\(\int \frac {1}{(d+e x+f \sqrt {a+b x+c x^2})^2} \, dx\) [28]

Optimal result

Integrand size = 23, antiderivative size = 394 \[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+c x^2}\right )^2} \, dx=-\frac {2 \left (2 b \left (c d^2+a e^2\right )-4 a^{3/2} c e f-b^2 \left (2 d e-\sqrt {a} e f\right )+\frac {\left (4 a^{3/2} e^2+4 \sqrt {a} d (c d-b e)+b^2 d f-4 a c d f\right ) \left (\sqrt {a}-\sqrt {a+b x+c x^2}\right )}{x}\right )}{\left (d-\sqrt {a} f\right ) \left (4 b d e-4 a e^2-b^2 f^2-4 c \left (d^2-a f^2\right )\right ) \left (b e-c \left (d+\sqrt {a} f\right )+\frac {\left (2 \sqrt {a} e-b f\right ) \left (\sqrt {a}-\sqrt {a+b x+c x^2}\right )}{x}+\frac {\left (d-\sqrt {a} f\right ) \left (\sqrt {a}-\sqrt {a+b x+c x^2}\right )^2}{x^2}\right )}+\frac {4 \left (b^2-4 a c\right ) f \text {arctanh}\left (\frac {2 a f+b f x+2 d \sqrt {a+x (b+c x)}-2 \sqrt {a} \left (d+e x+f \sqrt {a+x (b+c x)}\right )}{\sqrt {-4 b d e+4 a e^2+b^2 f^2+4 c \left (d^2-a f^2\right )} x}\right )}{\left (-4 b d e+4 a e^2+b^2 f^2+4 c \left (d^2-a f^2\right )\right )^{3/2}} \] Output:

(-4*b*(a*e^2+c*d^2)+8*a^(3/2)*c*e*f+2*b^2*(2*d*e-a^(1/2)*e*f)-2*(4*a^(3/2) 
*e^2+4*a^(1/2)*d*(-b*e+c*d)+b^2*d*f-4*a*c*d*f)*(a^(1/2)-(c*x^2+b*x+a)^(1/2 
))/x)/(d-a^(1/2)*f)/(4*b*d*e-4*a*e^2-b^2*f^2-4*c*(-a*f^2+d^2))/(b*e-c*(d+a 
^(1/2)*f)+(2*a^(1/2)*e-b*f)*(a^(1/2)-(c*x^2+b*x+a)^(1/2))/x+(d-a^(1/2)*f)* 
(a^(1/2)-(c*x^2+b*x+a)^(1/2))^2/x^2)+4*(-4*a*c+b^2)*f*arctanh((2*a*f+b*f*x 
+2*d*(a+x*(c*x+b))^(1/2)-2*a^(1/2)*(d+e*x+f*(a+x*(c*x+b))^(1/2)))/(-4*b*d* 
e+4*a*e^2+b^2*f^2+4*c*(-a*f^2+d^2))^(1/2)/x)/(-4*b*d*e+4*a*e^2+b^2*f^2+4*c 
*(-a*f^2+d^2))^(3/2)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(2162\) vs. \(2(394)=788\).

Time = 19.10 (sec) , antiderivative size = 2162, normalized size of antiderivative = 5.49 \[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+c x^2}\right )^2} \, dx=\text {Result too large to show} \] Input:

Integrate[(d + e*x + f*Sqrt[a + b*x + c*x^2])^(-2),x]
 

Output:

(-2*(2*c*d^3*e - 2*b*d^2*e^2 + 2*a*d*e^3 + b*c*d^2*f^2 - 4*a*c*d*e*f^2 + a 
*b*e^2*f^2 + 2*c*d^2*e^2*x - 2*b*d*e^3*x + 2*a*e^4*x + 2*c^2*d^2*f^2*x - 2 
*b*c*d*e*f^2*x + b^2*e^2*f^2*x - 2*a*c*e^2*f^2*x))/((e^2 - c*f^2)*(4*c*d^2 
 - 4*b*d*e + 4*a*e^2 + b^2*f^2 - 4*a*c*f^2)*(d^2 - a*f^2 + 2*d*e*x - b*f^2 
*x + e^2*x^2 - c*f^2*x^2)) + (2*(-(b*d*f) + 2*a*e*f - 2*c*d*f*x + b*e*f*x) 
*Sqrt[a + b*x + c*x^2])/((4*c*d^2 - 4*b*d*e + 4*a*e^2 + b^2*f^2 - 4*a*c*f^ 
2)*(d^2 - a*f^2 + 2*d*e*x - b*f^2*x + e^2*x^2 - c*f^2*x^2)) - (2*(-b^2 + 4 
*a*c)*f*ArcTanh[(2*d*e - b*f^2 + 2*e^2*x - 2*c*f^2*x)/(f*Sqrt[4*c*d^2 - 4* 
b*d*e + 4*a*e^2 + b^2*f^2 - 4*a*c*f^2])])/(4*c*d^2 - 4*b*d*e + 4*a*e^2 + b 
^2*f^2 - 4*a*c*f^2)^(3/2) - ((-b^2 + 4*a*c)*f*(-2*c*d*f + b*e*f + e*Sqrt[4 
*c*d^2 - 4*b*d*e + 4*a*e^2 + b^2*f^2 - 4*a*c*f^2])*Log[2*d*e - b*f^2 - f*S 
qrt[4*c*d^2 - 4*b*d*e + 4*a*e^2 + b^2*f^2 - 4*a*c*f^2] + 2*e^2*x - 2*c*f^2 
*x])/(Sqrt[2]*(4*c*d^2 - 4*b*d*e + 4*a*e^2 + b^2*f^2 - 4*a*c*f^2)^(3/2)*Sq 
rt[2*c*d^2*e^2 - 2*b*d*e^3 + 2*a*e^4 + 2*c^2*d^2*f^2 - 2*b*c*d*e*f^2 + b^2 
*e^2*f^2 - 2*a*c*e^2*f^2 - 2*c*d*e*f*Sqrt[4*c*d^2 - 4*b*d*e + 4*a*e^2 + b^ 
2*f^2 - 4*a*c*f^2] + b*e^2*f*Sqrt[4*c*d^2 - 4*b*d*e + 4*a*e^2 + b^2*f^2 - 
4*a*c*f^2]]) + ((-b^2 + 4*a*c)*f*(-2*c*d*f + b*e*f - e*Sqrt[4*c*d^2 - 4*b* 
d*e + 4*a*e^2 + b^2*f^2 - 4*a*c*f^2])*Log[2*d*e - b*f^2 + f*Sqrt[4*c*d^2 - 
 4*b*d*e + 4*a*e^2 + b^2*f^2 - 4*a*c*f^2] + 2*e^2*x - 2*c*f^2*x])/(Sqrt[2] 
*(4*c*d^2 - 4*b*d*e + 4*a*e^2 + b^2*f^2 - 4*a*c*f^2)^(3/2)*Sqrt[2*c*d^2...
 

Rubi [F]

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.

\(\displaystyle \int \frac {1}{\left (f \sqrt {a+b x+c x^2}+d+e x\right )^2} \, dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

Input:

Int[(d + e*x + f*Sqrt[a + b*x + c*x^2])^(-2),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 1.72 (sec) , antiderivative size = 616160, normalized size of antiderivative = 1563.86

method	result	size
default	\(\text {Expression too large to display}\)	\(616160\)

Input:

int(1/(d+e*x+f*(c*x^2+b*x+a)^(1/2))^2,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2867 vs. \(2 (359) = 718\).

Time = 19.40 (sec) , antiderivative size = 6899, normalized size of antiderivative = 17.51 \[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+c x^2}\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+c x^2}\right )^2} \, dx=\int \frac {1}{\left (d + e x + f \sqrt {a + b x + c x^{2}}\right )^{2}}\, dx \] Input:

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

Output:

Integral((d + e*x + f*sqrt(a + b*x + c*x**2))**(-2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+c x^2}\right )^2} \, dx=\int { \frac {1}{{\left (e x + \sqrt {c x^{2} + b x + a} f + d\right )}^{2}} \,d x } \] Input:

integrate(1/(d+e*x+f*(c*x^2+b*x+a)^(1/2))^2,x, algorithm="maxima")
 

Output:

integrate((e*x + sqrt(c*x^2 + b*x + a)*f + d)^(-2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+c x^2}\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+c x^2}\right )^2} \, dx=\int \frac {1}{{\left (d+e\,x+f\,\sqrt {c\,x^2+b\,x+a}\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+c x^2}\right )^2} \, dx=\int \frac {1}{\left (d +e x +f \sqrt {c \,x^{2}+b x +a}\right )^{2}}d x \] Input:

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

Output:

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