(d*x+c)^(1/2)/(b*(f*x+e)/(-a*f+b*e))^( 
1/2)+2/3*(a*d-b*c)^(1/2)*(-c*f+d*e)*(-3*B*b*f+4*C*a*f+2*C*b*e)*(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^(1/2)/f^2/(d 
*x+c)^(1/2)/(f*x+e)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 25.12 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2} \sqrt {e+f x}} \, dx=-\frac {2 \left (b^2 \sqrt {-a+\frac {b c}{d}} \left (-8 a^2 C d f^2+a b f (3 C d e+c C f+6 B d f)+b^2 (-3 d f (B e+A f)+C e (2 d e-c f))\right ) (c+d x) (e+f x)+b^2 \sqrt {-a+\frac {b c}{d}} d f (c+d x) (e+f x) \left (3 \left (A b^2+a (-b B+a C)\right ) f-C (b e-a f) (a+b x)\right )-i (b c-a d) f \left (8 a^2 C d f^2-a b f (3 C d e+c C f+6 B d f)+b^2 (3 d f (B e+A f)+C e (-2 d e+c f))\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )-i b f (d e-c f) \left (4 a^2 C d f+b^2 (c C e+3 A d f)-a b (C d e+c C f+3 B d f)\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )\right )}{3 b^4 \sqrt {-a+\frac {b c}{d}} d f^2 (b e-a f) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \] Input:

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

Output:

(-2*(b^2*Sqrt[-a + (b*c)/d]*(-8*a^2*C*d*f^2 + a*b*f*(3*C*d*e + c*C*f + 6*B 
*d*f) + b^2*(-3*d*f*(B*e + A*f) + C*e*(2*d*e - c*f)))*(c + d*x)*(e + f*x) 
+ b^2*Sqrt[-a + (b*c)/d]*d*f*(c + d*x)*(e + f*x)*(3*(A*b^2 + a*(-(b*B) + a 
*C))*f - C*(b*e - a*f)*(a + b*x)) - I*(b*c - a*d)*f*(8*a^2*C*d*f^2 - a*b*f 
*(3*C*d*e + c*C*f + 6*B*d*f) + b^2*(3*d*f*(B*e + A*f) + C*e*(-2*d*e + c*f) 
))*(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*f*(d*e - c*f)*(4*a^2*C*d*f + b^2*(c*C*e + 
 3*A*d*f) - a*b*(C*d*e + c*C*f + 3*B*d*f))*(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[S 
qrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)]))/(3*b^ 
4*Sqrt[-a + (b*c)/d]*d*f^2*(b*e - a*f)*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e 
+ f*x])
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2117, 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.

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

\(\Big \downarrow \) 2117

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 123

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 130

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

Input:

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

Output:

(-2*(A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*Sqrt[e + f*x])/(b*(b*c - a*d)* 
(b*e - a*f)*Sqrt[a + b*x]) + ((2*((4*a^2*C*d*f)/b + b*(c*C*e + 3*A*d*f) - 
a*(C*d*e + c*C*f + 3*B*d*f))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(3 
*f) + ((b*c - a*d)*((2*Sqrt[-(b*c) + a*d]*(8*a^2*C*d*f^2 - a*b*f*(3*C*d*e 
+ c*C*f + 6*B*d*f) + b^2*(3*d*f*(B*e + A*f) - C*e*(2*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*S 
qrt[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)*(2*b*C*e - 3*b*B*f + 4*a*C*f)*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*f))/(b*(b*c - a*d)*(b*e - a*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 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 [A] (verified)

Time = 6.11 (sec) , antiderivative size = 861, normalized size of antiderivative = 1.59

method	result	size
elliptic	\(\frac {\sqrt {\left (f x +e \right ) \left (b x +a \right ) \left (x d +c \right )}\, \left (\frac {2 \left (b d f \,x^{2}+b c f x +b d e x +b c e \right ) \left (b^{2} A -a b B +a^{2} C \right )}{b^{3} \left (a f -b e \right ) \sqrt {\left (x +\frac {a}{b}\right ) \left (b d f \,x^{2}+b c f x +b d e x +b c e \right )}}+\frac {2 C \sqrt {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}}{3 b^{2} f}+\frac {2 \left (\frac {b^{2} A d -a b B d +B \,b^{2} c +a^{2} C d -C a b c}{b^{3}}-\frac {\left (b^{2} A -a b B +a^{2} C \right ) \left (a d f -b c f -b d e \right )}{b^{3} \left (a f -b e \right )}-\frac {\left (b c f +b d e \right ) \left (b^{2} A -a b B +a^{2} C \right )}{b^{3} \left (a f -b e \right )}-\frac {2 C \left (\frac {1}{2} a c f +\frac {1}{2} a d e +\frac {1}{2} b c e \right )}{3 b^{2} f}\right ) \left (\frac {c}{d}-\frac {a}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {a}{b}}{-\frac {c}{d}+\frac {e}{f}}}\right )}{\sqrt {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}}+\frac {2 \left (\frac {B b d -C a d +C b c}{b^{2}}-\frac {\left (b^{2} A -a b B +a^{2} C \right ) d f}{b^{2} \left (a f -b e \right )}-\frac {2 C \left (a d f +b c f +b d e \right )}{3 b^{2} f}\right ) \left (\frac {c}{d}-\frac {a}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \left (\left (-\frac {c}{d}+\frac {e}{f}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {a}{b}}{-\frac {c}{d}+\frac {e}{f}}}\right )-\frac {e \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {a}{b}}{-\frac {c}{d}+\frac {e}{f}}}\right )}{f}\right )}{\sqrt {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}}\right )}{\sqrt {b x +a}\, \sqrt {x d +c}\, \sqrt {f x +e}}\)	\(861\)
default	\(\text {Expression too large to display}\)	\(5069\)

Input:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1336 vs. \(2 (487) = 974\).

Time = 0.15 (sec) , antiderivative size = 1336, normalized size of antiderivative = 2.47 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2} \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

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

Output:

2/9*(3*(C*a*b^3*d^2*e*f^2 - (4*C*a^2*b^2 - 3*B*a*b^3 + 3*A*b^4)*d^2*f^3 + 
(C*b^4*d^2*e*f^2 - C*a*b^3*d^2*f^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f* 
x + e) + (2*C*a*b^3*d^2*e^3 - (2*C*a*b^3*c*d - (2*C*a^2*b^2 - 3*B*a*b^3)*d 
^2)*e^2*f - (C*a*b^3*c^2 + 6*(C*a^2*b^2 - B*a*b^3)*c*d - (7*C*a^3*b - 6*B* 
a^2*b^2 + 6*A*a*b^3)*d^2)*e*f^2 + (C*a^2*b^2*c^2 + (5*C*a^3*b - 3*B*a^2*b^ 
2 - 3*A*a*b^3)*c*d - (8*C*a^4 - 6*B*a^3*b + 3*A*a^2*b^2)*d^2)*f^3 + (2*C*b 
^4*d^2*e^3 - (2*C*b^4*c*d - (2*C*a*b^3 - 3*B*b^4)*d^2)*e^2*f - (C*b^4*c^2 
+ 6*(C*a*b^3 - B*b^4)*c*d - (7*C*a^2*b^2 - 6*B*a*b^3 + 6*A*b^4)*d^2)*e*f^2 
 + (C*a*b^3*c^2 + (5*C*a^2*b^2 - 3*B*a*b^3 - 3*A*b^4)*c*d - (8*C*a^3*b - 6 
*B*a^2*b^2 + 3*A*a*b^3)*d^2)*f^3)*x)*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*(2*C*a*b^3*d^2*e^2*f - (C*a*b^3*c*d - 3*(C*a 
^2*b^2 - B*a*b^3)*d^2)*e*f^2 + (C*a^2*b^2*c*d - (8*C*a^3*b - 6*B*a^2*b^2 + 
 3*A*a*b^3)*d^2)*f^3 + (2*C*b^4*d^2*e^2*f - (C*b^4*c*d - 3*(C*a*b^3 - B*b^ 
4)*d^2)*e*f^2 + (C*a*b^3*c*d - (8*C*a^2*b^2 - 6*B*a*b^3 + 3*A*b^4)*d^2)*f^ 
3)*x)*sqrt(b*d*f)*weierstrassZeta(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...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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