Integrand size = 22, antiderivative size = 363 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (8 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d-\sqrt {b^2-4 a c} d-2 a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (8 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d+\sqrt {b^2-4 a c} d-2 a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]
-(b*d-2*a*e+(-b*e+2*c*d)*x)*(e*x+d)^(1/2)/(-4*a*c+b^2)/(c*x^2+b*x+a)+1/2*a rctanh(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 ))*(8*c^2*d^2+b*e^2*(b-(-4*a*c+b^2)^(1/2))-2*c*e*(4*b*d-2*a*e-d*(-4*a*c+b^ 2)^(1/2)))/(-4*a*c+b^2)^(3/2)*2^(1/2)/c^(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))*(8*c^2*d^2+b*e^2*(b+(-4*a*c+b^2)^(1/2))-2*c*e*(4*b*d-2* a*e+d*(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)*2^(1/2)/c^(1/2)/(2*c*d-e*(b+ (-4*a*c+b^2)^(1/2)))^(1/2)
Result contains complex when optimal does not.
Time = 3.39 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.13 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {d+e x} (-b d+2 a e-2 c d x+b e x)}{a+x (b+c x)}+\frac {\sqrt {2} \left (8 i c^2 d^2+b \left (i b+\sqrt {-b^2+4 a c}\right ) e^2+2 i c e \left (-4 b d+i \sqrt {-b^2+4 a c} d+2 a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {c} \sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\sqrt {2} \left (-8 i c^2 d^2+b \left (-i b+\sqrt {-b^2+4 a c}\right ) e^2-2 c e \left (-4 i b d+\sqrt {-b^2+4 a c} d+2 i a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {c} \sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{2 \left (b^2-4 a c\right )} \]
((2*Sqrt[d + e*x]*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x))/(a + x*(b + c*x)) + (Sqrt[2]*((8*I)*c^2*d^2 + b*(I*b + Sqrt[-b^2 + 4*a*c])*e^2 + (2*I)*c*e*(-4 *b*d + I*Sqrt[-b^2 + 4*a*c]*d + 2*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[c]*Sqrt[-b^2 + 4* a*c]*Sqrt[-2*c*d + (b - I*Sqrt[-b^2 + 4*a*c])*e]) + (Sqrt[2]*((-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 + (2*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[c]*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))
Time = 0.67 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1164, 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.
| \(\displaystyle \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1164 |
| \(\displaystyle -\frac {\int \frac {4 c d^2-3 b e d+2 a e^2+e (2 c d-b e) x}{2 \sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{b^2-4 a c}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {4 c d^2-e (3 b d-2 a e)+e (2 c d-b e) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{2 \left (b^2-4 a c\right )}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {\int \frac {e \left (2 \left (c d^2-b e d+a e^2\right )+(2 c d-b e) (d+e x)\right )}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{b^2-4 a c}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \int \frac {2 \left (c d^2-b e d+a e^2\right )+(2 c d-b e) (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{b^2-4 a c}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {e \left (\frac {1}{2} \left (\frac {-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2}{e \sqrt {b^2-4 a c}}-b e+2 c d\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}+\frac {1}{2} \left (-\frac {-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2}{e \sqrt {b^2-4 a c}}-b e+2 c d\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}\right )}{b^2-4 a c}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {e \left (-\frac {\left (\frac {-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2}{e \sqrt {b^2-4 a c}}-b e+2 c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (-\frac {-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2}{e \sqrt {b^2-4 a c}}-b e+2 c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{b^2-4 a c}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
-((Sqrt[d + e*x]*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2))) - (e*(-(((2*c*d - b*e + (8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a *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[c]*Sqrt[2*c*d - (b - Sqr t[b^2 - 4*a*c])*e])) - ((2*c*d - b*e - (8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sq rt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])))/(b^2 - 4*a*c)
3.23.97.3.1 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)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* c)) Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int QuadraticQ[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 = 0.54 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.25
| method | result | size |
| pseudoelliptic | \(-\frac {\sqrt {2}\, \left (c \,x^{2}+b x +a \right ) e \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (\left (-\frac {b e}{4}+\frac {c d}{2}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}+2 c^{2} d^{2}+\left (a \,e^{2}-2 b d e \right ) c +\frac {b^{2} e^{2}}{4}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (\sqrt {2}\, \left (c \,x^{2}+b x +a \right ) \left (\left (\frac {b e}{4}-\frac {c d}{2}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}+2 c^{2} d^{2}+\left (a \,e^{2}-2 b d e \right ) c +\frac {b^{2} e^{2}}{4}\right ) e \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )+\left (-c d x +\left (\frac {b x}{2}+a \right ) e -\frac {b d}{2}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {e x +d}\right )}{2 \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (a c -\frac {b^{2}}{4}\right ) \left (c \,x^{2}+b x +a \right )}\) | \(454\) |
| derivativedivides | \(2 e^{3} \left (\frac {-\frac {\left (b e -2 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{2 e^{2} \left (4 a c -b^{2}\right )}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}}{e^{2} \left (4 a c -b^{2}\right )}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}}+\frac {2 c \left (\frac {\left (-4 a c \,e^{2}-b^{2} e^{2}+8 b c d e -8 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (4 a c \,e^{2}+b^{2} e^{2}-8 b c d e +8 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e^{2} \left (4 a c -b^{2}\right )}\right )\) | \(481\) |
| default | \(2 e^{3} \left (\frac {-\frac {\left (b e -2 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{2 e^{2} \left (4 a c -b^{2}\right )}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}}{e^{2} \left (4 a c -b^{2}\right )}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}}+\frac {2 c \left (\frac {\left (-4 a c \,e^{2}-b^{2} e^{2}+8 b c d e -8 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (4 a c \,e^{2}+b^{2} e^{2}-8 b c d e +8 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e^{2} \left (4 a c -b^{2}\right )}\right )\) | \(481\) |
-1/2/(-4*(a*c-1/4*b^2)*e^2)^(1/2)/((-b*e+2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2 ))*c)^(1/2)*(2^(1/2)*(c*x^2+b*x+a)*e*((b*e-2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1 /2))*c)^(1/2)*((-1/4*b*e+1/2*c*d)*(-4*(a*c-1/4*b^2)*e^2)^(1/2)+2*c^2*d^2+( a*e^2-2*b*d*e)*c+1/4*b^2*e^2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d +(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2))+((-b*e+2*c*d+(-4*(a*c-1/4*b^2)*e^ 2)^(1/2))*c)^(1/2)*(2^(1/2)*(c*x^2+b*x+a)*((1/4*b*e-1/2*c*d)*(-4*(a*c-1/4* b^2)*e^2)^(1/2)+2*c^2*d^2+(a*e^2-2*b*d*e)*c+1/4*b^2*e^2)*e*arctan(c*(e*x+d )^(1/2)*2^(1/2)/((b*e-2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2))+(-c*d* x+(1/2*b*x+a)*e-1/2*b*d)*(-4*(a*c-1/4*b^2)*e^2)^(1/2)*((b*e-2*c*d+(-4*(a*c -1/4*b^2)*e^2)^(1/2))*c)^(1/2)*(e*x+d)^(1/2)))/((b*e-2*c*d+(-4*(a*c-1/4*b^ 2)*e^2)^(1/2))*c)^(1/2)/(a*c-1/4*b^2)/(c*x^2+b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 2396 vs. \(2 (314) = 628\).
Time = 0.46 (sec) , antiderivative size = 2396, normalized size of antiderivative = 6.60 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
1/2*(sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)* x)*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqr t(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6*c - 12 *a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*log(sqrt(1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^4 + 2*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d - (b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*e))*sqrt((32*c^3* d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48* a^2*b^2*c^3 - 64*a^3*c^4)) + (16*c^2*d^2*e^3 - 16*b*c*d*e^4 + (3*b^2 + 4*a *c)*e^5)*sqrt(e*x + d)) - sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x ^2 + (b^3 - 4*a*b*c)*x)*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4 *a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2* c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a ^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*log(-sqrt( 1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^4 + 2*sqrt(e^6/(b^6*c^2 - 12*a*b^4* c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^ 2*c^4 - 64*a^3*c^5)*d - (b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3...
Timed out. \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x + a\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1193 vs. \(2 (314) = 628\).
Time = 0.54 (sec) , antiderivative size = 1193, normalized size of antiderivative = 3.29 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
-(2*(e*x + d)^(3/2)*c*d*e - 2*sqrt(e*x + d)*c*d^2*e - (e*x + d)^(3/2)*b*e^ 2 + 2*sqrt(e*x + d)*b*d*e^2 - 2*sqrt(e*x + d)*a*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)*(b^2 - 4*a*c)) - 1/8 *(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(b^2*e - 4*a*c*e)^2*(2* c*d*e - b*e^2) + 4*(sqrt(b^2 - 4*a*c)*c^2*d^2*e - sqrt(b^2 - 4*a*c)*b*c*d* e^2 + sqrt(b^2 - 4*a*c)*a*c*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c )*c)*e)*abs(b^2*e - 4*a*c*e) - (16*(b^2*c^3 - 4*a*c^4)*d^3*e - 24*(b^3*c^2 - 4*a*b*c^3)*d^2*e^2 + 2*(5*b^4*c - 16*a*b^2*c^2 - 16*a^2*c^3)*d*e^3 - (b ^5 - 16*a^2*b*c^2)*e^4)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))* arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*b^2*c*d - 8*a*c^2*d - b^3*e + 4* a*b*c*e + sqrt((2*b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e)^2 - 4*(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)*(b^2*c - 4*a*c^2)))/(b^2*c - 4*a*c^2)))/(((b^2*c^2 - 4*a*c^3)*sqrt(b^2 - 4*a*c)*d ^2 - (b^3*c - 4*a*b*c^2)*sqrt(b^2 - 4*a*c)*d*e + (a*b^2*c - 4*a^2*c^2)*sqr t(b^2 - 4*a*c)*e^2)*abs(b^2*e - 4*a*c*e)*abs(c)) + 1/8*(sqrt(-4*c^2*d + 2* (b*c + sqrt(b^2 - 4*a*c)*c)*e)*(b^2*e - 4*a*c*e)^2*(2*c*d*e - b*e^2) - 4*( sqrt(b^2 - 4*a*c)*c^2*d^2*e - sqrt(b^2 - 4*a*c)*b*c*d*e^2 + sqrt(b^2 - 4*a *c)*a*c*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(b^2*e - 4*a*c*e) - (16*(b^2*c^3 - 4*a*c^4)*d^3*e - 24*(b^3*c^2 - 4*a*b*c^3)*d^2*e^ 2 + 2*(5*b^4*c - 16*a*b^2*c^2 - 16*a^2*c^3)*d*e^3 - (b^5 - 16*a^2*b*c^2...
Time = 6.25 (sec) , antiderivative size = 5326, normalized size of antiderivative = 14.67 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
log((c*e^3*(b*e - 2*c*d)*(16*c^3*d^4 + 3*a*b^2*e^4 + 4*a^2*c*e^4 - 3*b^3*d *e^3 + 20*a*c^2*d^2*e^2 + 19*b^2*c*d^2*e^2 - 32*b*c^2*d^3*e - 20*a*b*c*d*e ^3))/(4*a*c - b^2)^3 - (2^(1/2)*((2^(1/2)*(8*c^2*e^3*(a*e^2 + c*d^2 - b*d* e) - 2*2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-(b^9* e^3 + e^3*(-(4*a*c - b^2)^9)^(1/2) + 2048*a^3*c^6*d^3 - 32*b^6*c^3*d^3 + 3 84*a*b^4*c^4*d^3 - 768*a^4*b*c^4*e^3 + 1536*a^4*c^5*d*e^2 + 48*b^7*c^2*d^2 *e - 1536*a^2*b^2*c^5*d^3 - 96*a^2*b^5*c^2*e^3 + 512*a^3*b^3*c^3*e^3 - 18* b^8*c*d*e^2 - 576*a*b^5*c^3*d^2*e + 192*a*b^6*c^2*d*e^2 - 3072*a^3*b*c^5*d ^2*e + 2304*a^2*b^3*c^4*d^2*e - 576*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b^2)^6) )^(1/2))*(-(b^9*e^3 + e^3*(-(4*a*c - b^2)^9)^(1/2) + 2048*a^3*c^6*d^3 - 32 *b^6*c^3*d^3 + 384*a*b^4*c^4*d^3 - 768*a^4*b*c^4*e^3 + 1536*a^4*c^5*d*e^2 + 48*b^7*c^2*d^2*e - 1536*a^2*b^2*c^5*d^3 - 96*a^2*b^5*c^2*e^3 + 512*a^3*b ^3*c^3*e^3 - 18*b^8*c*d*e^2 - 576*a*b^5*c^3*d^2*e + 192*a*b^6*c^2*d*e^2 - 3072*a^3*b*c^5*d^2*e + 2304*a^2*b^3*c^4*d^2*e - 576*a^2*b^4*c^3*d*e^2)/(c* (4*a*c - b^2)^6))^(1/2))/4 + (2*c*e^2*(d + e*x)^(1/2)*(b^4*e^4 + 32*c^4*d^ 4 + 8*a^2*c^2*e^4 + 24*a*c^3*d^2*e^2 + 42*b^2*c^2*d^2*e^2 + 2*a*b^2*c*e^4 - 64*b*c^3*d^3*e - 10*b^3*c*d*e^3 - 24*a*b*c^2*d*e^3))/(4*a*c - b^2)^2)*(- (b^9*e^3 + e^3*(-(4*a*c - b^2)^9)^(1/2) + 2048*a^3*c^6*d^3 - 32*b^6*c^3*d^ 3 + 384*a*b^4*c^4*d^3 - 768*a^4*b*c^4*e^3 + 1536*a^4*c^5*d*e^2 + 48*b^7*c^ 2*d^2*e - 1536*a^2*b^2*c^5*d^3 - 96*a^2*b^5*c^2*e^3 + 512*a^3*b^3*c^3*e...