Integrand size = 26, antiderivative size = 386 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {e \left (2 A c^4 d^3+7 b^4 B e^3-b c^3 d^2 (B d+3 A e)-b^3 c e^2 (19 B d+5 A e)+b^2 c^2 d e (15 B d+11 A e)\right ) \sqrt {d+e x}}{b^2 c^4}+\frac {e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) (d+e x)^{3/2}}{3 b^2 c^3}+\frac {e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\frac {(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}-\frac {d^{7/2} (2 b B d-4 A c d+9 A b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {(c d-b e)^{7/2} \left (4 A c^2 d-7 b^2 B e-b c (2 B d-5 A e)\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{9/2}} \]
1/3*e*(6*A*c^3*d^2-7*b^3*B*e^2-3*b*c^2*d*(2*A*e+B*d)+b^2*c*e*(5*A*e+12*B*d ))*(e*x+d)^(3/2)/b^2/c^3+1/5*e*(10*A*c^2*d+7*b^2*B*e-5*b*c*(A*e+B*d))*(e*x +d)^(5/2)/b^2/c^2-(e*x+d)^(7/2)*(A*b*c*d+(2*A*c^2*d+b^2*B*e-b*c*(A*e+B*d)) *x)/b^2/c/(c*x^2+b*x)-d^(7/2)*(9*A*b*e-4*A*c*d+2*B*b*d)*arctanh((e*x+d)^(1 /2)/d^(1/2))/b^3-(-b*e+c*d)^(7/2)*(4*A*c^2*d-7*b^2*B*e-b*c*(-5*A*e+2*B*d)) *arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^3/c^(9/2)+e*(2*A*c^4*d^ 3+7*b^4*B*e^3-b*c^3*d^2*(3*A*e+B*d)-b^3*c*e^2*(5*A*e+19*B*d)+b^2*c^2*d*e*( 11*A*e+15*B*d))*(e*x+d)^(1/2)/b^2/c^4
Time = 0.88 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.87 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {\frac {b \sqrt {d+e x} \left (-5 A c \left (6 c^4 d^4 x+15 b^4 e^4 x+3 b c^3 d^3 (d-4 e x)+2 b^3 c e^3 x (-19 d+5 e x)-2 b^2 c^2 e^2 x \left (-9 d^2+13 d e x+e^2 x^2\right )\right )+b B x \left (15 c^4 d^4+105 b^4 e^4+10 b^3 c e^3 (-32 d+7 e x)-2 b^2 c^2 e^2 \left (-153 d^2+109 d e x+7 e^2 x^2\right )+6 b c^3 e \left (-10 d^3+36 d^2 e x+7 d e^2 x^2+e^3 x^3\right )\right )\right )}{c^4 x (b+c x)}+\frac {15 (-c d+b e)^{7/2} \left (-2 b B c d+4 A c^2 d-7 b^2 B e+5 A b c e\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{9/2}}-15 d^{7/2} (2 b B d-4 A c d+9 A b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{15 b^3} \]
((b*Sqrt[d + e*x]*(-5*A*c*(6*c^4*d^4*x + 15*b^4*e^4*x + 3*b*c^3*d^3*(d - 4 *e*x) + 2*b^3*c*e^3*x*(-19*d + 5*e*x) - 2*b^2*c^2*e^2*x*(-9*d^2 + 13*d*e*x + e^2*x^2)) + b*B*x*(15*c^4*d^4 + 105*b^4*e^4 + 10*b^3*c*e^3*(-32*d + 7*e *x) - 2*b^2*c^2*e^2*(-153*d^2 + 109*d*e*x + 7*e^2*x^2) + 6*b*c^3*e*(-10*d^ 3 + 36*d^2*e*x + 7*d*e^2*x^2 + e^3*x^3))))/(c^4*x*(b + c*x)) + (15*(-(c*d) + b*e)^(7/2)*(-2*b*B*c*d + 4*A*c^2*d - 7*b^2*B*e + 5*A*b*c*e)*ArcTan[(Sqr t[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/c^(9/2) - 15*d^(7/2)*(2*b*B*d - 4 *A*c*d + 9*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(15*b^3)
Time = 1.23 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1233, 27, 1196, 1196, 1196, 1197, 25, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {\int \frac {(d+e x)^{5/2} \left (c d (2 b B d-4 A c d+9 A b e)+e \left (7 B e b^2-5 c (B d+A e) b+10 A c^2 d\right ) x\right )}{2 \left (c x^2+b x\right )}dx}{b^2 c}-\frac {(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(d+e x)^{5/2} \left (c d (2 b B d-4 A c d+9 A b e)+e \left (7 B e b^2-5 c (B d+A e) b+10 A c^2 d\right ) x\right )}{c x^2+b x}dx}{2 b^2 c}-\frac {(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle \frac {\frac {\int \frac {(d+e x)^{3/2} \left (c^2 (2 b B d-4 A c d+9 A b e) d^2+e \left (-7 B e^2 b^3+c e (12 B d+5 A e) b^2-3 c^2 d (B d+2 A e) b+6 A c^3 d^2\right ) x\right )}{c x^2+b x}dx}{c}+\frac {2 e (d+e x)^{5/2} \left (-5 b c (A e+B d)+10 A c^2 d+7 b^2 B e\right )}{5 c}}{2 b^2 c}-\frac {(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {d+e x} \left (c^3 (2 b B d-4 A c d+9 A b e) d^3+e \left (7 B e^3 b^4-c e^2 (19 B d+5 A e) b^3+c^2 d e (15 B d+11 A e) b^2-c^3 d^2 (B d+3 A e) b+2 A c^4 d^3\right ) x\right )}{c x^2+b x}dx}{c}+\frac {2 e (d+e x)^{3/2} \left (b^2 c e (5 A e+12 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-7 b^3 B e^2\right )}{3 c}}{c}+\frac {2 e (d+e x)^{5/2} \left (-5 b c (A e+B d)+10 A c^2 d+7 b^2 B e\right )}{5 c}}{2 b^2 c}-\frac {(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {c^4 d^4 (2 b B d-4 A c d+9 A b e)-e \left (7 B e^4 b^5-c e^3 (26 B d+5 A e) b^4+2 c^2 d e^2 (17 B d+8 A e) b^3-2 c^3 d^2 e (8 B d+7 A e) b^2-c^4 d^3 (B d+4 A e) b+2 A c^5 d^4\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{c}+\frac {2 e \sqrt {d+e x} \left (-b^3 c e^2 (5 A e+19 B d)+b^2 c^2 d e (11 A e+15 B d)-b c^3 d^2 (3 A e+B d)+2 A c^4 d^3+7 b^4 B e^3\right )}{c}}{c}+\frac {2 e (d+e x)^{3/2} \left (b^2 c e (5 A e+12 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-7 b^3 B e^2\right )}{3 c}}{c}+\frac {2 e (d+e x)^{5/2} \left (-5 b c (A e+B d)+10 A c^2 d+7 b^2 B e\right )}{5 c}}{2 b^2 c}-\frac {(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \int -\frac {e \left (d (c d-b e) \left (7 B e^3 b^4-c e^2 (19 B d+5 A e) b^3+c^2 d e (15 B d+11 A e) b^2-c^3 d^2 (B d+3 A e) b+2 A c^4 d^3\right )+\left (7 B e^4 b^5-c e^3 (26 B d+5 A e) b^4+2 c^2 d e^2 (17 B d+8 A e) b^3-2 c^3 d^2 e (8 B d+7 A e) b^2-c^4 d^3 (B d+4 A e) b+2 A c^5 d^4\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{c}+\frac {2 e \sqrt {d+e x} \left (-b^3 c e^2 (5 A e+19 B d)+b^2 c^2 d e (11 A e+15 B d)-b c^3 d^2 (3 A e+B d)+2 A c^4 d^3+7 b^4 B e^3\right )}{c}}{c}+\frac {2 e (d+e x)^{3/2} \left (b^2 c e (5 A e+12 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-7 b^3 B e^2\right )}{3 c}}{c}+\frac {2 e (d+e x)^{5/2} \left (-5 b c (A e+B d)+10 A c^2 d+7 b^2 B e\right )}{5 c}}{2 b^2 c}-\frac {(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 e \sqrt {d+e x} \left (-b^3 c e^2 (5 A e+19 B d)+b^2 c^2 d e (11 A e+15 B d)-b c^3 d^2 (3 A e+B d)+2 A c^4 d^3+7 b^4 B e^3\right )}{c}-\frac {2 \int \frac {e \left (d (c d-b e) \left (7 B e^3 b^4-c e^2 (19 B d+5 A e) b^3+c^2 d e (15 B d+11 A e) b^2-c^3 d^2 (B d+3 A e) b+2 A c^4 d^3\right )+\left (7 B e^4 b^5-c e^3 (26 B d+5 A e) b^4+2 c^2 d e^2 (17 B d+8 A e) b^3-2 c^3 d^2 e (8 B d+7 A e) b^2-c^4 d^3 (B d+4 A e) b+2 A c^5 d^4\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{c}}{c}+\frac {2 e (d+e x)^{3/2} \left (b^2 c e (5 A e+12 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-7 b^3 B e^2\right )}{3 c}}{c}+\frac {2 e (d+e x)^{5/2} \left (-5 b c (A e+B d)+10 A c^2 d+7 b^2 B e\right )}{5 c}}{2 b^2 c}-\frac {(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 e \sqrt {d+e x} \left (-b^3 c e^2 (5 A e+19 B d)+b^2 c^2 d e (11 A e+15 B d)-b c^3 d^2 (3 A e+B d)+2 A c^4 d^3+7 b^4 B e^3\right )}{c}-\frac {2 e \int \frac {d (c d-b e) \left (7 B e^3 b^4-c e^2 (19 B d+5 A e) b^3+c^2 d e (15 B d+11 A e) b^2-c^3 d^2 (B d+3 A e) b+2 A c^4 d^3\right )+\left (7 B e^4 b^5-c e^3 (26 B d+5 A e) b^4+2 c^2 d e^2 (17 B d+8 A e) b^3-2 c^3 d^2 e (8 B d+7 A e) b^2-c^4 d^3 (B d+4 A e) b+2 A c^5 d^4\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{c}}{c}+\frac {2 e (d+e x)^{3/2} \left (b^2 c e (5 A e+12 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-7 b^3 B e^2\right )}{3 c}}{c}+\frac {2 e (d+e x)^{5/2} \left (-5 b c (A e+B d)+10 A c^2 d+7 b^2 B e\right )}{5 c}}{2 b^2 c}-\frac {(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 e \sqrt {d+e x} \left (-b^3 c e^2 (5 A e+19 B d)+b^2 c^2 d e (11 A e+15 B d)-b c^3 d^2 (3 A e+B d)+2 A c^4 d^3+7 b^4 B e^3\right )}{c}-\frac {2 e \left (-\frac {(c d-b e)^4 \left (-b c (2 B d-5 A e)+4 A c^2 d-7 b^2 B e\right ) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}-\frac {c^5 d^4 (9 A b e-4 A c d+2 b B d) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}\right )}{c}}{c}+\frac {2 e (d+e x)^{3/2} \left (b^2 c e (5 A e+12 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-7 b^3 B e^2\right )}{3 c}}{c}+\frac {2 e (d+e x)^{5/2} \left (-5 b c (A e+B d)+10 A c^2 d+7 b^2 B e\right )}{5 c}}{2 b^2 c}-\frac {(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 e \sqrt {d+e x} \left (-b^3 c e^2 (5 A e+19 B d)+b^2 c^2 d e (11 A e+15 B d)-b c^3 d^2 (3 A e+B d)+2 A c^4 d^3+7 b^4 B e^3\right )}{c}-\frac {2 e \left (\frac {(c d-b e)^{7/2} \left (-b c (2 B d-5 A e)+4 A c^2 d-7 b^2 B e\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b \sqrt {c} e}+\frac {c^4 d^{7/2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (9 A b e-4 A c d+2 b B d)}{b e}\right )}{c}}{c}+\frac {2 e (d+e x)^{3/2} \left (b^2 c e (5 A e+12 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-7 b^3 B e^2\right )}{3 c}}{c}+\frac {2 e (d+e x)^{5/2} \left (-5 b c (A e+B d)+10 A c^2 d+7 b^2 B e\right )}{5 c}}{2 b^2 c}-\frac {(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
-(((d + e*x)^(7/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/ (b^2*c*(b*x + c*x^2))) + ((2*e*(10*A*c^2*d + 7*b^2*B*e - 5*b*c*(B*d + A*e) )*(d + e*x)^(5/2))/(5*c) + ((2*e*(6*A*c^3*d^2 - 7*b^3*B*e^2 - 3*b*c^2*d*(B *d + 2*A*e) + b^2*c*e*(12*B*d + 5*A*e))*(d + e*x)^(3/2))/(3*c) + ((2*e*(2* A*c^4*d^3 + 7*b^4*B*e^3 - b*c^3*d^2*(B*d + 3*A*e) - b^3*c*e^2*(19*B*d + 5* A*e) + b^2*c^2*d*e*(15*B*d + 11*A*e))*Sqrt[d + e*x])/c - (2*e*((c^4*d^(7/2 )*(2*b*B*d - 4*A*c*d + 9*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*e) + (( c*d - b*e)^(7/2)*(4*A*c^2*d - 7*b^2*B*e - b*c*(2*B*d - 5*A*e))*ArcTanh[(Sq rt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*Sqrt[c]*e)))/c)/c)/c)/(2*b^2*c)
3.13.38.3.1 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_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c Int [(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] & & GtQ[m, 0]
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))*(a + b*x + c*x^2) ^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[m + 2*p + 3, 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 = 0.68 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.86
| method | result | size |
| pseudoelliptic | \(-\frac {-4 \sqrt {d}\, x \left (c x +b \right ) \left (A \,c^{2} d +\frac {5 \left (A e -\frac {2 B d}{5}\right ) b c}{4}-\frac {7 b^{2} B e}{4}\right ) \left (-b e +c d \right )^{4} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )+\sqrt {\left (b e -c d \right ) c}\, \left (9 c^{4} x \left (c x +b \right ) \left (-\frac {4 A c d}{9}+b \left (A e +\frac {2 B d}{9}\right )\right ) d^{4} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )+\sqrt {e x +d}\, \sqrt {d}\, \left (2 A \,c^{5} d^{4} x +d^{3} \left (\left (-B x +A \right ) d -4 A e x \right ) b \,c^{4}+6 x e \left (\frac {2 B \,d^{3}}{3}+e \left (-\frac {12 B x}{5}+A \right ) d^{2}-\frac {13 \left (\frac {21 B x}{65}+A \right ) x \,e^{2} d}{9}-\frac {x^{2} \left (\frac {3 B x}{5}+A \right ) e^{3}}{9}\right ) b^{2} c^{3}-\frac {38 x \left (\frac {153 B \,d^{2}}{95}+e \left (-\frac {109 B x}{95}+A \right ) d -\frac {5 x \left (\frac {7 B x}{25}+A \right ) e^{2}}{19}\right ) e^{2} b^{3} c^{2}}{3}+5 \left (\frac {64 B d}{15}+e \left (-\frac {14 B x}{15}+A \right )\right ) x \,e^{3} b^{4} c -7 B \,b^{5} e^{4} x \right ) b \right )}{\sqrt {\left (b e -c d \right ) c}\, \sqrt {d}\, c^{4} \left (c x +b \right ) b^{3} x}\) | \(333\) |
| derivativedivides | \(2 e^{2} \left (-\frac {-\frac {B \,c^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A \,c^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B b c e \left (e x +d \right )^{\frac {3}{2}}}{3}-B \,c^{2} d \left (e x +d \right )^{\frac {3}{2}}+2 A b c \,e^{2} \sqrt {e x +d}-4 A \,c^{2} d e \sqrt {e x +d}-3 B \,b^{2} e^{2} \sqrt {e x +d}+8 B b c d e \sqrt {e x +d}-6 B \,c^{2} d^{2} \sqrt {e x +d}}{c^{4}}+\frac {\frac {\left (-\frac {1}{2} A \,b^{5} c \,e^{5}+2 A \,b^{4} c^{2} d \,e^{4}-3 A \,b^{3} c^{3} d^{2} e^{3}+2 A \,b^{2} c^{4} d^{3} e^{2}-\frac {1}{2} A b \,c^{5} d^{4} e +\frac {1}{2} b^{6} B \,e^{5}-2 b^{5} B c d \,e^{4}+3 B \,b^{4} c^{2} d^{2} e^{3}-2 B \,b^{3} c^{3} d^{3} e^{2}+\frac {1}{2} B \,b^{2} c^{4} d^{4} e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (5 A \,b^{5} c \,e^{5}-16 A \,b^{4} c^{2} d \,e^{4}+14 A \,b^{3} c^{3} d^{2} e^{3}+4 A \,b^{2} c^{4} d^{3} e^{2}-11 A b \,c^{5} d^{4} e +4 A \,c^{6} d^{5}-7 b^{6} B \,e^{5}+26 b^{5} B c d \,e^{4}-34 B \,b^{4} c^{2} d^{2} e^{3}+16 B \,b^{3} c^{3} d^{3} e^{2}+B \,b^{2} c^{4} d^{4} e -2 B \,c^{5} d^{5} b \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}}{b^{3} e^{2} c^{4}}-\frac {d^{4} \left (\frac {A b \sqrt {e x +d}}{2 x}+\frac {\left (9 A b e -4 A c d +2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{e^{2} b^{3}}\right )\) | \(548\) |
| default | \(2 e^{2} \left (-\frac {-\frac {B \,c^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A \,c^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B b c e \left (e x +d \right )^{\frac {3}{2}}}{3}-B \,c^{2} d \left (e x +d \right )^{\frac {3}{2}}+2 A b c \,e^{2} \sqrt {e x +d}-4 A \,c^{2} d e \sqrt {e x +d}-3 B \,b^{2} e^{2} \sqrt {e x +d}+8 B b c d e \sqrt {e x +d}-6 B \,c^{2} d^{2} \sqrt {e x +d}}{c^{4}}+\frac {\frac {\left (-\frac {1}{2} A \,b^{5} c \,e^{5}+2 A \,b^{4} c^{2} d \,e^{4}-3 A \,b^{3} c^{3} d^{2} e^{3}+2 A \,b^{2} c^{4} d^{3} e^{2}-\frac {1}{2} A b \,c^{5} d^{4} e +\frac {1}{2} b^{6} B \,e^{5}-2 b^{5} B c d \,e^{4}+3 B \,b^{4} c^{2} d^{2} e^{3}-2 B \,b^{3} c^{3} d^{3} e^{2}+\frac {1}{2} B \,b^{2} c^{4} d^{4} e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (5 A \,b^{5} c \,e^{5}-16 A \,b^{4} c^{2} d \,e^{4}+14 A \,b^{3} c^{3} d^{2} e^{3}+4 A \,b^{2} c^{4} d^{3} e^{2}-11 A b \,c^{5} d^{4} e +4 A \,c^{6} d^{5}-7 b^{6} B \,e^{5}+26 b^{5} B c d \,e^{4}-34 B \,b^{4} c^{2} d^{2} e^{3}+16 B \,b^{3} c^{3} d^{3} e^{2}+B \,b^{2} c^{4} d^{4} e -2 B \,c^{5} d^{5} b \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}}{b^{3} e^{2} c^{4}}-\frac {d^{4} \left (\frac {A b \sqrt {e x +d}}{2 x}+\frac {\left (9 A b e -4 A c d +2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{e^{2} b^{3}}\right )\) | \(548\) |
| risch | \(-\frac {d^{4} A \sqrt {e x +d}}{b^{2} x}+\frac {e \left (\frac {2 b^{2} e \left (\frac {B \,c^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {A \,c^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {2 B b c e \left (e x +d \right )^{\frac {3}{2}}}{3}+B \,c^{2} d \left (e x +d \right )^{\frac {3}{2}}-2 A b c \,e^{2} \sqrt {e x +d}+4 A \,c^{2} d e \sqrt {e x +d}+3 B \,b^{2} e^{2} \sqrt {e x +d}-8 B b c d e \sqrt {e x +d}+6 B \,c^{2} d^{2} \sqrt {e x +d}\right )}{c^{4}}+\frac {\frac {2 \left (-\frac {1}{2} A \,b^{5} c \,e^{5}+2 A \,b^{4} c^{2} d \,e^{4}-3 A \,b^{3} c^{3} d^{2} e^{3}+2 A \,b^{2} c^{4} d^{3} e^{2}-\frac {1}{2} A b \,c^{5} d^{4} e +\frac {1}{2} b^{6} B \,e^{5}-2 b^{5} B c d \,e^{4}+3 B \,b^{4} c^{2} d^{2} e^{3}-2 B \,b^{3} c^{3} d^{3} e^{2}+\frac {1}{2} B \,b^{2} c^{4} d^{4} e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (5 A \,b^{5} c \,e^{5}-16 A \,b^{4} c^{2} d \,e^{4}+14 A \,b^{3} c^{3} d^{2} e^{3}+4 A \,b^{2} c^{4} d^{3} e^{2}-11 A b \,c^{5} d^{4} e +4 A \,c^{6} d^{5}-7 b^{6} B \,e^{5}+26 b^{5} B c d \,e^{4}-34 B \,b^{4} c^{2} d^{2} e^{3}+16 B \,b^{3} c^{3} d^{3} e^{2}+B \,b^{2} c^{4} d^{4} e -2 B \,c^{5} d^{5} b \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}}}{c^{4} b e}-\frac {d^{\frac {7}{2}} \left (9 A b e -4 A c d +2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e}\right )}{b^{2}}\) | \(552\) |
-1/((b*e-c*d)*c)^(1/2)*(-4*d^(1/2)*x*(c*x+b)*(A*c^2*d+5/4*(A*e-2/5*B*d)*b* c-7/4*b^2*B*e)*(-b*e+c*d)^4*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))+(( b*e-c*d)*c)^(1/2)*(9*c^4*x*(c*x+b)*(-4/9*A*c*d+b*(A*e+2/9*B*d))*d^4*arctan h((e*x+d)^(1/2)/d^(1/2))+(e*x+d)^(1/2)*d^(1/2)*(2*A*c^5*d^4*x+d^3*((-B*x+A )*d-4*A*e*x)*b*c^4+6*x*e*(2/3*B*d^3+e*(-12/5*B*x+A)*d^2-13/9*(21/65*B*x+A) *x*e^2*d-1/9*x^2*(3/5*B*x+A)*e^3)*b^2*c^3-38/3*x*(153/95*B*d^2+e*(-109/95* B*x+A)*d-5/19*x*(7/25*B*x+A)*e^2)*e^2*b^3*c^2+5*(64/15*B*d+e*(-14/15*B*x+A ))*x*e^3*b^4*c-7*B*b^5*e^4*x)*b))/d^(1/2)/c^4/(c*x+b)/b^3/x
Time = 189.23 (sec) , antiderivative size = 2712, normalized size of antiderivative = 7.03 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
[1/30*(15*((2*(B*b*c^5 - 2*A*c^6)*d^4 + (B*b^2*c^4 + 7*A*b*c^5)*d^3*e - 3* (5*B*b^3*c^3 - A*b^2*c^4)*d^2*e^2 + (19*B*b^4*c^2 - 11*A*b^3*c^3)*d*e^3 - (7*B*b^5*c - 5*A*b^4*c^2)*e^4)*x^2 + (2*(B*b^2*c^4 - 2*A*b*c^5)*d^4 + (B*b ^3*c^3 + 7*A*b^2*c^4)*d^3*e - 3*(5*B*b^4*c^2 - A*b^3*c^3)*d^2*e^2 + (19*B* b^5*c - 11*A*b^4*c^2)*d*e^3 - (7*B*b^6 - 5*A*b^5*c)*e^4)*x)*sqrt((c*d - b* e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c *x + b)) + 15*((9*A*b*c^5*d^3*e + 2*(B*b*c^5 - 2*A*c^6)*d^4)*x^2 + (9*A*b^ 2*c^4*d^3*e + 2*(B*b^2*c^4 - 2*A*b*c^5)*d^4)*x)*sqrt(d)*log((e*x - 2*sqrt( e*x + d)*sqrt(d) + 2*d)/x) + 2*(6*B*b^3*c^3*e^4*x^4 - 15*A*b^2*c^4*d^4 + 2 *(21*B*b^3*c^3*d*e^3 - (7*B*b^4*c^2 - 5*A*b^3*c^3)*e^4)*x^3 + 2*(108*B*b^3 *c^3*d^2*e^2 - (109*B*b^4*c^2 - 65*A*b^3*c^3)*d*e^3 + 5*(7*B*b^5*c - 5*A*b ^4*c^2)*e^4)*x^2 + (15*(B*b^2*c^4 - 2*A*b*c^5)*d^4 - 60*(B*b^3*c^3 - A*b^2 *c^4)*d^3*e + 18*(17*B*b^4*c^2 - 5*A*b^3*c^3)*d^2*e^2 - 10*(32*B*b^5*c - 1 9*A*b^4*c^2)*d*e^3 + 15*(7*B*b^6 - 5*A*b^5*c)*e^4)*x)*sqrt(e*x + d))/(b^3* c^5*x^2 + b^4*c^4*x), 1/30*(30*((2*(B*b*c^5 - 2*A*c^6)*d^4 + (B*b^2*c^4 + 7*A*b*c^5)*d^3*e - 3*(5*B*b^3*c^3 - A*b^2*c^4)*d^2*e^2 + (19*B*b^4*c^2 - 1 1*A*b^3*c^3)*d*e^3 - (7*B*b^5*c - 5*A*b^4*c^2)*e^4)*x^2 + (2*(B*b^2*c^4 - 2*A*b*c^5)*d^4 + (B*b^3*c^3 + 7*A*b^2*c^4)*d^3*e - 3*(5*B*b^4*c^2 - A*b^3* c^3)*d^2*e^2 + (19*B*b^5*c - 11*A*b^4*c^2)*d*e^3 - (7*B*b^6 - 5*A*b^5*c)*e ^4)*x)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c...
Timed out. \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 834 vs. \(2 (358) = 716\).
Time = 0.30 (sec) , antiderivative size = 834, normalized size of antiderivative = 2.16 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {{\left (2 \, B b d^{5} - 4 \, A c d^{5} + 9 \, A b d^{4} e\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} - \frac {{\left (2 \, B b c^{5} d^{5} - 4 \, A c^{6} d^{5} - B b^{2} c^{4} d^{4} e + 11 \, A b c^{5} d^{4} e - 16 \, B b^{3} c^{3} d^{3} e^{2} - 4 \, A b^{2} c^{4} d^{3} e^{2} + 34 \, B b^{4} c^{2} d^{2} e^{3} - 14 \, A b^{3} c^{3} d^{2} e^{3} - 26 \, B b^{5} c d e^{4} + 16 \, A b^{4} c^{2} d e^{4} + 7 \, B b^{6} e^{5} - 5 \, A b^{5} c e^{5}\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3} c^{4}} + \frac {{\left (e x + d\right )}^{\frac {3}{2}} B b c^{4} d^{4} e - 2 \, {\left (e x + d\right )}^{\frac {3}{2}} A c^{5} d^{4} e - \sqrt {e x + d} B b c^{4} d^{5} e + 2 \, \sqrt {e x + d} A c^{5} d^{5} e - 4 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{2} c^{3} d^{3} e^{2} + 4 \, {\left (e x + d\right )}^{\frac {3}{2}} A b c^{4} d^{3} e^{2} + 4 \, \sqrt {e x + d} B b^{2} c^{3} d^{4} e^{2} - 5 \, \sqrt {e x + d} A b c^{4} d^{4} e^{2} + 6 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{3} c^{2} d^{2} e^{3} - 6 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{2} c^{3} d^{2} e^{3} - 6 \, \sqrt {e x + d} B b^{3} c^{2} d^{3} e^{3} + 6 \, \sqrt {e x + d} A b^{2} c^{3} d^{3} e^{3} - 4 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{4} c d e^{4} + 4 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{3} c^{2} d e^{4} + 4 \, \sqrt {e x + d} B b^{4} c d^{2} e^{4} - 4 \, \sqrt {e x + d} A b^{3} c^{2} d^{2} e^{4} + {\left (e x + d\right )}^{\frac {3}{2}} B b^{5} e^{5} - {\left (e x + d\right )}^{\frac {3}{2}} A b^{4} c e^{5} - \sqrt {e x + d} B b^{5} d e^{5} + \sqrt {e x + d} A b^{4} c d e^{5}}{{\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )} b^{2} c^{4}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{8} e^{2} + 15 \, {\left (e x + d\right )}^{\frac {3}{2}} B c^{8} d e^{2} + 90 \, \sqrt {e x + d} B c^{8} d^{2} e^{2} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} B b c^{7} e^{3} + 5 \, {\left (e x + d\right )}^{\frac {3}{2}} A c^{8} e^{3} - 120 \, \sqrt {e x + d} B b c^{7} d e^{3} + 60 \, \sqrt {e x + d} A c^{8} d e^{3} + 45 \, \sqrt {e x + d} B b^{2} c^{6} e^{4} - 30 \, \sqrt {e x + d} A b c^{7} e^{4}\right )}}{15 \, c^{10}} \]
(2*B*b*d^5 - 4*A*c*d^5 + 9*A*b*d^4*e)*arctan(sqrt(e*x + d)/sqrt(-d))/(b^3* sqrt(-d)) - (2*B*b*c^5*d^5 - 4*A*c^6*d^5 - B*b^2*c^4*d^4*e + 11*A*b*c^5*d^ 4*e - 16*B*b^3*c^3*d^3*e^2 - 4*A*b^2*c^4*d^3*e^2 + 34*B*b^4*c^2*d^2*e^3 - 14*A*b^3*c^3*d^2*e^3 - 26*B*b^5*c*d*e^4 + 16*A*b^4*c^2*d*e^4 + 7*B*b^6*e^5 - 5*A*b^5*c*e^5)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2* d + b*c*e)*b^3*c^4) + ((e*x + d)^(3/2)*B*b*c^4*d^4*e - 2*(e*x + d)^(3/2)*A *c^5*d^4*e - sqrt(e*x + d)*B*b*c^4*d^5*e + 2*sqrt(e*x + d)*A*c^5*d^5*e - 4 *(e*x + d)^(3/2)*B*b^2*c^3*d^3*e^2 + 4*(e*x + d)^(3/2)*A*b*c^4*d^3*e^2 + 4 *sqrt(e*x + d)*B*b^2*c^3*d^4*e^2 - 5*sqrt(e*x + d)*A*b*c^4*d^4*e^2 + 6*(e* x + d)^(3/2)*B*b^3*c^2*d^2*e^3 - 6*(e*x + d)^(3/2)*A*b^2*c^3*d^2*e^3 - 6*s qrt(e*x + d)*B*b^3*c^2*d^3*e^3 + 6*sqrt(e*x + d)*A*b^2*c^3*d^3*e^3 - 4*(e* x + d)^(3/2)*B*b^4*c*d*e^4 + 4*(e*x + d)^(3/2)*A*b^3*c^2*d*e^4 + 4*sqrt(e* x + d)*B*b^4*c*d^2*e^4 - 4*sqrt(e*x + d)*A*b^3*c^2*d^2*e^4 + (e*x + d)^(3/ 2)*B*b^5*e^5 - (e*x + d)^(3/2)*A*b^4*c*e^5 - sqrt(e*x + d)*B*b^5*d*e^5 + s qrt(e*x + d)*A*b^4*c*d*e^5)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e *x + d)*b*e - b*d*e)*b^2*c^4) + 2/15*(3*(e*x + d)^(5/2)*B*c^8*e^2 + 15*(e* x + d)^(3/2)*B*c^8*d*e^2 + 90*sqrt(e*x + d)*B*c^8*d^2*e^2 - 10*(e*x + d)^( 3/2)*B*b*c^7*e^3 + 5*(e*x + d)^(3/2)*A*c^8*e^3 - 120*sqrt(e*x + d)*B*b*c^7 *d*e^3 + 60*sqrt(e*x + d)*A*c^8*d*e^3 + 45*sqrt(e*x + d)*B*b^2*c^6*e^4 - 3 0*sqrt(e*x + d)*A*b*c^7*e^4)/c^10
Time = 16.33 (sec) , antiderivative size = 12636, normalized size of antiderivative = 32.74 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
atan(((((20*A*b^10*c^6*d*e^7 - 28*B*b^11*c^5*d*e^7 + 8*A*b^6*c^10*d^5*e^3 - 20*A*b^7*c^9*d^4*e^4 + 56*A*b^8*c^8*d^3*e^5 - 64*A*b^9*c^7*d^2*e^6 - 4*B *b^7*c^9*d^5*e^3 + 64*B*b^8*c^8*d^4*e^4 - 136*B*b^9*c^7*d^3*e^5 + 104*B*b^ 10*c^6*d^2*e^6)/(b^6*c^7) - (2*(4*b^7*c^9*e^3 - 8*b^6*c^10*d*e^2)*(d + e*x )^(1/2)*((16*A^2*c^11*d^9 - 49*B^2*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2*b ^2*c^9*d^9 + 81*A^2*b^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*d^6*e^3 - 315*A^2*b^ 4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 + 147*A^2*b^6*c^5*d^3*e^6 - 261*A^ 2*b^7*c^4*d^2*e^7 - 63*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 + 189 *B^2*b^6*c^5*d^5*e^4 - 819*B^2*b^7*c^4*d^4*e^5 + 1155*B^2*b^8*c^3*d^3*e^6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^10*d^8*e + 315*B^2*b^10*c*d*e^8 + 1 35*A^2*b^8*c^3*d*e^8 - 16*A*B*b*c^10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*B*b^2* c^9*d^8*e - 414*A*B*b^9*c^2*d*e^8 + 126*A*B*b^3*c^8*d^7*e^2 - 546*A*B*b^4* c^7*d^6*e^3 + 630*A*B*b^5*c^6*d^5*e^4 + 126*A*B*b^6*c^5*d^4*e^5 - 966*A*B* b^7*c^4*d^3*e^6 + 954*A*B*b^8*c^3*d^2*e^7)/(4*b^6*c^9))^(1/2))/(b^4*c^7))* ((16*A^2*c^11*d^9 - 49*B^2*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2*b^2*c^9*d ^9 + 81*A^2*b^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*d^6*e^3 - 315*A^2*b^4*c^7*d^ 5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 + 147*A^2*b^6*c^5*d^3*e^6 - 261*A^2*b^7*c^ 4*d^2*e^7 - 63*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 + 189*B^2*b^6 *c^5*d^5*e^4 - 819*B^2*b^7*c^4*d^4*e^5 + 1155*B^2*b^8*c^3*d^3*e^6 - 837*B^ 2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^10*d^8*e + 315*B^2*b^10*c*d*e^8 + 135*A^...