\(\int \frac {(a+b x)^2 (A+B x+C x^2)}{\sqrt {c+d x} \sqrt {e+f x}} \, dx\) [76]

Optimal result

Integrand size = 36, antiderivative size = 723 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {\left (16 a^2 d^2 f^2 (3 C d e+5 c C f-4 B d f)+16 a b d f \left (2 d f (3 B d e+5 B c f-4 A d f)-C \left (5 d^2 e^2+8 c d e f+11 c^2 f^2\right )\right )+b^2 \left (C \left (35 d^3 e^3+55 c d^2 e^2 f+73 c^2 d e f^2+93 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+5 c f)-B \left (5 d^2 e^2+8 c d e f+11 c^2 f^2\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{64 d^4 f^4}+\frac {\left (48 a^2 C d^2 f^2-16 a b d f (5 C d e+13 c C f-6 B d f)-b^2 \left (8 d f (5 B d e+13 B c f-6 A d f)-C \left (35 d^2 e^2+90 c d e f+163 c^2 f^2\right )\right )\right ) (c+d x)^{3/2} \sqrt {e+f x}}{96 d^4 f^3}+\frac {b (16 a C d f-b (7 C d e+25 c C f-8 B d f)) (c+d x)^{5/2} \sqrt {e+f x}}{24 d^4 f^2}+\frac {b^2 C (c+d x)^{7/2} \sqrt {e+f x}}{4 d^4 f}+\frac {\left (16 a^2 d^2 f^2 \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )-16 a b d f \left (C \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )+2 d f \left (4 A d f (d e+c f)-B \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right )\right )+b^2 \left (C \left (35 d^4 e^4+20 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+20 c^3 d e f^3+35 c^4 f^4\right )+8 d f \left (2 A d f \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )-B \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{64 d^{9/2} f^{9/2}} \] Output:

-1/64*(16*a^2*d^2*f^2*(-4*B*d*f+5*C*c*f+3*C*d*e)+16*a*b*d*f*(2*d*f*(-4*A*d 
*f+5*B*c*f+3*B*d*e)-C*(11*c^2*f^2+8*c*d*e*f+5*d^2*e^2))+b^2*(C*(93*c^3*f^3 
+73*c^2*d*e*f^2+55*c*d^2*e^2*f+35*d^3*e^3)+8*d*f*(2*A*d*f*(5*c*f+3*d*e)-B* 
(11*c^2*f^2+8*c*d*e*f+5*d^2*e^2))))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/d^4/f^4+1/ 
96*(48*a^2*C*d^2*f^2-16*a*b*d*f*(-6*B*d*f+13*C*c*f+5*C*d*e)-b^2*(8*d*f*(-6 
*A*d*f+13*B*c*f+5*B*d*e)-C*(163*c^2*f^2+90*c*d*e*f+35*d^2*e^2)))*(d*x+c)^( 
3/2)*(f*x+e)^(1/2)/d^4/f^3+1/24*b*(16*a*C*d*f-b*(-8*B*d*f+25*C*c*f+7*C*d*e 
))*(d*x+c)^(5/2)*(f*x+e)^(1/2)/d^4/f^2+1/4*b^2*C*(d*x+c)^(7/2)*(f*x+e)^(1/ 
2)/d^4/f+1/64*(16*a^2*d^2*f^2*(C*(3*c^2*f^2+2*c*d*e*f+3*d^2*e^2)+4*d*f*(2* 
A*d*f-B*(c*f+d*e)))-16*a*b*d*f*(C*(5*c^3*f^3+3*c^2*d*e*f^2+3*c*d^2*e^2*f+5 
*d^3*e^3)+2*d*f*(4*A*d*f*(c*f+d*e)-B*(3*c^2*f^2+2*c*d*e*f+3*d^2*e^2)))+b^2 
*(C*(35*c^4*f^4+20*c^3*d*e*f^3+18*c^2*d^2*e^2*f^2+20*c*d^3*e^3*f+35*d^4*e^ 
4)+8*d*f*(2*A*d*f*(3*c^2*f^2+2*c*d*e*f+3*d^2*e^2)-B*(5*c^3*f^3+3*c^2*d*e*f 
^2+3*c*d^2*e^2*f+5*d^3*e^3))))*arctanh(f^(1/2)*(d*x+c)^(1/2)/d^(1/2)/(f*x+ 
e)^(1/2))/d^(9/2)/f^(9/2)
 

Mathematica [A] (verified)

Time = 2.52 (sec) , antiderivative size = 632, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {\sqrt {d} \sqrt {f} \sqrt {c+d x} \sqrt {e+f x} \left (48 a^2 d^2 f^2 (4 B d f+C (-3 d e-3 c f+2 d f x))+16 a b d f \left (6 d f (4 A d f+B (-3 d e-3 c f+2 d f x))+C \left (15 c^2 f^2+2 c d f (7 e-5 f x)+d^2 \left (15 e^2-10 e f x+8 f^2 x^2\right )\right )\right )+b^2 \left (-C \left (105 c^3 f^3+5 c^2 d f^2 (19 e-14 f x)+c d^2 f \left (95 e^2-68 e f x+56 f^2 x^2\right )+d^3 \left (105 e^3-70 e^2 f x+56 e f^2 x^2-48 f^3 x^3\right )\right )+8 d f \left (6 A d f (-3 d e-3 c f+2 d f x)+B \left (15 c^2 f^2+2 c d f (7 e-5 f x)+d^2 \left (15 e^2-10 e f x+8 f^2 x^2\right )\right )\right )\right )\right )+3 \left (16 a^2 d^2 f^2 \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )-16 a b d f \left (C \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )+2 d f \left (4 A d f (d e+c f)-B \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right )\right )+b^2 \left (C \left (35 d^4 e^4+20 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+20 c^3 d e f^3+35 c^4 f^4\right )+8 d f \left (2 A d f \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )-B \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {f} \sqrt {c+d x}}\right )}{192 d^{9/2} f^{9/2}} \] Input:

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

Output:

(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*Sqrt[e + f*x]*(48*a^2*d^2*f^2*(4*B*d*f + C* 
(-3*d*e - 3*c*f + 2*d*f*x)) + 16*a*b*d*f*(6*d*f*(4*A*d*f + B*(-3*d*e - 3*c 
*f + 2*d*f*x)) + C*(15*c^2*f^2 + 2*c*d*f*(7*e - 5*f*x) + d^2*(15*e^2 - 10* 
e*f*x + 8*f^2*x^2))) + b^2*(-(C*(105*c^3*f^3 + 5*c^2*d*f^2*(19*e - 14*f*x) 
 + c*d^2*f*(95*e^2 - 68*e*f*x + 56*f^2*x^2) + d^3*(105*e^3 - 70*e^2*f*x + 
56*e*f^2*x^2 - 48*f^3*x^3))) + 8*d*f*(6*A*d*f*(-3*d*e - 3*c*f + 2*d*f*x) + 
 B*(15*c^2*f^2 + 2*c*d*f*(7*e - 5*f*x) + d^2*(15*e^2 - 10*e*f*x + 8*f^2*x^ 
2))))) + 3*(16*a^2*d^2*f^2*(C*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) + 4*d*f* 
(2*A*d*f - B*(d*e + c*f))) - 16*a*b*d*f*(C*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3* 
c^2*d*e*f^2 + 5*c^3*f^3) + 2*d*f*(4*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 2*c 
*d*e*f + 3*c^2*f^2))) + b^2*(C*(35*d^4*e^4 + 20*c*d^3*e^3*f + 18*c^2*d^2*e 
^2*f^2 + 20*c^3*d*e*f^3 + 35*c^4*f^4) + 8*d*f*(2*A*d*f*(3*d^2*e^2 + 2*c*d* 
e*f + 3*c^2*f^2) - B*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^ 
3))))*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/(Sqrt[f]*Sqrt[c + d*x])])/(192*d^(9/ 
2)*f^(9/2))
 

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2118, 27, 170, 27, 164, 66, 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.

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

\(\Big \downarrow \) 2118

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 170

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 164

\(\displaystyle \frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f}-\frac {\frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (32 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-11 C (c f+d e))-16 a b^2 d f \left (6 d f (4 A d f-3 B (c f+d e))+C \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+2 b d f x (6 b d f (a c C f+a C d e-8 A b d f+6 b c C e)-(4 a d f-5 b (c f+d e)) (-2 a C d f+8 b B d f-7 b C (c f+d e)))+b^3 \left (8 d f \left (18 A d f (c f+d e)-B \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+5 C \left (21 c^3 f^3+19 c^2 d e f^2+19 c d^2 e^2 f+21 d^3 e^3\right )\right )\right )}{4 d^2 f^2}-\frac {3 b \left (16 a^2 d^2 f^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )-16 a b d f \left (2 d f \left (4 A d f (c f+d e)-B \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+C \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+b^2 \left (8 d f \left (2 A d f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )-B \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+C \left (35 c^4 f^4+20 c^3 d e f^3+18 c^2 d^2 e^2 f^2+20 c d^3 e^3 f+35 d^4 e^4\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}}dx}{8 d^2 f^2}}{6 d f}-\frac {(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+8 b B d f-7 b C (c f+d e))}{3 d f}}{8 b d f}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f}-\frac {\frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (32 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-11 C (c f+d e))-16 a b^2 d f \left (6 d f (4 A d f-3 B (c f+d e))+C \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+2 b d f x (6 b d f (a c C f+a C d e-8 A b d f+6 b c C e)-(4 a d f-5 b (c f+d e)) (-2 a C d f+8 b B d f-7 b C (c f+d e)))+b^3 \left (8 d f \left (18 A d f (c f+d e)-B \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+5 C \left (21 c^3 f^3+19 c^2 d e f^2+19 c d^2 e^2 f+21 d^3 e^3\right )\right )\right )}{4 d^2 f^2}-\frac {3 b \left (16 a^2 d^2 f^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )-16 a b d f \left (2 d f \left (4 A d f (c f+d e)-B \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+C \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+b^2 \left (8 d f \left (2 A d f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )-B \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+C \left (35 c^4 f^4+20 c^3 d e f^3+18 c^2 d^2 e^2 f^2+20 c d^3 e^3 f+35 d^4 e^4\right )\right )\right ) \int \frac {1}{d-\frac {f (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{4 d^2 f^2}}{6 d f}-\frac {(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+8 b B d f-7 b C (c f+d e))}{3 d f}}{8 b d f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f}-\frac {\frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (32 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-11 C (c f+d e))-16 a b^2 d f \left (6 d f (4 A d f-3 B (c f+d e))+C \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+2 b d f x (6 b d f (a c C f+a C d e-8 A b d f+6 b c C e)-(4 a d f-5 b (c f+d e)) (-2 a C d f+8 b B d f-7 b C (c f+d e)))+b^3 \left (8 d f \left (18 A d f (c f+d e)-B \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+5 C \left (21 c^3 f^3+19 c^2 d e f^2+19 c d^2 e^2 f+21 d^3 e^3\right )\right )\right )}{4 d^2 f^2}-\frac {3 b \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (16 a^2 d^2 f^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )-16 a b d f \left (2 d f \left (4 A d f (c f+d e)-B \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+C \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+b^2 \left (8 d f \left (2 A d f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )-B \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+C \left (35 c^4 f^4+20 c^3 d e f^3+18 c^2 d^2 e^2 f^2+20 c d^3 e^3 f+35 d^4 e^4\right )\right )\right )}{4 d^{5/2} f^{5/2}}}{6 d f}-\frac {(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+8 b B d f-7 b C (c f+d e))}{3 d f}}{8 b d f}\)

Input:

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

Output:

(C*(a + b*x)^3*Sqrt[c + d*x]*Sqrt[e + f*x])/(4*b*d*f) - (-1/3*((8*b*B*d*f 
- 2*a*C*d*f - 7*b*C*(d*e + c*f))*(a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x])/ 
(d*f) + ((Sqrt[c + d*x]*Sqrt[e + f*x]*(32*a^3*C*d^3*f^3 - 8*a^2*b*d^2*f^2* 
(16*B*d*f - 11*C*(d*e + c*f)) - 16*a*b^2*d*f*(C*(15*d^2*e^2 + 14*c*d*e*f + 
 15*c^2*f^2) + 6*d*f*(4*A*d*f - 3*B*(d*e + c*f))) + b^3*(5*C*(21*d^3*e^3 + 
 19*c*d^2*e^2*f + 19*c^2*d*e*f^2 + 21*c^3*f^3) + 8*d*f*(18*A*d*f*(d*e + c* 
f) - B*(15*d^2*e^2 + 14*c*d*e*f + 15*c^2*f^2))) + 2*b*d*f*(6*b*d*f*(6*b*c* 
C*e + a*C*d*e + a*c*C*f - 8*A*b*d*f) - (4*a*d*f - 5*b*(d*e + c*f))*(8*b*B* 
d*f - 2*a*C*d*f - 7*b*C*(d*e + c*f)))*x))/(4*d^2*f^2) - (3*b*(16*a^2*d^2*f 
^2*(C*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) + 4*d*f*(2*A*d*f - B*(d*e + c*f) 
)) - 16*a*b*d*f*(C*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3) 
 + 2*d*f*(4*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2))) + 
b^2*(C*(35*d^4*e^4 + 20*c*d^3*e^3*f + 18*c^2*d^2*e^2*f^2 + 20*c^3*d*e*f^3 
+ 35*c^4*f^4) + 8*d*f*(2*A*d*f*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) - B*(5* 
d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3))))*ArcTanh[(Sqrt[f]*S 
qrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(4*d^(5/2)*f^(5/2)))/(6*d*f))/(8*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 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 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

rule 170

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] 
 && IntegerQ[m]
 

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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2527\) vs. \(2(685)=1370\).

Time = 0.76 (sec) , antiderivative size = 2528, normalized size of antiderivative = 3.50

method	result	size
default	\(\text {Expression too large to display}\)	\(2528\)

Input:

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

Output:

1/384*(96*C*b^2*d^3*f^3*x^3*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)-320*C*(d*f 
)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*a*b*c*d^2*f^3*x-320*C*(d*f)^(1/2)*((f*x+e) 
*(d*x+c))^(1/2)*a*b*d^3*e*f^2*x-576*B*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)* 
a*b*d^3*e*f^2+224*B*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*b^2*c*d^2*e*f^2-72 
*B*ln(1/2*(2*d*f*x+2*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1 
/2))*b^2*c^2*d^2*e*f^3-72*B*ln(1/2*(2*d*f*x+2*((f*x+e)*(d*x+c))^(1/2)*(d*f 
)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^2*c*d^3*e^2*f^2+128*B*b^2*d^3*f^3*x^2*(d*f 
)^(1/2)*((f*x+e)*(d*x+c))^(1/2)+384*B*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)* 
a*b*d^3*f^3*x-160*B*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*b^2*c*d^2*f^3*x-16 
0*B*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*b^2*d^3*e*f^2*x+140*C*(d*f)^(1/2)* 
((f*x+e)*(d*x+c))^(1/2)*b^2*c^2*d*f^3*x+480*C*(d*f)^(1/2)*((f*x+e)*(d*x+c) 
)^(1/2)*a*b*d^3*e^2*f-190*C*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*b^2*c^2*d* 
e*f^2-190*C*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*b^2*c*d^2*e^2*f+240*B*(d*f 
)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*b^2*c^2*d*f^3+240*B*(d*f)^(1/2)*((f*x+e)*( 
d*x+c))^(1/2)*b^2*d^3*e^2*f-288*C*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*a^2* 
c*d^2*f^3-288*C*(d*f)^(1/2)*((f*x+e)*(d*x+c))^(1/2)*a^2*d^3*e*f^2-384*A*ln 
(1/2*(2*d*f*x+2*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))* 
a*b*c*d^3*f^4+192*B*ln(1/2*(2*d*f*x+2*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(1/2)+ 
c*f+d*e)/(d*f)^(1/2))*a*b*c*d^3*e*f^3-112*C*b^2*d^3*e*f^2*x^2*(d*f)^(1/2)* 
((f*x+e)*(d*x+c))^(1/2)+256*C*a*b*d^3*f^3*x^2*(d*f)^(1/2)*((f*x+e)*(d*x...
 

Fricas [A] (verification not implemented)

Time = 1.10 (sec) , antiderivative size = 1436, normalized size of antiderivative = 1.99 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/768*(3*(35*C*b^2*d^4*e^4 + 20*(C*b^2*c*d^3 - 2*(2*C*a*b + B*b^2)*d^4)*e 
^3*f + 6*(3*C*b^2*c^2*d^2 - 4*(2*C*a*b + B*b^2)*c*d^3 + 8*(C*a^2 + 2*B*a*b 
 + A*b^2)*d^4)*e^2*f^2 + 4*(5*C*b^2*c^3*d - 6*(2*C*a*b + B*b^2)*c^2*d^2 + 
8*(C*a^2 + 2*B*a*b + A*b^2)*c*d^3 - 16*(B*a^2 + 2*A*a*b)*d^4)*e*f^3 + (35* 
C*b^2*c^4 + 128*A*a^2*d^4 - 40*(2*C*a*b + B*b^2)*c^3*d + 48*(C*a^2 + 2*B*a 
*b + A*b^2)*c^2*d^2 - 64*(B*a^2 + 2*A*a*b)*c*d^3)*f^4)*sqrt(d*f)*log(8*d^2 
*f^2*x^2 + d^2*e^2 + 6*c*d*e*f + c^2*f^2 + 4*(2*d*f*x + d*e + c*f)*sqrt(d* 
f)*sqrt(d*x + c)*sqrt(f*x + e) + 8*(d^2*e*f + c*d*f^2)*x) + 4*(48*C*b^2*d^ 
4*f^4*x^3 - 105*C*b^2*d^4*e^3*f - 5*(19*C*b^2*c*d^3 - 24*(2*C*a*b + B*b^2) 
*d^4)*e^2*f^2 - (95*C*b^2*c^2*d^2 - 112*(2*C*a*b + B*b^2)*c*d^3 + 144*(C*a 
^2 + 2*B*a*b + A*b^2)*d^4)*e*f^3 - 3*(35*C*b^2*c^3*d - 40*(2*C*a*b + B*b^2 
)*c^2*d^2 + 48*(C*a^2 + 2*B*a*b + A*b^2)*c*d^3 - 64*(B*a^2 + 2*A*a*b)*d^4) 
*f^4 - 8*(7*C*b^2*d^4*e*f^3 + (7*C*b^2*c*d^3 - 8*(2*C*a*b + B*b^2)*d^4)*f^ 
4)*x^2 + 2*(35*C*b^2*d^4*e^2*f^2 + 2*(17*C*b^2*c*d^3 - 20*(2*C*a*b + B*b^2 
)*d^4)*e*f^3 + (35*C*b^2*c^2*d^2 - 40*(2*C*a*b + B*b^2)*c*d^3 + 48*(C*a^2 
+ 2*B*a*b + A*b^2)*d^4)*f^4)*x)*sqrt(d*x + c)*sqrt(f*x + e))/(d^5*f^5), -1 
/384*(3*(35*C*b^2*d^4*e^4 + 20*(C*b^2*c*d^3 - 2*(2*C*a*b + B*b^2)*d^4)*e^3 
*f + 6*(3*C*b^2*c^2*d^2 - 4*(2*C*a*b + B*b^2)*c*d^3 + 8*(C*a^2 + 2*B*a*b + 
 A*b^2)*d^4)*e^2*f^2 + 4*(5*C*b^2*c^3*d - 6*(2*C*a*b + B*b^2)*c^2*d^2 + 8* 
(C*a^2 + 2*B*a*b + A*b^2)*c*d^3 - 16*(B*a^2 + 2*A*a*b)*d^4)*e*f^3 + (35...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 946, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/192*(sqrt(d^2*e + (d*x + c)*d*f - c*d*f)*(2*(d*x + c)*(4*(d*x + c)*(6*(d 
*x + c)*C*b^2/(d^5*f) - (7*C*b^2*d^20*e*f^5 + 25*C*b^2*c*d^19*f^6 - 16*C*a 
*b*d^20*f^6 - 8*B*b^2*d^20*f^6)/(d^24*f^7)) + (35*C*b^2*d^21*e^2*f^4 + 90* 
C*b^2*c*d^20*e*f^5 - 80*C*a*b*d^21*e*f^5 - 40*B*b^2*d^21*e*f^5 + 163*C*b^2 
*c^2*d^19*f^6 - 208*C*a*b*c*d^20*f^6 - 104*B*b^2*c*d^20*f^6 + 48*C*a^2*d^2 
1*f^6 + 96*B*a*b*d^21*f^6 + 48*A*b^2*d^21*f^6)/(d^24*f^7)) - 3*(35*C*b^2*d 
^22*e^3*f^3 + 55*C*b^2*c*d^21*e^2*f^4 - 80*C*a*b*d^22*e^2*f^4 - 40*B*b^2*d 
^22*e^2*f^4 + 73*C*b^2*c^2*d^20*e*f^5 - 128*C*a*b*c*d^21*e*f^5 - 64*B*b^2* 
c*d^21*e*f^5 + 48*C*a^2*d^22*e*f^5 + 96*B*a*b*d^22*e*f^5 + 48*A*b^2*d^22*e 
*f^5 + 93*C*b^2*c^3*d^19*f^6 - 176*C*a*b*c^2*d^20*f^6 - 88*B*b^2*c^2*d^20* 
f^6 + 80*C*a^2*c*d^21*f^6 + 160*B*a*b*c*d^21*f^6 + 80*A*b^2*c*d^21*f^6 - 6 
4*B*a^2*d^22*f^6 - 128*A*a*b*d^22*f^6)/(d^24*f^7))*sqrt(d*x + c) - 3*(35*C 
*b^2*d^4*e^4 + 20*C*b^2*c*d^3*e^3*f - 80*C*a*b*d^4*e^3*f - 40*B*b^2*d^4*e^ 
3*f + 18*C*b^2*c^2*d^2*e^2*f^2 - 48*C*a*b*c*d^3*e^2*f^2 - 24*B*b^2*c*d^3*e 
^2*f^2 + 48*C*a^2*d^4*e^2*f^2 + 96*B*a*b*d^4*e^2*f^2 + 48*A*b^2*d^4*e^2*f^ 
2 + 20*C*b^2*c^3*d*e*f^3 - 48*C*a*b*c^2*d^2*e*f^3 - 24*B*b^2*c^2*d^2*e*f^3 
 + 32*C*a^2*c*d^3*e*f^3 + 64*B*a*b*c*d^3*e*f^3 + 32*A*b^2*c*d^3*e*f^3 - 64 
*B*a^2*d^4*e*f^3 - 128*A*a*b*d^4*e*f^3 + 35*C*b^2*c^4*f^4 - 80*C*a*b*c^3*d 
*f^4 - 40*B*b^2*c^3*d*f^4 + 48*C*a^2*c^2*d^2*f^4 + 96*B*a*b*c^2*d^2*f^4 + 
48*A*b^2*c^2*d^2*f^4 - 64*B*a^2*c*d^3*f^4 - 128*A*a*b*c*d^3*f^4 + 128*A...
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

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

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

Output:

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