\(\int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}} \, dx\) [31]

Optimal result

Integrand size = 51, antiderivative size = 402 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {2 D \sqrt {b c^2-b d^2 x^2}}{b d^2 f^2 \sqrt {c+d x}}-\frac {\left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \sqrt {b c^2-b d^2 x^2}}{b f^2 (d e-c f) (d e+c f) \sqrt {c+d x} (e+f x)}-\frac {\sqrt {2} \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {b c^2-b d^2 x^2}}{\sqrt {2} \sqrt {b} \sqrt {c} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {c} d^2 (d e-c f)^2}-\frac {\left (2 c^2 f^2 \left (3 D e^2-f (2 C e-B f)\right )-d^2 \left (3 D e^4-e f \left (C e^2+B e f-3 A f^2\right )\right )+c d f \left (D e^3-f \left (C e^2-B e f+A f^2\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c^2-b d^2 x^2}}{\sqrt {b} \sqrt {d e+c f} \sqrt {c+d x}}\right )}{\sqrt {b} f^{5/2} (d e-c f)^2 (d e+c f)^{3/2}} \] Output:

-2*D*(-b*d^2*x^2+b*c^2)^(1/2)/b/d^2/f^2/(d*x+c)^(1/2)-(D*e^3-f*(C*e^2-f*(- 
A*f+B*e)))*(-b*d^2*x^2+b*c^2)^(1/2)/b/f^2/(-c*f+d*e)/(c*f+d*e)/(d*x+c)^(1/ 
2)/(f*x+e)-2^(1/2)*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*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^2/(-c 
*f+d*e)^2-(2*c^2*f^2*(3*D*e^2-f*(-B*f+2*C*e))-d^2*(3*D*e^4-e*f*(-3*A*f^2+B 
*e*f+C*e^2))+c*d*f*(D*e^3-f*(A*f^2-B*e*f+C*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^(5/2)/(- 
c*f+d*e)^2/(c*f+d*e)^(3/2)
 

Mathematica [A] (verified)

Time = 2.06 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (\frac {\sqrt {c^2-d^2 x^2} \left (2 c^2 D f^2 (e+f x)+d^2 \left (f \left (C e^2-B e f+A f^2\right )-D e^2 (3 e+2 f x)\right )\right )}{d^2 f^2 \left (d^2 e^2-c^2 f^2\right ) \sqrt {c+d x} (e+f x)}+\frac {\left (-2 c^2 f^2 \left (3 D e^2+f (-2 C e+B f)\right )+d^2 \left (3 D e^4-e f \left (C e^2+B e f-3 A f^2\right )\right )+c d f \left (-D e^3+f \left (C e^2-B e f+A f^2\right )\right )\right ) \arctan \left (\frac {\sqrt {-d e-c f} \sqrt {c^2-d^2 x^2}}{\sqrt {f} (-c+d x) \sqrt {c+d x}}\right )}{f^{5/2} (-d e-c f)^{3/2} (d e-c f)^2}+\frac {\sqrt {2} \left (-c^2 C d+B c d^2-A d^3+c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{\sqrt {c} d^2 (d e-c f)^2}\right )}{\sqrt {b \left (c^2-d^2 x^2\right )}} \] Input:

Integrate[(A + B*x + C*x^2 + D*x^3)/(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]*((Sqrt[c^2 - d^2*x^2]*(2*c^2*D*f^2*(e + f*x) + d^2*(f 
*(C*e^2 - B*e*f + A*f^2) - D*e^2*(3*e + 2*f*x))))/(d^2*f^2*(d^2*e^2 - c^2* 
f^2)*Sqrt[c + d*x]*(e + f*x)) + ((-2*c^2*f^2*(3*D*e^2 + f*(-2*C*e + B*f)) 
+ d^2*(3*D*e^4 - e*f*(C*e^2 + B*e*f - 3*A*f^2)) + c*d*f*(-(D*e^3) + f*(C*e 
^2 - B*e*f + A*f^2)))*ArcTan[(Sqrt[-(d*e) - c*f]*Sqrt[c^2 - d^2*x^2])/(Sqr 
t[f]*(-c + d*x)*Sqrt[c + d*x])])/(f^(5/2)*(-(d*e) - c*f)^(3/2)*(d*e - c*f) 
^2) + (Sqrt[2]*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D)*ArcTanh[(Sqrt[2]*Sqr 
t[c]*Sqrt[c + d*x])/Sqrt[c^2 - d^2*x^2]])/(Sqrt[c]*d^2*(d*e - c*f)^2)))/Sq 
rt[b*(c^2 - d^2*x^2)]
 

Rubi [A] (verified)

Time = 2.80 (sec) , antiderivative size = 595, normalized size of antiderivative = 1.48, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {2349, 718, 114, 27, 174, 73, 221, 2349, 600, 458, 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.

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

\(\Big \downarrow \) 2349

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

\(\Big \downarrow \) 718

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 2349

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 458

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

\(\Big \downarrow \) 471

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 718

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

\(\Big \downarrow \) 97

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 458

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

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 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

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. \(2348\) vs. \(2(363)=726\).

Time = 0.21 (sec) , antiderivative size = 2349, normalized size of antiderivative = 5.84

method	result	size
default	\(\text {Expression too large to display}\)	\(2349\)

Input:

int((D*x^3+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 
,method=_RETURNVERBOSE)
 

Output:

(b*(-d^2*x^2+c^2))^(1/2)*(-B*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^3+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^2*e^2*f^3+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^2+D*(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^4*e*f 
^3-6*D*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^3*f^2-D*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^4*f-2*D*((-d*x+c)*b)^(1/2)*(b*(c*f+d*e)*f)^(1/2)*(b*c)^(1 
/2)*d^3*e^3*f*x+B*((-d*x+c)*b)^(1/2)*(b*(c*f+d*e)*f)^(1/2)*(b*c)^(1/2)*c*d 
^2*e*f^3-C*((-d*x+c)*b)^(1/2)*(b*(c*f+d*e)*f)^(1/2)*(b*c)^(1/2)*c*d^2*e^2* 
f^2+2*D*((-d*x+c)*b)^(1/2)*(b*(c*f+d*e)*f)^(1/2)*(b*c)^(1/2)*c^2*d*e^2*f^2 
+3*D*((-d*x+c)*b)^(1/2)*(b*(c*f+d*e)*f)^(1/2)*(b*c)^(1/2)*c*d^2*e^3*f+A*ar 
ctanh(f*((-d*x+c)*b)^(1/2)/(b*(c*f+d*e)*f)^(1/2))*(b*c)^(1/2)*b*c*d^3*f^5* 
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^ 
4*e*f^4*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^2*f^5*x-B*arctanh(f*((-d*x+c)*b)^(1/2)/(b*(c*f+d*e)*f)^(1/2))* 
(b*c)^(1/2)*b*d^4*e^2*f^3*x-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^3*f^2*x+D*(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^4*f^4*x+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*f^4-2*B*arc...
 

Fricas [F(-1)]

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

integrate((D*x^3+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:

Timed out
 

Sympy [F]

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

integrate((D*x**3+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 + D*x**3)/(sqrt(-b*(-c + d*x)*(c + d*x))*sqrt(c 
 + d*x)*(e + f*x)**2), x)
 

Maxima [F]

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

integrate((D*x^3+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((D*x^3 + 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 = 0.14 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.19 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {\frac {\sqrt {2} {\left (D c^{3} - C c^{2} d + B c d^{2} - A d^{3}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (d x + c\right )} b + 2 \, b c}}{2 \, \sqrt {-b c}}\right )}{{\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} \sqrt {-b c}} + \frac {{\left (3 \, D d^{3} e^{4} - D c d^{2} e^{3} f - C d^{3} e^{3} f - 6 \, D c^{2} d e^{2} f^{2} + C c d^{2} e^{2} f^{2} - B d^{3} e^{2} f^{2} + 4 \, C c^{2} d e f^{3} - B c d^{2} e f^{3} + 3 \, A d^{3} e f^{3} - 2 \, B c^{2} d f^{4} + A c d^{2} f^{4}\right )} \arctan \left (\frac {\sqrt {-{\left (d x + c\right )} b + 2 \, b c} f}{\sqrt {-b d e f - b c f^{2}}}\right )}{{\left (d^{3} e^{3} f^{2} - c d^{2} e^{2} f^{3} - c^{2} d e f^{4} + c^{3} f^{5}\right )} \sqrt {-b d e f - b c f^{2}}} + \frac {\sqrt {-{\left (d x + c\right )} b + 2 \, b c} D d^{2} e^{3} - \sqrt {-{\left (d x + c\right )} b + 2 \, b c} C d^{2} e^{2} f + \sqrt {-{\left (d x + c\right )} b + 2 \, b c} B d^{2} e f^{2} - \sqrt {-{\left (d x + c\right )} b + 2 \, b c} A d^{2} f^{3}}{{\left (d^{2} e^{2} f^{2} - c^{2} f^{4}\right )} {\left (b d e + b c f + {\left ({\left (d x + c\right )} b - 2 \, b c\right )} f\right )}} + \frac {2 \, \sqrt {-{\left (d x + c\right )} b + 2 \, b c} D}{b d f^{2}}}{d} \] Input:

integrate((D*x^3+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)*(D*c^3 - 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^2 - 2*c*d^2*e*f + c^2*d*f^2)*sqrt(-b 
*c)) + (3*D*d^3*e^4 - D*c*d^2*e^3*f - C*d^3*e^3*f - 6*D*c^2*d*e^2*f^2 + C* 
c*d^2*e^2*f^2 - B*d^3*e^2*f^2 + 4*C*c^2*d*e*f^3 - B*c*d^2*e*f^3 + 3*A*d^3* 
e*f^3 - 2*B*c^2*d*f^4 + A*c*d^2*f^4)*arctan(sqrt(-(d*x + c)*b + 2*b*c)*f/s 
qrt(-b*d*e*f - b*c*f^2))/((d^3*e^3*f^2 - c*d^2*e^2*f^3 - c^2*d*e*f^4 + c^3 
*f^5)*sqrt(-b*d*e*f - b*c*f^2)) + (sqrt(-(d*x + c)*b + 2*b*c)*D*d^2*e^3 - 
sqrt(-(d*x + c)*b + 2*b*c)*C*d^2*e^2*f + sqrt(-(d*x + c)*b + 2*b*c)*B*d^2* 
e*f^2 - sqrt(-(d*x + c)*b + 2*b*c)*A*d^2*f^3)/((d^2*e^2*f^2 - c^2*f^4)*(b* 
d*e + b*c*f + ((d*x + c)*b - 2*b*c)*f)) + 2*sqrt(-(d*x + c)*b + 2*b*c)*D/( 
b*d*f^2))/d
 

Mupad [F(-1)]

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

int((A + B*x + C*x^2 + x^3*D)/((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 + x^3*D)/((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 = 1928, normalized size of antiderivative = 4.80 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x)^2 \sqrt {b c^2-b d^2 x^2}} \, dx =\text {Too large to display} \] Input:

int((D*x^3+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**4*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**5*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**3*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**4*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**4*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**5*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**3*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**4*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**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*d**3*e**2*f**3*i*x - 8*sqrt(f)*sqr 
t(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*c**4*d*e* 
*2*f**3*i - 8*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sq 
rt(c*f + d*e)))*c**4*d*e*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)))*c**3*d**2*e**3*f**2*i + 10*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**3*i*x + 4*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i...