Integrand size = 28, antiderivative size = 322 \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {3 e (b+2 c x) \sqrt {d+e x}}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {3 \sqrt {c} e \left (4 c d-\left (2 b-\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {3 \sqrt {c} e \left (4 c d-\left (2 b+\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]
-1/2*(e*x+d)^(3/2)/(c*x^2+b*x+a)^2-3/4*e*(2*c*x+b)*(e*x+d)^(1/2)/(-4*a*c+b ^2)/(c*x^2+b*x+a)+3/4*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)*(4*c*d-e*(2*b-(-4*a*c+b^2)^(1/2)))/(-4 *a*c+b^2)^(3/2)*2^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)-3/4*e*arcta nh(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)*(4*c*d-e*(2*b+(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)*2^(1/2)/(2*c* d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
Time = 14.66 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.56 \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {\left (b^2-4 a c\right ) (d+e x)^{5/2} (-c d+b e+c e x)}{(a+x (b+c x))^2}-\frac {e (d+e x)^{5/2} \left (b^3 e^2+b^2 c e (-4 d+e x)+b c \left (-a e^2+3 c d (d-2 e x)\right )+2 c^2 \left (3 c d^2 x+a e (2 d+e x)\right )\right )}{2 \left (c d^2+e (-b d+a e)\right ) (a+x (b+c x))}+\frac {1}{2} e \left (-3 e (-2 c d+b e) \sqrt {d+e x}+\frac {e \left (6 c^2 d^2+b^2 e^2+2 c e (-3 b d+a e)\right ) (d+e x)^{3/2}}{c d^2+e (-b d+a e)}+\frac {3 \sqrt {2} \sqrt {c} \left (c d^2+e (-b d+a e)\right ) \left (\frac {\left (4 c d+\left (-2 b+\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e}}\right )}{\sqrt {2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (-4 c d+\left (2 b+\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c}}\right )}{2 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )} \]
(((b^2 - 4*a*c)*(d + e*x)^(5/2)*(-(c*d) + b*e + c*e*x))/(a + x*(b + c*x))^ 2 - (e*(d + e*x)^(5/2)*(b^3*e^2 + b^2*c*e*(-4*d + e*x) + b*c*(-(a*e^2) + 3 *c*d*(d - 2*e*x)) + 2*c^2*(3*c*d^2*x + a*e*(2*d + e*x))))/(2*(c*d^2 + e*(- (b*d) + a*e))*(a + x*(b + c*x))) + (e*(-3*e*(-2*c*d + b*e)*Sqrt[d + e*x] + (e*(6*c^2*d^2 + b^2*e^2 + 2*c*e*(-3*b*d + a*e))*(d + e*x)^(3/2))/(c*d^2 + e*(-(b*d) + a*e)) + (3*Sqrt[2]*Sqrt[c]*(c*d^2 + e*(-(b*d) + a*e))*(((4*c* d + (-2*b + Sqrt[b^2 - 4*a*c])*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/ Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4* a*c])*e] + ((-4*c*d + (2*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*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/Sqrt[b^2 - 4*a*c]))/2)/(2*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))
Time = 0.59 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1222, 1163, 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.
| \(\displaystyle \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1222 |
| \(\displaystyle \frac {3}{4} e \int \frac {\sqrt {d+e x}}{\left (c x^2+b x+a\right )^2}dx-\frac {(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1163 |
| \(\displaystyle \frac {3}{4} e \left (\frac {\int -\frac {4 c d-b e+2 c e x}{2 \sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{b^2-4 a c}-\frac {(b+2 c x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{4} e \left (-\frac {\int \frac {4 c d-b e+2 c e x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{2 \left (b^2-4 a c\right )}-\frac {(b+2 c x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {3}{4} e \left (-\frac {\int \frac {e (2 c d-b e+2 c (d+e x))}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{b^2-4 a c}-\frac {(b+2 c x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{4} e \left (-\frac {e \int \frac {2 c d-b e+2 c (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{b^2-4 a c}-\frac {(b+2 c x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {3}{4} e \left (-\frac {e \left (c \left (\frac {4 c d-2 b e}{e \sqrt {b^2-4 a c}}+1\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}+c \left (1-\frac {2 (2 c d-b e)}{e \sqrt {b^2-4 a c}}\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}\right )}{b^2-4 a c}-\frac {(b+2 c x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {3}{4} e \left (-\frac {e \left (-\frac {\sqrt {2} \sqrt {c} \left (\frac {4 c d-2 b e}{e \sqrt {b^2-4 a c}}+1\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\sqrt {2} \sqrt {c} \left (1-\frac {2 (2 c d-b e)}{e \sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{b^2-4 a c}-\frac {(b+2 c x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2}\) |
-1/2*(d + e*x)^(3/2)/(a + b*x + c*x^2)^2 + (3*e*(-(((b + 2*c*x)*Sqrt[d + e *x])/((b^2 - 4*a*c)*(a + b*x + c*x^2))) - (e*(-((Sqrt[2]*Sqrt[c]*(1 + (4*c *d - 2*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*c*d - (b - Sqrt[b^2 - 4* a*c])*e]) - (Sqrt[2]*Sqrt[c]*(1 - (2*(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*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/(b^2 - 4*a*c)))/4
3.17.25.3.1 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*(b + 2*c*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 - 1 )*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[ m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && 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[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] - Simp[e*g*(m/(2*c*(p + 1))) Int[(d + e*x)^(m - 1)* (a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ [2*c*f - b*g, 0] && LtQ[p, -1] && GtQ[m, 0]
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 = 0.60 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.34
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {3 \sqrt {2}\, e^{2} \left (c \,x^{2}+b x +a \right )^{2} \left (-2 b e +4 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, c \,\operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )}{16}+\frac {3 \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (c \,e^{2} \sqrt {2}\, \left (c \,x^{2}+b x +a \right )^{2} \left (2 b e -4 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (2 e \,x^{3} c^{2}+\left (\left (3 b \,x^{2}-\frac {2}{3} a x \right ) e -\frac {8 a d}{3}\right ) c +\left (\left (\frac {5 b x}{3}+a \right ) e +\frac {2 b d}{3}\right ) b \right ) \sqrt {e x +d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right )}{16}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (c \,x^{2}+b x +a \right )^{2} \left (a c -\frac {b^{2}}{4}\right )}\) | \(430\) |
| derivativedivides | \(2 e^{4} \left (\frac {\frac {3 c^{2} \left (e x +d \right )^{\frac {7}{2}}}{4 e^{2} \left (4 a c -b^{2}\right )}+\frac {9 \left (b e -2 c d \right ) c \left (e x +d \right )^{\frac {5}{2}}}{8 e^{2} \left (4 a c -b^{2}\right )}-\frac {\left (2 a c \,e^{2}-5 b^{2} e^{2}+18 b c d e -18 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{2} \left (4 a c -b^{2}\right )}+\frac {3 \left (a b \,e^{3}-2 a c d \,e^{2}-b^{2} d \,e^{2}+3 b c \,d^{2} e -2 c^{2} d^{3}\right ) \sqrt {e x +d}}{8 e^{2} \left (4 a c -b^{2}\right )}}{\left (c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}\right )^{2}}+\frac {3 c \left (\frac {\left (2 b e -4 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-2 b e +4 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 e^{2} \left (4 a c -b^{2}\right )}\right )\) | \(490\) |
| default | \(2 e^{4} \left (\frac {\frac {3 c^{2} \left (e x +d \right )^{\frac {7}{2}}}{4 e^{2} \left (4 a c -b^{2}\right )}+\frac {9 \left (b e -2 c d \right ) c \left (e x +d \right )^{\frac {5}{2}}}{8 e^{2} \left (4 a c -b^{2}\right )}-\frac {\left (2 a c \,e^{2}-5 b^{2} e^{2}+18 b c d e -18 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{2} \left (4 a c -b^{2}\right )}+\frac {3 \left (a b \,e^{3}-2 a c d \,e^{2}-b^{2} d \,e^{2}+3 b c \,d^{2} e -2 c^{2} d^{3}\right ) \sqrt {e x +d}}{8 e^{2} \left (4 a c -b^{2}\right )}}{\left (c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}\right )^{2}}+\frac {3 c \left (\frac {\left (2 b e -4 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-2 b e +4 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 e^{2} \left (4 a c -b^{2}\right )}\right )\) | \(490\) |
3/16/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*(-2^(1/2)*e^2*(c*x ^2+b*x+a)^2*(-2*b*e+4*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*((b*e-2*c*d+(-4*e^ 2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*c*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+ 2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2))+((-b*e+2*c*d+(-4*e^2*(a*c-1/ 4*b^2))^(1/2))*c)^(1/2)*(c*e^2*2^(1/2)*(c*x^2+b*x+a)^2*(2*b*e-4*c*d+(-4*e^ 2*(a*c-1/4*b^2))^(1/2))*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-4*e^2 *(a*c-1/4*b^2))^(1/2))*c)^(1/2))+(2*e*x^3*c^2+((3*b*x^2-2/3*a*x)*e-8/3*a*d )*c+((5/3*b*x+a)*e+2/3*b*d)*b)*(e*x+d)^(1/2)*((b*e-2*c*d+(-4*e^2*(a*c-1/4* b^2))^(1/2))*c)^(1/2)*(-4*e^2*(a*c-1/4*b^2))^(1/2)))/(-4*e^2*(a*c-1/4*b^2) )^(1/2)/((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)/(c*x^2+b*x+a)^ 2/(a*c-1/4*b^2)
Leaf count of result is larger than twice the leaf count of optimal. 6728 vs. \(2 (266) = 532\).
Time = 0.54 (sec) , antiderivative size = 6728, normalized size of antiderivative = 20.89 \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\int { \frac {{\left (2 \, c x + b\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x + a\right )}^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1169 vs. \(2 (266) = 532\).
Time = 1.17 (sec) , antiderivative size = 1169, normalized size of antiderivative = 3.63 \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
-3/16*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(b^2*e - 4*a*c*e)^ 2*e^2 + (2*sqrt(b^2 - 4*a*c)*c*d*e^2 - sqrt(b^2 - 4*a*c)*b*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) - 2*(4*(b^2*c^ 2 - 4*a*c^3)*d^2*e^2 - 4*(b^3*c - 4*a*b*c^2)*d*e^3 + (b^4 - 4*a*b^2*c)*e^4 )*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqr t(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 - 4*a*c^2)*sqrt(b^2 - 4*a*c)*d^2 - (b^3 - 4*a*b*c)*sqr t(b^2 - 4*a*c)*d*e + (a*b^2 - 4*a^2*c)*sqrt(b^2 - 4*a*c)*e^2)*abs(b^2*e - 4*a*c*e)*abs(c)) + 3/16*(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)* (b^2*e - 4*a*c*e)^2*e^2 - (2*sqrt(b^2 - 4*a*c)*c*d*e^2 - sqrt(b^2 - 4*a*c) *b*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) - 2*(4*(b^2*c^2 - 4*a*c^3)*d^2*e^2 - 4*(b^3*c - 4*a*b*c^2)*d*e^3 + (b^ 4 - 4*a*b^2*c)*e^4)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arct an(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 - 4*a*c^2)*sqrt(b^2 - 4*a*c)*d^2 - ( b^3 - 4*a*b*c)*sqrt(b^2 - 4*a*c)*d*e + (a*b^2 - 4*a^2*c)*sqrt(b^2 - 4*a...
Time = 15.73 (sec) , antiderivative size = 16393, normalized size of antiderivative = 50.91 \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
(((d + e*x)^(3/2)*(5*b^2*e^4 + 18*c^2*d^2*e^2 - 2*a*c*e^4 - 18*b*c*d*e^3)) /(4*(4*a*c - b^2)) - (3*(d + e*x)^(1/2)*(b^2*d*e^4 + 2*c^2*d^3*e^2 - a*b*e ^5 + 2*a*c*d*e^4 - 3*b*c*d^2*e^3))/(4*(4*a*c - b^2)) + (3*c^2*e^2*(d + e*x )^(7/2))/(2*(4*a*c - b^2)) + (9*c*e^2*(b*e - 2*c*d)*(d + e*x)^(5/2))/(4*(4 *a*c - b^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*( 32*b^7*c^2*e^5 - 384*a*b^5*c^3*e^5 - 2048*a^3*b*c^5*e^5 + 4096*a^3*c^6*d*e ^4 - 64*b^6*c^3*d*e^4 + 1536*a^2*b^3*c^4*e^5 + 768*a*b^4*c^4*d*e^4 - 3072* a^2*b^2*c^5*d*e^4))/(32*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) - ((d + e*x)^(1/2)*((9*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b *c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c *d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c ^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3 *d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6 *c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c *e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 38...