Integrand size = 28, antiderivative size = 364 \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx=-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}+\frac {\sqrt {c} e \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}-\frac {\sqrt {c} e \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )} \]
-((-4*a*c+b^2)*(-b*e+c*d)-c*(-4*a*c+b^2)*e*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*e*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)*(2*c*d-e*(b+(-4*a*c+b^2 )^(1/2)))/(a*e^2-b*d*e+c*d^2)*2^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b-(-4*a *c+b^2)^(1/2)))^(1/2)-1/2*e*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)*(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))/( a*e^2-b*d*e+c*d^2)*2^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/ 2)))^(1/2)
Time = 1.29 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.81 \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {d+e x} (-c d+b e+c e x)}{a+x (b+c x)}+\frac {\sqrt {2} \sqrt {c} e \left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} e \left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}}{2 \left (c d^2+e (-b d+a e)\right )} \]
((2*Sqrt[d + e*x]*(-(c*d) + b*e + c*e*x))/(a + x*(b + c*x)) + (Sqrt[2]*Sqr t[c]*e*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqr t[-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*Sqrt[c]*e*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b + Sqrt[ b^2 - 4*a*c])*e]))/(2*(c*d^2 + e*(-(b*d) + a*e)))
Time = 0.65 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {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.
| \(\displaystyle \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {\int \frac {\left (b^2-4 a c\right ) e (c d-b e-c e x)}{2 \sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \int \frac {c d-b e-c e x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {e \int \frac {e (2 c d-b e-c (d+e x))}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{a e^2-b d e+c d^2}-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e^2 \int \frac {2 c d-b e-c (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{a e^2-b d e+c d^2}-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {e^2 \left (\frac {c \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}}{2 e \sqrt {b^2-4 a c}}-\frac {1}{2} c \left (\frac {2 c d-b e}{e \sqrt {b^2-4 a c}}+1\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}\right )}{a e^2-b d e+c d^2}-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {e^2 \left (\frac {\sqrt {c} \left (\frac {2 c d-b e}{e \sqrt {b^2-4 a c}}+1\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {c} \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} e \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{a e^2-b d e+c d^2}-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
-((Sqrt[d + e*x]*((b^2 - 4*a*c)*(c*d - b*e) - c*(b^2 - 4*a*c)*e*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2))) - (e^2*(-((Sqrt[c]*(2 *c*d - (b + 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[b^2 - 4*a*c]*e*Sqr t[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e])) + (Sqrt[c]*(1 + (2*c*d - b*e)/(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] )))/(c*d^2 - b*d*e + a*e^2)
3.17.21.3.1 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[((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 = 0.50 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.96
| method | result | size |
| pseudoelliptic | \(\frac {4 c^{2} \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \left (\frac {\sqrt {e x +d}}{\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \left (2 x c e +b e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right )}+\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 \left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\frac {\sqrt {e x +d}}{-2 x c e -b e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}}\right )}{4 a c -b^{2}}\) | \(348\) |
| derivativedivides | \(32 e^{2} c^{2} \left (\frac {\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \left (\frac {\sqrt {e x +d}}{2 \left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \left (2 c \left (e x +d \right )+b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right )}+\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 e^{2} \left (4 a c -b^{2}\right )}-\frac {\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \left (\frac {\sqrt {e x +d}}{2 \left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \left (-2 c \left (e x +d \right )-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right )}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 e^{2} \left (4 a c -b^{2}\right )}\right )\) | \(433\) |
| default | \(32 e^{2} c^{2} \left (\frac {\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \left (\frac {\sqrt {e x +d}}{2 \left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \left (2 c \left (e x +d \right )+b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right )}+\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 e^{2} \left (4 a c -b^{2}\right )}-\frac {\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \left (\frac {\sqrt {e x +d}}{2 \left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \left (-2 c \left (e x +d \right )-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right )}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 e^{2} \left (4 a c -b^{2}\right )}\right )\) | \(433\) |
4*c^2*(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)*((e*x+d)^(1/2)/(b*e-2*c*d+(-e^2 *(4*a*c-b^2))^(1/2))/(2*x*c*e+b*e+(-e^2*(4*a*c-b^2))^(1/2))+1/2/(b*e-2*c*d +(-e^2*(4*a*c-b^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))-1/(-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*((e*x+d)^(1/2)/(-2*x* c*e-b*e+(-e^2*(4*a*c-b^2))^(1/2))+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))))
Leaf count of result is larger than twice the leaf count of optimal. 11482 vs. \(2 (318) = 636\).
Time = 0.73 (sec) , antiderivative size = 11482, normalized size of antiderivative = 31.54 \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx=\int { \frac {2 \, c x + b}{{\left (c x^{2} + b x + a\right )}^{2} \sqrt {e x + d}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1574 vs. \(2 (318) = 636\).
Time = 0.55 (sec) , antiderivative size = 1574, normalized size of antiderivative = 4.32 \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
((e*x + d)^(3/2)*c*e^2 - 2*sqrt(e*x + d)*c*d*e^2 + sqrt(e*x + d)*b*e^3)/(( (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)*( c*d^2 - b*d*e + a*e^2)) + 1/8*((c*d^2*e - b*d*e^2 + a*e^3)^2*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(b^2 - 4*a*c)*e^2 - 2*(2*sqrt(b^2 - 4* a*c)*c^2*d^3*e^2 - 3*sqrt(b^2 - 4*a*c)*b*c*d^2*e^3 - sqrt(b^2 - 4*a*c)*a*b *e^5 + (b^2 + 2*a*c)*sqrt(b^2 - 4*a*c)*d*e^4)*sqrt(-4*c^2*d + 2*(b*c - sqr t(b^2 - 4*a*c)*c)*e)*abs(c*d^2*e - b*d*e^2 + a*e^3) + (4*c^4*d^6*e^2 - 12* b*c^3*d^5*e^3 + a^2*b^2*e^8 + (13*b^2*c^2 + 8*a*c^3)*d^4*e^4 - 2*(3*b^3*c + 8*a*b*c^2)*d^3*e^5 + (b^4 + 10*a*b^2*c + 4*a^2*c^2)*d^2*e^6 - 2*(a*b^3 + 2*a^2*b*c)*d*e^7)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arcta n(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 + 2 *a*c*d*e^2 - a*b*e^3 + sqrt((2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 + 2*a*c*d *e^2 - a*b*e^3)^2 - 4*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*(c^2*d^2 - b*c*d*e + a*c*e^2)))/(c^2*d^2 - b*c*d *e + a*c*e^2)))/((sqrt(b^2 - 4*a*c)*c^3*d^6 - 3*sqrt(b^2 - 4*a*c)*b*c^2*d^ 5*e - 3*sqrt(b^2 - 4*a*c)*a^2*b*d*e^5 + sqrt(b^2 - 4*a*c)*a^3*e^6 + 3*(b^2 *c + a*c^2)*sqrt(b^2 - 4*a*c)*d^4*e^2 - (b^3 + 6*a*b*c)*sqrt(b^2 - 4*a*c)* d^3*e^3 + 3*(a*b^2 + a^2*c)*sqrt(b^2 - 4*a*c)*d^2*e^4)*abs(c*d^2*e - b*d*e ^2 + a*e^3)*abs(c)) - 1/8*((c*d^2*e - b*d*e^2 + a*e^3)^2*sqrt(-4*c^2*d + 2 *(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(b^2 - 4*a*c)*e^2 + 2*(2*sqrt(b^2 - 4*a...
Time = 14.06 (sec) , antiderivative size = 18615, normalized size of antiderivative = 51.14 \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
((c*e^2*(d + e*x)^(3/2))/(a*e^2 + c*d^2 - b*d*e) + (e^2*(b*e - 2*c*d)*(d + e*x)^(1/2))/(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(((((4*a*b^3*c^2*e^7 - 16*a^2*b*c^3*e^ 7 + 32*a*c^5*d^3*e^4 + 32*a^2*c^4*d*e^6 - 4*b^4*c^2*d*e^6 - 8*b^2*c^4*d^3* e^4 + 12*b^3*c^3*d^2*e^5 - 48*a*b*c^4*d^2*e^5 + 8*a*b^2*c^3*d*e^6)/(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) - ( 2*(d + e*x)^(1/2)*(-(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b *c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3* c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a*c *e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2 )^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4)/(8*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^ 6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c ^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^ 3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48 *a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b* c^3*d^3*e^3)))^(1/2)*(32*a*c^6*d^5*e^2 - 16*a^3*b*c^3*e^7 + 32*a^3*c^4*d*e ^6 + 4*a^2*b^3*c^2*e^7 + 64*a^2*c^5*d^3*e^4 - 8*b^2*c^5*d^5*e^2 + 20*b^3*c ^4*d^4*e^3 - 16*b^4*c^3*d^3*e^4 + 4*b^5*c^2*d^2*e^5 - 80*a*b*c^5*d^4*e^...