Integrand size = 22, antiderivative size = 128 \[ \int (a+b x)^2 (A+B x) (d+e x)^{5/2} \, dx=-\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{7/2}}{7 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{9/2}}{9 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{11/2}}{11 e^4}+\frac {2 b^2 B (d+e x)^{13/2}}{13 e^4} \]
-2/7*(-a*e+b*d)^2*(-A*e+B*d)*(e*x+d)^(7/2)/e^4+2/9*(-a*e+b*d)*(-2*A*b*e-B* a*e+3*B*b*d)*(e*x+d)^(9/2)/e^4-2/11*b*(-A*b*e-2*B*a*e+3*B*b*d)*(e*x+d)^(11 /2)/e^4+2/13*b^2*B*(e*x+d)^(13/2)/e^4
Time = 0.12 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.08 \[ \int (a+b x)^2 (A+B x) (d+e x)^{5/2} \, dx=\frac {2 (d+e x)^{7/2} \left (143 a^2 e^2 (-2 B d+9 A e+7 B e x)+26 a b e \left (11 A e (-2 d+7 e x)+B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )+b^2 \left (13 A e \left (8 d^2-28 d e x+63 e^2 x^2\right )+B \left (-48 d^3+168 d^2 e x-378 d e^2 x^2+693 e^3 x^3\right )\right )\right )}{9009 e^4} \]
(2*(d + e*x)^(7/2)*(143*a^2*e^2*(-2*B*d + 9*A*e + 7*B*e*x) + 26*a*b*e*(11* A*e*(-2*d + 7*e*x) + B*(8*d^2 - 28*d*e*x + 63*e^2*x^2)) + b^2*(13*A*e*(8*d ^2 - 28*d*e*x + 63*e^2*x^2) + B*(-48*d^3 + 168*d^2*e*x - 378*d*e^2*x^2 + 6 93*e^3*x^3))))/(9009*e^4)
Time = 0.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {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.
| \(\displaystyle \int (a+b x)^2 (A+B x) (d+e x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b (d+e x)^{9/2} (2 a B e+A b e-3 b B d)}{e^3}+\frac {(d+e x)^{7/2} (a e-b d) (a B e+2 A b e-3 b B d)}{e^3}+\frac {(d+e x)^{5/2} (a e-b d)^2 (A e-B d)}{e^3}+\frac {b^2 B (d+e x)^{11/2}}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b (d+e x)^{11/2} (-2 a B e-A b e+3 b B d)}{11 e^4}+\frac {2 (d+e x)^{9/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{9 e^4}-\frac {2 (d+e x)^{7/2} (b d-a e)^2 (B d-A e)}{7 e^4}+\frac {2 b^2 B (d+e x)^{13/2}}{13 e^4}\) |
(-2*(b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^4) + (2*(b*d - a*e)*(3 *b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(9/2))/(9*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(11/2))/(11*e^4) + (2*b^2*B*(d + e*x)^(13/2))/(13*e^ 4)
3.18.27.3.1 Defintions of rubi rules used
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]))))
Time = 0.97 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\left (\frac {7}{13} x^{3} B +\frac {7}{11} A \,x^{2}\right ) b^{2}+\frac {14 x \left (\frac {9 B x}{11}+A \right ) a b}{9}+a^{2} \left (\frac {7 B x}{9}+A \right )\right ) e^{3}-\frac {4 d \left (\frac {7 x \left (\frac {27 B x}{26}+A \right ) b^{2}}{11}+a \left (\frac {14 B x}{11}+A \right ) b +\frac {a^{2} B}{2}\right ) e^{2}}{9}+\frac {8 d^{2} b \left (\left (\frac {21 B x}{13}+A \right ) b +2 B a \right ) e}{99}-\frac {16 b^{2} B \,d^{3}}{429}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7 e^{4}}\) | \(120\) |
| derivativedivides | \(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (a e -b d \right ) b B +b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right )^{2} B +2 \left (a e -b d \right ) b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a e -b d \right )^{2} \left (A e -B d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{4}}\) | \(122\) |
| default | \(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (a e -b d \right ) b B +b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right )^{2} B +2 \left (a e -b d \right ) b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a e -b d \right )^{2} \left (A e -B d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{4}}\) | \(122\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (693 b^{2} B \,x^{3} e^{3}+819 A \,b^{2} e^{3} x^{2}+1638 B a b \,e^{3} x^{2}-378 B \,b^{2} d \,e^{2} x^{2}+2002 A a b \,e^{3} x -364 A \,b^{2} d \,e^{2} x +1001 B \,a^{2} e^{3} x -728 B a b d \,e^{2} x +168 B \,b^{2} d^{2} e x +1287 a^{2} A \,e^{3}-572 A a b d \,e^{2}+104 A \,b^{2} d^{2} e -286 B \,a^{2} d \,e^{2}+208 B a b \,d^{2} e -48 b^{2} B \,d^{3}\right )}{9009 e^{4}}\) | \(169\) |
| trager | \(\frac {2 \left (693 B \,b^{2} e^{6} x^{6}+819 A \,b^{2} e^{6} x^{5}+1638 B a b \,e^{6} x^{5}+1701 B \,b^{2} d \,e^{5} x^{5}+2002 A a b \,e^{6} x^{4}+2093 A \,b^{2} d \,e^{5} x^{4}+1001 B \,a^{2} e^{6} x^{4}+4186 B a b d \,e^{5} x^{4}+1113 B \,b^{2} d^{2} e^{4} x^{4}+1287 A \,a^{2} e^{6} x^{3}+5434 A a b d \,e^{5} x^{3}+1469 A \,b^{2} d^{2} e^{4} x^{3}+2717 B \,a^{2} d \,e^{5} x^{3}+2938 B a b \,d^{2} e^{4} x^{3}+15 B \,b^{2} d^{3} e^{3} x^{3}+3861 A \,a^{2} d \,e^{5} x^{2}+4290 A a b \,d^{2} e^{4} x^{2}+39 A \,b^{2} d^{3} e^{3} x^{2}+2145 B \,a^{2} d^{2} e^{4} x^{2}+78 B a b \,d^{3} e^{3} x^{2}-18 B \,b^{2} d^{4} e^{2} x^{2}+3861 A \,a^{2} d^{2} e^{4} x +286 A a b \,d^{3} e^{3} x -52 A \,b^{2} d^{4} e^{2} x +143 B \,a^{2} d^{3} e^{3} x -104 B a b \,d^{4} e^{2} x +24 B \,b^{2} d^{5} e x +1287 A \,a^{2} d^{3} e^{3}-572 A a b \,d^{4} e^{2}+104 A \,b^{2} d^{5} e -286 B \,a^{2} d^{4} e^{2}+208 B a b \,d^{5} e -48 B \,b^{2} d^{6}\right ) \sqrt {e x +d}}{9009 e^{4}}\) | \(429\) |
| risch | \(\frac {2 \left (693 B \,b^{2} e^{6} x^{6}+819 A \,b^{2} e^{6} x^{5}+1638 B a b \,e^{6} x^{5}+1701 B \,b^{2} d \,e^{5} x^{5}+2002 A a b \,e^{6} x^{4}+2093 A \,b^{2} d \,e^{5} x^{4}+1001 B \,a^{2} e^{6} x^{4}+4186 B a b d \,e^{5} x^{4}+1113 B \,b^{2} d^{2} e^{4} x^{4}+1287 A \,a^{2} e^{6} x^{3}+5434 A a b d \,e^{5} x^{3}+1469 A \,b^{2} d^{2} e^{4} x^{3}+2717 B \,a^{2} d \,e^{5} x^{3}+2938 B a b \,d^{2} e^{4} x^{3}+15 B \,b^{2} d^{3} e^{3} x^{3}+3861 A \,a^{2} d \,e^{5} x^{2}+4290 A a b \,d^{2} e^{4} x^{2}+39 A \,b^{2} d^{3} e^{3} x^{2}+2145 B \,a^{2} d^{2} e^{4} x^{2}+78 B a b \,d^{3} e^{3} x^{2}-18 B \,b^{2} d^{4} e^{2} x^{2}+3861 A \,a^{2} d^{2} e^{4} x +286 A a b \,d^{3} e^{3} x -52 A \,b^{2} d^{4} e^{2} x +143 B \,a^{2} d^{3} e^{3} x -104 B a b \,d^{4} e^{2} x +24 B \,b^{2} d^{5} e x +1287 A \,a^{2} d^{3} e^{3}-572 A a b \,d^{4} e^{2}+104 A \,b^{2} d^{5} e -286 B \,a^{2} d^{4} e^{2}+208 B a b \,d^{5} e -48 B \,b^{2} d^{6}\right ) \sqrt {e x +d}}{9009 e^{4}}\) | \(429\) |
2/7*(((7/13*x^3*B+7/11*A*x^2)*b^2+14/9*x*(9/11*B*x+A)*a*b+a^2*(7/9*B*x+A)) *e^3-4/9*d*(7/11*x*(27/26*B*x+A)*b^2+a*(14/11*B*x+A)*b+1/2*a^2*B)*e^2+8/99 *d^2*b*((21/13*B*x+A)*b+2*B*a)*e-16/429*b^2*B*d^3)*(e*x+d)^(7/2)/e^4
Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (112) = 224\).
Time = 0.22 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.78 \[ \int (a+b x)^2 (A+B x) (d+e x)^{5/2} \, dx=\frac {2 \, {\left (693 \, B b^{2} e^{6} x^{6} - 48 \, B b^{2} d^{6} + 1287 \, A a^{2} d^{3} e^{3} + 104 \, {\left (2 \, B a b + A b^{2}\right )} d^{5} e - 286 \, {\left (B a^{2} + 2 \, A a b\right )} d^{4} e^{2} + 63 \, {\left (27 \, B b^{2} d e^{5} + 13 \, {\left (2 \, B a b + A b^{2}\right )} e^{6}\right )} x^{5} + 7 \, {\left (159 \, B b^{2} d^{2} e^{4} + 299 \, {\left (2 \, B a b + A b^{2}\right )} d e^{5} + 143 \, {\left (B a^{2} + 2 \, A a b\right )} e^{6}\right )} x^{4} + {\left (15 \, B b^{2} d^{3} e^{3} + 1287 \, A a^{2} e^{6} + 1469 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{4} + 2717 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{5}\right )} x^{3} - 3 \, {\left (6 \, B b^{2} d^{4} e^{2} - 1287 \, A a^{2} d e^{5} - 13 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e^{3} - 715 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{4}\right )} x^{2} + {\left (24 \, B b^{2} d^{5} e + 3861 \, A a^{2} d^{2} e^{4} - 52 \, {\left (2 \, B a b + A b^{2}\right )} d^{4} e^{2} + 143 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{3}\right )} x\right )} \sqrt {e x + d}}{9009 \, e^{4}} \]
2/9009*(693*B*b^2*e^6*x^6 - 48*B*b^2*d^6 + 1287*A*a^2*d^3*e^3 + 104*(2*B*a *b + A*b^2)*d^5*e - 286*(B*a^2 + 2*A*a*b)*d^4*e^2 + 63*(27*B*b^2*d*e^5 + 1 3*(2*B*a*b + A*b^2)*e^6)*x^5 + 7*(159*B*b^2*d^2*e^4 + 299*(2*B*a*b + A*b^2 )*d*e^5 + 143*(B*a^2 + 2*A*a*b)*e^6)*x^4 + (15*B*b^2*d^3*e^3 + 1287*A*a^2* e^6 + 1469*(2*B*a*b + A*b^2)*d^2*e^4 + 2717*(B*a^2 + 2*A*a*b)*d*e^5)*x^3 - 3*(6*B*b^2*d^4*e^2 - 1287*A*a^2*d*e^5 - 13*(2*B*a*b + A*b^2)*d^3*e^3 - 71 5*(B*a^2 + 2*A*a*b)*d^2*e^4)*x^2 + (24*B*b^2*d^5*e + 3861*A*a^2*d^2*e^4 - 52*(2*B*a*b + A*b^2)*d^4*e^2 + 143*(B*a^2 + 2*A*a*b)*d^3*e^3)*x)*sqrt(e*x + d)/e^4
Leaf count of result is larger than twice the leaf count of optimal. 857 vs. \(2 (129) = 258\).
Time = 0.49 (sec) , antiderivative size = 857, normalized size of antiderivative = 6.70 \[ \int (a+b x)^2 (A+B x) (d+e x)^{5/2} \, dx=\begin {cases} \frac {2 A a^{2} d^{3} \sqrt {d + e x}}{7 e} + \frac {6 A a^{2} d^{2} x \sqrt {d + e x}}{7} + \frac {6 A a^{2} d e x^{2} \sqrt {d + e x}}{7} + \frac {2 A a^{2} e^{2} x^{3} \sqrt {d + e x}}{7} - \frac {8 A a b d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {4 A a b d^{3} x \sqrt {d + e x}}{63 e} + \frac {20 A a b d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {76 A a b d e x^{3} \sqrt {d + e x}}{63} + \frac {4 A a b e^{2} x^{4} \sqrt {d + e x}}{9} + \frac {16 A b^{2} d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {8 A b^{2} d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {2 A b^{2} d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {226 A b^{2} d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {46 A b^{2} d e x^{4} \sqrt {d + e x}}{99} + \frac {2 A b^{2} e^{2} x^{5} \sqrt {d + e x}}{11} - \frac {4 B a^{2} d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {2 B a^{2} d^{3} x \sqrt {d + e x}}{63 e} + \frac {10 B a^{2} d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {38 B a^{2} d e x^{3} \sqrt {d + e x}}{63} + \frac {2 B a^{2} e^{2} x^{4} \sqrt {d + e x}}{9} + \frac {32 B a b d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {16 B a b d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {4 B a b d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {452 B a b d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {92 B a b d e x^{4} \sqrt {d + e x}}{99} + \frac {4 B a b e^{2} x^{5} \sqrt {d + e x}}{11} - \frac {32 B b^{2} d^{6} \sqrt {d + e x}}{3003 e^{4}} + \frac {16 B b^{2} d^{5} x \sqrt {d + e x}}{3003 e^{3}} - \frac {4 B b^{2} d^{4} x^{2} \sqrt {d + e x}}{1001 e^{2}} + \frac {10 B b^{2} d^{3} x^{3} \sqrt {d + e x}}{3003 e} + \frac {106 B b^{2} d^{2} x^{4} \sqrt {d + e x}}{429} + \frac {54 B b^{2} d e x^{5} \sqrt {d + e x}}{143} + \frac {2 B b^{2} e^{2} x^{6} \sqrt {d + e x}}{13} & \text {for}\: e \neq 0 \\d^{\frac {5}{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**3*sqrt(d + e*x)/(7*e) + 6*A*a**2*d**2*x*sqrt(d + e* x)/7 + 6*A*a**2*d*e*x**2*sqrt(d + e*x)/7 + 2*A*a**2*e**2*x**3*sqrt(d + e*x )/7 - 8*A*a*b*d**4*sqrt(d + e*x)/(63*e**2) + 4*A*a*b*d**3*x*sqrt(d + e*x)/ (63*e) + 20*A*a*b*d**2*x**2*sqrt(d + e*x)/21 + 76*A*a*b*d*e*x**3*sqrt(d + e*x)/63 + 4*A*a*b*e**2*x**4*sqrt(d + e*x)/9 + 16*A*b**2*d**5*sqrt(d + e*x) /(693*e**3) - 8*A*b**2*d**4*x*sqrt(d + e*x)/(693*e**2) + 2*A*b**2*d**3*x** 2*sqrt(d + e*x)/(231*e) + 226*A*b**2*d**2*x**3*sqrt(d + e*x)/693 + 46*A*b* *2*d*e*x**4*sqrt(d + e*x)/99 + 2*A*b**2*e**2*x**5*sqrt(d + e*x)/11 - 4*B*a **2*d**4*sqrt(d + e*x)/(63*e**2) + 2*B*a**2*d**3*x*sqrt(d + e*x)/(63*e) + 10*B*a**2*d**2*x**2*sqrt(d + e*x)/21 + 38*B*a**2*d*e*x**3*sqrt(d + e*x)/63 + 2*B*a**2*e**2*x**4*sqrt(d + e*x)/9 + 32*B*a*b*d**5*sqrt(d + e*x)/(693*e **3) - 16*B*a*b*d**4*x*sqrt(d + e*x)/(693*e**2) + 4*B*a*b*d**3*x**2*sqrt(d + e*x)/(231*e) + 452*B*a*b*d**2*x**3*sqrt(d + e*x)/693 + 92*B*a*b*d*e*x** 4*sqrt(d + e*x)/99 + 4*B*a*b*e**2*x**5*sqrt(d + e*x)/11 - 32*B*b**2*d**6*s qrt(d + e*x)/(3003*e**4) + 16*B*b**2*d**5*x*sqrt(d + e*x)/(3003*e**3) - 4* B*b**2*d**4*x**2*sqrt(d + e*x)/(1001*e**2) + 10*B*b**2*d**3*x**3*sqrt(d + e*x)/(3003*e) + 106*B*b**2*d**2*x**4*sqrt(d + e*x)/429 + 54*B*b**2*d*e*x** 5*sqrt(d + e*x)/143 + 2*B*b**2*e**2*x**6*sqrt(d + e*x)/13, Ne(e, 0)), (d** (5/2)*(A*a**2*x + A*a*b*x**2 + A*b**2*x**3/3 + B*a**2*x**2/2 + 2*B*a*b*x** 3/3 + B*b**2*x**4/4), True))
Time = 0.19 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.24 \[ \int (a+b x)^2 (A+B x) (d+e x)^{5/2} \, dx=\frac {2 \, {\left (693 \, {\left (e x + d\right )}^{\frac {13}{2}} B b^{2} - 819 \, {\left (3 \, B b^{2} d - {\left (2 \, B a b + A b^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 1001 \, {\left (3 \, B b^{2} d^{2} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 1287 \, {\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{9009 \, e^{4}} \]
2/9009*(693*(e*x + d)^(13/2)*B*b^2 - 819*(3*B*b^2*d - (2*B*a*b + A*b^2)*e) *(e*x + d)^(11/2) + 1001*(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)^(9/2) - 1287*(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)^(7/2))/e^4
Leaf count of result is larger than twice the leaf count of optimal. 1293 vs. \(2 (112) = 224\).
Time = 0.30 (sec) , antiderivative size = 1293, normalized size of antiderivative = 10.10 \[ \int (a+b x)^2 (A+B x) (d+e x)^{5/2} \, dx=\text {Too large to display} \]
2/45045*(45045*sqrt(e*x + d)*A*a^2*d^3 + 45045*((e*x + d)^(3/2) - 3*sqrt(e *x + d)*d)*A*a^2*d^2 + 15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*B*a^2*d ^3/e + 30030*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a*b*d^3/e + 9009*(3*( e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a^2*d + 60 06*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*a*b *d^3/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^3/e^2 + 9009*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 1 5*sqrt(e*x + d)*d^2)*B*a^2*d^2/e + 18018*(3*(e*x + d)^(5/2) - 10*(e*x + d) ^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a*b*d^2/e + 1287*(5*(e*x + d)^(7/2) - 2 1*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*A*a^2 + 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^3/e^3 + 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*b*d^2/e ^2 + 3861*(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^2/e^2 + 3861*(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/e + 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)*A*a*b*d/e + 429*(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)*B*b^2*d^2/e^3 + 858*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(...
Time = 1.36 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.90 \[ \int (a+b x)^2 (A+B x) (d+e x)^{5/2} \, dx=\frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b^2\,e-6\,B\,b^2\,d+4\,B\,a\,b\,e\right )}{11\,e^4}+\frac {2\,B\,b^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^4}+\frac {2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b\,e+B\,a\,e-3\,B\,b\,d\right )}{9\,e^4}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4} \]