Integrand size = 28, antiderivative size = 398 \[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {(d+e x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {5 e \sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {5 e \left (8 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d-\sqrt {b^2-4 a c} d-2 a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {5 e \left (8 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d+\sqrt {b^2-4 a c} d-2 a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]
-1/2*(e*x+d)^(5/2)/(c*x^2+b*x+a)^2-5/4*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)+5/8*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))*(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)-5/8*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))*(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)
Leaf count is larger than twice the leaf count of optimal. \(1197\) vs. \(2(398)=796\).
Time = 16.62 (sec) , antiderivative size = 1197, normalized size of antiderivative = 3.01 \[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {(d+e x)^{7/2} \left (-2 a c (2 c d-b e)+b \left (b c d-b^2 e+2 a c e\right )+c (-2 c (b d-2 a e)+b (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 {-\frac {(d+e x)^{7/2} \left (-\frac {1}{2} a c \left (b^2-4 a c\right ) e^2 (2 c d-b e)-\frac {1}{2} \left (b^2-4 a c\right ) e (5 c d-3 b e) \left (b c d-b^2 e+2 a c e\right )+c \left (-\frac {1}{2} c \left (b^2-4 a c\right ) e^2 (b d-2 a e)-\frac {1}{2} \left (b^2-4 a c\right ) e (5 c d-3 b e) (2 c d-b e)\right ) 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 {\frac {1}{2} \left (b^2-4 a c\right ) e^2 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) (d+e x)^{5/2}+\frac {2 \left (\frac {25}{4} c \left (b^2-4 a c\right ) e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right ) (d+e x)^{3/2}+\frac {2 \left (\frac {75}{4} c^2 \left (b^2-4 a c\right ) e^2 \left (c d^2-e (b d-a e)\right )^2 \sqrt {d+e x}+\frac {4 \left (\frac {\sqrt {2 c d-b e-\sqrt {b^2-4 a c} e} \left (-\frac {75}{32} c^3 \left (b^2-4 a c\right ) e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right )^2-\frac {\frac {75}{32} c^3 \left (b^2-4 a c\right ) e^2 (2 c d-b e) (-2 c d+b e) \left (c d^2-e (b d-a e)\right )^2+2 c \left (\frac {75}{32} c^3 \left (b^2-4 a c\right ) d e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right )^2-\frac {75}{32} c^3 \left (b^2-4 a c\right ) e^2 \left (4 c d^2-e (3 b d-2 a e)\right ) \left (c d^2-e (b d-a e)\right )^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-b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {2} \sqrt {c} \left (-2 c d+b e+\sqrt {b^2-4 a c} e\right )}+\frac {\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e} \left (-\frac {75}{32} c^3 \left (b^2-4 a c\right ) e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right )^2+\frac {\frac {75}{32} c^3 \left (b^2-4 a c\right ) e^2 (2 c d-b e) (-2 c d+b e) \left (c d^2-e (b d-a e)\right )^2+2 c \left (\frac {75}{32} c^3 \left (b^2-4 a c\right ) d e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right )^2-\frac {75}{32} c^3 \left (b^2-4 a c\right ) e^2 \left (4 c d^2-e (3 b d-2 a e)\right ) \left (c d^2-e (b d-a e)\right )^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-b e+\sqrt {b^2-4 a c} e}}\right )}{\sqrt {2} \sqrt {c} \left (-2 c d+b e-\sqrt {b^2-4 a c} e\right )}\right )}{c}\right )}{3 c}\right )}{5 c}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )} \]
-1/2*((d + e*x)^(7/2)*(-2*a*c*(2*c*d - b*e) + b*(b*c*d - b^2*e + 2*a*c*e) + c*(-2*c*(b*d - 2*a*e) + b*(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) - (-(((d + e*x)^(7/2)*(-1/2*(a*c*(b^2 - 4 *a*c)*e^2*(2*c*d - b*e)) - ((b^2 - 4*a*c)*e*(5*c*d - 3*b*e)*(b*c*d - b^2*e + 2*a*c*e))/2 + c*(-1/2*(c*(b^2 - 4*a*c)*e^2*(b*d - 2*a*e)) - ((b^2 - 4*a *c)*e*(5*c*d - 3*b*e)*(2*c*d - b*e))/2)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2))) - (((b^2 - 4*a*c)*e^2*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*(d + e*x)^(5/2))/2 + (2*((25*c*(b^2 - 4*a*c)*e^2*( 2*c*d - b*e)*(c*d^2 - e*(b*d - a*e))*(d + e*x)^(3/2))/4 + (2*((75*c^2*(b^2 - 4*a*c)*e^2*(c*d^2 - e*(b*d - a*e))^2*Sqrt[d + e*x])/4 + (4*((Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]*((-75*c^3*(b^2 - 4*a*c)*e^2*(2*c*d - b*e)*(c *d^2 - e*(b*d - a*e))^2)/32 - ((75*c^3*(b^2 - 4*a*c)*e^2*(2*c*d - b*e)*(-2 *c*d + b*e)*(c*d^2 - e*(b*d - a*e))^2)/32 + 2*c*((75*c^3*(b^2 - 4*a*c)*d*e ^2*(2*c*d - b*e)*(c*d^2 - e*(b*d - a*e))^2)/32 - (75*c^3*(b^2 - 4*a*c)*e^2 *(4*c*d^2 - e*(3*b*d - 2*a*e))*(c*d^2 - e*(b*d - a*e))^2)/32))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e - Sqr t[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e + Sqrt[b^2 - 4*a*c]*e)) + (Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*((-75*c^3*(b^2 - 4*a*c)*e^2*(2 *c*d - b*e)*(c*d^2 - e*(b*d - a*e))^2)/32 + ((75*c^3*(b^2 - 4*a*c)*e^2*(2* c*d - b*e)*(-2*c*d + b*e)*(c*d^2 - e*(b*d - a*e))^2)/32 + 2*c*((75*c^3*...
Time = 0.73 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1222, 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.
| \(\displaystyle \int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1222 |
| \(\displaystyle \frac {5}{4} e \int \frac {(d+e x)^{3/2}}{\left (c x^2+b x+a\right )^2}dx-\frac {(d+e x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1164 |
| \(\displaystyle \frac {5}{4} e \left (-\frac {\int \frac {4 c d^2-3 b e d+2 a e^2+e (2 c d-b e) x}{2 \sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{b^2-4 a c}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{4} e \left (-\frac {\int \frac {4 c d^2-e (3 b d-2 a e)+e (2 c d-b e) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{2 \left (b^2-4 a c\right )}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {5}{4} e \left (-\frac {\int \frac {e \left (2 \left (c d^2-b e d+a e^2\right )+(2 c d-b e) (d+e x)\right )}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{b^2-4 a c}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{4} e \left (-\frac {e \int \frac {2 \left (c d^2-b e d+a e^2\right )+(2 c d-b e) (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{b^2-4 a c}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {5}{4} e \left (-\frac {e \left (\frac {1}{2} \left (\frac {-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2}{e \sqrt {b^2-4 a c}}-b e+2 c d\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}+\frac {1}{2} \left (-\frac {-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2}{e \sqrt {b^2-4 a c}}-b e+2 c d\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}\right )}{b^2-4 a c}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {5}{4} e \left (-\frac {e \left (-\frac {\left (\frac {-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2}{e \sqrt {b^2-4 a c}}-b e+2 c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (-\frac {-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2}{e \sqrt {b^2-4 a c}}-b e+2 c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{b^2-4 a c}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}\) |
-1/2*(d + e*x)^(5/2)/(a + b*x + c*x^2)^2 + (5*e*(-((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 - Sqrt[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])/Sqrt[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)))/4
3.17.24.3.1 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_))^(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 = 0.86 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.34
| method | result | size |
| pseudoelliptic | \(-\frac {5 \left (\left (\left (-\frac {b e}{4}+\frac {c d}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+2 c^{2} d^{2}+\left (e^{2} a -2 b d e \right ) c +\frac {b^{2} e^{2}}{4}\right ) \sqrt {2}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, e^{2} \left (c \,x^{2}+b x +a \right )^{2} \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (\sqrt {2}\, \left (\left (\frac {b e}{4}-\frac {c d}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+2 c^{2} d^{2}+\left (e^{2} a -2 b d e \right ) c +\frac {b^{2} e^{2}}{4}\right ) e^{2} \left (c \,x^{2}+b x +a \right )^{2} \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (-c^{2} d e \,x^{3}+\left (\left (\frac {1}{2} b \,x^{3}+\frac {9}{5} a \,x^{2}\right ) e^{2}+\frac {3 d \left (-\frac {5 b x}{2}+a \right ) x e}{5}+\frac {4 a \,d^{2}}{5}\right ) c +\left (\frac {3}{2} a b x +\frac {3}{10} b^{2} x^{2}+a^{2}\right ) e^{2}-\frac {d \left (\frac {9 b x}{5}+a \right ) b e}{2}-\frac {b^{2} d^{2}}{5}\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {e x +d}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\right )}{8 \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (c \,x^{2}+b x +a \right )^{2} \left (a c -\frac {b^{2}}{4}\right )}\) | \(532\) |
| derivativedivides | \(2 e^{4} \left (\frac {-\frac {5 \left (b e -2 c d \right ) c \left (e x +d \right )^{\frac {7}{2}}}{8 e^{2} \left (4 a c -b^{2}\right )}-\frac {3 \left (6 a c \,e^{2}+b^{2} e^{2}-10 b c d e +10 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{8 \left (4 a c -b^{2}\right ) e^{2}}-\frac {15 \left (a b \,e^{3}-2 a c d \,e^{2}-b^{2} d \,e^{2}+3 b c \,d^{2} e -2 c^{2} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{2} \left (4 a c -b^{2}\right )}-\frac {5 \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}}{4 e^{2} \left (4 a c -b^{2}\right )}}{\left (c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}\right )^{2}}+\frac {5 c \left (\frac {\left (-4 a c \,e^{2}-b^{2} e^{2}+8 b c d e -8 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (4 a c \,e^{2}+b^{2} e^{2}-8 b c d e +8 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 e^{2} \left (4 a c -b^{2}\right )}\right )\) | \(601\) |
| default | \(2 e^{4} \left (\frac {-\frac {5 \left (b e -2 c d \right ) c \left (e x +d \right )^{\frac {7}{2}}}{8 e^{2} \left (4 a c -b^{2}\right )}-\frac {3 \left (6 a c \,e^{2}+b^{2} e^{2}-10 b c d e +10 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{8 \left (4 a c -b^{2}\right ) e^{2}}-\frac {15 \left (a b \,e^{3}-2 a c d \,e^{2}-b^{2} d \,e^{2}+3 b c \,d^{2} e -2 c^{2} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{2} \left (4 a c -b^{2}\right )}-\frac {5 \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}}{4 e^{2} \left (4 a c -b^{2}\right )}}{\left (c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}\right )^{2}}+\frac {5 c \left (\frac {\left (-4 a c \,e^{2}-b^{2} e^{2}+8 b c d e -8 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (4 a c \,e^{2}+b^{2} e^{2}-8 b c d e +8 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 e^{2} \left (4 a c -b^{2}\right )}\right )\) | \(601\) |
-5/8/((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)*(((-1/4*b*e+1/2*c*d)*(-4*e^2*(a*c-1/4*b^2))^(1/2)+2*c^2*d^2+(a* e^2-2*b*d*e)*c+1/4*b^2*e^2)*2^(1/2)*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/ 2))*c)^(1/2)*e^2*(c*x^2+b*x+a)^2*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))+(2^(1/2)*((1/4*b*e-1/2*c*d)*(- 4*e^2*(a*c-1/4*b^2))^(1/2)+2*c^2*d^2+(a*e^2-2*b*d*e)*c+1/4*b^2*e^2)*e^2*(c *x^2+b*x+a)^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))+(-c^2*d*e*x^3+((1/2*b*x^3+9/5*a*x^2)*e^2+3/5*d*(-5/ 2*b*x+a)*x*e+4/5*a*d^2)*c+(3/2*a*b*x+3/10*b^2*x^2+a^2)*e^2-1/2*d*(9/5*b*x+ a)*b*e-1/5*b^2*d^2)*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*(e* x+d)^(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))/((-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. 2773 vs. \(2 (338) = 676\).
Time = 0.52 (sec) , antiderivative size = 2773, normalized size of antiderivative = 6.97 \[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
1/8*(5*sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c )*x)*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 + sqrt(e^10/(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))/(b ^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*log(125*sqrt(1/2)*((b^ 4 - 8*a*b^2*c + 16*a^2*c^2)*e^6 + 2*sqrt(e^10/(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*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - ( b^3 + 12*a*b*c)*e^5 + sqrt(e^10/(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))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)) + 125*(16*c^2*d^2*e^6 - 16 *b*c*d*e^7 + (3*b^2 + 4*a*c)*e^8)*sqrt(e*x + d)) - 5*sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a *b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 + sq rt(e^10/(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 - 1 2*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))/(b^6*c - 12*a*b^4*c^2 + 48*a^2 *b^2*c^3 - 64*a^3*c^4))*log(-125*sqrt(1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c...
Timed out. \[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=\int { \frac {{\left (2 \, c x + b\right )} {\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + b x + a\right )}^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1411 vs. \(2 (338) = 676\).
Time = 1.18 (sec) , antiderivative size = 1411, normalized size of antiderivative = 3.55 \[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
-5/32*((2*c*d*e^2 - b*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e )*(b^2*e - 4*a*c*e)^2 + 4*(sqrt(b^2 - 4*a*c)*c^2*d^2*e^2 - sqrt(b^2 - 4*a* c)*b*c*d*e^3 + sqrt(b^2 - 4*a*c)*a*c*e^4)*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^2 - 24*(b^3*c^2 - 4*a*b*c^3)*d^2*e^3 + 2*(5*b^4*c - 16*a*b^2*c^2 - 16*a^2*c^3) *d*e^4 - (b^5 - 16*a^2*b*c^2)*e^5)*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)*sqrt(b^2 - 4*a*c)*e^2)*abs(b^2*e - 4*a*c*e)*abs(c)) + 5/32*((2*c*d *e^2 - b*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(b^2*e - 4* a*c*e)^2 - 4*(sqrt(b^2 - 4*a*c)*c^2*d^2*e^2 - sqrt(b^2 - 4*a*c)*b*c*d*e^3 + sqrt(b^2 - 4*a*c)*a*c*e^4)*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^2 - 24*(b^3*c^2 - 4*a*b*c^3)*d^2*e^3 + 2*(5*b^4*c - 16*a*b^2*c^2 - 16*a^2*c^3)*d*e^4 - (b^5 - 16*a^2*b*c^2)*e^5)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*ar ctan(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...
Time = 16.24 (sec) , antiderivative size = 12750, normalized size of antiderivative = 32.04 \[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
- atan(((((5*(8192*a^4*c^5*e^6 - 128*a*b^6*c^2*e^6 + 128*b^7*c^2*d*e^5 + 1 536*a^2*b^4*c^3*e^6 - 6144*a^3*b^2*c^4*e^6 + 8192*a^3*c^6*d^2*e^4 - 128*b^ 6*c^3*d^2*e^4 - 6144*a^2*b^2*c^5*d^2*e^4 - 1536*a*b^5*c^3*d*e^5 - 8192*a^3 *b*c^5*d*e^5 + 1536*a*b^4*c^4*d^2*e^4 + 6144*a^2*b^3*c^4*d*e^5))/(64*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) - ((d + e*x)^(1/2)*(-(25*(b^9 *e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2) - 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*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 128 0*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2)*(64*b^7*c^2*e ^3 - 768*a*b^5*c^3*e^3 - 4096*a^3*b*c^5*e^3 + 8192*a^3*c^6*d*e^2 - 128*b^6 *c^3*d*e^2 + 3072*a^2*b^3*c^4*e^3 + 1536*a*b^4*c^4*d*e^2 - 6144*a^2*b^2*c^ 5*d*e^2))/(8*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(-(25*(b^9*e^5 + e^5*(-(4*a* c - b^2)^9)^(1/2) - 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*(b^...