Integrand size = 31, antiderivative size = 128 \[ \int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=-\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{9/2}}{9 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{11/2}}{11 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{13/2}}{13 e^4}+\frac {2 b^2 B (d+e x)^{15/2}}{15 e^4} \]
-2/9*(-a*e+b*d)^2*(-A*e+B*d)*(e*x+d)^(9/2)/e^4+2/11*(-a*e+b*d)*(-2*A*b*e-B *a*e+3*B*b*d)*(e*x+d)^(11/2)/e^4-2/13*b*(-A*b*e-2*B*a*e+3*B*b*d)*(e*x+d)^( 13/2)/e^4+2/15*b^2*B*(e*x+d)^(15/2)/e^4
Time = 0.05 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.08 \[ \int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 (d+e x)^{9/2} \left (65 a^2 e^2 (-2 B d+11 A e+9 B e x)+10 a b e \left (13 A e (-2 d+9 e x)+B \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )+b^2 \left (5 A e \left (8 d^2-36 d e x+99 e^2 x^2\right )+B \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )\right )\right )}{6435 e^4} \]
(2*(d + e*x)^(9/2)*(65*a^2*e^2*(-2*B*d + 11*A*e + 9*B*e*x) + 10*a*b*e*(13* A*e*(-2*d + 9*e*x) + B*(8*d^2 - 36*d*e*x + 99*e^2*x^2)) + b^2*(5*A*e*(8*d^ 2 - 36*d*e*x + 99*e^2*x^2) + B*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429 *e^3*x^3))))/(6435*e^4)
Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1184, 27, 86, 2009}
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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right ) (A+B x) (d+e x)^{7/2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^2 (a+b x)^2 (A+B x) (d+e x)^{7/2}dx}{b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^2 (A+B x) (d+e x)^{7/2}dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b (d+e x)^{11/2} (2 a B e+A b e-3 b B d)}{e^3}+\frac {(d+e x)^{9/2} (a e-b d) (a B e+2 A b e-3 b B d)}{e^3}+\frac {(d+e x)^{7/2} (a e-b d)^2 (A e-B d)}{e^3}+\frac {b^2 B (d+e x)^{13/2}}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b (d+e x)^{13/2} (-2 a B e-A b e+3 b B d)}{13 e^4}+\frac {2 (d+e x)^{11/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{11 e^4}-\frac {2 (d+e x)^{9/2} (b d-a e)^2 (B d-A e)}{9 e^4}+\frac {2 b^2 B (d+e x)^{15/2}}{15 e^4}\) |
(-2*(b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(9/2))/(9*e^4) + (2*(b*d - a*e)*(3 *b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(11/2))/(11*e^4) - (2*b*(3*b*B*d - A*b *e - 2*a*B*e)*(d + e*x)^(13/2))/(13*e^4) + (2*b^2*B*(d + e*x)^(15/2))/(15* e^4)
3.18.84.3.1 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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.37 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\left (\frac {3}{5} B \,x^{3}+\frac {9}{13} A \,x^{2}\right ) b^{2}+\frac {18 \left (\frac {11 B x}{13}+A \right ) x a b}{11}+a^{2} \left (\frac {9 B x}{11}+A \right )\right ) e^{3}-\frac {4 d \left (\frac {9 x \left (\frac {11 B x}{10}+A \right ) b^{2}}{13}+a \left (\frac {18 B x}{13}+A \right ) b +\frac {B \,a^{2}}{2}\right ) e^{2}}{11}+\frac {8 b \left (\left (\frac {9 B x}{5}+A \right ) b +2 B a \right ) d^{2} e}{143}-\frac {16 B \,b^{2} d^{3}}{715}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9 e^{4}}\) | \(120\) |
| derivativedivides | \(\frac {\frac {2 B \,b^{2} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (A e -B d \right ) b^{2}+B \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (A e -B d \right ) \left (2 a b e -2 b^{2} d \right )+B \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (A e -B d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{4}}\) | \(148\) |
| default | \(\frac {\frac {2 B \,b^{2} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (A e -B d \right ) b^{2}+B \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (A e -B d \right ) \left (2 a b e -2 b^{2} d \right )+B \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (A e -B d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{4}}\) | \(148\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (429 B \,b^{2} x^{3} e^{3}+495 A \,b^{2} e^{3} x^{2}+990 B \,x^{2} a b \,e^{3}-198 B \,x^{2} b^{2} d \,e^{2}+1170 A x a b \,e^{3}-180 A x \,b^{2} d \,e^{2}+585 B x \,a^{2} e^{3}-360 B x a b d \,e^{2}+72 B x \,b^{2} d^{2} e +715 A \,a^{2} e^{3}-260 A a b d \,e^{2}+40 A \,b^{2} d^{2} e -130 B \,a^{2} d \,e^{2}+80 B a b \,d^{2} e -16 B \,b^{2} d^{3}\right )}{6435 e^{4}}\) | \(169\) |
| trager | \(\frac {2 \left (429 B \,e^{7} b^{2} x^{7}+495 A \,b^{2} e^{7} x^{6}+990 B a b \,e^{7} x^{6}+1518 B \,b^{2} d \,e^{6} x^{6}+1170 A a b \,e^{7} x^{5}+1800 A \,b^{2} d \,e^{6} x^{5}+585 B \,a^{2} e^{7} x^{5}+3600 B a b d \,e^{6} x^{5}+1854 B \,b^{2} d^{2} e^{5} x^{5}+715 A \,a^{2} e^{7} x^{4}+4420 A a b d \,e^{6} x^{4}+2290 A \,b^{2} d^{2} e^{5} x^{4}+2210 B \,a^{2} d \,e^{6} x^{4}+4580 B a b \,d^{2} e^{5} x^{4}+800 B \,b^{2} d^{3} e^{4} x^{4}+2860 A \,a^{2} d \,e^{6} x^{3}+5980 A a b \,d^{2} e^{5} x^{3}+1060 A \,b^{2} d^{3} e^{4} x^{3}+2990 B \,a^{2} d^{2} e^{5} x^{3}+2120 B a b \,d^{3} e^{4} x^{3}+5 B \,b^{2} d^{4} e^{3} x^{3}+4290 A \,a^{2} d^{2} e^{5} x^{2}+3120 A a b \,d^{3} e^{4} x^{2}+15 A \,b^{2} d^{4} e^{3} x^{2}+1560 B \,a^{2} d^{3} e^{4} x^{2}+30 B a b \,d^{4} e^{3} x^{2}-6 B \,b^{2} d^{5} e^{2} x^{2}+2860 A \,a^{2} d^{3} e^{4} x +130 A a b \,d^{4} e^{3} x -20 A \,b^{2} d^{5} e^{2} x +65 B \,a^{2} d^{4} e^{3} x -40 B a b \,d^{5} e^{2} x +8 B \,b^{2} d^{6} e x +715 A \,a^{2} d^{4} e^{3}-260 A a b \,d^{5} e^{2}+40 A \,b^{2} d^{6} e -130 B \,a^{2} d^{5} e^{2}+80 B a b \,d^{6} e -16 B \,b^{2} d^{7}\right ) \sqrt {e x +d}}{6435 e^{4}}\) | \(517\) |
| risch | \(\frac {2 \left (429 B \,e^{7} b^{2} x^{7}+495 A \,b^{2} e^{7} x^{6}+990 B a b \,e^{7} x^{6}+1518 B \,b^{2} d \,e^{6} x^{6}+1170 A a b \,e^{7} x^{5}+1800 A \,b^{2} d \,e^{6} x^{5}+585 B \,a^{2} e^{7} x^{5}+3600 B a b d \,e^{6} x^{5}+1854 B \,b^{2} d^{2} e^{5} x^{5}+715 A \,a^{2} e^{7} x^{4}+4420 A a b d \,e^{6} x^{4}+2290 A \,b^{2} d^{2} e^{5} x^{4}+2210 B \,a^{2} d \,e^{6} x^{4}+4580 B a b \,d^{2} e^{5} x^{4}+800 B \,b^{2} d^{3} e^{4} x^{4}+2860 A \,a^{2} d \,e^{6} x^{3}+5980 A a b \,d^{2} e^{5} x^{3}+1060 A \,b^{2} d^{3} e^{4} x^{3}+2990 B \,a^{2} d^{2} e^{5} x^{3}+2120 B a b \,d^{3} e^{4} x^{3}+5 B \,b^{2} d^{4} e^{3} x^{3}+4290 A \,a^{2} d^{2} e^{5} x^{2}+3120 A a b \,d^{3} e^{4} x^{2}+15 A \,b^{2} d^{4} e^{3} x^{2}+1560 B \,a^{2} d^{3} e^{4} x^{2}+30 B a b \,d^{4} e^{3} x^{2}-6 B \,b^{2} d^{5} e^{2} x^{2}+2860 A \,a^{2} d^{3} e^{4} x +130 A a b \,d^{4} e^{3} x -20 A \,b^{2} d^{5} e^{2} x +65 B \,a^{2} d^{4} e^{3} x -40 B a b \,d^{5} e^{2} x +8 B \,b^{2} d^{6} e x +715 A \,a^{2} d^{4} e^{3}-260 A a b \,d^{5} e^{2}+40 A \,b^{2} d^{6} e -130 B \,a^{2} d^{5} e^{2}+80 B a b \,d^{6} e -16 B \,b^{2} d^{7}\right ) \sqrt {e x +d}}{6435 e^{4}}\) | \(517\) |
2/9*(((3/5*B*x^3+9/13*A*x^2)*b^2+18/11*(11/13*B*x+A)*x*a*b+a^2*(9/11*B*x+A ))*e^3-4/11*d*(9/13*x*(11/10*B*x+A)*b^2+a*(18/13*B*x+A)*b+1/2*B*a^2)*e^2+8 /143*b*((9/5*B*x+A)*b+2*B*a)*d^2*e-16/715*B*b^2*d^3)*(e*x+d)^(9/2)/e^4
Leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (112) = 224\).
Time = 0.34 (sec) , antiderivative size = 424, normalized size of antiderivative = 3.31 \[ \int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 \, {\left (429 \, B b^{2} e^{7} x^{7} - 16 \, B b^{2} d^{7} + 715 \, A a^{2} d^{4} e^{3} + 40 \, {\left (2 \, B a b + A b^{2}\right )} d^{6} e - 130 \, {\left (B a^{2} + 2 \, A a b\right )} d^{5} e^{2} + 33 \, {\left (46 \, B b^{2} d e^{6} + 15 \, {\left (2 \, B a b + A b^{2}\right )} e^{7}\right )} x^{6} + 9 \, {\left (206 \, B b^{2} d^{2} e^{5} + 200 \, {\left (2 \, B a b + A b^{2}\right )} d e^{6} + 65 \, {\left (B a^{2} + 2 \, A a b\right )} e^{7}\right )} x^{5} + 5 \, {\left (160 \, B b^{2} d^{3} e^{4} + 143 \, A a^{2} e^{7} + 458 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{5} + 442 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{6}\right )} x^{4} + 5 \, {\left (B b^{2} d^{4} e^{3} + 572 \, A a^{2} d e^{6} + 212 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e^{4} + 598 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{5}\right )} x^{3} - 3 \, {\left (2 \, B b^{2} d^{5} e^{2} - 1430 \, A a^{2} d^{2} e^{5} - 5 \, {\left (2 \, B a b + A b^{2}\right )} d^{4} e^{3} - 520 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{4}\right )} x^{2} + {\left (8 \, B b^{2} d^{6} e + 2860 \, A a^{2} d^{3} e^{4} - 20 \, {\left (2 \, B a b + A b^{2}\right )} d^{5} e^{2} + 65 \, {\left (B a^{2} + 2 \, A a b\right )} d^{4} e^{3}\right )} x\right )} \sqrt {e x + d}}{6435 \, e^{4}} \]
2/6435*(429*B*b^2*e^7*x^7 - 16*B*b^2*d^7 + 715*A*a^2*d^4*e^3 + 40*(2*B*a*b + A*b^2)*d^6*e - 130*(B*a^2 + 2*A*a*b)*d^5*e^2 + 33*(46*B*b^2*d*e^6 + 15* (2*B*a*b + A*b^2)*e^7)*x^6 + 9*(206*B*b^2*d^2*e^5 + 200*(2*B*a*b + A*b^2)* d*e^6 + 65*(B*a^2 + 2*A*a*b)*e^7)*x^5 + 5*(160*B*b^2*d^3*e^4 + 143*A*a^2*e ^7 + 458*(2*B*a*b + A*b^2)*d^2*e^5 + 442*(B*a^2 + 2*A*a*b)*d*e^6)*x^4 + 5* (B*b^2*d^4*e^3 + 572*A*a^2*d*e^6 + 212*(2*B*a*b + A*b^2)*d^3*e^4 + 598*(B* a^2 + 2*A*a*b)*d^2*e^5)*x^3 - 3*(2*B*b^2*d^5*e^2 - 1430*A*a^2*d^2*e^5 - 5* (2*B*a*b + A*b^2)*d^4*e^3 - 520*(B*a^2 + 2*A*a*b)*d^3*e^4)*x^2 + (8*B*b^2* d^6*e + 2860*A*a^2*d^3*e^4 - 20*(2*B*a*b + A*b^2)*d^5*e^2 + 65*(B*a^2 + 2* A*a*b)*d^4*e^3)*x)*sqrt(e*x + d)/e^4
Leaf count of result is larger than twice the leaf count of optimal. 1020 vs. \(2 (129) = 258\).
Time = 0.63 (sec) , antiderivative size = 1020, normalized size of antiderivative = 7.97 \[ \int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\begin {cases} \frac {2 A a^{2} d^{4} \sqrt {d + e x}}{9 e} + \frac {8 A a^{2} d^{3} x \sqrt {d + e x}}{9} + \frac {4 A a^{2} d^{2} e x^{2} \sqrt {d + e x}}{3} + \frac {8 A a^{2} d e^{2} x^{3} \sqrt {d + e x}}{9} + \frac {2 A a^{2} e^{3} x^{4} \sqrt {d + e x}}{9} - \frac {8 A a b d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {4 A a b d^{4} x \sqrt {d + e x}}{99 e} + \frac {32 A a b d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {184 A a b d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {136 A a b d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {4 A a b e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {16 A b^{2} d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 A b^{2} d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 A b^{2} d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 A b^{2} d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 A b^{2} d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 A b^{2} d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 A b^{2} e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {4 B a^{2} d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {2 B a^{2} d^{4} x \sqrt {d + e x}}{99 e} + \frac {16 B a^{2} d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {92 B a^{2} d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {68 B a^{2} d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {2 B a^{2} e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {32 B a b d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {16 B a b d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {4 B a b d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {848 B a b d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {1832 B a b d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {160 B a b d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {4 B a b e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {32 B b^{2} d^{7} \sqrt {d + e x}}{6435 e^{4}} + \frac {16 B b^{2} d^{6} x \sqrt {d + e x}}{6435 e^{3}} - \frac {4 B b^{2} d^{5} x^{2} \sqrt {d + e x}}{2145 e^{2}} + \frac {2 B b^{2} d^{4} x^{3} \sqrt {d + e x}}{1287 e} + \frac {320 B b^{2} d^{3} x^{4} \sqrt {d + e x}}{1287} + \frac {412 B b^{2} d^{2} e x^{5} \sqrt {d + e x}}{715} + \frac {92 B b^{2} d e^{2} x^{6} \sqrt {d + e x}}{195} + \frac {2 B b^{2} e^{3} x^{7} \sqrt {d + e x}}{15} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (A a^{2} x + A a b x^{2} + \frac {A b^{2} x^{3}}{3} + \frac {B a^{2} x^{2}}{2} + \frac {2 B a b x^{3}}{3} + \frac {B b^{2} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*A*a**2*d**4*sqrt(d + e*x)/(9*e) + 8*A*a**2*d**3*x*sqrt(d + e* x)/9 + 4*A*a**2*d**2*e*x**2*sqrt(d + e*x)/3 + 8*A*a**2*d*e**2*x**3*sqrt(d + e*x)/9 + 2*A*a**2*e**3*x**4*sqrt(d + e*x)/9 - 8*A*a*b*d**5*sqrt(d + e*x) /(99*e**2) + 4*A*a*b*d**4*x*sqrt(d + e*x)/(99*e) + 32*A*a*b*d**3*x**2*sqrt (d + e*x)/33 + 184*A*a*b*d**2*e*x**3*sqrt(d + e*x)/99 + 136*A*a*b*d*e**2*x **4*sqrt(d + e*x)/99 + 4*A*a*b*e**3*x**5*sqrt(d + e*x)/11 + 16*A*b**2*d**6 *sqrt(d + e*x)/(1287*e**3) - 8*A*b**2*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2 *A*b**2*d**4*x**2*sqrt(d + e*x)/(429*e) + 424*A*b**2*d**3*x**3*sqrt(d + e* x)/1287 + 916*A*b**2*d**2*e*x**4*sqrt(d + e*x)/1287 + 80*A*b**2*d*e**2*x** 5*sqrt(d + e*x)/143 + 2*A*b**2*e**3*x**6*sqrt(d + e*x)/13 - 4*B*a**2*d**5* sqrt(d + e*x)/(99*e**2) + 2*B*a**2*d**4*x*sqrt(d + e*x)/(99*e) + 16*B*a**2 *d**3*x**2*sqrt(d + e*x)/33 + 92*B*a**2*d**2*e*x**3*sqrt(d + e*x)/99 + 68* B*a**2*d*e**2*x**4*sqrt(d + e*x)/99 + 2*B*a**2*e**3*x**5*sqrt(d + e*x)/11 + 32*B*a*b*d**6*sqrt(d + e*x)/(1287*e**3) - 16*B*a*b*d**5*x*sqrt(d + e*x)/ (1287*e**2) + 4*B*a*b*d**4*x**2*sqrt(d + e*x)/(429*e) + 848*B*a*b*d**3*x** 3*sqrt(d + e*x)/1287 + 1832*B*a*b*d**2*e*x**4*sqrt(d + e*x)/1287 + 160*B*a *b*d*e**2*x**5*sqrt(d + e*x)/143 + 4*B*a*b*e**3*x**6*sqrt(d + e*x)/13 - 32 *B*b**2*d**7*sqrt(d + e*x)/(6435*e**4) + 16*B*b**2*d**6*x*sqrt(d + e*x)/(6 435*e**3) - 4*B*b**2*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 2*B*b**2*d**4*x **3*sqrt(d + e*x)/(1287*e) + 320*B*b**2*d**3*x**4*sqrt(d + e*x)/1287 + ...
Time = 0.19 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.24 \[ \int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 \, {\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} B b^{2} - 495 \, {\left (3 \, B b^{2} d - {\left (2 \, B a b + A b^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 585 \, {\left (3 \, B b^{2} d^{2} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 715 \, {\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{6435 \, e^{4}} \]
2/6435*(429*(e*x + d)^(15/2)*B*b^2 - 495*(3*B*b^2*d - (2*B*a*b + A*b^2)*e) *(e*x + d)^(13/2) + 585*(3*B*b^2*d^2 - 2*(2*B*a*b + A*b^2)*d*e + (B*a^2 + 2*A*a*b)*e^2)*(e*x + d)^(11/2) - 715*(B*b^2*d^3 - A*a^2*e^3 - (2*B*a*b + A *b^2)*d^2*e + (B*a^2 + 2*A*a*b)*d*e^2)*(e*x + d)^(9/2))/e^4
Leaf count of result is larger than twice the leaf count of optimal. 1804 vs. \(2 (112) = 224\).
Time = 0.34 (sec) , antiderivative size = 1804, normalized size of antiderivative = 14.09 \[ \int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\text {Too large to display} \]
2/45045*(45045*sqrt(e*x + d)*A*a^2*d^4 + 60060*((e*x + d)^(3/2) - 3*sqrt(e *x + d)*d)*A*a^2*d^3 + 15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*B*a^2*d ^4/e + 30030*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a*b*d^4/e + 18018*(3* (e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a^2*d^2 + 6006*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B* a*b*d^4/e^2 + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*b^2*d^4/e^2 + 12012*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*a^2*d^3/e + 24024*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a*b*d^3/e + 5148*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*A *a^2*d + 1287*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/ 2)*d^2 - 35*sqrt(e*x + d)*d^3)*B*b^2*d^4/e^3 + 10296*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*B*a* b*d^3/e^2 + 5148*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^ (3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*A*b^2*d^3/e^2 + 7722*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*B* a^2*d^2/e + 15444*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d) ^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*A*a*b*d^2/e + 143*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*A*a^2 + 572*(35*(e*x + d)^(9/2) - 180*(e*x + ...
Time = 10.91 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.90 \[ \int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,A\,b^2\,e-6\,B\,b^2\,d+4\,B\,a\,b\,e\right )}{13\,e^4}+\frac {2\,B\,b^2\,{\left (d+e\,x\right )}^{15/2}}{15\,e^4}+\frac {2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b\,e+B\,a\,e-3\,B\,b\,d\right )}{11\,e^4}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4} \]