\(\int \frac {\sqrt {c+d x} (A+B x+C x^2)}{(a+b x)^4 \sqrt {e+f x}} \, dx\) [75]

Optimal result

Integrand size = 36, antiderivative size = 685 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^4 \sqrt {e+f x}} \, dx=\frac {\left (4 a^3 C d f-b^3 (6 B c e-3 A d e-5 A c f)+a b^2 (12 c C e+3 B d e+B c f-8 A d f)-a^2 b (9 C d e+7 c C f-2 B d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b^2 (b c-a d) (b e-a f)^2 (a+b x)^2}-\frac {\left (8 a^4 C d^2 f^2-2 a^3 b d f (13 C d e+7 c C f-2 B d f)-b^4 \left (3 A d^2 e^2-2 c d e (3 B e-2 A f)-3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)+2 c d \left (30 C e^2-14 B e f+13 A f^2\right )\right )-a^2 b^2 \left (4 d f (4 B d e+B c f-2 A d f)-C \left (33 d^2 e^2+44 c d e f+3 c^2 f^2\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b^2 (b c-a d)^2 (b e-a f)^3 (a+b x)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}-\frac {(d e-c f) \left (b^2 \left (A d^2 e^2-2 c d e (B e-A f)+c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b \left (d^2 e (B e-4 A f)-c^2 f (4 C e-B f)-2 c d \left (6 C e^2-7 B e f+6 A f^2\right )\right )-a^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{8 (b c-a d)^{5/2} (b e-a f)^{7/2}} \] Output:

1/12*(4*a^3*C*d*f-b^3*(-5*A*c*f-3*A*d*e+6*B*c*e)+a*b^2*(-8*A*d*f+B*c*f+3*B 
*d*e+12*C*c*e)-a^2*b*(-2*B*d*f+7*C*c*f+9*C*d*e))*(d*x+c)^(1/2)*(f*x+e)^(1/ 
2)/b^2/(-a*d+b*c)/(-a*f+b*e)^2/(b*x+a)^2-1/24*(8*a^4*C*d^2*f^2-2*a^3*b*d*f 
*(-2*B*d*f+7*C*c*f+13*C*d*e)-b^4*(3*A*d^2*e^2-2*c*d*e*(-2*A*f+3*B*e)-3*c^2 
*(5*A*f^2-6*B*e*f+8*C*e^2))-a*b^3*(d^2*e*(-10*A*f+3*B*e)+3*c^2*f*(-B*f+4*C 
*e)+2*c*d*(13*A*f^2-14*B*e*f+30*C*e^2))-a^2*b^2*(4*d*f*(-2*A*d*f+B*c*f+4*B 
*d*e)-C*(3*c^2*f^2+44*c*d*e*f+33*d^2*e^2)))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^ 
2/(-a*d+b*c)^2/(-a*f+b*e)^3/(b*x+a)-1/3*(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)^3-1/8*(-c*f+d*e)*(b^2*(A*d^2 
*e^2-2*c*d*e*(-A*f+B*e)+c^2*(5*A*f^2-6*B*e*f+8*C*e^2))+a*b*(d^2*e*(-4*A*f+ 
B*e)-c^2*f*(-B*f+4*C*e)-2*c*d*(6*A*f^2-7*B*e*f+6*C*e^2))-a^2*(2*d*f*(-4*A* 
d*f+B*c*f+3*B*d*e)-C*(c^2*f^2+2*c*d*e*f+5*d^2*e^2)))*arctanh((-a*f+b*e)^(1 
/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2)/(f*x+e)^(1/2))/(-a*d+b*c)^(5/2)/(-a*f+b 
*e)^(7/2)
 

Mathematica [A] (verified)

Time = 15.14 (sec) , antiderivative size = 657, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^4 \sqrt {e+f x}} \, dx=-\frac {\frac {12 C \sqrt {c+d x} \sqrt {e+f x}}{(b e-a f) (a+b x)}+\frac {4 b \left (A b^2+a (-b B+a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{(b c-a d) (b e-a f) (a+b x)^3}+\frac {6 b (b B-2 a C) (c+d x)^{3/2} \sqrt {e+f x}}{(b c-a d) (b e-a f) (a+b x)^2}-\frac {12 C (-d e+c f) \text {arctanh}\left (\frac {\sqrt {-b e+a f} \sqrt {c+d x}}{\sqrt {-b c+a d} \sqrt {e+f x}}\right )}{\sqrt {-b c+a d} (-b e+a f)^{3/2}}+\frac {3 (b B-2 a C) (b d e+3 b c f-4 a d f) \left (\sqrt {-b c+a d} \sqrt {-b e+a f} \sqrt {c+d x} \sqrt {e+f x}-(d e-c f) (a+b x) \text {arctanh}\left (\frac {\sqrt {-b e+a f} \sqrt {c+d x}}{\sqrt {-b c+a d} \sqrt {e+f x}}\right )\right )}{(-b c+a d)^{3/2} (-b e+a f)^{5/2} (a+b x)}-\frac {\left (A b^2+a (-b B+a C)\right ) \left (\frac {2 b (3 b d e+5 b c f-8 a d f) (c+d x)^{3/2} \sqrt {e+f x}}{(a+b x)^2}+3 \left (8 a^2 d^2 f^2-4 a b d f (d e+3 c f)+b^2 \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )\right ) \left (\frac {\sqrt {c+d x} \sqrt {e+f x}}{(-b e+a f) (a+b x)}+\frac {(-d e+c f) \text {arctanh}\left (\frac {\sqrt {-b e+a f} \sqrt {c+d x}}{\sqrt {-b c+a d} \sqrt {e+f x}}\right )}{\sqrt {-b c+a d} (-b e+a f)^{3/2}}\right )\right )}{2 (b c-a d)^2 (b e-a f)^2}}{12 b^2} \] Input:

Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^4*Sqrt[e + f*x]),x]
 

Output:

-1/12*((12*C*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*e - a*f)*(a + b*x)) + (4*b*( 
A*b^2 + a*(-(b*B) + a*C))*(c + d*x)^(3/2)*Sqrt[e + f*x])/((b*c - a*d)*(b*e 
 - a*f)*(a + b*x)^3) + (6*b*(b*B - 2*a*C)*(c + d*x)^(3/2)*Sqrt[e + f*x])/( 
(b*c - a*d)*(b*e - a*f)*(a + b*x)^2) - (12*C*(-(d*e) + c*f)*ArcTanh[(Sqrt[ 
-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(Sqrt[-( 
b*c) + a*d]*(-(b*e) + a*f)^(3/2)) + (3*(b*B - 2*a*C)*(b*d*e + 3*b*c*f - 4* 
a*d*f)*(Sqrt[-(b*c) + a*d]*Sqrt[-(b*e) + a*f]*Sqrt[c + d*x]*Sqrt[e + f*x] 
- (d*e - c*f)*(a + b*x)*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[- 
(b*c) + a*d]*Sqrt[e + f*x])]))/((-(b*c) + a*d)^(3/2)*(-(b*e) + a*f)^(5/2)* 
(a + b*x)) - ((A*b^2 + a*(-(b*B) + a*C))*((2*b*(3*b*d*e + 5*b*c*f - 8*a*d* 
f)*(c + d*x)^(3/2)*Sqrt[e + f*x])/(a + b*x)^2 + 3*(8*a^2*d^2*f^2 - 4*a*b*d 
*f*(d*e + 3*c*f) + b^2*(d^2*e^2 + 2*c*d*e*f + 5*c^2*f^2))*((Sqrt[c + d*x]* 
Sqrt[e + f*x])/((-(b*e) + a*f)*(a + b*x)) + ((-(d*e) + c*f)*ArcTanh[(Sqrt[ 
-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(Sqrt[-( 
b*c) + a*d]*(-(b*e) + a*f)^(3/2)))))/(2*(b*c - a*d)^2*(b*e - a*f)^2))/b^2
 

Rubi [A] (verified)

Time = 1.72 (sec) , antiderivative size = 719, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2116, 27, 166, 27, 168, 27, 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.

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

\(\Big \downarrow \) 2116

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 166

\(\displaystyle \frac {\frac {\int -\frac {4 C d f (d e+c f) a^3+b \left (2 B d f (d e+c f)-C \left (9 d^2 e^2+20 c d f e+3 c^2 f^2\right )\right ) a^2+b^2 \left (3 f (4 C e-B f) c^2+4 d \left (9 C e^2-4 B f e+4 A f^2\right ) c+d^2 e (3 B e-8 A f)\right ) a+b^3 \left (-3 \left (8 C e^2-6 B f e+5 A f^2\right ) c^2-2 d e (3 B e-2 A f) c+3 A d^2 e^2\right )+2 d \left (4 C d f^2 a^3-b f (11 C d e+5 c C f-2 B d f) a^2+b^2 (12 C e (d e+c f)-f (7 B d e+B c f-4 A d f)) a-b^3 \left (12 c C e^2-A d f e-c f (6 B e-5 A f)\right )\right ) x}{2 (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}dx}{2 b (b e-a f)}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (4 a^3 C d f-a^2 b (-2 B d f+7 c C f+9 C d e)+a b^2 (-8 A d f+B c f+3 B d e+12 c C e)-b^3 (-5 A c f-3 A d e+6 B c e)\right )}{2 b (a+b x)^2 (b e-a f)}}{6 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^3 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (4 a^3 C d f-a^2 b (-2 B d f+7 c C f+9 C d e)+a b^2 (-8 A d f+B c f+3 B d e+12 c C e)-b^3 (-5 A c f-3 A d e+6 B c e)\right )}{2 b (a+b x)^2 (b e-a f)}-\frac {\int \frac {4 C d f (d e+c f) a^3+b \left (2 B d f (d e+c f)-C \left (9 d^2 e^2+20 c d f e+3 c^2 f^2\right )\right ) a^2+b^2 \left (3 f (4 C e-B f) c^2+4 d \left (9 C e^2-4 B f e+4 A f^2\right ) c+d^2 e (3 B e-8 A f)\right ) a+b^3 \left (-3 \left (8 C e^2-6 B f e+5 A f^2\right ) c^2-2 d e (3 B e-2 A f) c+3 A d^2 e^2\right )+2 d \left (4 C d f^2 a^3-b f (11 C d e+5 c C f-2 B d f) a^2+b^2 (12 C e (d e+c f)-f (7 B d e+B c f-4 A d f)) a-b^3 \left (12 c C e^2-A d f e-c f (6 B e-5 A f)\right )\right ) x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}dx}{4 b (b e-a f)}}{6 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^3 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (4 a^3 C d f-a^2 b (-2 B d f+7 c C f+9 C d e)+a b^2 (-8 A d f+B c f+3 B d e+12 c C e)-b^3 (-5 A c f-3 A d e+6 B c e)\right )}{2 b (a+b x)^2 (b e-a f)}-\frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (8 a^4 C d^2 f^2-2 a^3 b d f (-2 B d f+7 c C f+13 C d e)-a^2 b^2 \left (4 d f (-2 A d f+B c f+4 B d e)-C \left (3 c^2 f^2+44 c d e f+33 d^2 e^2\right )\right )-a b^3 \left (2 c d \left (13 A f^2-14 B e f+30 C e^2\right )+d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)\right )-b^4 \left (-3 c^2 \left (5 A f^2-6 B e f+8 C e^2\right )-2 c d e (3 B e-2 A f)+3 A d^2 e^2\right )\right )}{(a+b x) (b c-a d) (b e-a f)}-\frac {\int -\frac {3 b^2 (d e-c f) \left (\left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) a^2-b \left (-f (4 C e-B f) c^2-2 d \left (6 C e^2-f (7 B e-6 A f)\right ) c+d^2 e (B e-4 A f)\right ) a-b^2 \left (\left (8 C e^2-f (6 B e-5 A f)\right ) c^2-2 d e (B e-A f) c+A d^2 e^2\right )\right )}{2 (a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{4 b (b e-a f)}}{6 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^3 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (4 a^3 C d f-a^2 b (-2 B d f+7 c C f+9 C d e)+a b^2 (-8 A d f+B c f+3 B d e+12 c C e)-b^3 (-5 A c f-3 A d e+6 B c e)\right )}{2 b (a+b x)^2 (b e-a f)}-\frac {\frac {3 b^2 (d e-c f) \left (a^2 \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )-a b \left (-2 c d \left (6 C e^2-f (7 B e-6 A f)\right )+d^2 e (B e-4 A f)+c^2 (-f) (4 C e-B f)\right )-b^2 \left (c^2 \left (8 C e^2-f (6 B e-5 A f)\right )-2 c d e (B e-A f)+A d^2 e^2\right )\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (8 a^4 C d^2 f^2-2 a^3 b d f (-2 B d f+7 c C f+13 C d e)-a^2 b^2 \left (4 d f (-2 A d f+B c f+4 B d e)-C \left (3 c^2 f^2+44 c d e f+33 d^2 e^2\right )\right )-a b^3 \left (2 c d \left (13 A f^2-14 B e f+30 C e^2\right )+d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)\right )-b^4 \left (-3 c^2 \left (5 A f^2-6 B e f+8 C e^2\right )-2 c d e (3 B e-2 A f)+3 A d^2 e^2\right )\right )}{(a+b x) (b c-a d) (b e-a f)}}{4 b (b e-a f)}}{6 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^3 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (4 a^3 C d f-a^2 b (-2 B d f+7 c C f+9 C d e)+a b^2 (-8 A d f+B c f+3 B d e+12 c C e)-b^3 (-5 A c f-3 A d e+6 B c e)\right )}{2 b (a+b x)^2 (b e-a f)}-\frac {\frac {3 b^2 (d e-c f) \left (a^2 \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )-a b \left (-2 c d \left (6 C e^2-f (7 B e-6 A f)\right )+d^2 e (B e-4 A f)+c^2 (-f) (4 C e-B f)\right )-b^2 \left (c^2 \left (8 C e^2-f (6 B e-5 A f)\right )-2 c d e (B e-A f)+A d^2 e^2\right )\right ) \int \frac {1}{-b c+a d+\frac {(b e-a f) (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{(b c-a d) (b e-a f)}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (8 a^4 C d^2 f^2-2 a^3 b d f (-2 B d f+7 c C f+13 C d e)-a^2 b^2 \left (4 d f (-2 A d f+B c f+4 B d e)-C \left (3 c^2 f^2+44 c d e f+33 d^2 e^2\right )\right )-a b^3 \left (2 c d \left (13 A f^2-14 B e f+30 C e^2\right )+d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)\right )-b^4 \left (-3 c^2 \left (5 A f^2-6 B e f+8 C e^2\right )-2 c d e (3 B e-2 A f)+3 A d^2 e^2\right )\right )}{(a+b x) (b c-a d) (b e-a f)}}{4 b (b e-a f)}}{6 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^3 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (4 a^3 C d f-a^2 b (-2 B d f+7 c C f+9 C d e)+a b^2 (-8 A d f+B c f+3 B d e+12 c C e)-b^3 (-5 A c f-3 A d e+6 B c e)\right )}{2 b (a+b x)^2 (b e-a f)}-\frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (8 a^4 C d^2 f^2-2 a^3 b d f (-2 B d f+7 c C f+13 C d e)-a^2 b^2 \left (4 d f (-2 A d f+B c f+4 B d e)-C \left (3 c^2 f^2+44 c d e f+33 d^2 e^2\right )\right )-a b^3 \left (2 c d \left (13 A f^2-14 B e f+30 C e^2\right )+d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)\right )-b^4 \left (-3 c^2 \left (5 A f^2-6 B e f+8 C e^2\right )-2 c d e (3 B e-2 A f)+3 A d^2 e^2\right )\right )}{(a+b x) (b c-a d) (b e-a f)}-\frac {3 b^2 (d e-c f) \text {arctanh}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right ) \left (a^2 \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )-a b \left (-2 c d \left (6 C e^2-f (7 B e-6 A f)\right )+d^2 e (B e-4 A f)+c^2 (-f) (4 C e-B f)\right )-b^2 \left (c^2 \left (8 C e^2-f (6 B e-5 A f)\right )-2 c d e (B e-A f)+A d^2 e^2\right )\right )}{(b c-a d)^{3/2} (b e-a f)^{3/2}}}{4 b (b e-a f)}}{6 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^3 (b c-a d) (b e-a f)}\)

Input:

Int[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^4*Sqrt[e + f*x]),x]
 

Output:

-1/3*((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)*(a + b*x)^3) + (((4*a^3*C*d*f - b^3*(6*B*c*e - 3*A*d*e - 5*A 
*c*f) + a*b^2*(12*c*C*e + 3*B*d*e + B*c*f - 8*A*d*f) - a^2*b*(9*C*d*e + 7* 
c*C*f - 2*B*d*f))*Sqrt[c + d*x]*Sqrt[e + f*x])/(2*b*(b*e - a*f)*(a + b*x)^ 
2) - (((8*a^4*C*d^2*f^2 - 2*a^3*b*d*f*(13*C*d*e + 7*c*C*f - 2*B*d*f) - b^4 
*(3*A*d^2*e^2 - 2*c*d*e*(3*B*e - 2*A*f) - 3*c^2*(8*C*e^2 - 6*B*e*f + 5*A*f 
^2)) - a*b^3*(d^2*e*(3*B*e - 10*A*f) + 3*c^2*f*(4*C*e - B*f) + 2*c*d*(30*C 
*e^2 - 14*B*e*f + 13*A*f^2)) - a^2*b^2*(4*d*f*(4*B*d*e + B*c*f - 2*A*d*f) 
- C*(33*d^2*e^2 + 44*c*d*e*f + 3*c^2*f^2)))*Sqrt[c + d*x]*Sqrt[e + f*x])/( 
(b*c - a*d)*(b*e - a*f)*(a + b*x)) - (3*b^2*(d*e - c*f)*(a^2*(2*d*f*(3*B*d 
*e + B*c*f - 4*A*d*f) - C*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2)) - a*b*(d^2*e* 
(B*e - 4*A*f) - c^2*f*(4*C*e - B*f) - 2*c*d*(6*C*e^2 - f*(7*B*e - 6*A*f))) 
 - b^2*(A*d^2*e^2 - 2*c*d*e*(B*e - A*f) + c^2*(8*C*e^2 - f*(6*B*e - 5*A*f) 
)))*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x] 
)])/((b*c - a*d)^(3/2)*(b*e - a*f)^(3/2)))/(4*b*(b*e - a*f)))/(6*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 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 166

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - 
a*f)*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S 
imp[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*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

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. \(15989\) vs. \(2(653)=1306\).

Time = 1.10 (sec) , antiderivative size = 15990, normalized size of antiderivative = 23.34

method	result	size
default	\(\text {Expression too large to display}\)	\(15990\)

Input:

int((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^4/(f*x+e)^(1/2),x,method=_RETURNVE 
RBOSE)
 

Output:

result too large to display
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^4 \sqrt {e+f x}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^4/(f*x+e)^(1/2),x, algorithm 
="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^4 \sqrt {e+f x}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(1/2)*(C*x**2+B*x+A)/(b*x+a)**4/(f*x+e)**(1/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^4 \sqrt {e+f x}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^4/(f*x+e)^(1/2),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((a*d-b*c)>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 25338 vs. \(2 (653) = 1306\).

Time = 79.24 (sec) , antiderivative size = 25338, normalized size of antiderivative = 36.99 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^4 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^4/(f*x+e)^(1/2),x, algorithm 
="giac")
 

Output:

-1/8*(8*sqrt(d*f)*C*b^2*c^2*d^3*e^3 - 12*sqrt(d*f)*C*a*b*c*d^4*e^3 - 2*sqr 
t(d*f)*B*b^2*c*d^4*e^3 + 5*sqrt(d*f)*C*a^2*d^5*e^3 + sqrt(d*f)*B*a*b*d^5*e 
^3 + sqrt(d*f)*A*b^2*d^5*e^3 - 8*sqrt(d*f)*C*b^2*c^3*d^2*e^2*f + 8*sqrt(d* 
f)*C*a*b*c^2*d^3*e^2*f - 4*sqrt(d*f)*B*b^2*c^2*d^3*e^2*f - 3*sqrt(d*f)*C*a 
^2*c*d^4*e^2*f + 13*sqrt(d*f)*B*a*b*c*d^4*e^2*f + sqrt(d*f)*A*b^2*c*d^4*e^ 
2*f - 6*sqrt(d*f)*B*a^2*d^5*e^2*f - 4*sqrt(d*f)*A*a*b*d^5*e^2*f + 4*sqrt(d 
*f)*C*a*b*c^3*d^2*e*f^2 + 6*sqrt(d*f)*B*b^2*c^3*d^2*e*f^2 - sqrt(d*f)*C*a^ 
2*c^2*d^3*e*f^2 - 13*sqrt(d*f)*B*a*b*c^2*d^3*e*f^2 + 3*sqrt(d*f)*A*b^2*c^2 
*d^3*e*f^2 + 4*sqrt(d*f)*B*a^2*c*d^4*e*f^2 - 8*sqrt(d*f)*A*a*b*c*d^4*e*f^2 
 + 8*sqrt(d*f)*A*a^2*d^5*e*f^2 - sqrt(d*f)*C*a^2*c^3*d^2*f^3 - sqrt(d*f)*B 
*a*b*c^3*d^2*f^3 - 5*sqrt(d*f)*A*b^2*c^3*d^2*f^3 + 2*sqrt(d*f)*B*a^2*c^2*d 
^3*f^3 + 12*sqrt(d*f)*A*a*b*c^2*d^3*f^3 - 8*sqrt(d*f)*A*a^2*c*d^4*f^3)*arc 
tan(-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))/((b^5*c^2*e^3*abs(d) - 2*a*b^4*c*d*e^3*abs(d) 
 + a^2*b^3*d^2*e^3*abs(d) - 3*a*b^4*c^2*e^2*f*abs(d) + 6*a^2*b^3*c*d*e^2*f 
*abs(d) - 3*a^3*b^2*d^2*e^2*f*abs(d) + 3*a^2*b^3*c^2*e*f^2*abs(d) - 6*a^3* 
b^2*c*d*e*f^2*abs(d) + 3*a^4*b*d^2*e*f^2*abs(d) - a^3*b^2*c^2*f^3*abs(d) + 
 2*a^4*b*c*d*f^3*abs(d) - a^5*d^2*f^3*abs(d))*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) - 1/12*(24*sqrt(d*f)*C*b^7*c^2*d^13...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^4 \sqrt {e+f x}} \, dx=\text {Hanged} \] Input:

int(((c + d*x)^(1/2)*(A + B*x + C*x^2))/((e + f*x)^(1/2)*(a + b*x)^4),x)
 

Output:

\text{Hanged}
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^4 \sqrt {e+f x}} \, dx=\int \frac {\sqrt {d x +c}\, \left (C \,x^{2}+B x +A \right )}{\left (b x +a \right )^{4} \sqrt {f x +e}}d x \] Input:

int((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^4/(f*x+e)^(1/2),x)
 

Output:

int((d*x+c)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^4/(f*x+e)^(1/2),x)