\(\int \frac {A+B x}{(a+b x+c x^2)^{5/2} (d-f x^2)} \, dx\) [85]

Optimal result

Integrand size = 30, antiderivative size = 762 \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^{5/2} \left (d-f x^2\right )} \, dx=-\frac {2 \left (A b c (c d-a f)-(A b-a B) \left (2 c^2 d-b^2 f+2 a c f\right )+c \left (A b^2 f+b B (c d-a f)-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (a f \left (2 c^2 d-b^2 f+2 a c f\right ) \left (A b^3 f-4 A b c (2 c d-a f)+b^2 B (7 c d-a f)-12 a B c (c d+a f)\right )+b \left (c^2 d-b^2 f+3 a c f\right ) \left (3 b^3 B d f-7 A b^2 f (c d+a f)-4 b B \left (c^2 d^2+3 a c d f-a^2 f^2\right )+4 A c \left (2 c^2 d^2+7 a c d f+5 a^2 f^2\right )\right )+c \left (b f (c d-a f) \left (A b^3 f-4 A b c (2 c d-a f)+b^2 B (7 c d-a f)-12 a B c (c d+a f)\right )+\left (2 c^2 d-b^2 f+2 a c f\right ) \left (3 b^3 B d f-7 A b^2 f (c d+a f)-4 b B \left (c^2 d^2+3 a c d f-a^2 f^2\right )+4 A c \left (2 c^2 d^2+7 a c d f+5 a^2 f^2\right )\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c^2 d^2+2 a c d f-f \left (b^2 d-a^2 f\right )\right )^2 \sqrt {a+b x+c x^2}}-\frac {\left (B \sqrt {d}-A \sqrt {f}\right ) f^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{5/2}}+\frac {\left (B \sqrt {d}+A \sqrt {f}\right ) f^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{5/2}} \] Output:

1/3*(-2*A*b*c*(-a*f+c*d)+2*(A*b-B*a)*(2*a*c*f-b^2*f+2*c^2*d)-2*c*(A*b^2*f+ 
b*B*(-a*f+c*d)-2*A*c*(a*f+c*d))*x)/(-4*a*c+b^2)/(b^2*d*f-(a*f+c*d)^2)/(c*x 
^2+b*x+a)^(3/2)+2/3*(a*f*(2*a*c*f-b^2*f+2*c^2*d)*(A*b^3*f-4*A*b*c*(-a*f+2* 
c*d)+b^2*B*(-a*f+7*c*d)-12*a*B*c*(a*f+c*d))+b*(3*a*c*f-b^2*f+c^2*d)*(3*b^3 
*B*d*f-7*A*b^2*f*(a*f+c*d)-4*b*B*(-a^2*f^2+3*a*c*d*f+c^2*d^2)+4*A*c*(5*a^2 
*f^2+7*a*c*d*f+2*c^2*d^2))+c*(b*f*(-a*f+c*d)*(A*b^3*f-4*A*b*c*(-a*f+2*c*d) 
+b^2*B*(-a*f+7*c*d)-12*a*B*c*(a*f+c*d))+(2*a*c*f-b^2*f+2*c^2*d)*(3*b^3*B*d 
*f-7*A*b^2*f*(a*f+c*d)-4*b*B*(-a^2*f^2+3*a*c*d*f+c^2*d^2)+4*A*c*(5*a^2*f^2 
+7*a*c*d*f+2*c^2*d^2)))*x)/(-4*a*c+b^2)^2/(c^2*d^2+2*a*c*d*f-f*(-a^2*f+b^2 
*d))^2/(c*x^2+b*x+a)^(1/2)-1/2*(B*d^(1/2)-A*f^(1/2))*f^(3/2)*arctanh(1/2*( 
b*d^(1/2)-2*a*f^(1/2)+(2*c*d^(1/2)-b*f^(1/2))*x)/(c*d-b*d^(1/2)*f^(1/2)+a* 
f)^(1/2)/(c*x^2+b*x+a)^(1/2))/d^(1/2)/(c*d-b*d^(1/2)*f^(1/2)+a*f)^(5/2)+1/ 
2*(B*d^(1/2)+A*f^(1/2))*f^(3/2)*arctanh(1/2*(b*d^(1/2)+2*a*f^(1/2)+(2*c*d^ 
(1/2)+b*f^(1/2))*x)/(c*d+b*d^(1/2)*f^(1/2)+a*f)^(1/2)/(c*x^2+b*x+a)^(1/2)) 
/d^(1/2)/(c*d+b*d^(1/2)*f^(1/2)+a*f)^(5/2)
 

Mathematica [A] (verified)

Time = 14.37 (sec) , antiderivative size = 674, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^{5/2} \left (d-f x^2\right )} \, dx=\frac {2 \left (\frac {4 c \left (-A b^2 f+b B (-c d+a f)+2 A c (c d+a f)\right ) (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}}-\frac {3 f \left (b^4 B d f+2 c (c d+a f)^2 (-a B+A c x)+b^3 f (-A (c d+2 a f)+B c d x)+b c (c d+a f) (A c d+5 a A f-3 B c d x+a B f x)-b^2 \left (B \left (c^2 d^2+2 a c d f-a^2 f^2\right )+2 a A c f^2 x\right )\right )}{\left (c^2 d^2+2 a c d f+f \left (-b^2 d+a^2 f\right )\right ) \sqrt {a+x (b+c x)}}+\frac {A \left (b^3 f-b c (c d+3 a f)+b^2 c f x-2 c^2 (c d+a f) x\right )+B \left (2 a^2 c f+b c^2 d x+a \left (2 c^2 d-b^2 f-b c f x\right )\right )}{(a+x (b+c x))^{3/2}}+\frac {3 \left (b^2-4 a c\right ) f^{3/2} \left (\frac {\left (-B \sqrt {d}+A \sqrt {f}\right ) \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^2 \text {arctanh}\left (\frac {-2 a \sqrt {f}+2 c \sqrt {d} x+b \left (\sqrt {d}-\sqrt {f} x\right )}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c d-b \sqrt {d} \sqrt {f}+a f}}-\frac {\left (B \sqrt {d}+A \sqrt {f}\right ) \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^2 \text {arctanh}\left (\frac {-2 \left (a \sqrt {f}+c \sqrt {d} x\right )-b \left (\sqrt {d}+\sqrt {f} x\right )}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c d+b \sqrt {d} \sqrt {f}+a f}}\right )}{4 \sqrt {d} \left (-b^2 d f+(c d+a f)^2\right )}\right )}{3 \left (b^2-4 a c\right ) \left (-b^2 d f+(c d+a f)^2\right )} \] Input:

Integrate[(A + B*x)/((a + b*x + c*x^2)^(5/2)*(d - f*x^2)),x]
 

Output:

(2*((4*c*(-(A*b^2*f) + b*B*(-(c*d) + a*f) + 2*A*c*(c*d + a*f))*(b + 2*c*x) 
)/((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]) - (3*f*(b^4*B*d*f + 2*c*(c*d + a*f 
)^2*(-(a*B) + A*c*x) + b^3*f*(-(A*(c*d + 2*a*f)) + B*c*d*x) + b*c*(c*d + a 
*f)*(A*c*d + 5*a*A*f - 3*B*c*d*x + a*B*f*x) - b^2*(B*(c^2*d^2 + 2*a*c*d*f 
- a^2*f^2) + 2*a*A*c*f^2*x)))/((c^2*d^2 + 2*a*c*d*f + f*(-(b^2*d) + a^2*f) 
)*Sqrt[a + x*(b + c*x)]) + (A*(b^3*f - b*c*(c*d + 3*a*f) + b^2*c*f*x - 2*c 
^2*(c*d + a*f)*x) + B*(2*a^2*c*f + b*c^2*d*x + a*(2*c^2*d - b^2*f - b*c*f* 
x)))/(a + x*(b + c*x))^(3/2) + (3*(b^2 - 4*a*c)*f^(3/2)*(((-(B*Sqrt[d]) + 
A*Sqrt[f])*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^2*ArcTanh[(-2*a*Sqrt[f] + 2*c*S 
qrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f] 
*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f] - ((B*Sqrt[d 
] + A*Sqrt[f])*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^2*ArcTanh[(-2*(a*Sqrt[f] + 
c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + 
a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]))/(4*Sqr 
t[d]*(-(b^2*d*f) + (c*d + a*f)^2))))/(3*(b^2 - 4*a*c)*(-(b^2*d*f) + (c*d + 
 a*f)^2))
 

Rubi [A] (verified)

Time = 1.83 (sec) , antiderivative size = 895, normalized size of antiderivative = 1.17, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1351, 27, 2137, 27, 1366, 25, 27, 1154, 219}

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.

\(\displaystyle \int \frac {A+B x}{\left (d-f x^2\right ) \left (a+b x+c x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1351

\(\displaystyle \frac {2 \int \frac {3 B d f b^3-A f (7 c d+3 a f) b^2-4 B c d (c d+2 a f) b+4 c f \left (A f b^2+B (c d-a f) b-2 A c (c d+a f)\right ) x^2+4 A c \left (2 c^2 d^2+5 a c f d+3 a^2 f^2\right )+3 \left (b^2-4 a c\right ) f (A b f-B (c d+a f)) x}{2 \left (c x^2+b x+a\right )^{3/2} \left (d-f x^2\right )}dx}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(a f+c d)^2\right )}-\frac {2 \left (c x \left (-2 A c (a f+c d)+b B (c d-a f)+A b^2 f\right )-A b c (3 a f+c d)+a B \left (2 a c f+b^2 (-f)+2 c^2 d\right )+A b^3 f\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (b^2 d f-(a f+c d)^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {3 B d f b^3-A f (7 c d+3 a f) b^2-4 B c d (c d+2 a f) b+4 c f \left (A f b^2+B (c d-a f) b-2 A c (c d+a f)\right ) x^2+4 A c \left (2 c^2 d^2+5 a c f d+3 a^2 f^2\right )+3 \left (b^2-4 a c\right ) f (A b f-B (c d+a f)) x}{\left (c x^2+b x+a\right )^{3/2} \left (d-f x^2\right )}dx}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(a f+c d)^2\right )}-\frac {2 \left (c x \left (-2 A c (a f+c d)+b B (c d-a f)+A b^2 f\right )-A b c (3 a f+c d)+a B \left (2 a c f+b^2 (-f)+2 c^2 d\right )+A b^3 f\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (b^2 d f-(a f+c d)^2\right )}\)

\(\Big \downarrow \) 2137

\(\displaystyle \frac {\frac {2 \int \frac {3 \left (b^2-4 a c\right )^2 f^2 \left (A d f b^2-2 B d (c d+a f) b+A (c d+a f)^2-\left (2 A b f (c d+a f)-B \left (c^2 d^2+2 a c f d+f \left (f a^2+b^2 d\right )\right )\right ) x\right )}{2 \sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{\left (b^2-4 a c\right ) \left (b^2 d f-(a f+c d)^2\right )}-\frac {2 \left (A b^3 c f \left (43 a^2 f^2+46 a c d f+15 c^2 d^2\right )-b^4 B f \left (-3 a^2 f^2+14 a c d f+7 c^2 d^2\right )+24 a^2 B c^2 f (a f+c d)^2+c x \left (2 A b^2 c f \left (19 a^2 f^2+22 a c d f+15 c^2 d^2\right )-b^3 B f \left (-3 a^2 f^2+10 a c d f+17 c^2 d^2\right )+4 b B c \left (-5 a^3 f^3+4 a^2 c d f^2+11 a c^2 d^2 f+2 c^3 d^3\right )-2 A b^4 f^2 (3 a f+4 c d)-8 A c^2 (a f+c d)^2 (5 a f+2 c d)+3 b^5 B d f^2\right )-4 A b c^2 \left (17 a^3 f^3+24 a^2 c d f^2+9 a c^2 d^2 f+2 c^3 d^3\right )+2 b^2 B c \left (-11 a^3 f^3+4 a^2 c d f^2+5 a c^2 d^2 f+2 c^3 d^3\right )-A b^5 f^2 (6 a f+7 c d)+3 b^6 B d f^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(a f+c d)^2\right )}-\frac {2 \left (c x \left (-2 A c (a f+c d)+b B (c d-a f)+A b^2 f\right )-A b c (3 a f+c d)+a B \left (2 a c f+b^2 (-f)+2 c^2 d\right )+A b^3 f\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (b^2 d f-(a f+c d)^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 f^2 \left (b^2-4 a c\right ) \int \frac {A d f b^2-2 B d (c d+a f) b+A (c d+a f)^2-\left (2 A b f (c d+a f)-B \left (c^2 d^2+2 a c f d+f \left (f a^2+b^2 d\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{b^2 d f-(a f+c d)^2}-\frac {2 \left (A b^3 c f \left (43 a^2 f^2+46 a c d f+15 c^2 d^2\right )-b^4 B f \left (-3 a^2 f^2+14 a c d f+7 c^2 d^2\right )+24 a^2 B c^2 f (a f+c d)^2+c x \left (2 A b^2 c f \left (19 a^2 f^2+22 a c d f+15 c^2 d^2\right )-b^3 B f \left (-3 a^2 f^2+10 a c d f+17 c^2 d^2\right )+4 b B c \left (-5 a^3 f^3+4 a^2 c d f^2+11 a c^2 d^2 f+2 c^3 d^3\right )-2 A b^4 f^2 (3 a f+4 c d)-8 A c^2 (a f+c d)^2 (5 a f+2 c d)+3 b^5 B d f^2\right )-4 A b c^2 \left (17 a^3 f^3+24 a^2 c d f^2+9 a c^2 d^2 f+2 c^3 d^3\right )+2 b^2 B c \left (-11 a^3 f^3+4 a^2 c d f^2+5 a c^2 d^2 f+2 c^3 d^3\right )-A b^5 f^2 (6 a f+7 c d)+3 b^6 B d f^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(a f+c d)^2\right )}-\frac {2 \left (c x \left (-2 A c (a f+c d)+b B (c d-a f)+A b^2 f\right )-A b c (3 a f+c d)+a B \left (2 a c f+b^2 (-f)+2 c^2 d\right )+A b^3 f\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (b^2 d f-(a f+c d)^2\right )}\)

\(\Big \downarrow \) 1366

\(\displaystyle \frac {\frac {3 f^2 \left (b^2-4 a c\right ) \left (\frac {\left (A \sqrt {f}+B \sqrt {d}\right ) \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}+\frac {\left (B \sqrt {d}-A \sqrt {f}\right ) \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int -\frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}\right )}{b^2 d f-(a f+c d)^2}-\frac {2 \left (A b^3 c f \left (43 a^2 f^2+46 a c d f+15 c^2 d^2\right )-b^4 B f \left (-3 a^2 f^2+14 a c d f+7 c^2 d^2\right )+24 a^2 B c^2 f (a f+c d)^2+c x \left (2 A b^2 c f \left (19 a^2 f^2+22 a c d f+15 c^2 d^2\right )-b^3 B f \left (-3 a^2 f^2+10 a c d f+17 c^2 d^2\right )+4 b B c \left (-5 a^3 f^3+4 a^2 c d f^2+11 a c^2 d^2 f+2 c^3 d^3\right )-2 A b^4 f^2 (3 a f+4 c d)-8 A c^2 (a f+c d)^2 (5 a f+2 c d)+3 b^5 B d f^2\right )-4 A b c^2 \left (17 a^3 f^3+24 a^2 c d f^2+9 a c^2 d^2 f+2 c^3 d^3\right )+2 b^2 B c \left (-11 a^3 f^3+4 a^2 c d f^2+5 a c^2 d^2 f+2 c^3 d^3\right )-A b^5 f^2 (6 a f+7 c d)+3 b^6 B d f^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(a f+c d)^2\right )}-\frac {2 \left (c x \left (-2 A c (a f+c d)+b B (c d-a f)+A b^2 f\right )-A b c (3 a f+c d)+a B \left (2 a c f+b^2 (-f)+2 c^2 d\right )+A b^3 f\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (b^2 d f-(a f+c d)^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {3 f^2 \left (b^2-4 a c\right ) \left (\frac {\left (A \sqrt {f}+B \sqrt {d}\right ) \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}-\frac {\left (B \sqrt {d}-A \sqrt {f}\right ) \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}\right )}{b^2 d f-(a f+c d)^2}-\frac {2 \left (A b^3 c f \left (43 a^2 f^2+46 a c d f+15 c^2 d^2\right )-b^4 B f \left (-3 a^2 f^2+14 a c d f+7 c^2 d^2\right )+24 a^2 B c^2 f (a f+c d)^2+c x \left (2 A b^2 c f \left (19 a^2 f^2+22 a c d f+15 c^2 d^2\right )-b^3 B f \left (-3 a^2 f^2+10 a c d f+17 c^2 d^2\right )+4 b B c \left (-5 a^3 f^3+4 a^2 c d f^2+11 a c^2 d^2 f+2 c^3 d^3\right )-2 A b^4 f^2 (3 a f+4 c d)-8 A c^2 (a f+c d)^2 (5 a f+2 c d)+3 b^5 B d f^2\right )-4 A b c^2 \left (17 a^3 f^3+24 a^2 c d f^2+9 a c^2 d^2 f+2 c^3 d^3\right )+2 b^2 B c \left (-11 a^3 f^3+4 a^2 c d f^2+5 a c^2 d^2 f+2 c^3 d^3\right )-A b^5 f^2 (6 a f+7 c d)+3 b^6 B d f^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(a f+c d)^2\right )}-\frac {2 \left (c x \left (-2 A c (a f+c d)+b B (c d-a f)+A b^2 f\right )-A b c (3 a f+c d)+a B \left (2 a c f+b^2 (-f)+2 c^2 d\right )+A b^3 f\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (b^2 d f-(a f+c d)^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 f^2 \left (b^2-4 a c\right ) \left (\frac {\left (A \sqrt {f}+B \sqrt {d}\right ) \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int \frac {1}{\left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d} \sqrt {f}}-\frac {\left (B \sqrt {d}-A \sqrt {f}\right ) \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{\left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d} \sqrt {f}}\right )}{b^2 d f-(a f+c d)^2}-\frac {2 \left (A b^3 c f \left (43 a^2 f^2+46 a c d f+15 c^2 d^2\right )-b^4 B f \left (-3 a^2 f^2+14 a c d f+7 c^2 d^2\right )+24 a^2 B c^2 f (a f+c d)^2+c x \left (2 A b^2 c f \left (19 a^2 f^2+22 a c d f+15 c^2 d^2\right )-b^3 B f \left (-3 a^2 f^2+10 a c d f+17 c^2 d^2\right )+4 b B c \left (-5 a^3 f^3+4 a^2 c d f^2+11 a c^2 d^2 f+2 c^3 d^3\right )-2 A b^4 f^2 (3 a f+4 c d)-8 A c^2 (a f+c d)^2 (5 a f+2 c d)+3 b^5 B d f^2\right )-4 A b c^2 \left (17 a^3 f^3+24 a^2 c d f^2+9 a c^2 d^2 f+2 c^3 d^3\right )+2 b^2 B c \left (-11 a^3 f^3+4 a^2 c d f^2+5 a c^2 d^2 f+2 c^3 d^3\right )-A b^5 f^2 (6 a f+7 c d)+3 b^6 B d f^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(a f+c d)^2\right )}-\frac {2 \left (c x \left (-2 A c (a f+c d)+b B (c d-a f)+A b^2 f\right )-A b c (3 a f+c d)+a B \left (2 a c f+b^2 (-f)+2 c^2 d\right )+A b^3 f\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (b^2 d f-(a f+c d)^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {3 \left (b^2-4 a c\right ) f^2 \left (\frac {\left (B \sqrt {d}-A \sqrt {f}\right ) \left (\sqrt {d} \sqrt {f} b+c d+a f\right )^2 \int \frac {1}{4 \left (-\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )}{\sqrt {d} \sqrt {f}}-\frac {\left (\sqrt {f} A+B \sqrt {d}\right ) \left (-\sqrt {d} \sqrt {f} b+c d+a f\right )^2 \int \frac {1}{4 \left (\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )}{\sqrt {d} \sqrt {f}}\right )}{b^2 d f-(c d+a f)^2}-\frac {2 \left (3 B d f^2 b^6-A f^2 (7 c d+6 a f) b^5-B f \left (7 c^2 d^2+14 a c f d-3 a^2 f^2\right ) b^4+A c f \left (15 c^2 d^2+46 a c f d+43 a^2 f^2\right ) b^3+2 B c \left (2 c^3 d^3+5 a c^2 f d^2+4 a^2 c f^2 d-11 a^3 f^3\right ) b^2-4 A c^2 \left (2 c^3 d^3+9 a c^2 f d^2+24 a^2 c f^2 d+17 a^3 f^3\right ) b+24 a^2 B c^2 f (c d+a f)^2+c \left (3 B d f^2 b^5-2 A f^2 (4 c d+3 a f) b^4-B f \left (17 c^2 d^2+10 a c f d-3 a^2 f^2\right ) b^3+2 A c f \left (15 c^2 d^2+22 a c f d+19 a^2 f^2\right ) b^2+4 B c \left (2 c^3 d^3+11 a c^2 f d^2+4 a^2 c f^2 d-5 a^3 f^3\right ) b-8 A c^2 (c d+a f)^2 (2 c d+5 a f)\right ) x\right )}{\left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \sqrt {c x^2+b x+a}}}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right )}-\frac {2 \left (A f b^3-A c (c d+3 a f) b+a B \left (-f b^2+2 c^2 d+2 a c f\right )+c \left (A f b^2+B (c d-a f) b-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (c x^2+b x+a\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {3 \left (b^2-4 a c\right ) f^2 \left (\frac {\left (\sqrt {f} A+B \sqrt {d}\right ) \left (-\sqrt {d} \sqrt {f} b+c d+a f\right )^2 \text {arctanh}\left (\frac {2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}}{2 \sqrt {\sqrt {d} \sqrt {f} b+c d+a f} \sqrt {c x^2+b x+a}}\right )}{2 \sqrt {d} \sqrt {f} \sqrt {\sqrt {d} \sqrt {f} b+c d+a f}}-\frac {\left (B \sqrt {d}-A \sqrt {f}\right ) \left (\sqrt {d} \sqrt {f} b+c d+a f\right )^2 \text {arctanh}\left (\frac {-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}}{2 \sqrt {-\sqrt {d} \sqrt {f} b+c d+a f} \sqrt {c x^2+b x+a}}\right )}{2 \sqrt {d} \sqrt {f} \sqrt {-\sqrt {d} \sqrt {f} b+c d+a f}}\right )}{b^2 d f-(c d+a f)^2}-\frac {2 \left (3 B d f^2 b^6-A f^2 (7 c d+6 a f) b^5-B f \left (7 c^2 d^2+14 a c f d-3 a^2 f^2\right ) b^4+A c f \left (15 c^2 d^2+46 a c f d+43 a^2 f^2\right ) b^3+2 B c \left (2 c^3 d^3+5 a c^2 f d^2+4 a^2 c f^2 d-11 a^3 f^3\right ) b^2-4 A c^2 \left (2 c^3 d^3+9 a c^2 f d^2+24 a^2 c f^2 d+17 a^3 f^3\right ) b+24 a^2 B c^2 f (c d+a f)^2+c \left (3 B d f^2 b^5-2 A f^2 (4 c d+3 a f) b^4-B f \left (17 c^2 d^2+10 a c f d-3 a^2 f^2\right ) b^3+2 A c f \left (15 c^2 d^2+22 a c f d+19 a^2 f^2\right ) b^2+4 B c \left (2 c^3 d^3+11 a c^2 f d^2+4 a^2 c f^2 d-5 a^3 f^3\right ) b-8 A c^2 (c d+a f)^2 (2 c d+5 a f)\right ) x\right )}{\left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \sqrt {c x^2+b x+a}}}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right )}-\frac {2 \left (A f b^3-A c (c d+3 a f) b+a B \left (-f b^2+2 c^2 d+2 a c f\right )+c \left (A f b^2+B (c d-a f) b-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (c x^2+b x+a\right )^{3/2}}\)

Input:

Int[(A + B*x)/((a + b*x + c*x^2)^(5/2)*(d - f*x^2)),x]
 

Output:

(-2*(A*b^3*f - A*b*c*(c*d + 3*a*f) + a*B*(2*c^2*d - b^2*f + 2*a*c*f) + c*( 
A*b^2*f + b*B*(c*d - a*f) - 2*A*c*(c*d + a*f))*x))/(3*(b^2 - 4*a*c)*(b^2*d 
*f - (c*d + a*f)^2)*(a + b*x + c*x^2)^(3/2)) + ((-2*(3*b^6*B*d*f^2 + 24*a^ 
2*B*c^2*f*(c*d + a*f)^2 - A*b^5*f^2*(7*c*d + 6*a*f) - b^4*B*f*(7*c^2*d^2 + 
 14*a*c*d*f - 3*a^2*f^2) + A*b^3*c*f*(15*c^2*d^2 + 46*a*c*d*f + 43*a^2*f^2 
) + 2*b^2*B*c*(2*c^3*d^3 + 5*a*c^2*d^2*f + 4*a^2*c*d*f^2 - 11*a^3*f^3) - 4 
*A*b*c^2*(2*c^3*d^3 + 9*a*c^2*d^2*f + 24*a^2*c*d*f^2 + 17*a^3*f^3) + c*(3* 
b^5*B*d*f^2 - 2*A*b^4*f^2*(4*c*d + 3*a*f) - 8*A*c^2*(c*d + a*f)^2*(2*c*d + 
 5*a*f) - b^3*B*f*(17*c^2*d^2 + 10*a*c*d*f - 3*a^2*f^2) + 2*A*b^2*c*f*(15* 
c^2*d^2 + 22*a*c*d*f + 19*a^2*f^2) + 4*b*B*c*(2*c^3*d^3 + 11*a*c^2*d^2*f + 
 4*a^2*c*d*f^2 - 5*a^3*f^3))*x))/((b^2 - 4*a*c)*(b^2*d*f - (c*d + a*f)^2)* 
Sqrt[a + b*x + c*x^2]) + (3*(b^2 - 4*a*c)*f^2*(-1/2*((B*Sqrt[d] - A*Sqrt[f 
])*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^2*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2 
*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + 
 b*x + c*x^2])])/(Sqrt[d]*Sqrt[f]*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + ( 
(B*Sqrt[d] + A*Sqrt[f])*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^2*ArcTanh[(b*Sqrt[ 
d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sq 
rt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*Sqrt[f]*Sqrt[c*d + b*Sqrt 
[d]*Sqrt[f] + a*f])))/(b^2*d*f - (c*d + a*f)^2))/(3*(b^2 - 4*a*c)*(b^2*d*f 
 - (c*d + a*f)^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1351

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f 
_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(a + b*x + c*x^2)^(p + 1)*((d + f*x^2)^ 
(q + 1)/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)))*((g*c)*((-b)*(c* 
d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(2*a*f)) + c*(g*(2*c^2*d + b^2 
*f - c*(2*a*f)) - h*(b*c*d + a*b*f))*x), x] + Simp[1/((b^2 - 4*a*c)*(b^2*d* 
f + (c*d - a*f)^2)*(p + 1))   Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q*S 
imp[(b*h - 2*g*c)*((c*d - a*f)^2 - (b*d)*((-b)*f))*(p + 1) + (b^2*(g*f) - b 
*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2*f* 
((g*c)*((-b)*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(2*a*f)))*(p + 
 q + 2) - (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(b*f*(p + 1 
)))*x - c*f*(b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(2*p + 2* 
q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}, x] && NeQ[b^2 - 4 
*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] &&  !( !IntegerQ[ 
p] && ILtQ[q, -1])
 

rule 1366

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + ( 
f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Simp[(h/2 + c*(g/(2*q 
)))   Int[1/((-q + c*x)*Sqrt[d + e*x + f*x^2]), x], x] + Simp[(h/2 - c*(g/( 
2*q)))   Int[1/((q + c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d 
, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && PosQ[(-a)*c]
 

rule 2137

Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_ 
), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[P 
x, x, 2]}, Simp[(a + b*x + c*x^2)^(p + 1)*((d + f*x^2)^(q + 1)/((b^2 - 4*a* 
c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)))*((A*c - a*C)*((-b)*(c*d + a*f)) + (A 
*b - a*B)*(2*c^2*d + b^2*f - c*(2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(2*a*f) 
) - B*(b*c*d + a*b*f) + C*(b^2*d - 2*a*(c*d - a*f)))*x), x] + Simp[1/((b^2 
- 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1))   Int[(a + b*x + c*x^2)^(p + 1) 
*(d + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d)*((-b)*f))* 
(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c* 
C*d - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2*f*((A*c - a*C)*((-b)*(c*d + 
 a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(2*a*f)))*(p + q + 2) - (b^2*(C*d 
 + A*f) - b*(B*c*d + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(b*f 
*(p + 1)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + a*B*f) + 2*(A*c*(c*d - a*f 
) - a*(c*C*d - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]] /; FreeQ[{a, b, c, 
 d, f, q}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f) 
^2, 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1765\) vs. \(2(688)=1376\).

Time = 2.49 (sec) , antiderivative size = 1766, normalized size of antiderivative = 2.32

method	result	size
default	\(\text {Expression too large to display}\)	\(1766\)

Input:

int((B*x+A)/(c*x^2+b*x+a)^(5/2)/(-f*x^2+d),x,method=_RETURNVERBOSE)
 

Output:

-1/2*(A*f+B*(d*f)^(1/2))/(d*f)^(1/2)/f*(1/3/(b*(d*f)^(1/2)+a*f+c*d)*f/(c*( 
x-(d*f)^(1/2)/f)^2+(2*c*(d*f)^(1/2)+f*b)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2 
)+a*f+c*d)/f)^(3/2)-1/2*(2*c*(d*f)^(1/2)+f*b)/(b*(d*f)^(1/2)+a*f+c*d)*(2/3 
*(2*c*(x-(d*f)^(1/2)/f)+(2*c*(d*f)^(1/2)+f*b)/f)/(4*c*(b*(d*f)^(1/2)+a*f+c 
*d)/f-(2*c*(d*f)^(1/2)+f*b)^2/f^2)/(c*(x-(d*f)^(1/2)/f)^2+(2*c*(d*f)^(1/2) 
+f*b)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+a*f+c*d)/f)^(3/2)+16/3*c/(4*c*(b* 
(d*f)^(1/2)+a*f+c*d)/f-(2*c*(d*f)^(1/2)+f*b)^2/f^2)^2*(2*c*(x-(d*f)^(1/2)/ 
f)+(2*c*(d*f)^(1/2)+f*b)/f)/(c*(x-(d*f)^(1/2)/f)^2+(2*c*(d*f)^(1/2)+f*b)/f 
*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+a*f+c*d)/f)^(1/2))+1/(b*(d*f)^(1/2)+a*f+ 
c*d)*f*(1/(b*(d*f)^(1/2)+a*f+c*d)*f/(c*(x-(d*f)^(1/2)/f)^2+(2*c*(d*f)^(1/2 
)+f*b)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+a*f+c*d)/f)^(1/2)-(2*c*(d*f)^(1/ 
2)+f*b)/(b*(d*f)^(1/2)+a*f+c*d)*(2*c*(x-(d*f)^(1/2)/f)+(2*c*(d*f)^(1/2)+f* 
b)/f)/(4*c*(b*(d*f)^(1/2)+a*f+c*d)/f-(2*c*(d*f)^(1/2)+f*b)^2/f^2)/(c*(x-(d 
*f)^(1/2)/f)^2+(2*c*(d*f)^(1/2)+f*b)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+a* 
f+c*d)/f)^(1/2)-1/(b*(d*f)^(1/2)+a*f+c*d)*f/((b*(d*f)^(1/2)+a*f+c*d)/f)^(1 
/2)*ln((2*(b*(d*f)^(1/2)+a*f+c*d)/f+(2*c*(d*f)^(1/2)+f*b)/f*(x-(d*f)^(1/2) 
/f)+2*((b*(d*f)^(1/2)+a*f+c*d)/f)^(1/2)*(c*(x-(d*f)^(1/2)/f)^2+(2*c*(d*f)^ 
(1/2)+f*b)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+a*f+c*d)/f)^(1/2))/(x-(d*f)^ 
(1/2)/f))))+1/2*(A*f-B*(d*f)^(1/2))/(d*f)^(1/2)/f*(1/3*f/(-b*(d*f)^(1/2)+a 
*f+c*d)/(c*(x+(d*f)^(1/2)/f)^2+1/f*(-2*c*(d*f)^(1/2)+f*b)*(x+(d*f)^(1/2...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^{5/2} \left (d-f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)/(c*x^2+b*x+a)^(5/2)/(-f*x^2+d),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^{5/2} \left (d-f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)/(c*x**2+b*x+a)**(5/2)/(-f*x**2+d),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^{5/2} \left (d-f x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)/(c*x^2+b*x+a)^(5/2)/(-f*x^2+d),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(((c*sqrt(4*d*f))/(2*f^2)>0)', se 
e `assume?
 

Giac [F(-1)]

Timed out. \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^{5/2} \left (d-f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)/(c*x^2+b*x+a)^(5/2)/(-f*x^2+d),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^{5/2} \left (d-f x^2\right )} \, dx=\int \frac {A+B\,x}{\left (d-f\,x^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \] Input:

int((A + B*x)/((d - f*x^2)*(a + b*x + c*x^2)^(5/2)),x)
 

Output:

int((A + B*x)/((d - f*x^2)*(a + b*x + c*x^2)^(5/2)), x)
 

Reduce [B] (verification not implemented)

Time = 79.81 (sec) , antiderivative size = 112712, normalized size of antiderivative = 147.92 \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^{5/2} \left (d-f x^2\right )} \, dx =\text {Too large to display} \] Input:

int((B*x+A)/(c*x^2+b*x+a)^(5/2)/(-f*x^2+d),x)
 

Output:

(48*sqrt(d)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*atan((sqrt(d)*sqrt(a + b*x 
 + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*a*b*f - 2*sqrt(d)*sqrt(a + 
b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*a*c*f*x + sqrt(d)*sqrt(a 
 + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*b**2*f*x - sqrt(d)*sq 
rt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*b*c*d - 2*sqrt(d) 
*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*c**2*d*x + 2*s 
qrt(f)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*a**2*f + 
 sqrt(f)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*a*b*f* 
x + 2*sqrt(f)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*a 
*c*d - sqrt(f)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)* 
b**2*d - sqrt(f)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d 
)*b*c*d*x)/(2*a**3*f**2 + 2*a**2*b*f**2*x + 4*a**2*c*d*f + 2*a**2*c*f**2*x 
**2 - 2*a*b**2*d*f + 4*a*b*c*d*f*x + 2*a*c**2*d**2 + 4*a*c**2*d*f*x**2 - 2 
*b**3*d*f*x - 2*b**2*c*d*f*x**2 + 2*b*c**2*d**2*x + 2*c**3*d**2*x**2))*a** 
8*c**2*f**5 - 24*sqrt(d)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*atan((sqrt(d) 
*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*a*b*f - 2*sqrt 
(d)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*a*c*f*x + s 
qrt(d)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*b**2*f*x 
 - sqrt(d)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*b*c* 
d - 2*sqrt(d)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d...