Integrand size = 27, antiderivative size = 789 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )^2} \, dx=\frac {\left (f \left (b e^2-2 b d f-a e f\right )-c \left (e^3-3 d e f\right )+f \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{\left (e^2-4 d f\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (d+e x+f x^2\right )}+\frac {\left (f (2 c d-b e+2 a f) (c e-b f) \left (e-\sqrt {e^2-4 d f}\right )-2 f \left (2 c^2 d \left (e^2-4 d f\right )+f \left (3 a b e f-4 a^2 f^2+b^2 \left (e^2-6 d f\right )\right )-c \left (4 a f \left (e^2-3 d f\right )+b \left (e^3-5 d e f\right )\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {2} \left (e^2-4 d f\right )^{3/2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}-\frac {\left (f (2 c d-b e+2 a f) (c e-b f) \left (e+\sqrt {e^2-4 d f}\right )-2 f \left (2 c^2 d \left (e^2-4 d f\right )+f \left (3 a b e f-4 a^2 f^2+b^2 \left (e^2-6 d f\right )\right )-c \left (4 a f \left (e^2-3 d f\right )+b \left (e^3-5 d e f\right )\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {2} \left (e^2-4 d f\right )^{3/2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}} \]
(f*(-a*e*f-2*b*d*f+b*e^2)-c*(-3*d*e*f+e^3)+f*(f*(-2*a*f+b*e)-c*(-2*d*f+e^2 ))*x)*(c*x^2+b*x+a)^(1/2)/(-4*d*f+e^2)/((-a*f+c*d)^2-(-a*e+b*d)*(-b*f+c*e) )/(f*x^2+e*x+d)+1/4*arctanh(1/4*(4*a*f+2*x*(b*f-c*(e-(-4*d*f+e^2)^(1/2)))- b*(e-(-4*d*f+e^2)^(1/2)))*2^(1/2)/(c*x^2+b*x+a)^(1/2)/(c*e^2-2*c*d*f-b*e*f +2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2))*(-2*f*(2*c^2*d*(-4*d*f+e^2) +f*(3*a*b*e*f-4*a^2*f^2+b^2*(-6*d*f+e^2))-c*(4*a*f*(-3*d*f+e^2)+b*(-5*d*e* f+e^3)))+f*(2*a*f-b*e+2*c*d)*(-b*f+c*e)*(e-(-4*d*f+e^2)^(1/2)))/(-4*d*f+e^ 2)^(3/2)/((-a*f+c*d)^2-(-a*e+b*d)*(-b*f+c*e))*2^(1/2)/(c*e^2-2*c*d*f-b*e*f +2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)-1/4*arctanh(1/4*(4*a*f-b*(e+ (-4*d*f+e^2)^(1/2))+2*x*(b*f-c*(e+(-4*d*f+e^2)^(1/2))))*2^(1/2)/(c*x^2+b*x +a)^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2 ))*(-2*f*(2*c^2*d*(-4*d*f+e^2)+f*(3*a*b*e*f-4*a^2*f^2+b^2*(-6*d*f+e^2))-c* (4*a*f*(-3*d*f+e^2)+b*(-5*d*e*f+e^3)))+f*(2*a*f-b*e+2*c*d)*(-b*f+c*e)*(e+( -4*d*f+e^2)^(1/2)))/(-4*d*f+e^2)^(3/2)/((-a*f+c*d)^2-(-a*e+b*d)*(-b*f+c*e) )*2^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2 )
Time = 16.76 (sec) , antiderivative size = 1377, normalized size of antiderivative = 1.75 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )^2} \, dx=-\frac {8 f^3 \left (a+b x+c x^2\right )}{\left (e^2-4 d f\right ) \left (4 a f^2-2 b f \left (e-\sqrt {e^2-4 d f}\right )+c \left (e-\sqrt {e^2-4 d f}\right )^2\right ) \left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+x (b+c x)}}-\frac {8 f^3 \left (a+b x+c x^2\right )}{\left (e^2-4 d f\right ) \left (4 a f^2-2 b f \left (e+\sqrt {e^2-4 d f}\right )+c \left (e+\sqrt {e^2-4 d f}\right )^2\right ) \left (e+\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+x (b+c x)}}+\frac {2 \sqrt {2} f^2 \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+f \left (2 a f-b \left (e-\sqrt {e^2-4 d f}\right )\right )} \sqrt {a+b x+c x^2}}\right )}{\left (e^2-4 d f\right )^{3/2} \sqrt {c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+f \left (2 a f-b \left (e-\sqrt {e^2-4 d f}\right )\right )} \sqrt {a+x (b+c x)}}-\frac {8 \sqrt {2} f^2 \sqrt {c e^2-2 c d f-b e f+2 a f^2-c e \sqrt {e^2-4 d f}+b f \sqrt {e^2-4 d f}} \left (2 b f+2 c \left (-e+\sqrt {e^2-4 d f}\right )\right ) \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {-4 a f-b \left (-e+\sqrt {e^2-4 d f}\right )-\left (2 b f+2 c \left (-e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-c e \sqrt {e^2-4 d f}+b f \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\left (e^2-4 d f\right ) \left (4 a f^2+2 b f \left (-e+\sqrt {e^2-4 d f}\right )+c \left (-e+\sqrt {e^2-4 d f}\right )^2\right ) \left (16 a f^2+8 b f \left (-e+\sqrt {e^2-4 d f}\right )+4 c \left (-e+\sqrt {e^2-4 d f}\right )^2\right ) \sqrt {a+x (b+c x)}}-\frac {2 \sqrt {2} f^2 \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )+f \left (2 a f-b \left (e+\sqrt {e^2-4 d f}\right )\right )} \sqrt {a+b x+c x^2}}\right )}{\left (e^2-4 d f\right )^{3/2} \sqrt {c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )+f \left (2 a f-b \left (e+\sqrt {e^2-4 d f}\right )\right )} \sqrt {a+x (b+c x)}}-\frac {8 \sqrt {2} f^2 \sqrt {c e^2-2 c d f-b e f+2 a f^2+c e \sqrt {e^2-4 d f}-b f \sqrt {e^2-4 d f}} \left (-2 b f+2 c \left (e+\sqrt {e^2-4 d f}\right )\right ) \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )-\left (-2 b f+2 c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+c e \sqrt {e^2-4 d f}-b f \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\left (e^2-4 d f\right ) \left (4 a f^2-2 b f \left (e+\sqrt {e^2-4 d f}\right )+c \left (e+\sqrt {e^2-4 d f}\right )^2\right ) \left (16 a f^2-8 b f \left (e+\sqrt {e^2-4 d f}\right )+4 c \left (e+\sqrt {e^2-4 d f}\right )^2\right ) \sqrt {a+x (b+c x)}} \]
(-8*f^3*(a + b*x + c*x^2))/((e^2 - 4*d*f)*(4*a*f^2 - 2*b*f*(e - Sqrt[e^2 - 4*d*f]) + c*(e - Sqrt[e^2 - 4*d*f])^2)*(e - Sqrt[e^2 - 4*d*f] + 2*f*x)*Sq rt[a + x*(b + c*x)]) - (8*f^3*(a + b*x + c*x^2))/((e^2 - 4*d*f)*(4*a*f^2 - 2*b*f*(e + Sqrt[e^2 - 4*d*f]) + c*(e + Sqrt[e^2 - 4*d*f])^2)*(e + Sqrt[e^ 2 - 4*d*f] + 2*f*x)*Sqrt[a + x*(b + c*x)]) + (2*Sqrt[2]*f^2*Sqrt[a + b*x + c*x^2]*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[ e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e - Sqrt[e^2 - 4*d*f]))]*Sqrt[a + b*x + c*x^2])])/((e^2 - 4* d*f)^(3/2)*Sqrt[c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e - Sqrt[e^2 - 4*d*f]))]*Sqrt[a + x*(b + c*x)]) - (8*Sqrt[2]*f^2*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]] *(2*b*f + 2*c*(-e + Sqrt[e^2 - 4*d*f]))*Sqrt[a + b*x + c*x^2]*ArcTanh[(-4* a*f - b*(-e + Sqrt[e^2 - 4*d*f]) - (2*b*f + 2*c*(-e + Sqrt[e^2 - 4*d*f]))* x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d* f] + b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/((e^2 - 4*d*f)*(4*a*f ^2 + 2*b*f*(-e + Sqrt[e^2 - 4*d*f]) + c*(-e + Sqrt[e^2 - 4*d*f])^2)*(16*a* f^2 + 8*b*f*(-e + Sqrt[e^2 - 4*d*f]) + 4*c*(-e + Sqrt[e^2 - 4*d*f])^2)*Sqr t[a + x*(b + c*x)]) - (2*Sqrt[2]*f^2*Sqrt[a + b*x + c*x^2]*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sq rt[2]*Sqrt[c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e + Sq...
Time = 1.41 (sec) , antiderivative size = 764, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1305, 27, 1365, 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.
| \(\displaystyle \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1305 |
| \(\displaystyle \frac {\int \frac {-4 a^2 f^3+3 a b e f^2+b^2 \left (e^2-6 d f\right ) f-4 a c \left (e^2-3 d f\right ) f+(2 c d-b e+2 a f) (c e-b f) x f+2 c^2 d \left (e^2-4 d f\right )-b c \left (e^3-5 d e f\right )}{2 \sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{\left (e^2-4 d f\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}+\frac {\sqrt {a+b x+c x^2} \left (f x \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+f \left (-a e f-2 b d f+b e^2\right )-c \left (e^3-3 d e f\right )\right )}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-4 a^2 f^3+3 a b e f^2+b^2 \left (e^2-6 d f\right ) f-4 a c \left (e^2-3 d f\right ) f+(2 c d-b e+2 a f) (c e-b f) x f+2 c^2 d \left (e^2-4 d f\right )-b c \left (e^3-5 d e f\right )}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{2 \left (e^2-4 d f\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}+\frac {\sqrt {a+b x+c x^2} \left (f x \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+f \left (-a e f-2 b d f+b e^2\right )-c \left (e^3-3 d e f\right )\right )}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 1365 |
| \(\displaystyle \frac {\frac {f \left (\left (\sqrt {e^2-4 d f}+e\right ) (c e-b f) (2 a f-b e+2 c d)-2 \left (-4 a^2 f^3+3 a b e f^2-4 a c f \left (e^2-3 d f\right )+b^2 f \left (e^2-6 d f\right )-b c \left (e^3-5 d e f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \int \frac {1}{\left (e+2 f x+\sqrt {e^2-4 d f}\right ) \sqrt {c x^2+b x+a}}dx}{\sqrt {e^2-4 d f}}-\frac {f \left (\left (e-\sqrt {e^2-4 d f}\right ) (c e-b f) (2 a f-b e+2 c d)-2 \left (-4 a^2 f^3+3 a b e f^2-4 a c f \left (e^2-3 d f\right )+b^2 f \left (e^2-6 d f\right )-b c \left (e^3-5 d e f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \int \frac {1}{\left (e+2 f x-\sqrt {e^2-4 d f}\right ) \sqrt {c x^2+b x+a}}dx}{\sqrt {e^2-4 d f}}}{2 \left (e^2-4 d f\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}+\frac {\sqrt {a+b x+c x^2} \left (f x \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+f \left (-a e f-2 b d f+b e^2\right )-c \left (e^3-3 d e f\right )\right )}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {2 f \left (\left (e-\sqrt {e^2-4 d f}\right ) (c e-b f) (2 a f-b e+2 c d)-2 \left (-4 a^2 f^3+3 a b e f^2-4 a c f \left (e^2-3 d f\right )+b^2 f \left (e^2-6 d f\right )-b c \left (e^3-5 d e f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \int \frac {1}{4 \left (4 a f^2-2 b \left (e-\sqrt {e^2-4 d f}\right ) f+c \left (e-\sqrt {e^2-4 d f}\right )^2\right )-\frac {\left (4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x\right )^2}{c x^2+b x+a}}d\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {c x^2+b x+a}}}{\sqrt {e^2-4 d f}}-\frac {2 f \left (\left (\sqrt {e^2-4 d f}+e\right ) (c e-b f) (2 a f-b e+2 c d)-2 \left (-4 a^2 f^3+3 a b e f^2-4 a c f \left (e^2-3 d f\right )+b^2 f \left (e^2-6 d f\right )-b c \left (e^3-5 d e f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \int \frac {1}{4 \left (4 a f^2-2 b \left (e+\sqrt {e^2-4 d f}\right ) f+c \left (e+\sqrt {e^2-4 d f}\right )^2\right )-\frac {\left (4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x\right )^2}{c x^2+b x+a}}d\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {c x^2+b x+a}}}{\sqrt {e^2-4 d f}}}{2 \left (e^2-4 d f\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}+\frac {\sqrt {a+b x+c x^2} \left (f x \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+f \left (-a e f-2 b d f+b e^2\right )-c \left (e^3-3 d e f\right )\right )}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {f \left (\left (e-\sqrt {e^2-4 d f}\right ) (c e-b f) (2 a f-b e+2 c d)-2 \left (-4 a^2 f^3+3 a b e f^2-4 a c f \left (e^2-3 d f\right )+b^2 f \left (e^2-6 d f\right )-b c \left (e^3-5 d e f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \text {arctanh}\left (\frac {4 a f+2 x \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right )-b \left (e-\sqrt {e^2-4 d f}\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac {f \left (\left (\sqrt {e^2-4 d f}+e\right ) (c e-b f) (2 a f-b e+2 c d)-2 \left (-4 a^2 f^3+3 a b e f^2-4 a c f \left (e^2-3 d f\right )+b^2 f \left (e^2-6 d f\right )-b c \left (e^3-5 d e f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \text {arctanh}\left (\frac {4 a f+2 x \left (b f-c \left (\sqrt {e^2-4 d f}+e\right )\right )-b \left (\sqrt {e^2-4 d f}+e\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}}{2 \left (e^2-4 d f\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}+\frac {\sqrt {a+b x+c x^2} \left (f x \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+f \left (-a e f-2 b d f+b e^2\right )-c \left (e^3-3 d e f\right )\right )}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
((f*(b*e^2 - 2*b*d*f - a*e*f) - c*(e^3 - 3*d*e*f) + f*(f*(b*e - 2*a*f) - c *(e^2 - 2*d*f))*x)*Sqrt[a + b*x + c*x^2])/((e^2 - 4*d*f)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(d + e*x + f*x^2)) + ((f*((2*c*d - b*e + 2*a*f)*( c*e - b*f)*(e - Sqrt[e^2 - 4*d*f]) - 2*(3*a*b*e*f^2 - 4*a^2*f^3 + b^2*f*(e ^2 - 6*d*f) + 2*c^2*d*(e^2 - 4*d*f) - 4*a*c*f*(e^2 - 3*d*f) - b*c*(e^3 - 5 *d*e*f)))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqr t[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c *e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d *f]]) - (f*((2*c*d - b*e + 2*a*f)*(c*e - b*f)*(e + Sqrt[e^2 - 4*d*f]) - 2* (3*a*b*e*f^2 - 4*a^2*f^3 + b^2*f*(e^2 - 6*d*f) + 2*c^2*d*(e^2 - 4*d*f) - 4 *a*c*f*(e^2 - 3*d*f) - b*c*(e^3 - 5*d*e*f)))*ArcTanh[(4*a*f - b*(e + Sqrt[ e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e ^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b *x + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2 *a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]))/(2*(e^2 - 4*d*f)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f)))
3.2.13.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x _)^2)^(q_), x_Symbol] :> Simp[(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a *f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p + 1)*(( d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*( c*e - b*f))*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Si mp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f *(p + 1) - c*e*(2*p + q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))* (2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b ^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q , 0]
Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim p[(2*c*g - h*(b - q))/q Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] , x] - Simp[(2*c*g - h*(b + q))/q Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f *x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0 ] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]
Leaf count of result is larger than twice the leaf count of optimal. \(2107\) vs. \(2(735)=1470\).
Time = 1.13 (sec) , antiderivative size = 2108, normalized size of antiderivative = 2.67
-1/(4*d*f-e^2)*(-2/(-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2 -b*e*f-2*c*d*f+c*e^2)*f^2/(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)*((x+1/2*(e+(-4* d*f+e^2)^(1/2))/f)^2*c+1/f*(-c*(-4*d*f+e^2)^(1/2)+b*f-c*e)*(x+1/2*(e+(-4*d *f+e^2)^(1/2))/f)+1/2*(-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a* f^2-b*e*f-2*c*d*f+c*e^2)/f^2)^(1/2)+f*(-c*(-4*d*f+e^2)^(1/2)+b*f-c*e)/(-b* f*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)*2 ^(1/2)/((-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c* d*f+c*e^2)/f^2)^(1/2)*ln(((-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+ 2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2+1/f*(-c*(-4*d*f+e^2)^(1/2)+b*f-c*e)*(x+1/ 2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*2^(1/2)*((-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+ e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2)^(1/2)*(4*(x+1/2*(e+(-4*d* f+e^2)^(1/2))/f)^2*c+4/f*(-c*(-4*d*f+e^2)^(1/2)+b*f-c*e)*(x+1/2*(e+(-4*d*f +e^2)^(1/2))/f)+2*(-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2- b*e*f-2*c*d*f+c*e^2)/f^2)^(1/2))/(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)))-1/(4*d *f-e^2)*(-2/(b*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2 *c*d*f+c*e^2)*f^2/(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))*((x-1/2/f*(-e+(-4*d*f+ e^2)^(1/2)))^2*c+(c*(-4*d*f+e^2)^(1/2)+b*f-c*e)/f*(x-1/2/f*(-e+(-4*d*f+e^2 )^(1/2)))+1/2*(b*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f -2*c*d*f+c*e^2)/f^2)^(1/2)+(c*(-4*d*f+e^2)^(1/2)+b*f-c*e)*f/(b*f*(-4*d*f+e ^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)*2^(1/2)/(...
Timed out. \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )^2} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} {\left (f x^{2} + e x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )^2} \, dx=\int \frac {1}{\sqrt {c\,x^2+b\,x+a}\,{\left (f\,x^2+e\,x+d\right )}^2} \,d x \]