\(\int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}} \, dx\) [28]

Optimal result

Integrand size = 46, antiderivative size = 318 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {\left (C e^2-B e f+A f^2\right ) \sqrt {b c^2-b d^2 x^2}}{b f \left (d^2 e^2-c^2 f^2\right ) \sqrt {c+d x} (e+f x)}-\frac {\sqrt {2} \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {\sqrt {b c^2-b d^2 x^2}}{\sqrt {2} \sqrt {b} \sqrt {c} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {c} d (d e-c f)^2}-\frac {\left (C e \left (d^2 e^2-c d e f-4 c^2 f^2\right )-f \left (A d f (3 d e+c f)-B \left (d^2 e^2+c d e f+2 c^2 f^2\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c^2-b d^2 x^2}}{\sqrt {b} \sqrt {d e+c f} \sqrt {c+d x}}\right )}{\sqrt {b} f^{3/2} (d e-c f)^2 (d e+c f)^{3/2}} \] Output:

(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^(1/2)*(A*d^2-B*c*d+C*c^2)*arctanh(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)/d/(-c*f+d*e) 
^2-(C*e*(-4*c^2*f^2-c*d*e*f+d^2*e^2)-f*(A*d*f*(c*f+3*d*e)-B*(2*c^2*f^2+c*d 
*e*f+d^2*e^2)))*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)^2/(c*f+d*e)^(3/2)
 

Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (\frac {(d e-c f) \left (C e^2+f (-B e+A f)\right ) \sqrt {c^2-d^2 x^2}}{f (d e+c f) \sqrt {c+d x} (e+f x)}-\frac {\left (C e \left (d^2 e^2-c d e f-4 c^2 f^2\right )+f \left (-A d f (3 d e+c f)+B \left (d^2 e^2+c d e f+2 c^2 f^2\right )\right )\right ) \arctan \left (\frac {\sqrt {-d e-c f} \sqrt {c^2-d^2 x^2}}{\sqrt {f} (-c+d x) \sqrt {c+d x}}\right )}{f^{3/2} (-d e-c f)^{3/2}}-\frac {\sqrt {2} \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{\sqrt {c} d}\right )}{(d e-c f)^2 \sqrt {b \left (c^2-d^2 x^2\right )}} \] Input:

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

Output:

(Sqrt[c^2 - d^2*x^2]*(((d*e - c*f)*(C*e^2 + f*(-(B*e) + A*f))*Sqrt[c^2 - d 
^2*x^2])/(f*(d*e + c*f)*Sqrt[c + d*x]*(e + f*x)) - ((C*e*(d^2*e^2 - c*d*e* 
f - 4*c^2*f^2) + f*(-(A*d*f*(3*d*e + c*f)) + B*(d^2*e^2 + c*d*e*f + 2*c^2* 
f^2)))*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)^(3/2)) - (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]])/(S 
qrt[c]*d)))/((d*e - c*f)^2*Sqrt[b*(c^2 - d^2*x^2)])
 

Rubi [A] (verified)

Time = 2.53 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.63, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.326, Rules used = {2349, 718, 114, 27, 174, 73, 221, 2349, 27, 471, 221, 718, 97, 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.

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

\(\Big \downarrow \) 2349

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

\(\Big \downarrow \) 718

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 2349

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 471

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 718

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

\(\Big \downarrow \) 97

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-(((2*C*e - B*f)*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]*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])))/(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])/(b*(d^2*e^2 - c^2*f^2)*(e + f*x)) + (d*((-2*Sqrt[2]*(d*e + c*f)*Ar 
cTanh[Sqrt[b*c - b*d*x]/(Sqrt[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])/(Sq 
rt[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))))/Sqrt[b*c^2 - b*d^2*x^2] - (Sqrt[2]*C*ArcTanh[Sqrt[b*c^2 - 
 b*d^2*x^2]/(Sqrt[2]*Sqrt[b]*Sqrt[c]*Sqrt[c + d*x])])/(Sqrt[b]*Sqrt[c]*d*f 
^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 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 471

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[2*d   Subst[Int[1/(2*b*c + d^2*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x] 
], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
 

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. \(1549\) vs. \(2(283)=566\).

Time = 0.21 (sec) , antiderivative size = 1550, normalized size of antiderivative = 4.87

method	result	size
default	\(\text {Expression too large to display}\)	\(1550\)

Input:

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

Output:

(b*(-d^2*x^2+c^2))^(1/2)*(4*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*e^2*f^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^2*e^3*f+B*((-d*x+c)*b)^(1/2)*(b*(c*f+d*e 
)*f)^(1/2)*(b*c)^(1/2)*c*d*e*f^2-C*((-d*x+c)*b)^(1/2)*(b*(c*f+d*e)*f)^(1/2 
)*(b*c)^(1/2)*c*d*e^2*f-C*arctanh(f*((-d*x+c)*b)^(1/2)/(b*(c*f+d*e)*f)^(1/ 
2))*(b*c)^(1/2)*b*d^3*e^3*f*x-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^3*e^2*f+A*arctanh(f*((-d*x+c)* 
b)^(1/2)/(b*(c*f+d*e)*f)^(1/2))*(b*c)^(1/2)*b*c*d^2*e*f^3-2*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*e*f^3-B*arctan 
h(f*((-d*x+c)*b)^(1/2)/(b*(c*f+d*e)*f)^(1/2))*(b*c)^(1/2)*b*c*d^2*e^2*f^2- 
C*(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^3*e*f^2+A*arctanh(f*((-d*x+c)*b)^(1/2)/(b*(c*f+d*e)*f)^(1/2) 
)*(b*c)^(1/2)*b*c*d^2*f^4*x+3*A*arctanh(f*((-d*x+c)*b)^(1/2)/(b*(c*f+d*e)* 
f)^(1/2))*(b*c)^(1/2)*b*d^3*e*f^3*x-2*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*f^4*x-B*arctanh(f*((-d*x+c)*b)^(1/2) 
/(b*(c*f+d*e)*f)^(1/2))*(b*c)^(1/2)*b*d^3*e^2*f^2*x-C*(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^3*f^3*x+ 
B*(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*d^2*e*f^2*x-C*(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*e*f^2*x+3*A*arctanh(f*((-d*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1010 vs. \(2 (283) = 566\).

Time = 26.84 (sec) , antiderivative size = 4145, normalized size of antiderivative = 13.03 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.15 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {\frac {\sqrt {2} {\left (C c^{2} - B c d + A d^{2}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (d x + c\right )} b + 2 \, b c}}{2 \, \sqrt {-b c}}\right )}{{\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} \sqrt {-b c}} + \frac {{\left (C d^{3} e^{3} - C c d^{2} e^{2} f + B d^{3} e^{2} f - 4 \, C c^{2} d e f^{2} + B c d^{2} e f^{2} - 3 \, A d^{3} e f^{2} + 2 \, B c^{2} d f^{3} - A c d^{2} f^{3}\right )} \arctan \left (\frac {\sqrt {-{\left (d x + c\right )} b + 2 \, b c} f}{\sqrt {-b d e f - b c f^{2}}}\right )}{{\left (d^{3} e^{3} f - c d^{2} e^{2} f^{2} - c^{2} d e f^{3} + c^{3} f^{4}\right )} \sqrt {-b d e f - b c f^{2}}} + \frac {\sqrt {-{\left (d x + c\right )} b + 2 \, b c} C d^{2} e^{2} - \sqrt {-{\left (d x + c\right )} b + 2 \, b c} B d^{2} e f + \sqrt {-{\left (d x + c\right )} b + 2 \, b c} A d^{2} f^{2}}{{\left (d^{2} e^{2} f - c^{2} f^{3}\right )} {\left (b d e + b c f + {\left ({\left (d x + c\right )} b - 2 \, b c\right )} f\right )}}}{d} \] Input:

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

Output:

(sqrt(2)*(C*c^2 - B*c*d + A*d^2)*arctan(1/2*sqrt(2)*sqrt(-(d*x + c)*b + 2* 
b*c)/sqrt(-b*c))/((d^2*e^2 - 2*c*d*e*f + c^2*f^2)*sqrt(-b*c)) + (C*d^3*e^3 
 - C*c*d^2*e^2*f + B*d^3*e^2*f - 4*C*c^2*d*e*f^2 + B*c*d^2*e*f^2 - 3*A*d^3 
*e*f^2 + 2*B*c^2*d*f^3 - A*c*d^2*f^3)*arctan(sqrt(-(d*x + c)*b + 2*b*c)*f/ 
sqrt(-b*d*e*f - b*c*f^2))/((d^3*e^3*f - c*d^2*e^2*f^2 - c^2*d*e*f^3 + c^3* 
f^4)*sqrt(-b*d*e*f - b*c*f^2)) + (sqrt(-(d*x + c)*b + 2*b*c)*C*d^2*e^2 - s 
qrt(-(d*x + c)*b + 2*b*c)*B*d^2*e*f + sqrt(-(d*x + c)*b + 2*b*c)*A*d^2*f^2 
)/((d^2*e^2*f - c^2*f^3)*(b*d*e + b*c*f + ((d*x + c)*b - 2*b*c)*f)))/d
 

Mupad [F(-1)]

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

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 2073, normalized size of antiderivative = 6.52 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(b)*( - 2*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*s 
qrt(c*f + d*e)))*a*c**2*d**2*e*f**3*i - 2*sqrt(f)*sqrt(c*f + d*e)*atan((sq 
rt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*a*c**2*d**2*f**4*i*x - 6*sqrt( 
f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*a*c 
*d**3*e**2*f**2*i - 6*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sq 
rt(f)*sqrt(c*f + d*e)))*a*c*d**3*e*f**3*i*x + 4*sqrt(f)*sqrt(c*f + d*e)*at 
an((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*b*c**3*d*e*f**3*i + 4*sq 
rt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))* 
b*c**3*d*f**4*i*x + 2*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sq 
rt(f)*sqrt(c*f + d*e)))*b*c**2*d**2*e**2*f**2*i + 2*sqrt(f)*sqrt(c*f + d*e 
)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*b*c**2*d**2*e*f**3*i 
*x + 2*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f 
+ d*e)))*b*c*d**3*e**3*f*i + 2*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x) 
*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*b*c*d**3*e**2*f**2*i*x - 8*sqrt(f)*sqrt(c 
*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*c**4*d*e**2* 
f**2*i - 8*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt( 
c*f + d*e)))*c**4*d*e*f**3*i*x - 2*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - 
d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*c**3*d**2*e**3*f*i - 2*sqrt(f)*sqrt(c 
*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*c**3*d**2*e* 
*2*f**2*i*x + 2*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(...