Integrand size = 22, antiderivative size = 428 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx=-\frac {\sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}+\frac {\sqrt {c} \left (8 c^2 d^2-b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d-\sqrt {b^2-4 a c} d-6 a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}-\frac {\sqrt {c} \left (8 c^2 d^2-b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d+\sqrt {b^2-4 a c} d-6 a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )} \]
-(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)+1/2*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)*(8*c^2*d^2-b*e^2*(b+(-4*a*c+b^2 )^(1/2))-2*c*e*(4*b*d-6*a*e-d*(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)/(a*e ^2-b*d*e+c*d^2)*2^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)-1/2*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)*(8*c^2*d^2-b*e^2*(b-(-4*a*c+b^2)^(1/2))-2*c*e*(4*b*d-6*a*e+d*(-4*a*c+ b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)/(a*e^2-b*d*e+c*d^2)*2^(1/2)/(2*c*d-e*(b+(- 4*a*c+b^2)^(1/2)))^(1/2)
Result contains complex when optimal does not.
Time = 2.91 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {d+e x} \left (-b^2 e+2 c (a e+c d x)+b c (d-e x)\right )}{a+x (b+c x)}+\frac {\sqrt {2} \sqrt {c} \left (-8 i c^2 d^2-b \left (-i b+\sqrt {-b^2+4 a c}\right ) e^2+2 c e \left (4 i b d+\sqrt {-b^2+4 a c} d-6 i a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} \left (8 i c^2 d^2-b \left (i b+\sqrt {-b^2+4 a c}\right ) e^2+2 c e \left (-4 i b d+\sqrt {-b^2+4 a c} d+6 i a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{2 \left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right )} \]
((2*Sqrt[d + e*x]*(-(b^2*e) + 2*c*(a*e + c*d*x) + b*c*(d - e*x)))/(a + x*( b + c*x)) + (Sqrt[2]*Sqrt[c]*((-8*I)*c^2*d^2 - b*((-I)*b + Sqrt[-b^2 + 4*a *c])*e^2 + 2*c*e*((4*I)*b*d + Sqrt[-b^2 + 4*a*c]*d - (6*I)*a*e))*ArcTan[(S qrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]] )/(Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + (b - I*Sqrt[-b^2 + 4*a*c])*e]) + (Sqrt [2]*Sqrt[c]*((8*I)*c^2*d^2 - b*(I*b + Sqrt[-b^2 + 4*a*c])*e^2 + 2*c*e*((-4 *I)*b*d + Sqrt[-b^2 + 4*a*c]*d + (6*I)*a*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[ d + e*x])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt[-b^2 + 4*a*c ]*Sqrt[-2*c*d + (b + I*Sqrt[-b^2 + 4*a*c])*e]))/(2*(b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e)))
Time = 0.81 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1165, 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.
| \(\displaystyle \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle -\frac {\int \frac {4 c^2 d^2-b^2 e^2-3 c e (b d-2 a e)+c e (2 c d-b e) x}{2 \sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {4 c^2 d^2-b^2 e^2-3 c e (b d-2 a e)+c e (2 c d-b e) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {\int \frac {e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)+c (2 c d-b e) (d+e x)\right )}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \int \frac {2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)+c (2 c d-b e) (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {e \left (\frac {c \left (-2 c e \left (-d \sqrt {b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (\sqrt {b^2-4 a c}+b\right )+8 c^2 d^2\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}}{2 e \sqrt {b^2-4 a c}}-\frac {c \left (-2 c e \left (d \sqrt {b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (b-\sqrt {b^2-4 a c}\right )+8 c^2 d^2\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}}{2 e \sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {c} \left (-2 c e \left (d \sqrt {b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (b-\sqrt {b^2-4 a c}\right )+8 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} e \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {c} \left (-2 c e \left (-d \sqrt {b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (\sqrt {b^2-4 a c}+b\right )+8 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} e \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
-((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))) - (e*(-((Sqrt[c]*(8*c^2*d ^2 - b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(4*b*d - Sqrt[b^2 - 4*a*c]*d - 6*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]*(8*c^2*d^2 - b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e *(4*b*d + Sqrt[b^2 - 4*a*c]*d - 6*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])))/((b^2 - 4*a*c)*(c*d^2 - b*d* e + a*e^2))
3.23.99.3.1 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_)^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 = 0.62 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.24
| method | result | size |
| derivativedivides | \(32 e^{3} c^{2} \left (-\frac {\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{4 c \left (-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \left (-e x -\frac {b e}{2 c}+\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )}+\frac {\left (-2 b e +4 c d +3 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 \left (-2 b e +4 c d +2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{2} \left (4 a c -b^{2}\right )}-\frac {\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{4 c \left (-b e +2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \left (-e x -\frac {b e}{2 c}-\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )}+\frac {\left (2 b e -4 c d +3 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 \left (2 b e -4 c d +2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{2} \left (4 a c -b^{2}\right )}\right )\) | \(532\) |
| default | \(32 e^{3} c^{2} \left (-\frac {\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{4 c \left (-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \left (-e x -\frac {b e}{2 c}+\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )}+\frac {\left (-2 b e +4 c d +3 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 \left (-2 b e +4 c d +2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{2} \left (4 a c -b^{2}\right )}-\frac {\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{4 c \left (-b e +2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \left (-e x -\frac {b e}{2 c}-\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )}+\frac {\left (2 b e -4 c d +3 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 \left (2 b e -4 c d +2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{2} \left (4 a c -b^{2}\right )}\right )\) | \(532\) |
| pseudoelliptic | \(\frac {8 e^{3} c^{2} \left (-\frac {\sqrt {e x +d}}{2 \left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \left (2 c e x +b e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) e^{2}}-\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right ) b}{2 \left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}\, e \sqrt {-e^{2} \left (4 a c -b^{2}\right )}}+\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right ) c d}{\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}\, \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{2}}-\frac {3 \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}\, e^{2}}-\frac {\frac {\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {e x +d}}{-2 c e x -b e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}}+\frac {\left (-2 b e +4 c d +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{2} \left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right )}\right )}{4 a c -b^{2}}\) | \(654\) |
32*e^3*c^2*(-1/4/(-e^2*(4*a*c-b^2))^(1/2)/e^2/(4*a*c-b^2)*(1/4/c/(-b*e+2*c *d+(-4*a*c*e^2+b^2*e^2)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2)/(- e*x-1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))+1/2*(-2*b*e+4*c*d+3*(-4*a*c* e^2+b^2*e^2)^(1/2))/(-2*b*e+4*c*d+2*(-4*a*c*e^2+b^2*e^2)^(1/2))*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)))-1/4/(-e^2*(4*a*c-b^2 ))^(1/2)/e^2/(4*a*c-b^2)*(1/4/c/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*(- 4*a*c*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2)/(-e*x-1/2*b*e/c-1/2/c*(e^2*(-4*a*c+ b^2))^(1/2))+1/2*(2*b*e-4*c*d+3*(-4*a*c*e^2+b^2*e^2)^(1/2))/(2*b*e-4*c*d+2 *(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2)) *c)^(1/2)*arctan(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))))
Leaf count of result is larger than twice the leaf count of optimal. 20436 vs. \(2 (379) = 758\).
Time = 4.03 (sec) , antiderivative size = 20436, normalized size of antiderivative = 47.75 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{2} \sqrt {e x + d}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 2866 vs. \(2 (379) = 758\).
Time = 0.60 (sec) , antiderivative size = 2866, normalized size of antiderivative = 6.70 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
-(2*(e*x + d)^(3/2)*c^2*d*e - 2*sqrt(e*x + d)*c^2*d^2*e - (e*x + d)^(3/2)* b*c*e^2 + 2*sqrt(e*x + d)*b*c*d*e^2 - sqrt(e*x + d)*b^2*e^3 + 2*sqrt(e*x + d)*a*c*e^3)/((b^2*c*d^2 - 4*a*c^2*d^2 - b^3*d*e + 4*a*b*c*d*e + a*b^2*e^2 - 4*a^2*c*e^2)*((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d*e + a*e^2)) - 1/8*((b^2*c*d^2*e - 4*a*c^2*d^2*e - b^3*d*e^2 + 4*a*b*c *d*e^2 + a*b^2*e^3 - 4*a^2*c*e^3)^2*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4* a*c)*c)*e)*(2*c*d*e - b*e^2) + 2*(2*sqrt(b^2 - 4*a*c)*c^3*d^4*e - 4*sqrt(b ^2 - 4*a*c)*b*c^2*d^3*e^2 + (b^2*c + 8*a*c^2)*sqrt(b^2 - 4*a*c)*d^2*e^3 + (b^3 - 8*a*b*c)*sqrt(b^2 - 4*a*c)*d*e^4 - (a*b^2 - 6*a^2*c)*sqrt(b^2 - 4*a *c)*e^5)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(b^2*c*d^2*e - 4*a*c^2*d^2*e - b^3*d*e^2 + 4*a*b*c*d*e^2 + a*b^2*e^3 - 4*a^2*c*e^3) - ( 16*(b^2*c^5 - 4*a*c^6)*d^7*e - 56*(b^3*c^4 - 4*a*b*c^5)*d^6*e^2 + 14*(5*b^ 4*c^3 - 16*a*b^2*c^4 - 16*a^2*c^5)*d^5*e^3 - 35*(b^5*c^2 - 16*a^2*b*c^4)*d ^4*e^4 + 4*(b^6*c + 23*a*b^4*c^2 - 92*a^2*b^2*c^3 - 64*a^3*c^4)*d^3*e^5 + (b^7 - 26*a*b^5*c - 8*a^2*b^3*c^2 + 384*a^3*b*c^3)*d^2*e^6 - 2*(a*b^6 - 19 *a^2*b^4*c + 48*a^3*b^2*c^2 + 48*a^4*c^3)*d*e^7 + (a^2*b^5 - 16*a^3*b^3*c + 48*a^4*b*c^2)*e^8)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arc tan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*b^2*c^2*d^3 - 8*a*c^3*d^3 - 3*b^3*c *d^2*e + 12*a*b*c^2*d^2*e + b^4*d*e^2 - 2*a*b^2*c*d*e^2 - 8*a^2*c^2*d*e^2 - a*b^3*e^3 + 4*a^2*b*c*e^3 + sqrt((2*b^2*c^2*d^3 - 8*a*c^3*d^3 - 3*b^3...
Time = 16.23 (sec) , antiderivative size = 45676, normalized size of antiderivative = 106.72 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
- (((d + e*x)^(1/2)*(b^2*e^3 + 2*c^2*d^2*e - 2*a*c*e^3 - 2*b*c*d*e^2))/((4 *a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)) + (c*(b*e^2 - 2*c*d*e)*(d + e*x)^(3/2 ))/((4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)))/((b*e - 2*c*d)*(d + e*x) + c*( d + e*x)^2 + a*e^2 + c*d^2 - b*d*e) - atan(((((1536*a^5*c^6*e^7 + 4*a*b^8* c^2*e^7 - 4*b^9*c^2*d*e^6 - 72*a^2*b^6*c^3*e^7 + 480*a^3*b^4*c^4*e^7 - 140 8*a^4*b^2*c^5*e^7 + 512*a^3*c^8*d^4*e^3 + 2048*a^4*c^7*d^2*e^5 - 8*b^6*c^5 *d^4*e^3 + 16*b^7*c^4*d^3*e^4 - 4*b^8*c^3*d^2*e^5 - 384*a^2*b^2*c^7*d^4*e^ 3 + 768*a^2*b^3*c^6*d^3*e^4 + 192*a^2*b^4*c^5*d^2*e^5 - 1280*a^3*b^2*c^6*d ^2*e^5 + 80*a*b^7*c^3*d*e^6 - 2048*a^4*b*c^6*d*e^6 + 96*a*b^4*c^6*d^4*e^3 - 192*a*b^5*c^5*d^3*e^4 + 16*a*b^6*c^4*d^2*e^5 - 576*a^2*b^5*c^4*d*e^6 - 1 024*a^3*b*c^7*d^3*e^4 + 1792*a^3*b^3*c^5*d*e^6)/(a^2*b^6*e^4 - 64*a^3*c^5* d^4 - 64*a^5*c^3*e^4 + b^6*c^2*d^4 + b^8*d^2*e^2 - 12*a*b^4*c^3*d^4 - 12*a ^3*b^4*c*e^4 + 48*a^2*b^2*c^4*d^4 + 48*a^4*b^2*c^2*e^4 - 128*a^4*c^4*d^2*e ^2 - 2*a*b^7*d*e^3 - 2*b^7*c*d^3*e + 24*a^2*b^4*c^2*d^2*e^2 + 32*a^3*b^2*c ^3*d^2*e^2 + 24*a*b^5*c^2*d^3*e - 10*a*b^6*c*d^2*e^2 + 24*a^2*b^5*c*d*e^3 + 128*a^3*b*c^4*d^3*e + 128*a^4*b*c^3*d*e^3 - 96*a^2*b^3*c^3*d^3*e - 96*a^ 3*b^3*c^2*d*e^3) - (2*(d + e*x)^(1/2)*(-(32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840* a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d ^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 ...