\(\int \frac {\sqrt {c+d x} \sqrt {e+f x} (A+B x+C x^2)}{(a+b x)^3} \, dx\) [68]

Optimal result

Integrand size = 36, antiderivative size = 660 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^3} \, dx=-\frac {\left (12 a^3 C d^2 f-a^2 b d (11 C d e+17 c C f+4 B d f)-b^3 \left (4 c^2 C e-A d^2 e+c d (4 B e+A f)\right )+a b^2 \left (3 B d^2 e+4 c^2 C f+c d (16 C e+5 B f)\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^3 (b c-a d)^2 (b e-a f)}+\frac {\left (6 a^3 C d f-b^3 (4 B c e-A d e-A c f)+a b^2 (8 c C e+3 B d e+3 B c f-2 A d f)-a^2 b (2 B d f+7 C (d e+c f))\right ) (c+d x)^{3/2} \sqrt {e+f x}}{4 b^2 (b c-a d)^2 (b e-a f) (a+b x)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{2 b (b c-a d) (b e-a f) (a+b x)^2}-\frac {(6 a C d f-b (C d e+c C f+2 B d f)) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{b^4 \sqrt {d} \sqrt {f}}-\frac {\left (24 a^4 C d^2 f^2-3 a b^3 \left (B d^2 e^2+c^2 f (8 C e+B f)+2 c d e (4 C e+3 B f)\right )-8 a^3 b d f (B d f+5 C (d e+c f))-b^4 \left (A d^2 e^2-2 c d e (2 B e+A f)-c^2 \left (8 C e^2+4 B e f-A f^2\right )\right )+3 a^2 b^2 \left (4 B d f (d e+c f)+C \left (5 d^2 e^2+22 c d e f+5 c^2 f^2\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{4 b^4 (b c-a d)^{3/2} (b e-a f)^{3/2}} \] Output:

-1/4*(12*a^3*C*d^2*f-a^2*b*d*(4*B*d*f+17*C*c*f+11*C*d*e)-b^3*(4*c^2*C*e-A* 
d^2*e+c*d*(A*f+4*B*e))+a*b^2*(3*B*d^2*e+4*c^2*C*f+c*d*(5*B*f+16*C*e)))*(d* 
x+c)^(1/2)*(f*x+e)^(1/2)/b^3/(-a*d+b*c)^2/(-a*f+b*e)+1/4*(6*a^3*C*d*f-b^3* 
(-A*c*f-A*d*e+4*B*c*e)+a*b^2*(-2*A*d*f+3*B*c*f+3*B*d*e+8*C*c*e)-a^2*b*(2*B 
*d*f+7*C*(c*f+d*e)))*(d*x+c)^(3/2)*(f*x+e)^(1/2)/b^2/(-a*d+b*c)^2/(-a*f+b* 
e)/(b*x+a)-1/2*(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)^2-(6*a*C*d*f-b*(2*B*d*f+C*c*f+C*d*e))*arctanh(f^(1/2) 
*(d*x+c)^(1/2)/d^(1/2)/(f*x+e)^(1/2))/b^4/d^(1/2)/f^(1/2)-1/4*(24*a^4*C*d^ 
2*f^2-3*a*b^3*(B*d^2*e^2+c^2*f*(B*f+8*C*e)+2*c*d*e*(3*B*f+4*C*e))-8*a^3*b* 
d*f*(B*d*f+5*C*(c*f+d*e))-b^4*(A*d^2*e^2-2*c*d*e*(A*f+2*B*e)-c^2*(-A*f^2+4 
*B*e*f+8*C*e^2))+3*a^2*b^2*(4*B*d*f*(c*f+d*e)+C*(5*c^2*f^2+22*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))/b^4/(-a*d+b*c)^(3/2)/(-a*f+b*e)^(3/2)
 

Mathematica [A] (verified)

Time = 4.62 (sec) , antiderivative size = 536, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^3} \, dx=\frac {\frac {b \sqrt {c+d x} \sqrt {e+f x} \left (12 a^4 C d f+4 b^4 c e x (-B+C x)+A b^3 (a c f+a d (e+2 f x)-b (2 c e+d e x+c f x))+a b^3 (-4 C x (-4 c e+d e x+c f x)+B (-2 c e+5 d e x+5 c f x))+a^2 b^2 (3 B d (e-2 f x)+C d x (-17 e+4 f x)+c (10 C e+3 B f-17 C f x))-a^3 b (4 B d f+C (11 d e+11 c f-18 d f x))\right )}{(b c-a d) (b e-a f) (a+b x)^2}-\frac {\left (24 a^4 C d^2 f^2-3 a b^3 \left (B d^2 e^2+c^2 f (8 C e+B f)+2 c d e (4 C e+3 B f)\right )-8 a^3 b d f (B d f+5 C (d e+c f))+b^4 \left (-A d^2 e^2+2 c d e (2 B e+A f)+c^2 \left (8 C e^2+4 B e f-A f^2\right )\right )+3 a^2 b^2 \left (4 B d f (d e+c f)+C \left (5 d^2 e^2+22 c d e f+5 c^2 f^2\right )\right )\right ) \arctan \left (\frac {\sqrt {b c-a d} \sqrt {e+f x}}{\sqrt {-b e+a f} \sqrt {c+d x}}\right )}{(b c-a d)^{3/2} (-b e+a f)^{3/2}}+\frac {4 (-6 a C d f+b (C d e+c C f+2 B d f)) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {f} \sqrt {c+d x}}\right )}{\sqrt {d} \sqrt {f}}}{4 b^4} \] Input:

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

Output:

((b*Sqrt[c + d*x]*Sqrt[e + f*x]*(12*a^4*C*d*f + 4*b^4*c*e*x*(-B + C*x) + A 
*b^3*(a*c*f + a*d*(e + 2*f*x) - b*(2*c*e + d*e*x + c*f*x)) + a*b^3*(-4*C*x 
*(-4*c*e + d*e*x + c*f*x) + B*(-2*c*e + 5*d*e*x + 5*c*f*x)) + a^2*b^2*(3*B 
*d*(e - 2*f*x) + C*d*x*(-17*e + 4*f*x) + c*(10*C*e + 3*B*f - 17*C*f*x)) - 
a^3*b*(4*B*d*f + C*(11*d*e + 11*c*f - 18*d*f*x))))/((b*c - a*d)*(b*e - a*f 
)*(a + b*x)^2) - ((24*a^4*C*d^2*f^2 - 3*a*b^3*(B*d^2*e^2 + c^2*f*(8*C*e + 
B*f) + 2*c*d*e*(4*C*e + 3*B*f)) - 8*a^3*b*d*f*(B*d*f + 5*C*(d*e + c*f)) + 
b^4*(-(A*d^2*e^2) + 2*c*d*e*(2*B*e + A*f) + c^2*(8*C*e^2 + 4*B*e*f - A*f^2 
)) + 3*a^2*b^2*(4*B*d*f*(d*e + c*f) + C*(5*d^2*e^2 + 22*c*d*e*f + 5*c^2*f^ 
2)))*ArcTan[(Sqrt[b*c - a*d]*Sqrt[e + f*x])/(Sqrt[-(b*e) + a*f]*Sqrt[c + d 
*x])])/((b*c - a*d)^(3/2)*(-(b*e) + a*f)^(3/2)) + (4*(-6*a*C*d*f + b*(C*d* 
e + c*C*f + 2*B*d*f))*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/(Sqrt[f]*Sqrt[c + d* 
x])])/(Sqrt[d]*Sqrt[f]))/(4*b^4)
 

Rubi [A] (verified)

Time = 1.64 (sec) , antiderivative size = 695, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2116, 27, 166, 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.

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

\(\Big \downarrow \) 2116

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 166

\(\displaystyle \frac {\frac {\int -\frac {\sqrt {e+f x} \left (6 C d f (d e+3 c f) a^3-b \left (2 B d f (d e+3 c f)+C \left (7 d^2 e^2+34 c d f e+15 c^2 f^2\right )\right ) a^2+b^2 \left (3 f (8 C e+B f) c^2+2 d \left (8 C e^2+5 B f e+A f^2\right ) c+d^2 e (3 B e-2 A f)\right ) a+b^3 \left (-\left (\left (8 C e^2+4 B f e-A f^2\right ) c^2\right )-2 d e (2 B e+A f) c+A d^2 e^2\right )+2 d \left (12 C d f^2 a^3-b f (17 C d e+11 c C f+4 B d f) a^2+b^2 (B f (5 d e+3 c f)+4 C e (d e+4 c f)) a-b^3 \left (4 c C e^2+A d f e+c f (4 B e-A f)\right )\right ) x\right )}{2 (a+b x) \sqrt {c+d x}}dx}{b (b e-a f)}+\frac {\sqrt {c+d x} (e+f x)^{3/2} \left (6 a^3 C d f-a^2 b (2 B d f+7 C (c f+d e))+a b^2 (-2 A d f+3 B c f+3 B d e+8 c C e)-b^3 (-A c f-A d e+4 B c e)\right )}{b (a+b x) (b e-a f)}}{4 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {c+d x} (e+f x)^{3/2} \left (6 a^3 C d f-a^2 b (2 B d f+7 C (c f+d e))+a b^2 (-2 A d f+3 B c f+3 B d e+8 c C e)-b^3 (-A c f-A d e+4 B c e)\right )}{b (a+b x) (b e-a f)}-\frac {\int \frac {\sqrt {e+f x} \left (6 C d f (d e+3 c f) a^3-b \left (2 B d f (d e+3 c f)+C \left (7 d^2 e^2+34 c d f e+15 c^2 f^2\right )\right ) a^2+b^2 \left (3 f (8 C e+B f) c^2+2 d \left (8 C e^2+5 B f e+A f^2\right ) c+d^2 e (3 B e-2 A f)\right ) a+b^3 \left (-\left (\left (8 C e^2+4 B f e-A f^2\right ) c^2\right )-2 d e (2 B e+A f) c+A d^2 e^2\right )+2 d \left (12 C d f^2 a^3-b f (17 C d e+11 c C f+4 B d f) a^2+b^2 (B f (5 d e+3 c f)+4 C e (d e+4 c f)) a-b^3 \left (4 c C e^2+A d f e+c f (4 B e-A f)\right )\right ) x\right )}{(a+b x) \sqrt {c+d x}}dx}{2 b (b e-a f)}}{4 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {\frac {\sqrt {c+d x} (e+f x)^{3/2} \left (6 a^3 C d f-a^2 b (2 B d f+7 C (c f+d e))+a b^2 (-2 A d f+3 B c f+3 B d e+8 c C e)-b^3 (-A c f-A d e+4 B c e)\right )}{b (a+b x) (b e-a f)}-\frac {\frac {\int \frac {d (b e-a f) \left (12 C d f (d e+c f) a^3-b \left (4 B d f (d e+c f)+C \left (11 d^2 e^2+34 c d f e+11 c^2 f^2\right )\right ) a^2+b^2 \left (f (20 C e+3 B f) c^2+10 d e (2 C e+B f) c+3 B d^2 e^2\right ) a+b^3 \left (-\left (\left (8 C e^2+4 B f e-A f^2\right ) c^2\right )-2 d e (2 B e+A f) c+A d^2 e^2\right )+4 (b c-a d) (b e-a f) (6 a C d f-b (C d e+c C f+2 B d f)) x\right )}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b d}+\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (12 a^3 C d f^2-a^2 b f (4 B d f+11 c C f+17 C d e)+a b^2 (B f (3 c f+5 d e)+4 C e (4 c f+d e))-b^3 \left (c f (4 B e-A f)+A d e f+4 c C e^2\right )\right )}{b}}{2 b (b e-a f)}}{4 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {c+d x} (e+f x)^{3/2} \left (6 a^3 C d f-a^2 b (2 B d f+7 C (c f+d e))+a b^2 (-2 A d f+3 B c f+3 B d e+8 c C e)-b^3 (-A c f-A d e+4 B c e)\right )}{b (a+b x) (b e-a f)}-\frac {\frac {(b e-a f) \int \frac {12 C d f (d e+c f) a^3-b \left (4 B d f (d e+c f)+C \left (11 d^2 e^2+34 c d f e+11 c^2 f^2\right )\right ) a^2+b^2 \left (f (20 C e+3 B f) c^2+10 d e (2 C e+B f) c+3 B d^2 e^2\right ) a+b^3 \left (-\left (\left (8 C e^2+4 B f e-A f^2\right ) c^2\right )-2 d e (2 B e+A f) c+A d^2 e^2\right )+4 (b c-a d) (b e-a f) (6 a C d f-b (C d e+c C f+2 B d f)) x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b}+\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (12 a^3 C d f^2-a^2 b f (4 B d f+11 c C f+17 C d e)+a b^2 (B f (3 c f+5 d e)+4 C e (4 c f+d e))-b^3 \left (c f (4 B e-A f)+A d e f+4 c C e^2\right )\right )}{b}}{2 b (b e-a f)}}{4 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {\frac {\sqrt {c+d x} (e+f x)^{3/2} \left (6 a^3 C d f-a^2 b (2 B d f+7 C (c f+d e))+a b^2 (-2 A d f+3 B c f+3 B d e+8 c C e)-b^3 (-A c f-A d e+4 B c e)\right )}{b (a+b x) (b e-a f)}-\frac {\frac {(b e-a f) \left (\frac {4 (b c-a d) (b e-a f) (6 a C d f-b (2 B d f+c C f+C d e)) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}}dx}{b}-\frac {\left (24 a^4 C d^2 f^2-8 a^3 b d f (B d f+5 C (c f+d e))+3 a^2 b^2 \left (4 B d f (c f+d e)+C \left (5 c^2 f^2+22 c d e f+5 d^2 e^2\right )\right )-3 a b^3 \left (c^2 f (B f+8 C e)+2 c d e (3 B f+4 C e)+B d^2 e^2\right )-b^4 \left (-\left (c^2 \left (-A f^2+4 B e f+8 C e^2\right )\right )-2 c d e (A f+2 B e)+A d^2 e^2\right )\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b}\right )}{b}+\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (12 a^3 C d f^2-a^2 b f (4 B d f+11 c C f+17 C d e)+a b^2 (B f (3 c f+5 d e)+4 C e (4 c f+d e))-b^3 \left (c f (4 B e-A f)+A d e f+4 c C e^2\right )\right )}{b}}{2 b (b e-a f)}}{4 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {\frac {\sqrt {c+d x} (e+f x)^{3/2} \left (6 a^3 C d f-a^2 b (2 B d f+7 C (c f+d e))+a b^2 (-2 A d f+3 B c f+3 B d e+8 c C e)-b^3 (-A c f-A d e+4 B c e)\right )}{b (a+b x) (b e-a f)}-\frac {\frac {(b e-a f) \left (\frac {8 (b c-a d) (b e-a f) (6 a C d f-b (2 B d f+c C f+C d e)) \int \frac {1}{d-\frac {f (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}-\frac {\left (24 a^4 C d^2 f^2-8 a^3 b d f (B d f+5 C (c f+d e))+3 a^2 b^2 \left (4 B d f (c f+d e)+C \left (5 c^2 f^2+22 c d e f+5 d^2 e^2\right )\right )-3 a b^3 \left (c^2 f (B f+8 C e)+2 c d e (3 B f+4 C e)+B d^2 e^2\right )-b^4 \left (-\left (c^2 \left (-A f^2+4 B e f+8 C e^2\right )\right )-2 c d e (A f+2 B e)+A d^2 e^2\right )\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b}\right )}{b}+\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (12 a^3 C d f^2-a^2 b f (4 B d f+11 c C f+17 C d e)+a b^2 (B f (3 c f+5 d e)+4 C e (4 c f+d e))-b^3 \left (c f (4 B e-A f)+A d e f+4 c C e^2\right )\right )}{b}}{2 b (b e-a f)}}{4 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\frac {\sqrt {c+d x} (e+f x)^{3/2} \left (6 a^3 C d f-a^2 b (2 B d f+7 C (c f+d e))+a b^2 (-2 A d f+3 B c f+3 B d e+8 c C e)-b^3 (-A c f-A d e+4 B c e)\right )}{b (a+b x) (b e-a f)}-\frac {\frac {(b e-a f) \left (\frac {8 (b c-a d) (b e-a f) (6 a C d f-b (2 B d f+c C f+C d e)) \int \frac {1}{d-\frac {f (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}-\frac {2 \left (24 a^4 C d^2 f^2-8 a^3 b d f (B d f+5 C (c f+d e))+3 a^2 b^2 \left (4 B d f (c f+d e)+C \left (5 c^2 f^2+22 c d e f+5 d^2 e^2\right )\right )-3 a b^3 \left (c^2 f (B f+8 C e)+2 c d e (3 B f+4 C e)+B d^2 e^2\right )-b^4 \left (-\left (c^2 \left (-A f^2+4 B e f+8 C e^2\right )\right )-2 c d e (A f+2 B e)+A d^2 e^2\right )\right ) \int \frac {1}{-b c+a d+\frac {(b e-a f) (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}\right )}{b}+\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (12 a^3 C d f^2-a^2 b f (4 B d f+11 c C f+17 C d e)+a b^2 (B f (3 c f+5 d e)+4 C e (4 c f+d e))-b^3 \left (c f (4 B e-A f)+A d e f+4 c C e^2\right )\right )}{b}}{2 b (b e-a f)}}{4 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\sqrt {c+d x} (e+f x)^{3/2} \left (6 a^3 C d f-a^2 b (2 B d f+7 C (c f+d e))+a b^2 (-2 A d f+3 B c f+3 B d e+8 c C e)-b^3 (-A c f-A d e+4 B c e)\right )}{b (a+b x) (b e-a f)}-\frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (12 a^3 C d f^2-a^2 b f (4 B d f+11 c C f+17 C d e)+a b^2 (B f (3 c f+5 d e)+4 C e (4 c f+d e))-b^3 \left (c f (4 B e-A f)+A d e f+4 c C e^2\right )\right )}{b}+\frac {(b e-a f) \left (\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right ) \left (24 a^4 C d^2 f^2-8 a^3 b d f (B d f+5 C (c f+d e))+3 a^2 b^2 \left (4 B d f (c f+d e)+C \left (5 c^2 f^2+22 c d e f+5 d^2 e^2\right )\right )-3 a b^3 \left (c^2 f (B f+8 C e)+2 c d e (3 B f+4 C e)+B d^2 e^2\right )-b^4 \left (-\left (c^2 \left (-A f^2+4 B e f+8 C e^2\right )\right )-2 c d e (A f+2 B e)+A d^2 e^2\right )\right )}{b \sqrt {b c-a d} \sqrt {b e-a f}}+\frac {8 (b c-a d) (b e-a f) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) (6 a C d f-b (2 B d f+c C f+C d e))}{b \sqrt {d} \sqrt {f}}\right )}{b}}{2 b (b e-a f)}}{4 b (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}\)

Input:

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

Output:

-1/2*((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)^2) + (((6*a^3*C*d*f - b^3*(4*B*c*e - A*d*e - A*c 
*f) + a*b^2*(8*c*C*e + 3*B*d*e + 3*B*c*f - 2*A*d*f) - a^2*b*(2*B*d*f + 7*C 
*(d*e + c*f)))*Sqrt[c + d*x]*(e + f*x)^(3/2))/(b*(b*e - a*f)*(a + b*x)) - 
((2*(12*a^3*C*d*f^2 - a^2*b*f*(17*C*d*e + 11*c*C*f + 4*B*d*f) - b^3*(4*c*C 
*e^2 + A*d*e*f + c*f*(4*B*e - A*f)) + a*b^2*(B*f*(5*d*e + 3*c*f) + 4*C*e*( 
d*e + 4*c*f)))*Sqrt[c + d*x]*Sqrt[e + f*x])/b + ((b*e - a*f)*((8*(b*c - a* 
d)*(b*e - a*f)*(6*a*C*d*f - b*(C*d*e + c*C*f + 2*B*d*f))*ArcTanh[(Sqrt[f]* 
Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(b*Sqrt[d]*Sqrt[f]) + (2*(24*a^4* 
C*d^2*f^2 - 3*a*b^3*(B*d^2*e^2 + c^2*f*(8*C*e + B*f) + 2*c*d*e*(4*C*e + 3* 
B*f)) - 8*a^3*b*d*f*(B*d*f + 5*C*(d*e + c*f)) - b^4*(A*d^2*e^2 - 2*c*d*e*( 
2*B*e + A*f) - c^2*(8*C*e^2 + 4*B*e*f - A*f^2)) + 3*a^2*b^2*(4*B*d*f*(d*e 
+ c*f) + C*(5*d^2*e^2 + 22*c*d*e*f + 5*c^2*f^2)))*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])))/b)/(2*b*(b*e - a*f)))/(4*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 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 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. \(11203\) vs. \(2(616)=1232\).

Time = 0.88 (sec) , antiderivative size = 11204, normalized size of antiderivative = 16.98

method	result	size
default	\(\text {Expression too large to display}\)	\(11204\)

Input:

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

Output:

result too large to display
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((d*x+c)^(1/2)*(f*x+e)^(1/2)*(C*x^2+B*x+A)/(b*x+a)^3,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. 8241 vs. \(2 (615) = 1230\).

Time = 4.82 (sec) , antiderivative size = 8241, normalized size of antiderivative = 12.49 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/4*(8*sqrt(d*f)*C*b^4*c^2*e^2*abs(d) - 24*sqrt(d*f)*C*a*b^3*c*d*e^2*abs( 
d) + 4*sqrt(d*f)*B*b^4*c*d*e^2*abs(d) + 15*sqrt(d*f)*C*a^2*b^2*d^2*e^2*abs 
(d) - 3*sqrt(d*f)*B*a*b^3*d^2*e^2*abs(d) - sqrt(d*f)*A*b^4*d^2*e^2*abs(d) 
- 24*sqrt(d*f)*C*a*b^3*c^2*e*f*abs(d) + 4*sqrt(d*f)*B*b^4*c^2*e*f*abs(d) + 
 66*sqrt(d*f)*C*a^2*b^2*c*d*e*f*abs(d) - 18*sqrt(d*f)*B*a*b^3*c*d*e*f*abs( 
d) + 2*sqrt(d*f)*A*b^4*c*d*e*f*abs(d) - 40*sqrt(d*f)*C*a^3*b*d^2*e*f*abs(d 
) + 12*sqrt(d*f)*B*a^2*b^2*d^2*e*f*abs(d) + 15*sqrt(d*f)*C*a^2*b^2*c^2*f^2 
*abs(d) - 3*sqrt(d*f)*B*a*b^3*c^2*f^2*abs(d) - sqrt(d*f)*A*b^4*c^2*f^2*abs 
(d) - 40*sqrt(d*f)*C*a^3*b*c*d*f^2*abs(d) + 12*sqrt(d*f)*B*a^2*b^2*c*d*f^2 
*abs(d) + 24*sqrt(d*f)*C*a^4*d^2*f^2*abs(d) - 8*sqrt(d*f)*B*a^3*b*d^2*f^2* 
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))/((b^6*c*e - a*b^5*d*e - a*b^5*c*f 
+ a^2*b^4*d*f)*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) + 1/2*(8*sqrt(d*f)*C*a*b^4*c*d^7*e^5*abs(d) - 4*sqrt(d*f)*B*b^5*c*d^7 
*e^5*abs(d) - 9*sqrt(d*f)*C*a^2*b^3*d^8*e^5*abs(d) + 5*sqrt(d*f)*B*a*b^4*d 
^8*e^5*abs(d) - sqrt(d*f)*A*b^5*d^8*e^5*abs(d) - 32*sqrt(d*f)*C*a*b^4*c^2* 
d^6*e^4*f*abs(d) + 16*sqrt(d*f)*B*b^5*c^2*d^6*e^4*f*abs(d) + 27*sqrt(d*f)* 
C*a^2*b^3*c*d^7*e^4*f*abs(d) - 15*sqrt(d*f)*B*a*b^4*c*d^7*e^4*f*abs(d) + 3 
*sqrt(d*f)*A*b^5*c*d^7*e^4*f*abs(d) + 10*sqrt(d*f)*C*a^3*b^2*d^8*e^4*f*...
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

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

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

Output:

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