\(\int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x)^3 \sqrt {b c^2-b d^2 x^2}} \, dx\) [29]

Optimal result

Integrand size = 46, antiderivative size = 510 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x)^3 \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {\left (C e^2-B e f+A f^2\right ) \sqrt {b c^2-b d^2 x^2}}{2 b f \left (d^2 e^2-c^2 f^2\right ) \sqrt {c+d x} (e+f x)^2}-\frac {\left (C e \left (d^2 e^2-c d e f-8 c^2 f^2\right )-f \left (A d f (7 d e+c f)-B \left (3 d^2 e^2+c d e f+4 c^2 f^2\right )\right )\right ) \sqrt {b c^2-b d^2 x^2}}{4 b f (d e-c f)^2 (d e+c f)^2 \sqrt {c+d x} (e+f x)}-\frac {\sqrt {2} \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {\sqrt {b c^2-b d^2 x^2}}{\sqrt {2} \sqrt {b} \sqrt {c} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)^3}-\frac {\left (C \left (d^4 e^4-2 c d^3 e^3 f-15 c^2 d^2 e^2 f^2-8 c^3 d e f^3-8 c^4 f^4\right )-d f \left (A d f \left (15 d^2 e^2+10 c d e f+7 c^2 f^2\right )-B \left (3 d^3 e^3+6 c d^2 e^2 f+19 c^2 d e f^2+4 c^3 f^3\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c^2-b d^2 x^2}}{\sqrt {b} \sqrt {d e+c f} \sqrt {c+d x}}\right )}{4 \sqrt {b} f^{3/2} (d e-c f)^3 (d e+c f)^{5/2}} \] Output:

1/2*(A*f^2-B*e*f+C*e^2)*(-b*d^2*x^2+b*c^2)^(1/2)/b/f/(-c^2*f^2+d^2*e^2)/(d 
*x+c)^(1/2)/(f*x+e)^2-1/4*(C*e*(-8*c^2*f^2-c*d*e*f+d^2*e^2)-f*(A*d*f*(c*f+ 
7*d*e)-B*(4*c^2*f^2+c*d*e*f+3*d^2*e^2)))*(-b*d^2*x^2+b*c^2)^(1/2)/b/f/(-c* 
f+d*e)^2/(c*f+d*e)^2/(d*x+c)^(1/2)/(f*x+e)-2^(1/2)*(A*d^2-B*c*d+C*c^2)*arc 
tanh(1/2*(-b*d^2*x^2+b*c^2)^(1/2)*2^(1/2)/b^(1/2)/c^(1/2)/(d*x+c)^(1/2))/b 
^(1/2)/c^(1/2)/(-c*f+d*e)^3-1/4*(C*(-8*c^4*f^4-8*c^3*d*e*f^3-15*c^2*d^2*e^ 
2*f^2-2*c*d^3*e^3*f+d^4*e^4)-d*f*(A*d*f*(7*c^2*f^2+10*c*d*e*f+15*d^2*e^2)- 
B*(4*c^3*f^3+19*c^2*d*e*f^2+6*c*d^2*e^2*f+3*d^3*e^3)))*arctanh(f^(1/2)*(-b 
*d^2*x^2+b*c^2)^(1/2)/b^(1/2)/(c*f+d*e)^(1/2)/(d*x+c)^(1/2))/b^(1/2)/f^(3/ 
2)/(-c*f+d*e)^3/(c*f+d*e)^(5/2)
 

Mathematica [A] (verified)

Time = 2.49 (sec) , antiderivative size = 500, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x)^3 \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (\frac {\sqrt {c^2-d^2 x^2} \left (C e \left (d^2 e^2 (e-f x)+c d e f (e+f x)+2 c^2 f^2 (3 e+4 f x)\right )+f \left (-B \left (c d e f (e+f x)+2 c^2 f^2 (e+2 f x)+d^2 e^2 (5 e+3 f x)\right )+A f \left (-2 c^2 f^2+c d f (e+f x)+d^2 e (9 e+7 f x)\right )\right )\right )}{f (d e-c f)^2 (d e+c f)^2 \sqrt {c+d x} (e+f x)^2}+\frac {\left (C \left (d^4 e^4-2 c d^3 e^3 f-15 c^2 d^2 e^2 f^2-8 c^3 d e f^3-8 c^4 f^4\right )+d f \left (-A d f \left (15 d^2 e^2+10 c d e f+7 c^2 f^2\right )+B \left (3 d^3 e^3+6 c d^2 e^2 f+19 c^2 d e f^2+4 c^3 f^3\right )\right )\right ) \arctan \left (\frac {\sqrt {-d e-c f} \sqrt {c^2-d^2 x^2}}{\sqrt {f} (-c+d x) \sqrt {c+d x}}\right )}{f^{3/2} (-d e-c f)^{5/2} (d e-c f)^3}+\frac {4 \sqrt {2} \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{\sqrt {c} (-d e+c f)^3}\right )}{4 \sqrt {b \left (c^2-d^2 x^2\right )}} \] Input:

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

Output:

(Sqrt[c^2 - d^2*x^2]*((Sqrt[c^2 - d^2*x^2]*(C*e*(d^2*e^2*(e - f*x) + c*d*e 
*f*(e + f*x) + 2*c^2*f^2*(3*e + 4*f*x)) + f*(-(B*(c*d*e*f*(e + f*x) + 2*c^ 
2*f^2*(e + 2*f*x) + d^2*e^2*(5*e + 3*f*x))) + A*f*(-2*c^2*f^2 + c*d*f*(e + 
 f*x) + d^2*e*(9*e + 7*f*x)))))/(f*(d*e - c*f)^2*(d*e + c*f)^2*Sqrt[c + d* 
x]*(e + f*x)^2) + ((C*(d^4*e^4 - 2*c*d^3*e^3*f - 15*c^2*d^2*e^2*f^2 - 8*c^ 
3*d*e*f^3 - 8*c^4*f^4) + d*f*(-(A*d*f*(15*d^2*e^2 + 10*c*d*e*f + 7*c^2*f^2 
)) + B*(3*d^3*e^3 + 6*c*d^2*e^2*f + 19*c^2*d*e*f^2 + 4*c^3*f^3)))*ArcTan[( 
Sqrt[-(d*e) - c*f]*Sqrt[c^2 - d^2*x^2])/(Sqrt[f]*(-c + d*x)*Sqrt[c + d*x]) 
])/(f^(3/2)*(-(d*e) - c*f)^(5/2)*(d*e - c*f)^3) + (4*Sqrt[2]*(c^2*C - B*c* 
d + A*d^2)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[c + d*x])/Sqrt[c^2 - d^2*x^2]])/( 
Sqrt[c]*(-(d*e) + c*f)^3)))/(4*Sqrt[b*(c^2 - d^2*x^2)])
 

Rubi [A] (verified)

Time = 3.15 (sec) , antiderivative size = 800, normalized size of antiderivative = 1.57, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.413, Rules used = {2349, 718, 114, 27, 168, 27, 174, 73, 221, 2349, 27, 718, 97, 73, 114, 27, 174, 73, 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.

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

\(\Big \downarrow \) 2349

\(\displaystyle \left (A+\frac {e (C e-B f)}{f^2}\right ) \int \frac {1}{\sqrt {c+d x} (e+f x)^3 \sqrt {b c^2-b d^2 x^2}}dx+\int \frac {\frac {B}{f}+\frac {C x}{f}-\frac {C e}{f^2}}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}}dx\)

\(\Big \downarrow \) 718

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (A+\frac {e (C e-B f)}{f^2}\right ) \int \frac {1}{(c+d x) \sqrt {b c-b d x} (e+f x)^3}dx}{\sqrt {b c^2-b d^2 x^2}}+\int \frac {\frac {B}{f}+\frac {C x}{f}-\frac {C e}{f^2}}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}}dx\)

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (\frac {d \left (\frac {\int \frac {b d \left (8 d^2 e^2+9 c d f e+7 c^2 f^2-d f (7 d e+c f) x\right )}{2 (c+d x) \sqrt {b c-b d x} (e+f x)}dx}{b \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x} (c f+7 d e)}{b (e+f x) (d e-c f) (c f+d e)}\right )}{4 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{2 b (e+f x)^2 \left (d^2 e^2-c^2 f^2\right )}\right )}{\sqrt {b c^2-b d^2 x^2}}+\int \frac {\frac {B}{f}+\frac {C x}{f}-\frac {C e}{f^2}}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (\frac {d \left (\frac {d \int \frac {8 d^2 e^2+9 c d f e+7 c^2 f^2-d f (7 d e+c f) x}{(c+d x) \sqrt {b c-b d x} (e+f x)}dx}{2 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x} (c f+7 d e)}{b (e+f x) (d e-c f) (c f+d e)}\right )}{4 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{2 b (e+f x)^2 \left (d^2 e^2-c^2 f^2\right )}\right )}{\sqrt {b c^2-b d^2 x^2}}+\int \frac {\frac {B}{f}+\frac {C x}{f}-\frac {C e}{f^2}}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}}dx\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (\frac {d \left (\frac {d \left (\frac {8 d (c f+d e)^2 \int \frac {1}{(c+d x) \sqrt {b c-b d x}}dx}{d e-c f}-\frac {f \left (7 c^2 f^2+10 c d e f+15 d^2 e^2\right ) \int \frac {1}{\sqrt {b c-b d x} (e+f x)}dx}{d e-c f}\right )}{2 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x} (c f+7 d e)}{b (e+f x) (d e-c f) (c f+d e)}\right )}{4 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{2 b (e+f x)^2 \left (d^2 e^2-c^2 f^2\right )}\right )}{\sqrt {b c^2-b d^2 x^2}}+\int \frac {\frac {B}{f}+\frac {C x}{f}-\frac {C e}{f^2}}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}}dx\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (\frac {d \left (\frac {d \left (\frac {2 f \left (7 c^2 f^2+10 c d e f+15 d^2 e^2\right ) \int \frac {1}{e+\frac {c f}{d}-\frac {f (b c-b d x)}{b d}}d\sqrt {b c-b d x}}{b d (d e-c f)}-\frac {16 (c f+d e)^2 \int \frac {1}{2 c-\frac {b c-b d x}{b}}d\sqrt {b c-b d x}}{b (d e-c f)}\right )}{2 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x} (c f+7 d e)}{b (e+f x) (d e-c f) (c f+d e)}\right )}{4 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{2 b (e+f x)^2 \left (d^2 e^2-c^2 f^2\right )}\right )}{\sqrt {b c^2-b d^2 x^2}}+\int \frac {\frac {B}{f}+\frac {C x}{f}-\frac {C e}{f^2}}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}}dx\)

\(\Big \downarrow \) 221

\(\displaystyle \int \frac {\frac {B}{f}+\frac {C x}{f}-\frac {C e}{f^2}}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}}dx+\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (\frac {d \left (\frac {d \left (\frac {2 \sqrt {f} \left (7 c^2 f^2+10 c d e f+15 d^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {8 \sqrt {2} (c f+d e)^2 \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{2 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x} (c f+7 d e)}{b (e+f x) (d e-c f) (c f+d e)}\right )}{4 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{2 b (e+f x)^2 \left (d^2 e^2-c^2 f^2\right )}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 2349

\(\displaystyle -\frac {(2 C e-B f) \int \frac {1}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}}dx}{f^2}+\int \frac {C}{f^2 \sqrt {c+d x} (e+f x) \sqrt {b c^2-b d^2 x^2}}dx+\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (\frac {d \left (\frac {d \left (\frac {2 \sqrt {f} \left (7 c^2 f^2+10 c d e f+15 d^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {8 \sqrt {2} (c f+d e)^2 \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{2 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x} (c f+7 d e)}{b (e+f x) (d e-c f) (c f+d e)}\right )}{4 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{2 b (e+f x)^2 \left (d^2 e^2-c^2 f^2\right )}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(2 C e-B f) \int \frac {1}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}}dx}{f^2}+\frac {C \int \frac {1}{\sqrt {c+d x} (e+f x) \sqrt {b c^2-b d^2 x^2}}dx}{f^2}+\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (\frac {d \left (\frac {d \left (\frac {2 \sqrt {f} \left (7 c^2 f^2+10 c d e f+15 d^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {8 \sqrt {2} (c f+d e)^2 \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{2 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x} (c f+7 d e)}{b (e+f x) (d e-c f) (c f+d e)}\right )}{4 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{2 b (e+f x)^2 \left (d^2 e^2-c^2 f^2\right )}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 718

\(\displaystyle -\frac {\sqrt {c+d x} \sqrt {b c-b d x} (2 C e-B f) \int \frac {1}{(c+d x) \sqrt {b c-b d x} (e+f x)^2}dx}{f^2 \sqrt {b c^2-b d^2 x^2}}+\frac {C \sqrt {c+d x} \sqrt {b c-b d x} \int \frac {1}{(c+d x) \sqrt {b c-b d x} (e+f x)}dx}{f^2 \sqrt {b c^2-b d^2 x^2}}+\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (\frac {d \left (\frac {d \left (\frac {2 \sqrt {f} \left (7 c^2 f^2+10 c d e f+15 d^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {8 \sqrt {2} (c f+d e)^2 \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{2 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x} (c f+7 d e)}{b (e+f x) (d e-c f) (c f+d e)}\right )}{4 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{2 b (e+f x)^2 \left (d^2 e^2-c^2 f^2\right )}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 97

\(\displaystyle -\frac {\sqrt {c+d x} \sqrt {b c-b d x} (2 C e-B f) \int \frac {1}{(c+d x) \sqrt {b c-b d x} (e+f x)^2}dx}{f^2 \sqrt {b c^2-b d^2 x^2}}+\frac {C \sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {d \int \frac {1}{(c+d x) \sqrt {b c-b d x}}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b c-b d x} (e+f x)}dx}{d e-c f}\right )}{f^2 \sqrt {b c^2-b d^2 x^2}}+\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (\frac {d \left (\frac {d \left (\frac {2 \sqrt {f} \left (7 c^2 f^2+10 c d e f+15 d^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {8 \sqrt {2} (c f+d e)^2 \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{2 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x} (c f+7 d e)}{b (e+f x) (d e-c f) (c f+d e)}\right )}{4 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{2 b (e+f x)^2 \left (d^2 e^2-c^2 f^2\right )}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\sqrt {c+d x} \sqrt {b c-b d x} (2 C e-B f) \int \frac {1}{(c+d x) \sqrt {b c-b d x} (e+f x)^2}dx}{f^2 \sqrt {b c^2-b d^2 x^2}}+\frac {C \sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {2 f \int \frac {1}{e+\frac {c f}{d}-\frac {f (b c-b d x)}{b d}}d\sqrt {b c-b d x}}{b d (d e-c f)}-\frac {2 \int \frac {1}{2 c-\frac {b c-b d x}{b}}d\sqrt {b c-b d x}}{b (d e-c f)}\right )}{f^2 \sqrt {b c^2-b d^2 x^2}}+\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (\frac {d \left (\frac {d \left (\frac {2 \sqrt {f} \left (7 c^2 f^2+10 c d e f+15 d^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {8 \sqrt {2} (c f+d e)^2 \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{2 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x} (c f+7 d e)}{b (e+f x) (d e-c f) (c f+d e)}\right )}{4 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{2 b (e+f x)^2 \left (d^2 e^2-c^2 f^2\right )}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 114

\(\displaystyle -\frac {\sqrt {c+d x} \sqrt {b c-b d x} (2 C e-B f) \left (\frac {\int \frac {b d (2 d e+c f-d f x)}{2 (c+d x) \sqrt {b c-b d x} (e+f x)}dx}{b \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{b (e+f x) \left (d^2 e^2-c^2 f^2\right )}\right )}{f^2 \sqrt {b c^2-b d^2 x^2}}+\frac {C \sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {2 f \int \frac {1}{e+\frac {c f}{d}-\frac {f (b c-b d x)}{b d}}d\sqrt {b c-b d x}}{b d (d e-c f)}-\frac {2 \int \frac {1}{2 c-\frac {b c-b d x}{b}}d\sqrt {b c-b d x}}{b (d e-c f)}\right )}{f^2 \sqrt {b c^2-b d^2 x^2}}+\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (\frac {d \left (\frac {d \left (\frac {2 \sqrt {f} \left (7 c^2 f^2+10 c d e f+15 d^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {8 \sqrt {2} (c f+d e)^2 \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{2 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x} (c f+7 d e)}{b (e+f x) (d e-c f) (c f+d e)}\right )}{4 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{2 b (e+f x)^2 \left (d^2 e^2-c^2 f^2\right )}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {c+d x} \sqrt {b c-b d x} (2 C e-B f) \left (\frac {d \int \frac {2 d e+c f-d f x}{(c+d x) \sqrt {b c-b d x} (e+f x)}dx}{2 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{b (e+f x) \left (d^2 e^2-c^2 f^2\right )}\right )}{f^2 \sqrt {b c^2-b d^2 x^2}}+\frac {C \sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {2 f \int \frac {1}{e+\frac {c f}{d}-\frac {f (b c-b d x)}{b d}}d\sqrt {b c-b d x}}{b d (d e-c f)}-\frac {2 \int \frac {1}{2 c-\frac {b c-b d x}{b}}d\sqrt {b c-b d x}}{b (d e-c f)}\right )}{f^2 \sqrt {b c^2-b d^2 x^2}}+\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (\frac {d \left (\frac {d \left (\frac {2 \sqrt {f} \left (7 c^2 f^2+10 c d e f+15 d^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {8 \sqrt {2} (c f+d e)^2 \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{2 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x} (c f+7 d e)}{b (e+f x) (d e-c f) (c f+d e)}\right )}{4 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{2 b (e+f x)^2 \left (d^2 e^2-c^2 f^2\right )}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {\sqrt {c+d x} \sqrt {b c-b d x} (2 C e-B f) \left (\frac {d \left (\frac {2 d (c f+d e) \int \frac {1}{(c+d x) \sqrt {b c-b d x}}dx}{d e-c f}-\frac {f (c f+3 d e) \int \frac {1}{\sqrt {b c-b d x} (e+f x)}dx}{d e-c f}\right )}{2 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{b (e+f x) \left (d^2 e^2-c^2 f^2\right )}\right )}{f^2 \sqrt {b c^2-b d^2 x^2}}+\frac {C \sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {2 f \int \frac {1}{e+\frac {c f}{d}-\frac {f (b c-b d x)}{b d}}d\sqrt {b c-b d x}}{b d (d e-c f)}-\frac {2 \int \frac {1}{2 c-\frac {b c-b d x}{b}}d\sqrt {b c-b d x}}{b (d e-c f)}\right )}{f^2 \sqrt {b c^2-b d^2 x^2}}+\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (\frac {d \left (\frac {d \left (\frac {2 \sqrt {f} \left (7 c^2 f^2+10 c d e f+15 d^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {8 \sqrt {2} (c f+d e)^2 \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{2 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x} (c f+7 d e)}{b (e+f x) (d e-c f) (c f+d e)}\right )}{4 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{2 b (e+f x)^2 \left (d^2 e^2-c^2 f^2\right )}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\sqrt {c+d x} \sqrt {b c-b d x} (2 C e-B f) \left (\frac {d \left (\frac {2 f (c f+3 d e) \int \frac {1}{e+\frac {c f}{d}-\frac {f (b c-b d x)}{b d}}d\sqrt {b c-b d x}}{b d (d e-c f)}-\frac {4 (c f+d e) \int \frac {1}{2 c-\frac {b c-b d x}{b}}d\sqrt {b c-b d x}}{b (d e-c f)}\right )}{2 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{b (e+f x) \left (d^2 e^2-c^2 f^2\right )}\right )}{f^2 \sqrt {b c^2-b d^2 x^2}}+\frac {C \sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {2 f \int \frac {1}{e+\frac {c f}{d}-\frac {f (b c-b d x)}{b d}}d\sqrt {b c-b d x}}{b d (d e-c f)}-\frac {2 \int \frac {1}{2 c-\frac {b c-b d x}{b}}d\sqrt {b c-b d x}}{b (d e-c f)}\right )}{f^2 \sqrt {b c^2-b d^2 x^2}}+\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (\frac {d \left (\frac {d \left (\frac {2 \sqrt {f} \left (7 c^2 f^2+10 c d e f+15 d^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {8 \sqrt {2} (c f+d e)^2 \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{2 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x} (c f+7 d e)}{b (e+f x) (d e-c f) (c f+d e)}\right )}{4 \left (d^2 e^2-c^2 f^2\right )}+\frac {f \sqrt {b c-b d x}}{2 b (e+f x)^2 \left (d^2 e^2-c^2 f^2\right )}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {C \sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {d e+c f}}\right )}{\sqrt {b} (d e-c f) \sqrt {d e+c f}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{f^2 \sqrt {b c^2-b d^2 x^2}}-\frac {(2 C e-B f) \sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\sqrt {b c-b d x} f}{b \left (d^2 e^2-c^2 f^2\right ) (e+f x)}+\frac {d \left (\frac {2 \sqrt {f} (3 d e+c f) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {d e+c f}}\right )}{\sqrt {b} (d e-c f) \sqrt {d e+c f}}-\frac {2 \sqrt {2} (d e+c f) \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{2 \left (d^2 e^2-c^2 f^2\right )}\right )}{f^2 \sqrt {b c^2-b d^2 x^2}}+\frac {\left (A+\frac {e (C e-B f)}{f^2}\right ) \sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\sqrt {b c-b d x} f}{2 b \left (d^2 e^2-c^2 f^2\right ) (e+f x)^2}+\frac {d \left (\frac {f \sqrt {b c-b d x} (7 d e+c f)}{b (d e-c f) (d e+c f) (e+f x)}+\frac {d \left (\frac {2 \sqrt {f} \left (15 d^2 e^2+10 c d f e+7 c^2 f^2\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {d e+c f}}\right )}{\sqrt {b} (d e-c f) \sqrt {d e+c f}}-\frac {8 \sqrt {2} (d e+c f)^2 \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{2 \left (d^2 e^2-c^2 f^2\right )}\right )}{4 \left (d^2 e^2-c^2 f^2\right )}\right )}{\sqrt {b c^2-b d^2 x^2}}\)

Input:

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

Output:

(C*Sqrt[c + d*x]*Sqrt[b*c - b*d*x]*(-((Sqrt[2]*ArcTanh[Sqrt[b*c - b*d*x]/( 
Sqrt[2]*Sqrt[b]*Sqrt[c])])/(Sqrt[b]*Sqrt[c]*(d*e - c*f))) + (2*Sqrt[f]*Arc 
Tanh[(Sqrt[f]*Sqrt[b*c - b*d*x])/(Sqrt[b]*Sqrt[d*e + c*f])])/(Sqrt[b]*(d*e 
 - c*f)*Sqrt[d*e + c*f])))/(f^2*Sqrt[b*c^2 - b*d^2*x^2]) - ((2*C*e - B*f)* 
Sqrt[c + d*x]*Sqrt[b*c - b*d*x]*((f*Sqrt[b*c - b*d*x])/(b*(d^2*e^2 - c^2*f 
^2)*(e + f*x)) + (d*((-2*Sqrt[2]*(d*e + c*f)*ArcTanh[Sqrt[b*c - b*d*x]/(Sq 
rt[2]*Sqrt[b]*Sqrt[c])])/(Sqrt[b]*Sqrt[c]*(d*e - c*f)) + (2*Sqrt[f]*(3*d*e 
 + c*f)*ArcTanh[(Sqrt[f]*Sqrt[b*c - b*d*x])/(Sqrt[b]*Sqrt[d*e + c*f])])/(S 
qrt[b]*(d*e - c*f)*Sqrt[d*e + c*f])))/(2*(d^2*e^2 - c^2*f^2))))/(f^2*Sqrt[ 
b*c^2 - b*d^2*x^2]) + ((A + (e*(C*e - B*f))/f^2)*Sqrt[c + d*x]*Sqrt[b*c - 
b*d*x]*((f*Sqrt[b*c - b*d*x])/(2*b*(d^2*e^2 - c^2*f^2)*(e + f*x)^2) + (d*( 
(f*(7*d*e + c*f)*Sqrt[b*c - b*d*x])/(b*(d*e - c*f)*(d*e + c*f)*(e + f*x)) 
+ (d*((-8*Sqrt[2]*(d*e + c*f)^2*ArcTanh[Sqrt[b*c - b*d*x]/(Sqrt[2]*Sqrt[b] 
*Sqrt[c])])/(Sqrt[b]*Sqrt[c]*(d*e - c*f)) + (2*Sqrt[f]*(15*d^2*e^2 + 10*c* 
d*e*f + 7*c^2*f^2)*ArcTanh[(Sqrt[f]*Sqrt[b*c - b*d*x])/(Sqrt[b]*Sqrt[d*e + 
 c*f])])/(Sqrt[b]*(d*e - c*f)*Sqrt[d*e + c*f])))/(2*(d^2*e^2 - c^2*f^2)))) 
/(4*(d^2*e^2 - c^2*f^2))))/Sqrt[b*c^2 - b*d^2*x^2]
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 97

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[b/(b*c - a*d)   Int[(e + f*x)^p/(a + b*x), x], x] - Simp[d/(b*c 
 - a*d)   Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p}, 
 x] &&  !IntegerQ[p]
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(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] + Simp[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*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 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 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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 718

Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_) 
^2)^(p_), x_Symbol] :> Simp[(a + c*x^2)^FracPart[p]/((d + e*x)^FracPart[p]* 
(a/d + (c*x)/e)^FracPart[p])   Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c/ 
e)*x)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && EqQ[c*d^2 + a*e^2, 
 0]
 

rule 2349

Int[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. 
)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(c + d 
*x)^(m + 1)*(e + f*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c 
+ d*x, x]   Int[(c + d*x)^m*(e + f*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, 
b, c, d, e, f, n, p}, x] && PolynomialQ[Px, x] && LtQ[m, 0] &&  !IntegerQ[n 
] && IntegersQ[2*m, 2*n, 2*p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3917\) vs. \(2(465)=930\).

Time = 0.26 (sec) , antiderivative size = 3918, normalized size of antiderivative = 7.68

method	result	size
default	\(\text {Expression too large to display}\)	\(3918\)

Input:

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

Output:

-1/4*(b*(-d^2*x^2+c^2))^(1/2)/b*(-4*C*(b*(c*f+d*e)*f)^(1/2)*2^(1/2)*arctan 
h(1/2*((-d*x+c)*b)^(1/2)*2^(1/2)/(b*c)^(1/2))*b*c^2*d^2*e^4*f+8*C*arctanh( 
f*((-d*x+c)*b)^(1/2)/(b*(c*f+d*e)*f)^(1/2))*(b*c)^(1/2)*b*c^3*d*e*f^5*x^2- 
4*A*(b*(c*f+d*e)*f)^(1/2)*2^(1/2)*arctanh(1/2*((-d*x+c)*b)^(1/2)*2^(1/2)/( 
b*c)^(1/2))*b*c^2*d^2*f^5*x^2+15*C*arctanh(f*((-d*x+c)*b)^(1/2)/(b*(c*f+d* 
e)*f)^(1/2))*(b*c)^(1/2)*b*c^2*d^2*e^2*f^4*x^2+2*C*arctanh(f*((-d*x+c)*b)^ 
(1/2)/(b*(c*f+d*e)*f)^(1/2))*(b*c)^(1/2)*b*c*d^3*e^3*f^3*x^2-8*A*(b*(c*f+d 
*e)*f)^(1/2)*2^(1/2)*arctanh(1/2*((-d*x+c)*b)^(1/2)*2^(1/2)/(b*c)^(1/2))*b 
*d^4*e^3*f^2*x+14*A*arctanh(f*((-d*x+c)*b)^(1/2)/(b*(c*f+d*e)*f)^(1/2))*(b 
*c)^(1/2)*b*c^2*d^2*e*f^5*x+20*A*arctanh(f*((-d*x+c)*b)^(1/2)/(b*(c*f+d*e) 
*f)^(1/2))*(b*c)^(1/2)*b*c*d^3*e^2*f^4*x-8*B*arctanh(f*((-d*x+c)*b)^(1/2)/ 
(b*(c*f+d*e)*f)^(1/2))*(b*c)^(1/2)*b*c^3*d*e*f^5*x-38*B*arctanh(f*((-d*x+c 
)*b)^(1/2)/(b*(c*f+d*e)*f)^(1/2))*(b*c)^(1/2)*b*c^2*d^2*e^2*f^4*x-12*B*arc 
tanh(f*((-d*x+c)*b)^(1/2)/(b*(c*f+d*e)*f)^(1/2))*(b*c)^(1/2)*b*c*d^3*e^3*f 
^3*x+16*C*arctanh(f*((-d*x+c)*b)^(1/2)/(b*(c*f+d*e)*f)^(1/2))*(b*c)^(1/2)* 
b*c^3*d*e^2*f^4*x+2*A*((-d*x+c)*b)^(1/2)*(b*(c*f+d*e)*f)^(1/2)*(b*c)^(1/2) 
*c^3*f^5+C*((-d*x+c)*b)^(1/2)*(b*(c*f+d*e)*f)^(1/2)*(b*c)^(1/2)*d^3*e^5-C* 
arctanh(f*((-d*x+c)*b)^(1/2)/(b*(c*f+d*e)*f)^(1/2))*(b*c)^(1/2)*b*d^4*e^6+ 
15*C*arctanh(f*((-d*x+c)*b)^(1/2)/(b*(c*f+d*e)*f)^(1/2))*(b*c)^(1/2)*b*c^2 
*d^2*e^4*f^2+2*C*arctanh(f*((-d*x+c)*b)^(1/2)/(b*(c*f+d*e)*f)^(1/2))*(b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2002 vs. \(2 (465) = 930\).

Time = 133.72 (sec) , antiderivative size = 8113, normalized size of antiderivative = 15.91 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x)^3 \sqrt {b c^2-b d^2 x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

integrate((C*x^2 + B*x + A)/(sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x + c)*(f*x + 
 e)^3), x)
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 908, normalized size of antiderivative = 1.78 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x)^3 \sqrt {b c^2-b d^2 x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*(4*sqrt(2)*(C*c^2*d - B*c*d^2 + A*d^3)*arctan(1/2*sqrt(2)*sqrt(-(d*x + 
 c)*b + 2*b*c)/sqrt(-b*c))/((d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3 
*f^3)*sqrt(-b*c)) + (C*d^5*e^4 - 2*C*c*d^4*e^3*f + 3*B*d^5*e^3*f - 15*C*c^ 
2*d^3*e^2*f^2 + 6*B*c*d^4*e^2*f^2 - 15*A*d^5*e^2*f^2 - 8*C*c^3*d^2*e*f^3 + 
 19*B*c^2*d^3*e*f^3 - 10*A*c*d^4*e*f^3 - 8*C*c^4*d*f^4 + 4*B*c^3*d^2*f^4 - 
 7*A*c^2*d^3*f^4)*arctan(sqrt(-(d*x + c)*b + 2*b*c)*f/sqrt(-b*d*e*f - b*c* 
f^2))/((d^5*e^5*f - c*d^4*e^4*f^2 - 2*c^2*d^3*e^3*f^3 + 2*c^3*d^2*e^2*f^4 
+ c^4*d*e*f^5 - c^5*f^6)*sqrt(-b*d*e*f - b*c*f^2)) + (sqrt(-(d*x + c)*b + 
2*b*c)*C*b*d^5*e^4 - 5*sqrt(-(d*x + c)*b + 2*b*c)*B*b*d^5*e^3*f + 7*sqrt(- 
(d*x + c)*b + 2*b*c)*C*b*c^2*d^3*e^2*f^2 - 4*sqrt(-(d*x + c)*b + 2*b*c)*B* 
b*c*d^4*e^2*f^2 + 9*sqrt(-(d*x + c)*b + 2*b*c)*A*b*d^5*e^2*f^2 + 8*sqrt(-( 
d*x + c)*b + 2*b*c)*C*b*c^3*d^2*e*f^3 - 3*sqrt(-(d*x + c)*b + 2*b*c)*B*b*c 
^2*d^3*e*f^3 + 8*sqrt(-(d*x + c)*b + 2*b*c)*A*b*c*d^4*e*f^3 - 4*sqrt(-(d*x 
 + c)*b + 2*b*c)*B*b*c^3*d^2*f^4 - sqrt(-(d*x + c)*b + 2*b*c)*A*b*c^2*d^3* 
f^4 + (-(d*x + c)*b + 2*b*c)^(3/2)*C*d^4*e^3*f - (-(d*x + c)*b + 2*b*c)^(3 
/2)*C*c*d^3*e^2*f^2 + 3*(-(d*x + c)*b + 2*b*c)^(3/2)*B*d^4*e^2*f^2 - 8*(-( 
d*x + c)*b + 2*b*c)^(3/2)*C*c^2*d^2*e*f^3 + (-(d*x + c)*b + 2*b*c)^(3/2)*B 
*c*d^3*e*f^3 - 7*(-(d*x + c)*b + 2*b*c)^(3/2)*A*d^4*e*f^3 + 4*(-(d*x + c)* 
b + 2*b*c)^(3/2)*B*c^2*d^2*f^4 - (-(d*x + c)*b + 2*b*c)^(3/2)*A*c*d^3*f^4) 
/((d^4*e^4*f - 2*c^2*d^2*e^2*f^3 + c^4*f^5)*(b*d*e + b*c*f + ((d*x + c)...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 4913, normalized size of antiderivative = 9.63 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x)^3 \sqrt {b c^2-b d^2 x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(b)*(7*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt 
(c*f + d*e)))*a*c**3*d**2*e**2*f**4*i + 14*sqrt(f)*sqrt(c*f + d*e)*atan((s 
qrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*a*c**3*d**2*e*f**5*i*x + 7*sq 
rt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))* 
a*c**3*d**2*f**6*i*x**2 + 10*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f 
*i)/(sqrt(f)*sqrt(c*f + d*e)))*a*c**2*d**3*e**3*f**3*i + 20*sqrt(f)*sqrt(c 
*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*a*c**2*d**3* 
e**2*f**4*i*x + 10*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt( 
f)*sqrt(c*f + d*e)))*a*c**2*d**3*e*f**5*i*x**2 + 15*sqrt(f)*sqrt(c*f + d*e 
)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*a*c*d**4*e**4*f**2*i 
 + 30*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + 
 d*e)))*a*c*d**4*e**3*f**3*i*x + 15*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - 
 d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*a*c*d**4*e**2*f**4*i*x**2 - 4*sqrt(f 
)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*b*c* 
*4*d*e**2*f**4*i - 8*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqr 
t(f)*sqrt(c*f + d*e)))*b*c**4*d*e*f**5*i*x - 4*sqrt(f)*sqrt(c*f + d*e)*ata 
n((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*b*c**4*d*f**6*i*x**2 - 19 
*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e) 
))*b*c**3*d**2*e**3*f**3*i - 38*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x 
)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*b*c**3*d**2*e**2*f**4*i*x - 19*sqrt(f...