Integrand size = 22, antiderivative size = 835 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (12 c^2 d^2-3 b^2 e^2-7 c e (b d-2 a e)\right )-3 c (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \sqrt {c} \left (32 c^4 d^4+b^3 \left (b+\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (8 b d-\sqrt {b^2-4 a c} d-9 a e\right )+2 b c e^3 \left (b^2 d+b \sqrt {b^2-4 a c} d-5 a b e-4 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (15 b^2 d^2-6 b d \left (\sqrt {b^2-4 a c} d+6 a e\right )+4 a e \left (2 \sqrt {b^2-4 a c} d+7 a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )^2}+\frac {3 \sqrt {c} \left (32 c^4 d^4+b^3 \left (b-\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (8 b d+\sqrt {b^2-4 a c} d-9 a e\right )+2 c^2 e^2 \left (15 b^2 d^2-4 a e \left (2 \sqrt {b^2-4 a c} d-7 a e\right )+6 b d \left (\sqrt {b^2-4 a c} d-6 a e\right )\right )+2 b c e^3 \left (b^2 d+4 a \sqrt {b^2-4 a c} e-b \left (\sqrt {b^2-4 a c} d+5 a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-e (b d-a e)\right )^2} \]
-1/2*(b*c*d-b^2*e+2*a*c*e+c*(-b*e+2*c*d)*x)*(e*x+d)^(1/2)/(-4*a*c+b^2)/(a* e^2-b*d*e+c*d^2)/(c*x^2+b*x+a)^2-1/4*(5*a*c*e*(-b*e+2*c*d)^2-(2*a*c*e-b^2* e+b*c*d)*(12*c^2*d^2-3*b^2*e^2-7*c*e*(-2*a*e+b*d))-3*c*(-b*e+2*c*d)*(4*c^2 *d^2-b^2*e^2-4*c*e*(-2*a*e+b*d))*x)*(e*x+d)^(1/2)/(-4*a*c+b^2)^2/(a*e^2-b* d*e+c*d^2)^2/(c*x^2+b*x+a)-3/8*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c* d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2))*c^(1/2)*(32*c^4*d^4+b^3*e^4*(b+(-4*a*c+ b^2)^(1/2))-8*c^3*d^2*e*(8*b*d-9*a*e-d*(-4*a*c+b^2)^(1/2))+2*b*c*e^3*(b^2* d-5*a*b*e+b*d*(-4*a*c+b^2)^(1/2)-4*a*e*(-4*a*c+b^2)^(1/2))+2*c^2*e^2*(15*b ^2*d^2-6*b*d*(6*a*e+d*(-4*a*c+b^2)^(1/2))+4*a*e*(7*a*e+2*d*(-4*a*c+b^2)^(1 /2))))/(-4*a*c+b^2)^(5/2)/(a*e^2-b*d*e+c*d^2)^2*2^(1/2)/(2*c*d-e*(b-(-4*a* c+b^2)^(1/2)))^(1/2)+3/8*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b +(-4*a*c+b^2)^(1/2)))^(1/2))*c^(1/2)*(32*c^4*d^4+b^3*e^4*(b-(-4*a*c+b^2)^( 1/2))-8*c^3*d^2*e*(8*b*d-9*a*e+d*(-4*a*c+b^2)^(1/2))+2*b*c*e^3*(b^2*d+4*a* e*(-4*a*c+b^2)^(1/2)-b*(5*a*e+d*(-4*a*c+b^2)^(1/2)))+2*c^2*e^2*(15*b^2*d^2 +6*b*d*(-6*a*e+d*(-4*a*c+b^2)^(1/2))-4*a*e*(-7*a*e+2*d*(-4*a*c+b^2)^(1/2)) ))/(-4*a*c+b^2)^(5/2)/(a*e^2-b*d*e+c*d^2)^2*2^(1/2)/(2*c*d-e*(b+(-4*a*c+b^ 2)^(1/2)))^(1/2)
Time = 13.36 (sec) , antiderivative size = 799, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {\sqrt {d+e x} \left (b^2 e-2 c (a e+c d x)+b c (-d+e x)\right )}{(a+x (b+c x))^2}-\frac {\sqrt {d+e x} \left (3 b^4 e^3+b^3 c e^2 (4 d+3 e x)+4 b c^2 \left (a e^2 (5 d-6 e x)+3 c d^2 (d-3 e x)\right )+b^2 c e \left (-25 a e^2+c d (-19 d+6 e x)\right )+4 c^2 \left (7 a^2 e^3+6 c^2 d^3 x+a c d e (d+12 e x)\right )\right )}{2 \left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (a+x (b+c x))}+\frac {3 \sqrt {c} \left (-\frac {\left (32 c^4 d^4+b^3 \left (b+\sqrt {b^2-4 a c}\right ) e^4+8 c^3 d^2 e \left (-8 b d+\sqrt {b^2-4 a c} d+9 a e\right )+2 b c e^3 \left (b^2 d+b \sqrt {b^2-4 a c} d-5 a b e-4 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (15 b^2 d^2-6 b d \left (\sqrt {b^2-4 a c} d+6 a e\right )+4 a e \left (2 \sqrt {b^2-4 a c} d+7 a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e}}\right )}{\sqrt {2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (32 c^4 d^4+b^3 \left (b-\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (8 b d+\sqrt {b^2-4 a c} d-9 a e\right )+2 b c e^3 \left (b^2 d+4 a \sqrt {b^2-4 a c} e-b \left (\sqrt {b^2-4 a c} d+5 a e\right )\right )+2 c^2 e^2 \left (15 b^2 d^2+6 b d \left (\sqrt {b^2-4 a c} d-6 a e\right )+4 a e \left (-2 \sqrt {b^2-4 a c} d+7 a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \left (c d^2+e (-b d+a e)\right )}}{2 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )} \]
((Sqrt[d + e*x]*(b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x)))/(a + x*(b + c*x))^2 - (Sqrt[d + e*x]*(3*b^4*e^3 + b^3*c*e^2*(4*d + 3*e*x) + 4*b*c^2*(a *e^2*(5*d - 6*e*x) + 3*c*d^2*(d - 3*e*x)) + b^2*c*e*(-25*a*e^2 + c*d*(-19* d + 6*e*x)) + 4*c^2*(7*a^2*e^3 + 6*c^2*d^3*x + a*c*d*e*(d + 12*e*x))))/(2* (b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(a + x*(b + c*x))) + (3*Sqrt[c]*( -(((32*c^4*d^4 + b^3*(b + Sqrt[b^2 - 4*a*c])*e^4 + 8*c^3*d^2*e*(-8*b*d + S qrt[b^2 - 4*a*c]*d + 9*a*e) + 2*b*c*e^3*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 5 *a*b*e - 4*a*Sqrt[b^2 - 4*a*c]*e) + 2*c^2*e^2*(15*b^2*d^2 - 6*b*d*(Sqrt[b^ 2 - 4*a*c]*d + 6*a*e) + 4*a*e*(2*Sqrt[b^2 - 4*a*c]*d + 7*a*e)))*ArcTanh[(S qrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sq rt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + ((32*c^4*d^4 + b^3*(b - Sqrt[b^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(8*b*d + Sqrt[b^2 - 4*a*c]*d - 9*a*e) + 2*b*c *e^3*(b^2*d + 4*a*Sqrt[b^2 - 4*a*c]*e - b*(Sqrt[b^2 - 4*a*c]*d + 5*a*e)) + 2*c^2*e^2*(15*b^2*d^2 + 6*b*d*(Sqrt[b^2 - 4*a*c]*d - 6*a*e) + 4*a*e*(-2*S qrt[b^2 - 4*a*c]*d + 7*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt [2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c]) *e]))/(2*Sqrt[2]*(b^2 - 4*a*c)^(3/2)*(c*d^2 + e*(-(b*d) + a*e))))/(2*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))
Time = 2.33 (sec) , antiderivative size = 807, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1165, 27, 1235, 27, 1197, 27, 1480, 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.
| \(\displaystyle \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle -\frac {\int \frac {12 c^2 d^2-3 b^2 e^2-7 c e (b d-2 a e)+5 c e (2 c d-b e) x}{2 \sqrt {d+e x} \left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {12 c^2 d^2-3 b^2 e^2-7 c e (b d-2 a e)+5 c e (2 c d-b e) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )^2}dx}{4 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {\frac {\sqrt {d+e x} \left (-3 c x (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-7 c e (b d-2 a e)-3 b^2 e^2+12 c^2 d^2\right )+5 a c e (2 c d-b e)^2\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {3 \left (16 c^4 d^4-4 c^3 e (7 b d-9 a e) d^2+b^4 e^4+b^2 c e^3 (2 b d-9 a e)+c^2 e^2 \left (9 b^2 d^2-28 a b e d+28 a^2 e^2\right )+c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{2 \sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{4 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\sqrt {d+e x} \left (-3 c x (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-7 c e (b d-2 a e)-3 b^2 e^2+12 c^2 d^2\right )+5 a c e (2 c d-b e)^2\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {3 \int \frac {16 c^4 d^4-4 c^3 e (7 b d-9 a e) d^2+b^4 e^4+b^2 c e^3 (2 b d-9 a e)+c^2 e^2 \left (9 b^2 d^2-28 a b e d+28 a^2 e^2\right )+c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{4 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {\frac {\sqrt {d+e x} \left (-3 c x (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-7 c e (b d-2 a e)-3 b^2 e^2+12 c^2 d^2\right )+5 a c e (2 c d-b e)^2\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {3 \int \frac {e \left (8 c^4 d^4-4 c^3 e (4 b d-5 a e) d^2+b^4 e^4+b^2 c e^3 (b d-9 a e)+c^2 e^2 \left (7 b^2 d^2-20 a b e d+28 a^2 e^2\right )+c (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) (d+e x)\right )}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{4 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\sqrt {d+e x} \left (-3 c x (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-7 c e (b d-2 a e)-3 b^2 e^2+12 c^2 d^2\right )+5 a c e (2 c d-b e)^2\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {3 e \int \frac {8 c^4 d^4-4 c^3 e (4 b d-5 a e) d^2+b^4 e^4+b^2 c e^3 (b d-9 a e)+c^2 e^2 \left (7 b^2 d^2-20 a b e d+28 a^2 e^2\right )+c (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{4 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {\frac {\sqrt {d+e x} \left (-3 c x (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-7 c e (b d-2 a e)-3 b^2 e^2+12 c^2 d^2\right )+5 a c e (2 c d-b e)^2\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {3 e \left (\frac {1}{2} c \left ((2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )-\frac {2 c^2 e^2 \left (28 a^2 e^2-36 a b d e+15 b^2 d^2\right )+2 b^2 c e^3 (b d-5 a e)-8 c^3 d^2 e (8 b d-9 a e)+b^4 e^4+32 c^4 d^4}{e \sqrt {b^2-4 a c}}\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}+\frac {c \left (-8 c^3 d^2 e \left (-d \sqrt {b^2-4 a c}-9 a e+8 b d\right )+2 c^2 e^2 \left (-6 b d \left (d \sqrt {b^2-4 a c}+6 a e\right )+4 a e \left (2 d \sqrt {b^2-4 a c}+7 a e\right )+15 b^2 d^2\right )+2 b c e^3 \left (b d \sqrt {b^2-4 a c}-4 a e \sqrt {b^2-4 a c}-5 a b e+b^2 d\right )+b^3 e^4 \left (\sqrt {b^2-4 a c}+b\right )+32 c^4 d^4\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}}{2 e \sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{4 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\sqrt {d+e x} \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac {\frac {\sqrt {d+e x} \left (5 a c e (2 c d-b e)^2-3 c \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (12 c^2 d^2-3 b^2 e^2-7 c e (b d-2 a e)\right )\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac {3 e \left (-\frac {\sqrt {c} \left (32 c^4 d^4-8 c^3 e \left (8 b d-\sqrt {b^2-4 a c} d-9 a e\right ) d^2+b^3 \left (b+\sqrt {b^2-4 a c}\right ) e^4+2 b c e^3 \left (d b^2+\sqrt {b^2-4 a c} d b-5 a e b-4 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (15 b^2 d^2-6 b \left (\sqrt {b^2-4 a c} d+6 a e\right ) d+4 a e \left (2 \sqrt {b^2-4 a c} d+7 a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} e \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {c} \left ((2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )-\frac {32 c^4 d^4-8 c^3 e (8 b d-9 a e) d^2+b^4 e^4+2 b^2 c e^3 (b d-5 a e)+2 c^2 e^2 \left (15 b^2 d^2-36 a b e d+28 a^2 e^2\right )}{\sqrt {b^2-4 a c} e}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{4 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
-1/2*(Sqrt[d + e*x]*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - ((Sqrt[d + e*x]*(5* a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(12*c^2*d^2 - 3*b^2*e^2 - 7*c*e*(b*d - 2*a*e)) - 3*c*(2*c*d - b*e)*(4*c^2*d^2 - b^2*e^2 - 4*c*e*(b *d - 2*a*e))*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)) - (3*e*(-((Sqrt[c]*(32*c^4*d^4 + b^3*(b + Sqrt[b^2 - 4*a*c])*e^4 - 8*c^3* d^2*e*(8*b*d - Sqrt[b^2 - 4*a*c]*d - 9*a*e) + 2*b*c*e^3*(b^2*d + b*Sqrt[b^ 2 - 4*a*c]*d - 5*a*b*e - 4*a*Sqrt[b^2 - 4*a*c]*e) + 2*c^2*e^2*(15*b^2*d^2 - 6*b*d*(Sqrt[b^2 - 4*a*c]*d + 6*a*e) + 4*a*e*(2*Sqrt[b^2 - 4*a*c]*d + 7*a *e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*e*Sqrt[2*c*d - (b - Sqrt[b^2 - 4* a*c])*e])) - (Sqrt[c]*((2*c*d - b*e)*(4*c^2*d^2 - b^2*e^2 - 4*c*e*(b*d - 2 *a*e)) - (32*c^4*d^4 + b^4*e^4 - 8*c^3*d^2*e*(8*b*d - 9*a*e) + 2*b^2*c*e^3 *(b*d - 5*a*e) + 2*c^2*e^2*(15*b^2*d^2 - 36*a*b*d*e + 28*a^2*e^2))/(Sqrt[b ^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])) )/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)))/(4*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))
3.24.6.3.1 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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) *x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 46.46 (sec) , antiderivative size = 1027, normalized size of antiderivative = 1.23
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1027\) |
| default | \(\text {Expression too large to display}\) | \(1027\) |
| pseudoelliptic | \(\text {Expression too large to display}\) | \(3679\) |
128*e^5*c^3*(1/16/(-e^2*(4*a*c-b^2))^(1/2)/e^4/(4*a*c-b^2)^2*((3/16*(-2*b* e+4*c*d+3*(-4*a*c*e^2+b^2*e^2)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)/c/(-2*a*c *e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2-b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)+2*d*(-4*a* c*e^2+b^2*e^2)^(1/2)*c)*(e*x+d)^(3/2)-1/16*(-6*b*e+12*c*d+11*(-4*a*c*e^2+b ^2*e^2)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2) ^(1/2))/c^2*(e*x+d)^(1/2))/(-e*x-1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2)) ^2-3/4*(28*a*c*e^2-11*b^2*e^2+16*b*c*d*e-16*c^2*d^2+10*b*e*(-4*a*c*e^2+b^2 *e^2)^(1/2)-20*d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)/(8*a*c*e^2-4*b^2*e^2+8*b*c* d*e-8*c^2*d^2+4*b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)-8*d*(-4*a*c*e^2+b^2*e^2)^(1 /2)*c)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*( e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)))-1/1 6/(-e^2*(4*a*c-b^2))^(1/2)/e^4/(4*a*c-b^2)^2*((-3/16*(-2*b*e+4*c*d-3*(-4*a *c*e^2+b^2*e^2)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)/c/(-2*a*c*e^2+b^2*e^2-2* b*c*d*e+2*c^2*d^2+b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)-2*d*(-4*a*c*e^2+b^2*e^2)^ (1/2)*c)*(e*x+d)^(3/2)+1/16*(-6*b*e+12*c*d-11*(-4*a*c*e^2+b^2*e^2)^(1/2))* (-4*a*c*e^2+b^2*e^2)^(1/2)/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))/c^2*(e* x+d)^(1/2))/(-e*x-1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2+3/4*(-28*a*c *e^2+11*b^2*e^2-16*b*c*d*e+16*c^2*d^2+10*b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)-20 *d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)/(-8*a*c*e^2+4*b^2*e^2-8*b*c*d*e+8*c^2*d^2 +4*b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)-8*d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)*2^(...
Leaf count of result is larger than twice the leaf count of optimal. 55168 vs. \(2 (759) = 1518\).
Time = 135.89 (sec) , antiderivative size = 55168, normalized size of antiderivative = 66.07 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^3} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{3} \sqrt {e x + d}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 9101 vs. \(2 (759) = 1518\).
Time = 2.43 (sec) , antiderivative size = 9101, normalized size of antiderivative = 10.90 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
3/32*((b^4*c^2*d^4*e - 8*a*b^2*c^3*d^4*e + 16*a^2*c^4*d^4*e - 2*b^5*c*d^3* e^2 + 16*a*b^3*c^2*d^3*e^2 - 32*a^2*b*c^3*d^3*e^2 + b^6*d^2*e^3 - 6*a*b^4* c*d^2*e^3 + 32*a^3*c^3*d^2*e^3 - 2*a*b^5*d*e^4 + 16*a^2*b^3*c*d*e^4 - 32*a ^3*b*c^2*d*e^4 + a^2*b^4*e^5 - 8*a^3*b^2*c*e^5 + 16*a^4*c^2*e^5)^2*(8*c^3* d^3*e - 12*b*c^2*d^2*e^2 + 2*(b^2*c + 8*a*c^2)*d*e^3 + (b^3 - 8*a*b*c)*e^4 )*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e) + 2*(8*(b^2*c^6 - 4*a*c ^7)*sqrt(b^2 - 4*a*c)*d^8*e - 32*(b^3*c^5 - 4*a*b*c^6)*sqrt(b^2 - 4*a*c)*d ^7*e^2 + (47*b^4*c^4 - 152*a*b^2*c^5 - 144*a^2*c^6)*sqrt(b^2 - 4*a*c)*d^6* e^3 - (29*b^5*c^3 - 8*a*b^3*c^4 - 432*a^2*b*c^5)*sqrt(b^2 - 4*a*c)*d^5*e^4 + (6*b^6*c^2 + 73*a*b^4*c^3 - 312*a^2*b^2*c^4 - 304*a^3*c^5)*sqrt(b^2 - 4 *a*c)*d^4*e^5 - (b^7*c + 10*a*b^5*c^2 + 96*a^2*b^3*c^3 - 608*a^3*b*c^4)*sq rt(b^2 - 4*a*c)*d^3*e^6 + (b^8 - 13*a*b^6*c + 93*a^2*b^4*c^2 - 152*a^3*b^2 *c^3 - 304*a^4*c^4)*sqrt(b^2 - 4*a*c)*d^2*e^7 - (2*a*b^7 - 27*a^2*b^5*c + 152*a^3*b^3*c^2 - 304*a^4*b*c^3)*sqrt(b^2 - 4*a*c)*d*e^8 + (a^2*b^6 - 13*a ^3*b^4*c + 64*a^4*b^2*c^2 - 112*a^5*c^3)*sqrt(b^2 - 4*a*c)*e^9)*sqrt(-4*c^ 2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(b^4*c^2*d^4*e - 8*a*b^2*c^3*d^4 *e + 16*a^2*c^4*d^4*e - 2*b^5*c*d^3*e^2 + 16*a*b^3*c^2*d^3*e^2 - 32*a^2*b* c^3*d^3*e^2 + b^6*d^2*e^3 - 6*a*b^4*c*d^2*e^3 + 32*a^3*c^3*d^2*e^3 - 2*a*b ^5*d*e^4 + 16*a^2*b^3*c*d*e^4 - 32*a^3*b*c^2*d*e^4 + a^2*b^4*e^5 - 8*a^3*b ^2*c*e^5 + 16*a^4*c^2*e^5) - (64*(b^6*c^9 - 12*a*b^4*c^10 + 48*a^2*b^2*...
Time = 26.91 (sec) , antiderivative size = 132771, normalized size of antiderivative = 159.01 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
((e*(d + e*x)^(5/2)*(6*b^4*c*e^4 - 72*c^5*d^4 + 28*a^2*c^3*e^4 - 49*a*b^2* c^2*e^4 - 140*a*c^4*d^2*e^2 + b^3*c^2*d*e^3 - 73*b^2*c^3*d^2*e^2 + 144*b*c ^4*d^3*e + 140*a*b*c^3*d*e^3))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)*(a*e^2 + c*d^2 - b*d*e)^2) - ((d + e*x)^(1/2)*(24*c^4*d^4*e - 5*b^4*e^5 - 44*a^2*c^ 2*e^5 + 60*a*c^3*d^2*e^3 - 48*b*c^3*d^3*e^2 + 21*b^2*c^2*d^2*e^3 + 37*a*b^ 2*c*e^5 + 3*b^3*c*d*e^4 - 60*a*b*c^2*d*e^4))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^ 2*c)*(a*e^2 + c*d^2 - b*d*e)) + (e*(d + e*x)^(3/2)*(3*b^5*e^5 + 72*c^5*d^5 - 4*a^2*b*c^2*e^5 + 176*a*c^4*d^3*e^2 + 8*a^2*c^3*d*e^4 + 136*b^2*c^3*d^3 *e^2 - 24*b^3*c^2*d^2*e^3 - 20*a*b^3*c*e^5 - 180*b*c^4*d^4*e - 10*b^4*c*d* e^4 - 264*a*b*c^3*d^2*e^3 + 128*a*b^2*c^2*d*e^4))/(4*(b^4 + 16*a^2*c^2 - 8 *a*b^2*c)*(a*e^2 + c*d^2 - b*d*e)^2) + (3*c*e*(d + e*x)^(7/2)*(8*c^4*d^3 + b^3*c*e^3 + 2*b^2*c^2*d*e^2 - 8*a*b*c^2*e^3 + 16*a*c^3*d*e^2 - 12*b*c^3*d ^2*e))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)*(a*e^2 + c*d^2 - b*d*e)^2))/(c^2* (d + e*x)^4 - (d + e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 2*a*b*e^3 + 4*a*c*d*e^2 - 6*b*c*d^2*e) - (4*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c^2*d^2 + 2*a*c*e^2 - 6*b*c*d*e) + a^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2* a*b*d*e^3 - 2*b*c*d^3*e + 2*a*c*d^2*e^2) + atan(((((3*(1835008*a^9*c^9*e^1 1 - 64*a^2*b^14*c^2*e^11 + 1856*a^3*b^12*c^3*e^11 - 23552*a^4*b^10*c^4*e^1 1 + 168960*a^5*b^8*c^5*e^11 - 737280*a^6*b^6*c^6*e^11 + 1949696*a^7*b^4*c^ 7*e^11 - 2883584*a^8*b^2*c^8*e^11 + 524288*a^5*c^13*d^8*e^3 + 2359296*a...