\(\int \frac {\sqrt {c+d x} \sqrt {e+f x} (A+B x+C x^2)}{(a+b x)^2} \, dx\) [67]

Optimal result

Integrand size = 36, antiderivative size = 524 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^2} \, dx=\frac {\left (12 a^2 C d^2 f-a b d (C d e+7 c C f+8 B d f)-b^2 \left (c^2 C f-4 A d^2 f-c d (C e+4 B f)\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^3 d (b c-a d) f}+\frac {\left (3 a^2 C d f+b^2 (c C e+2 A d f)-a b (C d e+c C f+2 B d f)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{2 b^2 d (b c-a d) (b e-a f)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) (a+b x)}+\frac {\left (24 a^2 C d^2 f^2-8 a b d f (C d e+c C f+2 B d f)-b^2 \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{4 b^4 d^{3/2} f^{3/2}}+\frac {\left (6 a^3 C d f-b^3 (2 B c e+A d e+A c f)+a b^2 (4 c C e+3 B d e+3 B c f+2 A d f)-a^2 b (4 B d f+5 C (d e+c f))\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{b^4 \sqrt {b c-a d} \sqrt {b e-a f}} \] Output:

1/4*(12*a^2*C*d^2*f-a*b*d*(8*B*d*f+7*C*c*f+C*d*e)-b^2*(c^2*C*f-4*A*d^2*f-c 
*d*(4*B*f+C*e)))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^3/d/(-a*d+b*c)/f+1/2*(3*a^2 
*C*d*f+b^2*(2*A*d*f+C*c*e)-a*b*(2*B*d*f+C*c*f+C*d*e))*(d*x+c)^(3/2)*(f*x+e 
)^(1/2)/b^2/d/(-a*d+b*c)/(-a*f+b*e)-(A*b^2-a*(B*b-C*a))*(d*x+c)^(3/2)*(f*x 
+e)^(3/2)/b/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)+1/4*(24*a^2*C*d^2*f^2-8*a*b*d*f* 
(2*B*d*f+C*c*f+C*d*e)-b^2*(C*(-c*f+d*e)^2-4*d*f*(2*A*d*f+B*c*f+B*d*e)))*ar 
ctanh(f^(1/2)*(d*x+c)^(1/2)/d^(1/2)/(f*x+e)^(1/2))/b^4/d^(3/2)/f^(3/2)+(6* 
a^3*C*d*f-b^3*(A*c*f+A*d*e+2*B*c*e)+a*b^2*(2*A*d*f+3*B*c*f+3*B*d*e+4*C*c*e 
)-a^2*b*(4*B*d*f+5*C*(c*f+d*e)))*arctanh((-a*f+b*e)^(1/2)*(d*x+c)^(1/2)/(- 
a*d+b*c)^(1/2)/(f*x+e)^(1/2))/b^4/(-a*d+b*c)^(1/2)/(-a*f+b*e)^(1/2)
 

Mathematica [A] (verified)

Time = 2.05 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^2} \, dx=\frac {\frac {b \sqrt {c+d x} \sqrt {e+f x} \left (-12 a^2 C d f+a b (c C f+8 B d f+C d (e-6 f x))+b^2 (-4 A d f+x (c C f+4 B d f+C d (e+2 f x)))\right )}{d f (a+b x)}+\frac {4 \left (-6 a^3 C d f+b^3 (2 B c e+A d e+A c f)-a b^2 (4 c C e+3 B d e+3 B c f+2 A d f)+a^2 b (4 B d f+5 C (d e+c f))\right ) \arctan \left (\frac {\sqrt {b c-a d} \sqrt {e+f x}}{\sqrt {-b e+a f} \sqrt {c+d x}}\right )}{\sqrt {b c-a d} \sqrt {-b e+a f}}-\frac {\left (-24 a^2 C d^2 f^2+8 a b d f (C d e+c C f+2 B d f)+b^2 \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {f} \sqrt {c+d x}}\right )}{d^{3/2} f^{3/2}}}{4 b^4} \] Input:

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

Output:

((b*Sqrt[c + d*x]*Sqrt[e + f*x]*(-12*a^2*C*d*f + a*b*(c*C*f + 8*B*d*f + C* 
d*(e - 6*f*x)) + b^2*(-4*A*d*f + x*(c*C*f + 4*B*d*f + C*d*(e + 2*f*x)))))/ 
(d*f*(a + b*x)) + (4*(-6*a^3*C*d*f + b^3*(2*B*c*e + A*d*e + A*c*f) - a*b^2 
*(4*c*C*e + 3*B*d*e + 3*B*c*f + 2*A*d*f) + a^2*b*(4*B*d*f + 5*C*(d*e + c*f 
)))*ArcTan[(Sqrt[b*c - a*d]*Sqrt[e + f*x])/(Sqrt[-(b*e) + a*f]*Sqrt[c + d* 
x])])/(Sqrt[b*c - a*d]*Sqrt[-(b*e) + a*f]) - ((-24*a^2*C*d^2*f^2 + 8*a*b*d 
*f*(C*d*e + c*C*f + 2*B*d*f) + b^2*(C*(d*e - c*f)^2 - 4*d*f*(B*d*e + B*c*f 
 + 2*A*d*f)))*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/(Sqrt[f]*Sqrt[c + d*x])])/(d 
^(3/2)*f^(3/2)))/(4*b^4)
 

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2116, 27, 171, 27, 171, 27, 175, 66, 104, 221}

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.

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

\(\Big \downarrow \) 2116

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

\(\displaystyle \frac {\frac {(b c-a d) \left (\frac {\int \frac {(b e-a f) \left (12 C d f (d e+c f) a^2-b \left (8 B d f (d e+c f)+C \left (d^2 e^2+14 c d f e+c^2 f^2\right )\right ) a+4 b^2 d f (2 B c e+A d e+A c f)+\left (-\left (\left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right ) b^2\right )-8 a d f (C d e+c C f+2 B d f) b+24 a^2 C d^2 f^2\right ) x\right )}{2 (a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b d}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (12 a^2 C d f^2-a b f (8 B d f+c C f+7 C d e)+b^2 (4 d f (A f+B e)-C e (d e-c f))\right )}{b d}\right )}{2 b f}+\frac {\sqrt {c+d x} (e+f x)^{3/2} \left (\frac {3 a^2 C d f}{b}-a (2 B d f+c C f+C d e)+b (2 A d f+c C e)\right )}{f}}{2 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {(b c-a d) \left (\frac {(b e-a f) \int \frac {12 C d f (d e+c f) a^2-b \left (8 B d f (d e+c f)+C \left (d^2 e^2+14 c d f e+c^2 f^2\right )\right ) a+4 b^2 d f (2 B c e+A d e+A c f)+\left (-\left (\left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right ) b^2\right )-8 a d f (C d e+c C f+2 B d f) b+24 a^2 C d^2 f^2\right ) x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{2 b d}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (12 a^2 C d f^2-a b f (8 B d f+c C f+7 C d e)+b^2 (4 d f (A f+B e)-C e (d e-c f))\right )}{b d}\right )}{2 b f}+\frac {\sqrt {c+d x} (e+f x)^{3/2} \left (\frac {3 a^2 C d f}{b}-a (2 B d f+c C f+C d e)+b (2 A d f+c C e)\right )}{f}}{2 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {\frac {(b c-a d) \left (\frac {(b e-a f) \left (\frac {\left (24 a^2 C d^2 f^2-8 a b d f (2 B d f+c C f+C d e)-\left (b^2 \left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}}dx}{b}-\frac {4 d f \left (6 a^3 C d f-a^2 b (4 B d f+5 C (c f+d e))+a b^2 (2 A d f+3 B c f+3 B d e+4 c C e)-b^3 (A c f+A d e+2 B c e)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b}\right )}{2 b d}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (12 a^2 C d f^2-a b f (8 B d f+c C f+7 C d e)+b^2 (4 d f (A f+B e)-C e (d e-c f))\right )}{b d}\right )}{2 b f}+\frac {\sqrt {c+d x} (e+f x)^{3/2} \left (\frac {3 a^2 C d f}{b}-a (2 B d f+c C f+C d e)+b (2 A d f+c C e)\right )}{f}}{2 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {\frac {(b c-a d) \left (\frac {(b e-a f) \left (\frac {2 \left (24 a^2 C d^2 f^2-8 a b d f (2 B d f+c C f+C d e)-\left (b^2 \left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )\right )\right ) \int \frac {1}{d-\frac {f (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}-\frac {4 d f \left (6 a^3 C d f-a^2 b (4 B d f+5 C (c f+d e))+a b^2 (2 A d f+3 B c f+3 B d e+4 c C e)-b^3 (A c f+A d e+2 B c e)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b}\right )}{2 b d}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (12 a^2 C d f^2-a b f (8 B d f+c C f+7 C d e)+b^2 (4 d f (A f+B e)-C e (d e-c f))\right )}{b d}\right )}{2 b f}+\frac {\sqrt {c+d x} (e+f x)^{3/2} \left (\frac {3 a^2 C d f}{b}-a (2 B d f+c C f+C d e)+b (2 A d f+c C e)\right )}{f}}{2 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\frac {(b c-a d) \left (\frac {(b e-a f) \left (\frac {2 \left (24 a^2 C d^2 f^2-8 a b d f (2 B d f+c C f+C d e)-\left (b^2 \left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )\right )\right ) \int \frac {1}{d-\frac {f (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}-\frac {8 d f \left (6 a^3 C d f-a^2 b (4 B d f+5 C (c f+d e))+a b^2 (2 A d f+3 B c f+3 B d e+4 c C e)-b^3 (A c f+A d e+2 B c e)\right ) \int \frac {1}{-b c+a d+\frac {(b e-a f) (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}\right )}{2 b d}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (12 a^2 C d f^2-a b f (8 B d f+c C f+7 C d e)+b^2 (4 d f (A f+B e)-C e (d e-c f))\right )}{b d}\right )}{2 b f}+\frac {\sqrt {c+d x} (e+f x)^{3/2} \left (\frac {3 a^2 C d f}{b}-a (2 B d f+c C f+C d e)+b (2 A d f+c C e)\right )}{f}}{2 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\sqrt {c+d x} (e+f x)^{3/2} \left (\frac {3 a^2 C d f}{b}-a (2 B d f+c C f+C d e)+b (2 A d f+c C e)\right )}{f}+\frac {(b c-a d) \left (\frac {\sqrt {c+d x} \sqrt {e+f x} \left (12 a^2 C d f^2-a b f (8 B d f+c C f+7 C d e)+b^2 (4 d f (A f+B e)-C e (d e-c f))\right )}{b d}+\frac {(b e-a f) \left (\frac {2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (24 a^2 C d^2 f^2-8 a b d f (2 B d f+c C f+C d e)-\left (b^2 \left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )\right )\right )}{b \sqrt {d} \sqrt {f}}+\frac {8 d f \text {arctanh}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right ) \left (6 a^3 C d f-a^2 b (4 B d f+5 C (c f+d e))+a b^2 (2 A d f+3 B c f+3 B d e+4 c C e)-b^3 (A c f+A d e+2 B c e)\right )}{b \sqrt {b c-a d} \sqrt {b e-a f}}\right )}{2 b d}\right )}{2 b f}}{2 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}\)

Input:

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

Output:

-(((A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(b*(b*c - a*d) 
*(b*e - a*f)*(a + b*x))) + ((((3*a^2*C*d*f)/b + b*(c*C*e + 2*A*d*f) - a*(C 
*d*e + c*C*f + 2*B*d*f))*Sqrt[c + d*x]*(e + f*x)^(3/2))/f + ((b*c - a*d)*( 
((12*a^2*C*d*f^2 - a*b*f*(7*C*d*e + c*C*f + 8*B*d*f) + b^2*(4*d*f*(B*e + A 
*f) - C*e*(d*e - c*f)))*Sqrt[c + d*x]*Sqrt[e + f*x])/(b*d) + ((b*e - a*f)* 
((2*(24*a^2*C*d^2*f^2 - 8*a*b*d*f*(C*d*e + c*C*f + 2*B*d*f) - b^2*(C*(d*e 
- c*f)^2 - 4*d*f*(B*d*e + B*c*f + 2*A*d*f)))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x 
])/(Sqrt[d]*Sqrt[e + f*x])])/(b*Sqrt[d]*Sqrt[f]) + (8*d*f*(6*a^3*C*d*f - b 
^3*(2*B*c*e + A*d*e + A*c*f) + a*b^2*(4*c*C*e + 3*B*d*e + 3*B*c*f + 2*A*d* 
f) - a^2*b*(4*B*d*f + 5*C*(d*e + c*f)))*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[c + 
d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/(b*Sqrt[b*c - a*d]*Sqrt[b*e - a*f] 
)))/(2*b*d)))/(2*b*f))/(2*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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*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 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(b*g - a*h)/b   Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 2116

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] && ILtQ[m, 
-1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4679\) vs. \(2(482)=964\).

Time = 0.84 (sec) , antiderivative size = 4680, normalized size of antiderivative = 8.93

method	result	size
default	\(\text {Expression too large to display}\)	\(4680\)

Input:

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

Output:

1/8*(8*B*b^4*d*f*x*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*((a^2*d*f-a*b*c*f-a 
*b*d*e+b^2*c*e)/b^2)^(1/2)+2*C*b^4*c*f*x*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/ 
2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+2*C*b^4*d*e*x*(d*f)^(1/2) 
*((f*x+e)*(d*x+c))^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+16* 
B*a*b^3*d*f*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+ 
b^2*c*e)/b^2)^(1/2)+4*C*b^4*d*f*x^2*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*(( 
a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-24*C*a^2*b^2*d*f*(d*f)^(1/2)*( 
(f*x+e)*(d*x+c))^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+2*C*a 
*b^3*c*f*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2 
*c*e)/b^2)^(1/2)+2*C*a*b^3*d*e*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*((a^2*d 
*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+8*A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+ 
2*((f*x+e)*(d*x+c))^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b- 
a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a*b^3*d^2*f^2*x*(d*f)^(1/2)-4*A*ln((-2*a*d*f 
*x+b*c*f*x+b*d*e*x+2*((f*x+e)*(d*x+c))^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2 
*c*e)/b^2)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*b^4*c*d*f^2*x*(d*f)^(1/2) 
+24*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((f*x+e)*(d*x+c))^(1/2)*((a^2*d*f-a 
*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^4*d^2 
*f^2*(d*f)^(1/2)-4*A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((f*x+e)*(d*x+c))^(1 
/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/( 
b*x+a))*a*b^3*d^2*e*f*(d*f)^(1/2)+12*B*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(2*a*d*f-b*c*f>0)', see `assume?` 
 for more
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1507 vs. \(2 (481) = 962\).

Time = 1.39 (sec) , antiderivative size = 1507, normalized size of antiderivative = 2.88 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*sqrt(d^2*e + (d*x + c)*d*f - c*d*f)*sqrt(d*x + c)*(2*(d*x + c)*C*abs(d 
)/(b^2*d^3) + (C*b^7*d^4*e*f*abs(d) - C*b^7*c*d^3*f^2*abs(d) - 8*C*a*b^6*d 
^4*f^2*abs(d) + 4*B*b^7*d^4*f^2*abs(d))/(b^9*d^6*f^2)) + (4*sqrt(d*f)*C*a* 
b^2*c*e*abs(d) - 2*sqrt(d*f)*B*b^3*c*e*abs(d) - 5*sqrt(d*f)*C*a^2*b*d*e*ab 
s(d) + 3*sqrt(d*f)*B*a*b^2*d*e*abs(d) - sqrt(d*f)*A*b^3*d*e*abs(d) - 5*sqr 
t(d*f)*C*a^2*b*c*f*abs(d) + 3*sqrt(d*f)*B*a*b^2*c*f*abs(d) - sqrt(d*f)*A*b 
^3*c*f*abs(d) + 6*sqrt(d*f)*C*a^3*d*f*abs(d) - 4*sqrt(d*f)*B*a^2*b*d*f*abs 
(d) + 2*sqrt(d*f)*A*a*b^2*d*f*abs(d))*arctan(-1/2*(b*d^2*e + b*c*d*f - 2*a 
*d^2*f - (sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2 
*b)/(sqrt(-b^2*c*d*e*f + a*b*d^2*e*f + a*b*c*d*f^2 - a^2*d^2*f^2)*d))/(sqr 
t(-b^2*c*d*e*f + a*b*d^2*e*f + a*b*c*d*f^2 - a^2*d^2*f^2)*b^4*d) - 2*(sqrt 
(d*f)*C*a^2*b*d^3*e^2*abs(d) - sqrt(d*f)*B*a*b^2*d^3*e^2*abs(d) + sqrt(d*f 
)*A*b^3*d^3*e^2*abs(d) - 2*sqrt(d*f)*C*a^2*b*c*d^2*e*f*abs(d) + 2*sqrt(d*f 
)*B*a*b^2*c*d^2*e*f*abs(d) - 2*sqrt(d*f)*A*b^3*c*d^2*e*f*abs(d) + sqrt(d*f 
)*C*a^2*b*c^2*d*f^2*abs(d) - sqrt(d*f)*B*a*b^2*c^2*d*f^2*abs(d) + sqrt(d*f 
)*A*b^3*c^2*d*f^2*abs(d) - sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e 
 + (d*x + c)*d*f - c*d*f))^2*C*a^2*b*d*e*abs(d) + sqrt(d*f)*(sqrt(d*f)*sqr 
t(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*B*a*b^2*d*e*abs(d) - s 
qrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2 
*A*b^3*d*e*abs(d) - sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (...
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

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

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

Output:

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