\(\int \frac {A+B x+C x^2}{(a+b x)^4 \sqrt {c+d x} \sqrt {e+f x}} \, dx\) [82]

Optimal result

Integrand size = 36, antiderivative size = 770 \[ \int \frac {A+B x+C x^2}{(a+b x)^4 \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac {\left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac {\left (4 a^4 C d^2 f^2+8 a^3 b d f (B d f-2 C (d e+c f))-b^4 \left (15 A d^2 e^2-2 c d e (9 B e-7 A f)+3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-44 A f)-3 c^2 f (4 C e-B f)-2 c d \left (6 C e^2-29 B e f+22 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-34 c d e f+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}-\frac {\left (2 d f (2 b c e-a (d e+c f)) \left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right )-(b d e+b c f-2 a d f) \left ((4 a d f-3 b (d e+c f)) \left (a^2 C (d e+c f)-a b (6 c C e+B d e+B c f-6 A d f)+b^2 (6 B c e-5 A (d e+c f))\right )+2 (4 b c e-a (d e+c f)) \left (a^2 C d f+b^2 (3 c C e-2 A d f)+a b (2 B d f-3 C (d e+c f))\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{24 b (b c-a d)^{7/2} (b e-a f)^{7/2}} \] Output:

-1/3*(A*b^2-a*(B*b-C*a))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/(-a*d+b*c)/(-a*f+b* 
e)/(b*x+a)^3+1/12*(2*a^3*C*d*f+a*b^2*(-10*A*d*f+B*c*f+B*d*e+12*C*c*e)-b^3* 
(6*B*c*e-5*A*(c*f+d*e))+a^2*b*(4*B*d*f-7*C*(c*f+d*e)))*(d*x+c)^(1/2)*(f*x+ 
e)^(1/2)/b/(-a*d+b*c)^2/(-a*f+b*e)^2/(b*x+a)^2+1/24*(4*a^4*C*d^2*f^2+8*a^3 
*b*d*f*(B*d*f-2*C*(c*f+d*e))-b^4*(15*A*d^2*e^2-2*c*d*e*(-7*A*f+9*B*e)+3*c^ 
2*(5*A*f^2-6*B*e*f+8*C*e^2))-a*b^3*(d^2*e*(-44*A*f+3*B*e)-3*c^2*f*(-B*f+4* 
C*e)-2*c*d*(22*A*f^2-29*B*e*f+6*C*e^2))-a^2*b^2*(C*(3*c^2*f^2-34*c*d*e*f+3 
*d^2*e^2)+2*d*f*(22*A*d*f-5*B*(c*f+d*e))))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/( 
-a*d+b*c)^3/(-a*f+b*e)^3/(b*x+a)-1/24*(2*d*f*(2*b*c*e-a*(c*f+d*e))*(2*a^3* 
C*d*f+a*b^2*(-10*A*d*f+B*c*f+B*d*e+12*C*c*e)-b^3*(6*B*c*e-5*A*(c*f+d*e))+a 
^2*b*(4*B*d*f-7*C*(c*f+d*e)))-(-2*a*d*f+b*c*f+b*d*e)*((4*a*d*f-3*b*(c*f+d* 
e))*(a^2*C*(c*f+d*e)-a*b*(-6*A*d*f+B*c*f+B*d*e+6*C*c*e)+b^2*(6*B*c*e-5*A*( 
c*f+d*e)))+2*(4*b*c*e-a*(c*f+d*e))*(a^2*C*d*f+b^2*(-2*A*d*f+3*C*c*e)+a*b*( 
2*B*d*f-3*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/(-a*d+b*c)^(7/2)/(-a*f+b*e)^(7/2)
 

Mathematica [A] (verified)

Time = 9.26 (sec) , antiderivative size = 1036, normalized size of antiderivative = 1.35 \[ \int \frac {A+B x+C x^2}{(a+b x)^4 \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (6 b^5 c e x (4 c C e x+B (2 c e-3 d e x-3 c f x))+6 a^5 d f (-4 B d f+C (3 d e+3 c f-2 d f x))+a b^4 \left (-12 c C e x (-2 c e+d e x+c f x)+B \left (3 d^2 e^2 x^2+2 c d e x (-25 e+29 f x)+c^2 \left (4 e^2-50 e f x+3 f^2 x^2\right )\right )\right )+a^4 b \left (12 B d f (c f+d (e-2 f x))-C \left (3 c^2 f^2+2 c d f (29 e-25 f x)+d^2 \left (3 e^2-50 e f x+4 f^2 x^2\right )\right )\right )+a^2 b^3 \left (d^2 e x (8 B e+3 C e x-10 B f x)+c^2 \left (8 B f (-2 e+f x)+C \left (8 e^2+14 e f x+3 f^2 x^2\right )\right )-2 c d \left (C e x (-7 e+17 f x)+B \left (8 e^2-62 e f x+5 f^2 x^2\right )\right )\right )+a^3 b^2 \left (c^2 f (10 C e-3 B f-8 C f x)+2 c d \left (B f (17 e-7 f x)+C \left (5 e^2-62 e f x+8 f^2 x^2\right )\right )-d^2 \left (8 C e x (e-2 f x)+B \left (3 e^2+14 e f x+8 f^2 x^2\right )\right )\right )+A b \left (72 a^4 d^2 f^2+18 a^3 b d f (-5 d e-5 c f+6 d f x)+b^4 \left (15 d^2 e^2 x^2+2 c d e x (-5 e+7 f x)+c^2 \left (8 e^2-10 e f x+15 f^2 x^2\right )\right )-2 a b^3 \left (c^2 f (13 e-20 f x)+2 d^2 e x (-10 e+11 f x)+c d \left (13 e^2-34 e f x+22 f^2 x^2\right )\right )+a^2 b^2 \left (33 c^2 f^2+2 c d f (43 e-59 f x)+d^2 \left (33 e^2-118 e f x+44 f^2 x^2\right )\right )\right )\right )}{24 (b c-a d)^3 (b e-a f)^3 (a+b x)^3}+\frac {\left (a b^2 \left (d^3 e^2 (B e-18 A f)+c^3 f^2 (-4 C e+B f)+c^2 d f \left (-40 C e^2+23 B e f-18 A f^2\right )+c d^2 e \left (-4 C e^2+23 B e f-12 A f^2\right )\right )+b^3 \left (5 A d^3 e^3+3 c d^2 e^2 (-2 B e+A f)+c^2 d e \left (8 C e^2-4 B e f+3 A f^2\right )+c^3 f \left (8 C e^2-6 B e f+5 A f^2\right )\right )-2 a^3 d f \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 )+a^2 b \left (C \left (d^3 e^3+23 c d^2 e^2 f+23 c^2 d e f^2+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d e f+c^2 f^2\right )\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 )}{8 (b c-a d)^{7/2} (-b e+a f)^{7/2}} \] Input:

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

Output:

-1/24*(Sqrt[c + d*x]*Sqrt[e + f*x]*(6*b^5*c*e*x*(4*c*C*e*x + B*(2*c*e - 3* 
d*e*x - 3*c*f*x)) + 6*a^5*d*f*(-4*B*d*f + C*(3*d*e + 3*c*f - 2*d*f*x)) + a 
*b^4*(-12*c*C*e*x*(-2*c*e + d*e*x + c*f*x) + B*(3*d^2*e^2*x^2 + 2*c*d*e*x* 
(-25*e + 29*f*x) + c^2*(4*e^2 - 50*e*f*x + 3*f^2*x^2))) + a^4*b*(12*B*d*f* 
(c*f + d*(e - 2*f*x)) - C*(3*c^2*f^2 + 2*c*d*f*(29*e - 25*f*x) + d^2*(3*e^ 
2 - 50*e*f*x + 4*f^2*x^2))) + a^2*b^3*(d^2*e*x*(8*B*e + 3*C*e*x - 10*B*f*x 
) + c^2*(8*B*f*(-2*e + f*x) + C*(8*e^2 + 14*e*f*x + 3*f^2*x^2)) - 2*c*d*(C 
*e*x*(-7*e + 17*f*x) + B*(8*e^2 - 62*e*f*x + 5*f^2*x^2))) + a^3*b^2*(c^2*f 
*(10*C*e - 3*B*f - 8*C*f*x) + 2*c*d*(B*f*(17*e - 7*f*x) + C*(5*e^2 - 62*e* 
f*x + 8*f^2*x^2)) - d^2*(8*C*e*x*(e - 2*f*x) + B*(3*e^2 + 14*e*f*x + 8*f^2 
*x^2))) + A*b*(72*a^4*d^2*f^2 + 18*a^3*b*d*f*(-5*d*e - 5*c*f + 6*d*f*x) + 
b^4*(15*d^2*e^2*x^2 + 2*c*d*e*x*(-5*e + 7*f*x) + c^2*(8*e^2 - 10*e*f*x + 1 
5*f^2*x^2)) - 2*a*b^3*(c^2*f*(13*e - 20*f*x) + 2*d^2*e*x*(-10*e + 11*f*x) 
+ c*d*(13*e^2 - 34*e*f*x + 22*f^2*x^2)) + a^2*b^2*(33*c^2*f^2 + 2*c*d*f*(4 
3*e - 59*f*x) + d^2*(33*e^2 - 118*e*f*x + 44*f^2*x^2)))))/((b*c - a*d)^3*( 
b*e - a*f)^3*(a + b*x)^3) + ((a*b^2*(d^3*e^2*(B*e - 18*A*f) + c^3*f^2*(-4* 
C*e + B*f) + c^2*d*f*(-40*C*e^2 + 23*B*e*f - 18*A*f^2) + c*d^2*e*(-4*C*e^2 
 + 23*B*e*f - 12*A*f^2)) + b^3*(5*A*d^3*e^3 + 3*c*d^2*e^2*(-2*B*e + A*f) + 
 c^2*d*e*(8*C*e^2 - 4*B*e*f + 3*A*f^2) + c^3*f*(8*C*e^2 - 6*B*e*f + 5*A*f^ 
2)) - 2*a^3*d*f*(C*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) + 4*d*f*(2*A*d*f...
 

Rubi [A] (verified)

Time = 1.76 (sec) , antiderivative size = 869, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2116, 27, 168, 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.

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

\(\Big \downarrow \) 2116

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (2 a^3 C d f+a^2 b (4 B d f-7 C (c f+d e))+a b^2 (-10 A d f+B c f+B d e+12 c C e)-b^3 (6 B c e-5 A (c f+d e))\right )}{2 (a+b x)^2 (b c-a d) (b e-a f)}-\frac {\int -\frac {2 C d f (d e+c f) a^3-b \left (8 d f (B d e+B c f-3 A d f)-C \left (3 d^2 e^2-10 c d f e+3 c^2 f^2\right )\right ) a^2+b^2 \left (-3 f (4 C e-B f) c^2-2 d \left (6 C e^2-23 B f e+17 A f^2\right ) c+d^2 e (3 B e-34 A f)\right ) a+b^3 \left (3 \left (8 C e^2-6 B f e+5 A f^2\right ) c^2-2 d e (9 B e-7 A f) c+15 A d^2 e^2\right )+2 d f \left (2 C d f a^3+b (4 B d f-7 C (d e+c f)) a^2+b^2 (12 c C e+B d e+B c f-10 A d f) a-b^3 (6 B c e-5 A (d e+c f))\right ) x}{2 (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}}{6 b (b c-a d) (b e-a f)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^3 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {2 C d f (d e+c f) a^3+b \left (C \left (3 d^2 e^2-10 c d f e+3 c^2 f^2\right )+8 d f (3 A d f-B (d e+c f))\right ) a^2+b^2 \left (-3 f (4 C e-B f) c^2-2 d \left (6 C e^2-23 B f e+17 A f^2\right ) c+d^2 e (3 B e-34 A f)\right ) a+b^3 \left (3 \left (8 C e^2-6 B f e+5 A f^2\right ) c^2-2 d e (9 B e-7 A f) c+15 A d^2 e^2\right )+2 d f \left (2 C d f a^3+b (4 B d f-7 C (d e+c f)) a^2+b^2 (12 c C e+B d e+B c f-10 A d f) a-b^3 (6 B c e-5 A (d e+c f))\right ) x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}dx}{4 (b c-a d) (b e-a f)}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (2 a^3 C d f+a^2 b (4 B d f-7 C (c f+d e))+a b^2 (-10 A d f+B c f+B d e+12 c C e)-b^3 (6 B c e-5 A (c f+d e))\right )}{2 (a+b x)^2 (b c-a d) (b e-a f)}}{6 b (b c-a d) (b e-a f)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^3 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (2 C d f a^3+b (4 B d f-7 C (d e+c f)) a^2+b^2 (12 c C e+B d e+B c f-10 A d f) a-b^3 (6 B c e-5 A (d e+c f))\right )}{2 (b c-a d) (b e-a f) (a+b x)^2}+\frac {\frac {\left (4 C d^2 f^2 a^4+8 b d f (B d f-2 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-34 c d f e+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right ) a^2-b^3 \left (-3 f (4 C e-B f) c^2-2 d \left (6 C e^2-29 B f e+22 A f^2\right ) c+d^2 e (3 B e-44 A f)\right ) a-b^4 \left (3 \left (8 C e^2-6 B f e+5 A f^2\right ) c^2-2 d e (9 B e-7 A f) c+15 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) (a+b x)}-\frac {\int \frac {3 b \left (-2 d f \left (C \left (3 d^2 e^2+2 c d f e+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right ) a^3+b \left (C \left (d^3 e^3+23 c d^2 f e^2+23 c^2 d f^2 e+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d f e+c^2 f^2\right )\right )\right ) a^2+b^2 \left (-f^2 (4 C e-B f) c^3-d f \left (40 C e^2-23 B f e+18 A f^2\right ) c^2-d^2 e \left (4 C e^2-23 B f e+12 A f^2\right ) c+d^3 e^2 (B e-18 A f)\right ) a+b^3 \left (f \left (8 C e^2-6 B f e+5 A f^2\right ) c^3+d e \left (8 C e^2-4 B f e+3 A f^2\right ) c^2-3 d^2 e^2 (2 B e-A f) c+5 A d^3 e^3\right )\right )}{2 (a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d) (b e-a f)}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (2 C d f a^3+b (4 B d f-7 C (d e+c f)) a^2+b^2 (12 c C e+B d e+B c f-10 A d f) a-b^3 (6 B c e-5 A (d e+c f))\right )}{2 (b c-a d) (b e-a f) (a+b x)^2}+\frac {\frac {\left (4 C d^2 f^2 a^4+8 b d f (B d f-2 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-34 c d f e+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right ) a^2-b^3 \left (-3 f (4 C e-B f) c^2-2 d \left (6 C e^2-29 B f e+22 A f^2\right ) c+d^2 e (3 B e-44 A f)\right ) a-b^4 \left (3 \left (8 C e^2-6 B f e+5 A f^2\right ) c^2-2 d e (9 B e-7 A f) c+15 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) (a+b x)}-\frac {3 b \left (-2 d f \left (C \left (3 d^2 e^2+2 c d f e+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right ) a^3+b \left (C \left (d^3 e^3+23 c d^2 f e^2+23 c^2 d f^2 e+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d f e+c^2 f^2\right )\right )\right ) a^2+b^2 \left (-f^2 (4 C e-B f) c^3-d f \left (40 C e^2-23 B f e+18 A f^2\right ) c^2-d^2 e \left (4 C e^2-23 B f e+12 A f^2\right ) c+d^3 e^2 (B e-18 A f)\right ) a+b^3 \left (f \left (8 C e^2-6 B f e+5 A f^2\right ) c^3+d e \left (8 C e^2-4 B f e+3 A f^2\right ) c^2-3 d^2 e^2 (2 B e-A f) c+5 A d^3 e^3\right )\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d) (b e-a f)}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (2 C d f a^3+b (4 B d f-7 C (d e+c f)) a^2+b^2 (12 c C e+B d e+B c f-10 A d f) a-b^3 (6 B c e-5 A (d e+c f))\right )}{2 (b c-a d) (b e-a f) (a+b x)^2}+\frac {\frac {\left (4 C d^2 f^2 a^4+8 b d f (B d f-2 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-34 c d f e+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right ) a^2-b^3 \left (-3 f (4 C e-B f) c^2-2 d \left (6 C e^2-29 B f e+22 A f^2\right ) c+d^2 e (3 B e-44 A f)\right ) a-b^4 \left (3 \left (8 C e^2-6 B f e+5 A f^2\right ) c^2-2 d e (9 B e-7 A f) c+15 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) (a+b x)}-\frac {3 b \left (-2 d f \left (C \left (3 d^2 e^2+2 c d f e+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right ) a^3+b \left (C \left (d^3 e^3+23 c d^2 f e^2+23 c^2 d f^2 e+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d f e+c^2 f^2\right )\right )\right ) a^2+b^2 \left (-f^2 (4 C e-B f) c^3-d f \left (40 C e^2-23 B f e+18 A f^2\right ) c^2-d^2 e \left (4 C e^2-23 B f e+12 A f^2\right ) c+d^3 e^2 (B e-18 A f)\right ) a+b^3 \left (f \left (8 C e^2-6 B f e+5 A f^2\right ) c^3+d e \left (8 C e^2-4 B f e+3 A f^2\right ) c^2-3 d^2 e^2 (2 B e-A f) c+5 A d^3 e^3\right )\right ) \int \frac {1}{-b c+a d+\frac {(b e-a f) (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{(b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d) (b e-a f)}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (2 C d f a^3+b (4 B d f-7 C (d e+c f)) a^2+b^2 (12 c C e+B d e+B c f-10 A d f) a-b^3 (6 B c e-5 A (d e+c f))\right )}{2 (b c-a d) (b e-a f) (a+b x)^2}+\frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (4 C d^2 f^2 a^4+8 b d f (B d f-2 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-34 c d f e+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right ) a^2-b^3 \left (-3 f (4 C e-B f) c^2-2 d \left (6 C e^2-29 B f e+22 A f^2\right ) c+d^2 e (3 B e-44 A f)\right ) a-b^4 \left (3 \left (8 C e^2-6 B f e+5 A f^2\right ) c^2-2 d e (9 B e-7 A f) c+15 A d^2 e^2\right )\right )}{(b c-a d) (b e-a f) (a+b x)}+\frac {3 b \left (-2 d f \left (C \left (3 d^2 e^2+2 c d f e+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right ) a^3+b \left (C \left (d^3 e^3+23 c d^2 f e^2+23 c^2 d f^2 e+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d f e+c^2 f^2\right )\right )\right ) a^2+b^2 \left (-f^2 (4 C e-B f) c^3-d f \left (40 C e^2-23 B f e+18 A f^2\right ) c^2-d^2 e \left (4 C e^2-23 B f e+12 A f^2\right ) c+d^3 e^2 (B e-18 A f)\right ) a+b^3 \left (f \left (8 C e^2-6 B f e+5 A f^2\right ) c^3+d e \left (8 C e^2-4 B f e+3 A f^2\right ) c^2-3 d^2 e^2 (2 B e-A f) c+5 A d^3 e^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{(b c-a d)^{3/2} (b e-a f)^{3/2}}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d) (b e-a f)}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}\)

Input:

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

Output:

-1/3*((A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(b*(b*c - a*d)* 
(b*e - a*f)*(a + b*x)^3) + (((2*a^3*C*d*f + a*b^2*(12*c*C*e + B*d*e + B*c* 
f - 10*A*d*f) - b^3*(6*B*c*e - 5*A*(d*e + c*f)) + a^2*b*(4*B*d*f - 7*C*(d* 
e + c*f)))*Sqrt[c + d*x]*Sqrt[e + f*x])/(2*(b*c - a*d)*(b*e - a*f)*(a + b* 
x)^2) + (((4*a^4*C*d^2*f^2 + 8*a^3*b*d*f*(B*d*f - 2*C*(d*e + c*f)) - b^4*( 
15*A*d^2*e^2 - 2*c*d*e*(9*B*e - 7*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 - 44*A*f) - 3*c^2*f*(4*C*e - B*f) - 2*c*d*(6*C*e 
^2 - 29*B*e*f + 22*A*f^2)) - a^2*b^2*(C*(3*d^2*e^2 - 34*c*d*e*f + 3*c^2*f^ 
2) + 2*d*f*(22*A*d*f - 5*B*(d*e + c*f))))*Sqrt[c + d*x]*Sqrt[e + f*x])/((b 
*c - a*d)*(b*e - a*f)*(a + b*x)) + (3*b*(b^3*(5*A*d^3*e^3 - 3*c*d^2*e^2*(2 
*B*e - A*f) + c^2*d*e*(8*C*e^2 - 4*B*e*f + 3*A*f^2) + c^3*f*(8*C*e^2 - 6*B 
*e*f + 5*A*f^2)) + a*b^2*(d^3*e^2*(B*e - 18*A*f) - c^3*f^2*(4*C*e - B*f) - 
 c*d^2*e*(4*C*e^2 - 23*B*e*f + 12*A*f^2) - c^2*d*f*(40*C*e^2 - 23*B*e*f + 
18*A*f^2)) - 2*a^3*d*f*(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))) + a^2*b*(C*(d^3*e^3 + 23*c*d^2*e^2*f + 23*c^2*d*e*f 
^2 + c^3*f^3) + 4*d*f*(6*A*d*f*(d*e + c*f) - B*(d^2*e^2 + 10*c*d*e*f + c^2 
*f^2))))*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*c - a*d)*(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 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. \(18801\) vs. \(2(738)=1476\).

Time = 1.37 (sec) , antiderivative size = 18802, normalized size of antiderivative = 24.42

method	result	size
default	\(\text {Expression too large to display}\)	\(18802\)

Input:

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

Output:

result too large to display
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((C*x^2+B*x+A)/(b*x+a)^4/(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((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. 25632 vs. \(2 (737) = 1474\).

Time = 78.86 (sec) , antiderivative size = 25632, normalized size of antiderivative = 33.29 \[ \int \frac {A+B x+C x^2}{(a+b x)^4 \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*(8*sqrt(d*f)*C*b^3*c^2*d^3*e^3 - 4*sqrt(d*f)*C*a*b^2*c*d^4*e^3 - 6*sqr 
t(d*f)*B*b^3*c*d^4*e^3 + sqrt(d*f)*C*a^2*b*d^5*e^3 + sqrt(d*f)*B*a*b^2*d^5 
*e^3 + 5*sqrt(d*f)*A*b^3*d^5*e^3 + 8*sqrt(d*f)*C*b^3*c^3*d^2*e^2*f - 40*sq 
rt(d*f)*C*a*b^2*c^2*d^3*e^2*f - 4*sqrt(d*f)*B*b^3*c^2*d^3*e^2*f + 23*sqrt( 
d*f)*C*a^2*b*c*d^4*e^2*f + 23*sqrt(d*f)*B*a*b^2*c*d^4*e^2*f + 3*sqrt(d*f)* 
A*b^3*c*d^4*e^2*f - 6*sqrt(d*f)*C*a^3*d^5*e^2*f - 4*sqrt(d*f)*B*a^2*b*d^5* 
e^2*f - 18*sqrt(d*f)*A*a*b^2*d^5*e^2*f - 4*sqrt(d*f)*C*a*b^2*c^3*d^2*e*f^2 
 - 6*sqrt(d*f)*B*b^3*c^3*d^2*e*f^2 + 23*sqrt(d*f)*C*a^2*b*c^2*d^3*e*f^2 + 
23*sqrt(d*f)*B*a*b^2*c^2*d^3*e*f^2 + 3*sqrt(d*f)*A*b^3*c^2*d^3*e*f^2 - 4*s 
qrt(d*f)*C*a^3*c*d^4*e*f^2 - 40*sqrt(d*f)*B*a^2*b*c*d^4*e*f^2 - 12*sqrt(d* 
f)*A*a*b^2*c*d^4*e*f^2 + 8*sqrt(d*f)*B*a^3*d^5*e*f^2 + 24*sqrt(d*f)*A*a^2* 
b*d^5*e*f^2 + sqrt(d*f)*C*a^2*b*c^3*d^2*f^3 + sqrt(d*f)*B*a*b^2*c^3*d^2*f^ 
3 + 5*sqrt(d*f)*A*b^3*c^3*d^2*f^3 - 6*sqrt(d*f)*C*a^3*c^2*d^3*f^3 - 4*sqrt 
(d*f)*B*a^2*b*c^2*d^3*f^3 - 18*sqrt(d*f)*A*a*b^2*c^2*d^3*f^3 + 8*sqrt(d*f) 
*B*a^3*c*d^4*f^3 + 24*sqrt(d*f)*A*a^2*b*c*d^4*f^3 - 16*sqrt(d*f)*A*a^3*d^5 
*f^3)*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^3*e^3*abs(d) - 3*a*b^5*c^2*d 
*e^3*abs(d) + 3*a^2*b^4*c*d^2*e^3*abs(d) - a^3*b^3*d^3*e^3*abs(d) - 3*a*b^ 
5*c^3*e^2*f*abs(d) + 9*a^2*b^4*c^2*d*e^2*f*abs(d) - 9*a^3*b^3*c*d^2*e^2...
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

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

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

Output:

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