\(\int \frac {\sqrt {c+d x} (A+B x+C x^2)}{\sqrt {a+b x} \sqrt {e+f x}} \, dx\) [91]

Optimal result

Integrand size = 38, antiderivative size = 529 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {e+f x}} \, dx=-\frac {2 (4 a C d f+b (4 C d e+2 c C f-5 B d f)) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{15 b^2 d f^2}+\frac {2 C \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x}}{5 b d f}-\frac {2 \sqrt {-b c+a d} \left (3 d f (b c C e+3 a C d e+a c C f-5 A b d f)-\frac {(2 b d e-b c f+2 a d f) (4 a C d f+b (4 C d e+2 c C f-5 B d f))}{b}\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^2 d^{3/2} f^3 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {-b c+a d} (d e-c f) \left (4 a^2 C d f^2+a b f (3 C d e-c C f-5 B d f)-b^2 (5 d f (2 B e-3 A f)-C e (8 d e+c f))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 d^{3/2} f^3 \sqrt {c+d x} \sqrt {e+f x}} \] Output:

-2/15*(4*a*C*d*f+b*(-5*B*d*f+2*C*c*f+4*C*d*e))*(b*x+a)^(1/2)*(d*x+c)^(1/2) 
*(f*x+e)^(1/2)/b^2/d/f^2+2/5*C*(b*x+a)^(1/2)*(d*x+c)^(3/2)*(f*x+e)^(1/2)/b 
/d/f-2/15*(a*d-b*c)^(1/2)*(3*d*f*(-5*A*b*d*f+C*a*c*f+3*C*a*d*e+C*b*c*e)-(2 
*a*d*f-b*c*f+2*b*d*e)*(4*a*C*d*f+b*(-5*B*d*f+2*C*c*f+4*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^(3/2)/f^3/(d*x+c)^(1/2)/ 
(b*(f*x+e)/(-a*f+b*e))^(1/2)-2/15*(a*d-b*c)^(1/2)*(-c*f+d*e)*(4*a^2*C*d*f^ 
2+a*b*f*(-5*B*d*f-C*c*f+3*C*d*e)-b^2*(5*d*f*(-3*A*f+2*B*e)-C*e*(c*f+8*d*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^(3/2)/f^3/(d*x+c)^(1/2)/(f*x+e)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 27.04 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {e+f x}} \, dx=\frac {2 \sqrt {a+b x} \left (\frac {b^2 \left (8 a^2 C d^2 f^2+a b d f (7 C d e-3 c C f-10 B d f)+b^2 \left (5 d f (-2 B d e+B c f+3 A d f)+C \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )\right )\right ) (c+d x) (e+f x)}{a+b x}+b^2 d f (c+d x) (e+f x) (5 b B d f-4 a C d f+b C (-4 d e+c f+3 d f x))+\frac {i (b c-a d) f \left (8 a^2 C d^2 f^2+a b d f (7 C d e-3 c C f-10 B d f)+b^2 \left (5 d f (-2 B d e+B c f+3 A d f)+C \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )\right )\right ) \sqrt {a+b x} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )}{\sqrt {-a+\frac {b c}{d}}}+i b \sqrt {-a+\frac {b c}{d}} d f (d e-c f) (5 b B d f-4 a C d f-2 b C (2 d e+c f)) \sqrt {a+b x} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )\right )}{15 b^4 d^2 f^3 \sqrt {c+d x} \sqrt {e+f x}} \] Input:

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

Output:

(2*Sqrt[a + b*x]*((b^2*(8*a^2*C*d^2*f^2 + a*b*d*f*(7*C*d*e - 3*c*C*f - 10* 
B*d*f) + b^2*(5*d*f*(-2*B*d*e + B*c*f + 3*A*d*f) + C*(8*d^2*e^2 - 3*c*d*e* 
f - 2*c^2*f^2)))*(c + d*x)*(e + f*x))/(a + b*x) + b^2*d*f*(c + d*x)*(e + f 
*x)*(5*b*B*d*f - 4*a*C*d*f + b*C*(-4*d*e + c*f + 3*d*f*x)) + (I*(b*c - a*d 
)*f*(8*a^2*C*d^2*f^2 + a*b*d*f*(7*C*d*e - 3*c*C*f - 10*B*d*f) + b^2*(5*d*f 
*(-2*B*d*e + B*c*f + 3*A*d*f) + C*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2)))*Sq 
rt[a + b*x]*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)])/Sqrt[-a + (b*c)/d] + I*b*Sqrt[-a + (b*c)/d]*d*f*(d*e 
- c*f)*(5*b*B*d*f - 4*a*C*d*f - 2*b*C*(2*d*e + c*f))*Sqrt[a + b*x]*Sqrt[(b 
*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*A 
rcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)] 
))/(15*b^4*d^2*f^3*Sqrt[c + d*x]*Sqrt[e + f*x])
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 545, 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 = {2118, 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.

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

\(\Big \downarrow \) 2118

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 123

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 130

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

Input:

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

Output:

(2*C*Sqrt[a + b*x]*(c + d*x)^(3/2)*Sqrt[e + f*x])/(5*b*d*f) - ((2*(4*a*C*d 
*f + b*(4*C*d*e + 2*c*C*f - 5*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*(b*c*C*e + 3*a*C*d*e + a* 
c*C*f - 5*A*b*d*f) - (2*b*d*e - b*c*f + 2*a*d*f)*(4*a*C*d*f + b*(4*C*d*e + 
 2*c*C*f - 5*B*d*f)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*Ellipt 
icE[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]*(d*e - c*f)*(4*a^2*C*d*f^2 + a*b*f*(3*C*d*e - c* 
C*f - 5*B*d*f) - b^2*(5*d*f*(2*B*e - 3*A*f) - C*e*(8*d*e + 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))/(5*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 = 4.43 (sec) , antiderivative size = 812, normalized size of antiderivative = 1.53

method	result	size
elliptic	\(\frac {\sqrt {\left (f x +e \right ) \left (b x +a \right ) \left (x d +c \right )}\, \left (\frac {2 C x \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}}{5 f b}+\frac {2 \left (B d +C c -\frac {2 C \left (2 a d f +2 b c f +2 b d e \right )}{5 f b}\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}}{3 b d f}+\frac {2 \left (A c -\frac {2 C a c e}{5 f b}-\frac {2 \left (B d +C c -\frac {2 C \left (2 a d f +2 b c f +2 b d e \right )}{5 f b}\right ) \left (\frac {1}{2} a c f +\frac {1}{2} a d e +\frac {1}{2} b c e \right )}{3 b d 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 (A d +B c -\frac {2 C \left (\frac {3}{2} a c f +\frac {3}{2} a d e +\frac {3}{2} b c e \right )}{5 f b}-\frac {2 \left (B d +C c -\frac {2 C \left (2 a d f +2 b c f +2 b d e \right )}{5 f b}\right ) \left (a d f +b c f +b d e \right )}{3 b d 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}}\)	\(812\)
default	\(\text {Expression too large to display}\)	\(6266\)

Input:

int((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^(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)* 
(2/5*C/f/b*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*d+C*c-2/5*C/f/b*(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-2/5*C/f/b*a*c*e-2/3*(B*d+C*c-2/5*C/f/b*(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*d+B*c-2/5*C/f/b 
*(3/2*a*c*f+3/2*a*d*e+3/2*b*c*e)-2/3*(B*d+C*c-2/5*C/f/b*(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*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)*Ellipt 
icE(((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. 1036 vs. \(2 (473) = 946\).

Time = 0.12 (sec) , antiderivative size = 1036, normalized size of antiderivative = 1.96 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

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

Output:

2/45*(3*(3*C*b^3*d^3*f^3*x - 4*C*b^3*d^3*e*f^2 + (C*b^3*c*d^2 - (4*C*a*b^2 
 - 5*B*b^3)*d^3)*f^3)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e) - (8*C*b^3 
*d^3*e^3 - (7*C*b^3*c*d^2 - (3*C*a*b^2 - 10*B*b^3)*d^3)*e^2*f - (2*C*b^3*c 
^2*d + 2*(C*a*b^2 - 5*B*b^3)*c*d^2 - (3*C*a^2*b - 5*B*a*b^2 + 15*A*b^3)*d^ 
3)*e*f^2 - (2*C*b^3*c^3 + (2*C*a*b^2 - 5*B*b^3)*c^2*d + (7*C*a^2*b - 10*B* 
a*b^2 + 30*A*b^3)*c*d^2 - (8*C*a^3 - 10*B*a^2*b + 15*A*a*b^2)*d^3)*f^3)*sq 
rt(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^3* 
d^3*e^2*f - (3*C*b^3*c*d^2 - (7*C*a*b^2 - 10*B*b^3)*d^3)*e*f^2 - (2*C*b^3* 
c^2*d + (3*C*a*b^2 - 5*B*b^3)*c*d^2 - (8*C*a^2*b - 10*B*a*b^2 + 15*A*b^3)* 
d^3)*f^3)*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*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), 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^...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(10*sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*a*b*d*f - 6*sqrt(e + f*x)*sq 
rt(c + d*x)*sqrt(a + b*x)*a*c**2*f - 6*sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a 
+ b*x)*a*c*d*e + 4*sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*a*c*d*f*x + 1 
0*sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*b**2*c*f - 6*sqrt(e + f*x)*sqr 
t(c + d*x)*sqrt(a + b*x)*b*c**2*e + 4*sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + 
 b*x)*b*c**2*f*x + 4*sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*b*c*d*e*x - 
 8*int((sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*x**2)/(a**2*c*d*e*f + a* 
*2*c*d*f**2*x + a**2*d**2*e*f*x + a**2*d**2*f**2*x**2 + a*b*c**2*e*f + a*b 
*c**2*f**2*x + a*b*c*d*e**2 + 3*a*b*c*d*e*f*x + 2*a*b*c*d*f**2*x**2 + a*b* 
d**2*e**2*x + 2*a*b*d**2*e*f*x**2 + a*b*d**2*f**2*x**3 + b**2*c**2*e*f*x + 
 b**2*c**2*f**2*x**2 + b**2*c*d*e**2*x + 2*b**2*c*d*e*f*x**2 + b**2*c*d*f* 
*2*x**3 + b**2*d**2*e**2*x**2 + b**2*d**2*e*f*x**3),x)*a**3*c*d**3*f**3 - 
5*int((sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*x**2)/(a**2*c*d*e*f + a** 
2*c*d*f**2*x + a**2*d**2*e*f*x + a**2*d**2*f**2*x**2 + a*b*c**2*e*f + a*b* 
c**2*f**2*x + a*b*c*d*e**2 + 3*a*b*c*d*e*f*x + 2*a*b*c*d*f**2*x**2 + a*b*d 
**2*e**2*x + 2*a*b*d**2*e*f*x**2 + a*b*d**2*f**2*x**3 + b**2*c**2*e*f*x + 
b**2*c**2*f**2*x**2 + b**2*c*d*e**2*x + 2*b**2*c*d*e*f*x**2 + b**2*c*d*f** 
2*x**3 + b**2*d**2*e**2*x**2 + b**2*d**2*e*f*x**3),x)*a**2*b**2*d**3*f**3 
- 5*int((sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*x**2)/(a**2*c*d*e*f + a 
**2*c*d*f**2*x + a**2*d**2*e*f*x + a**2*d**2*f**2*x**2 + a*b*c**2*e*f +...