Integrand size = 33, antiderivative size = 218 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=-\frac {2 (b d-a e)^4 (B d-A e) (d+e x)^{7/2}}{7 e^6}+\frac {2 (b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)^{9/2}}{9 e^6}-\frac {4 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) (d+e x)^{11/2}}{11 e^6}+\frac {4 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^{13/2}}{13 e^6}-\frac {2 b^3 (5 b B d-A b e-4 a B e) (d+e x)^{15/2}}{15 e^6}+\frac {2 b^4 B (d+e x)^{17/2}}{17 e^6} \]
-2/7*(-a*e+b*d)^4*(-A*e+B*d)*(e*x+d)^(7/2)/e^6+2/9*(-a*e+b*d)^3*(-4*A*b*e- B*a*e+5*B*b*d)*(e*x+d)^(9/2)/e^6-4/11*b*(-a*e+b*d)^2*(-3*A*b*e-2*B*a*e+5*B *b*d)*(e*x+d)^(11/2)/e^6+4/13*b^2*(-a*e+b*d)*(-2*A*b*e-3*B*a*e+5*B*b*d)*(e *x+d)^(13/2)/e^6-2/15*b^3*(-A*b*e-4*B*a*e+5*B*b*d)*(e*x+d)^(15/2)/e^6+2/17 *b^4*B*(e*x+d)^(17/2)/e^6
Time = 0.28 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.56 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 (d+e x)^{7/2} \left (12155 a^4 e^4 (-2 B d+9 A e+7 B e x)+4420 a^3 b e^3 \left (11 A e (-2 d+7 e x)+B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-510 a^2 b^2 e^2 \left (-13 A e \left (8 d^2-28 d e x+63 e^2 x^2\right )+3 B \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )\right )+68 a b^3 e \left (15 A e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+B \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )+b^4 \left (17 A e \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )-5 B \left (256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right )\right )\right )}{765765 e^6} \]
(2*(d + e*x)^(7/2)*(12155*a^4*e^4*(-2*B*d + 9*A*e + 7*B*e*x) + 4420*a^3*b* e^3*(11*A*e*(-2*d + 7*e*x) + B*(8*d^2 - 28*d*e*x + 63*e^2*x^2)) - 510*a^2* b^2*e^2*(-13*A*e*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 3*B*(16*d^3 - 56*d^2*e* x + 126*d*e^2*x^2 - 231*e^3*x^3)) + 68*a*b^3*e*(15*A*e*(-16*d^3 + 56*d^2*e *x - 126*d*e^2*x^2 + 231*e^3*x^3) + B*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^ 2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)) + b^4*(17*A*e*(128*d^4 - 448*d^3*e *x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) - 5*B*(256*d^5 - 89 6*d^4*e*x + 2016*d^3*e^2*x^2 - 3696*d^2*e^3*x^3 + 6006*d*e^4*x^4 - 9009*e^ 5*x^5))))/(765765*e^6)
Time = 0.38 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^2 (A+B x) (d+e x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^4 (a+b x)^4 (A+B x) (d+e x)^{5/2}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^4 (A+B x) (d+e x)^{5/2}dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b^3 (d+e x)^{13/2} (4 a B e+A b e-5 b B d)}{e^5}-\frac {2 b^2 (d+e x)^{11/2} (b d-a e) (3 a B e+2 A b e-5 b B d)}{e^5}+\frac {2 b (d+e x)^{9/2} (b d-a e)^2 (2 a B e+3 A b e-5 b B d)}{e^5}+\frac {(d+e x)^{7/2} (a e-b d)^3 (a B e+4 A b e-5 b B d)}{e^5}+\frac {(d+e x)^{5/2} (a e-b d)^4 (A e-B d)}{e^5}+\frac {b^4 B (d+e x)^{15/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^3 (d+e x)^{15/2} (-4 a B e-A b e+5 b B d)}{15 e^6}+\frac {4 b^2 (d+e x)^{13/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{13 e^6}-\frac {4 b (d+e x)^{11/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{11 e^6}+\frac {2 (d+e x)^{9/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{9 e^6}-\frac {2 (d+e x)^{7/2} (b d-a e)^4 (B d-A e)}{7 e^6}+\frac {2 b^4 B (d+e x)^{17/2}}{17 e^6}\) |
(-2*(b*d - a*e)^4*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^6) + (2*(b*d - a*e)^3* (5*b*B*d - 4*A*b*e - a*B*e)*(d + e*x)^(9/2))/(9*e^6) - (4*b*(b*d - a*e)^2* (5*b*B*d - 3*A*b*e - 2*a*B*e)*(d + e*x)^(11/2))/(11*e^6) + (4*b^2*(b*d - a *e)*(5*b*B*d - 2*A*b*e - 3*a*B*e)*(d + e*x)^(13/2))/(13*e^6) - (2*b^3*(5*b *B*d - A*b*e - 4*a*B*e)*(d + e*x)^(15/2))/(15*e^6) + (2*b^4*B*(d + e*x)^(1 7/2))/(17*e^6)
3.18.93.3.1 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 = 0.34 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.30
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\left (\frac {7}{17} B \,x^{5}+\frac {7}{15} A \,x^{4}\right ) b^{4}+\frac {28 \left (\frac {13 B x}{15}+A \right ) x^{3} a \,b^{3}}{13}+\frac {42 \left (\frac {11 B x}{13}+A \right ) x^{2} a^{2} b^{2}}{11}+\frac {28 \left (\frac {9 B x}{11}+A \right ) x \,a^{3} b}{9}+a^{4} \left (\frac {7 B x}{9}+A \right )\right ) e^{5}-\frac {8 \left (\frac {21 \left (\frac {65 B x}{68}+A \right ) x^{3} b^{4}}{65}+\frac {189 x^{2} \left (\frac {44 B x}{45}+A \right ) a \,b^{3}}{143}+\frac {21 x \left (\frac {27 B x}{26}+A \right ) a^{2} b^{2}}{11}+a^{3} \left (\frac {14 B x}{11}+A \right ) b +\frac {B \,a^{4}}{4}\right ) d \,e^{4}}{9}+\frac {16 b \left (\frac {21 \left (\frac {55 B x}{51}+A \right ) x^{2} b^{3}}{65}+\frac {14 x \left (\frac {6 B x}{5}+A \right ) a \,b^{2}}{13}+a^{2} \left (\frac {21 B x}{13}+A \right ) b +\frac {2 B \,a^{3}}{3}\right ) d^{2} e^{3}}{33}-\frac {64 b^{2} d^{3} \left (\frac {7 x \left (\frac {45 B x}{34}+A \right ) b^{2}}{15}+a \left (\frac {28 B x}{15}+A \right ) b +\frac {3 B \,a^{2}}{2}\right ) e^{2}}{429}+\frac {128 b^{3} d^{4} \left (\left (\frac {35 B x}{17}+A \right ) b +4 B a \right ) e}{6435}-\frac {256 b^{4} B \,d^{5}}{21879}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7 e^{6}}\) | \(283\) |
| derivativedivides | \(\frac {\frac {2 b^{4} B \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (A e -B d \right ) b^{4}+2 B \left (2 a b e -2 b^{2} d \right ) b^{2}\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (2 \left (A e -B d \right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+B \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (A e -B d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 B \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (A e -B d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )+B \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (A e -B d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{6}}\) | \(352\) |
| default | \(\frac {\frac {2 b^{4} B \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (A e -B d \right ) b^{4}+2 B \left (2 a b e -2 b^{2} d \right ) b^{2}\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (2 \left (A e -B d \right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+B \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (A e -B d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 B \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (A e -B d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )+B \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (A e -B d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{6}}\) | \(352\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (45045 B \,x^{5} b^{4} e^{5}+51051 A \,b^{4} e^{5} x^{4}+204204 B \,x^{4} a \,b^{3} e^{5}-30030 B \,x^{4} b^{4} d \,e^{4}+235620 A \,x^{3} a \,b^{3} e^{5}-31416 A \,x^{3} b^{4} d \,e^{4}+353430 B \,x^{3} a^{2} b^{2} e^{5}-125664 B \,x^{3} a \,b^{3} d \,e^{4}+18480 B \,x^{3} b^{4} d^{2} e^{3}+417690 A \,x^{2} a^{2} b^{2} e^{5}-128520 A \,x^{2} a \,b^{3} d \,e^{4}+17136 A \,x^{2} b^{4} d^{2} e^{3}+278460 B \,x^{2} a^{3} b \,e^{5}-192780 B \,x^{2} a^{2} b^{2} d \,e^{4}+68544 B \,x^{2} a \,b^{3} d^{2} e^{3}-10080 B \,x^{2} b^{4} d^{3} e^{2}+340340 A x \,a^{3} b \,e^{5}-185640 A x \,a^{2} b^{2} d \,e^{4}+57120 A x a \,b^{3} d^{2} e^{3}-7616 A x \,b^{4} d^{3} e^{2}+85085 B x \,a^{4} e^{5}-123760 B x \,a^{3} b d \,e^{4}+85680 B x \,a^{2} b^{2} d^{2} e^{3}-30464 B x a \,b^{3} d^{3} e^{2}+4480 B x \,b^{4} d^{4} e +109395 A \,a^{4} e^{5}-97240 A \,a^{3} b d \,e^{4}+53040 A \,a^{2} b^{2} d^{2} e^{3}-16320 A a \,b^{3} d^{3} e^{2}+2176 A \,b^{4} d^{4} e -24310 B \,a^{4} d \,e^{4}+35360 B \,a^{3} b \,d^{2} e^{3}-24480 B \,a^{2} b^{2} d^{3} e^{2}+8704 B a \,b^{3} d^{4} e -1280 b^{4} B \,d^{5}\right )}{765765 e^{6}}\) | \(469\) |
| trager | \(\frac {2 \left (45045 B \,e^{8} b^{4} x^{8}+51051 A \,b^{4} e^{8} x^{7}+204204 B a \,b^{3} e^{8} x^{7}+105105 B \,b^{4} d \,e^{7} x^{7}+235620 A a \,b^{3} e^{8} x^{6}+121737 A \,b^{4} d \,e^{7} x^{6}+353430 B \,a^{2} b^{2} e^{8} x^{6}+486948 B a \,b^{3} d \,e^{7} x^{6}+63525 B \,b^{4} d^{2} e^{6} x^{6}+417690 A \,a^{2} b^{2} e^{8} x^{5}+578340 A a \,b^{3} d \,e^{7} x^{5}+76041 A \,b^{4} d^{2} e^{6} x^{5}+278460 B \,a^{3} b \,e^{8} x^{5}+867510 B \,a^{2} b^{2} d \,e^{7} x^{5}+304164 B a \,b^{3} d^{2} e^{6} x^{5}+315 B \,b^{4} d^{3} e^{5} x^{5}+340340 A \,a^{3} b \,e^{8} x^{4}+1067430 A \,a^{2} b^{2} d \,e^{7} x^{4}+378420 A a \,b^{3} d^{2} e^{6} x^{4}+595 A \,b^{4} d^{3} e^{5} x^{4}+85085 B \,a^{4} e^{8} x^{4}+711620 B \,a^{3} b d \,e^{7} x^{4}+567630 B \,a^{2} b^{2} d^{2} e^{6} x^{4}+2380 B a \,b^{3} d^{3} e^{5} x^{4}-350 B \,b^{4} d^{4} e^{4} x^{4}+109395 A \,a^{4} e^{8} x^{3}+923780 A \,a^{3} b d \,e^{7} x^{3}+749190 A \,a^{2} b^{2} d^{2} e^{6} x^{3}+5100 A a \,b^{3} d^{3} e^{5} x^{3}-680 A \,b^{4} d^{4} e^{4} x^{3}+230945 B \,a^{4} d \,e^{7} x^{3}+499460 B \,a^{3} b \,d^{2} e^{6} x^{3}+7650 B \,a^{2} b^{2} d^{3} e^{5} x^{3}-2720 B a \,b^{3} d^{4} e^{4} x^{3}+400 B \,b^{4} d^{5} e^{3} x^{3}+328185 A \,a^{4} d \,e^{7} x^{2}+729300 A \,a^{3} b \,d^{2} e^{6} x^{2}+19890 A \,a^{2} b^{2} d^{3} e^{5} x^{2}-6120 A a \,b^{3} d^{4} e^{4} x^{2}+816 A \,b^{4} d^{5} e^{3} x^{2}+182325 B \,a^{4} d^{2} e^{6} x^{2}+13260 B \,a^{3} b \,d^{3} e^{5} x^{2}-9180 B \,a^{2} b^{2} d^{4} e^{4} x^{2}+3264 B a \,b^{3} d^{5} e^{3} x^{2}-480 B \,b^{4} d^{6} e^{2} x^{2}+328185 A \,a^{4} d^{2} e^{6} x +48620 A \,a^{3} b \,d^{3} e^{5} x -26520 A \,a^{2} b^{2} d^{4} e^{4} x +8160 A a \,b^{3} d^{5} e^{3} x -1088 A \,b^{4} d^{6} e^{2} x +12155 B \,a^{4} d^{3} e^{5} x -17680 B \,a^{3} b \,d^{4} e^{4} x +12240 B \,a^{2} b^{2} d^{5} e^{3} x -4352 B a \,b^{3} d^{6} e^{2} x +640 B \,b^{4} d^{7} e x +109395 A \,a^{4} d^{3} e^{5}-97240 A \,a^{3} b \,d^{4} e^{4}+53040 A \,a^{2} b^{2} d^{5} e^{3}-16320 A a \,b^{3} d^{6} e^{2}+2176 A \,b^{4} d^{7} e -24310 B \,a^{4} d^{4} e^{4}+35360 B \,a^{3} b \,d^{5} e^{3}-24480 B \,a^{2} b^{2} d^{6} e^{2}+8704 B a \,b^{3} d^{7} e -1280 B \,b^{4} d^{8}\right ) \sqrt {e x +d}}{765765 e^{6}}\) | \(945\) |
| risch | \(\frac {2 \left (45045 B \,e^{8} b^{4} x^{8}+51051 A \,b^{4} e^{8} x^{7}+204204 B a \,b^{3} e^{8} x^{7}+105105 B \,b^{4} d \,e^{7} x^{7}+235620 A a \,b^{3} e^{8} x^{6}+121737 A \,b^{4} d \,e^{7} x^{6}+353430 B \,a^{2} b^{2} e^{8} x^{6}+486948 B a \,b^{3} d \,e^{7} x^{6}+63525 B \,b^{4} d^{2} e^{6} x^{6}+417690 A \,a^{2} b^{2} e^{8} x^{5}+578340 A a \,b^{3} d \,e^{7} x^{5}+76041 A \,b^{4} d^{2} e^{6} x^{5}+278460 B \,a^{3} b \,e^{8} x^{5}+867510 B \,a^{2} b^{2} d \,e^{7} x^{5}+304164 B a \,b^{3} d^{2} e^{6} x^{5}+315 B \,b^{4} d^{3} e^{5} x^{5}+340340 A \,a^{3} b \,e^{8} x^{4}+1067430 A \,a^{2} b^{2} d \,e^{7} x^{4}+378420 A a \,b^{3} d^{2} e^{6} x^{4}+595 A \,b^{4} d^{3} e^{5} x^{4}+85085 B \,a^{4} e^{8} x^{4}+711620 B \,a^{3} b d \,e^{7} x^{4}+567630 B \,a^{2} b^{2} d^{2} e^{6} x^{4}+2380 B a \,b^{3} d^{3} e^{5} x^{4}-350 B \,b^{4} d^{4} e^{4} x^{4}+109395 A \,a^{4} e^{8} x^{3}+923780 A \,a^{3} b d \,e^{7} x^{3}+749190 A \,a^{2} b^{2} d^{2} e^{6} x^{3}+5100 A a \,b^{3} d^{3} e^{5} x^{3}-680 A \,b^{4} d^{4} e^{4} x^{3}+230945 B \,a^{4} d \,e^{7} x^{3}+499460 B \,a^{3} b \,d^{2} e^{6} x^{3}+7650 B \,a^{2} b^{2} d^{3} e^{5} x^{3}-2720 B a \,b^{3} d^{4} e^{4} x^{3}+400 B \,b^{4} d^{5} e^{3} x^{3}+328185 A \,a^{4} d \,e^{7} x^{2}+729300 A \,a^{3} b \,d^{2} e^{6} x^{2}+19890 A \,a^{2} b^{2} d^{3} e^{5} x^{2}-6120 A a \,b^{3} d^{4} e^{4} x^{2}+816 A \,b^{4} d^{5} e^{3} x^{2}+182325 B \,a^{4} d^{2} e^{6} x^{2}+13260 B \,a^{3} b \,d^{3} e^{5} x^{2}-9180 B \,a^{2} b^{2} d^{4} e^{4} x^{2}+3264 B a \,b^{3} d^{5} e^{3} x^{2}-480 B \,b^{4} d^{6} e^{2} x^{2}+328185 A \,a^{4} d^{2} e^{6} x +48620 A \,a^{3} b \,d^{3} e^{5} x -26520 A \,a^{2} b^{2} d^{4} e^{4} x +8160 A a \,b^{3} d^{5} e^{3} x -1088 A \,b^{4} d^{6} e^{2} x +12155 B \,a^{4} d^{3} e^{5} x -17680 B \,a^{3} b \,d^{4} e^{4} x +12240 B \,a^{2} b^{2} d^{5} e^{3} x -4352 B a \,b^{3} d^{6} e^{2} x +640 B \,b^{4} d^{7} e x +109395 A \,a^{4} d^{3} e^{5}-97240 A \,a^{3} b \,d^{4} e^{4}+53040 A \,a^{2} b^{2} d^{5} e^{3}-16320 A a \,b^{3} d^{6} e^{2}+2176 A \,b^{4} d^{7} e -24310 B \,a^{4} d^{4} e^{4}+35360 B \,a^{3} b \,d^{5} e^{3}-24480 B \,a^{2} b^{2} d^{6} e^{2}+8704 B a \,b^{3} d^{7} e -1280 B \,b^{4} d^{8}\right ) \sqrt {e x +d}}{765765 e^{6}}\) | \(945\) |
2/7*(((7/17*B*x^5+7/15*A*x^4)*b^4+28/13*(13/15*B*x+A)*x^3*a*b^3+42/11*(11/ 13*B*x+A)*x^2*a^2*b^2+28/9*(9/11*B*x+A)*x*a^3*b+a^4*(7/9*B*x+A))*e^5-8/9*( 21/65*(65/68*B*x+A)*x^3*b^4+189/143*x^2*(44/45*B*x+A)*a*b^3+21/11*x*(27/26 *B*x+A)*a^2*b^2+a^3*(14/11*B*x+A)*b+1/4*B*a^4)*d*e^4+16/33*b*(21/65*(55/51 *B*x+A)*x^2*b^3+14/13*x*(6/5*B*x+A)*a*b^2+a^2*(21/13*B*x+A)*b+2/3*B*a^3)*d ^2*e^3-64/429*b^2*d^3*(7/15*x*(45/34*B*x+A)*b^2+a*(28/15*B*x+A)*b+3/2*B*a^ 2)*e^2+128/6435*b^3*d^4*((35/17*B*x+A)*b+4*B*a)*e-256/21879*b^4*B*d^5)*(e* x+d)^(7/2)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 772 vs. \(2 (194) = 388\).
Time = 0.33 (sec) , antiderivative size = 772, normalized size of antiderivative = 3.54 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (45045 \, B b^{4} e^{8} x^{8} - 1280 \, B b^{4} d^{8} + 109395 \, A a^{4} d^{3} e^{5} + 2176 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{7} e - 8160 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{6} e^{2} + 17680 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{5} e^{3} - 24310 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{4} e^{4} + 3003 \, {\left (35 \, B b^{4} d e^{7} + 17 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{8}\right )} x^{7} + 231 \, {\left (275 \, B b^{4} d^{2} e^{6} + 527 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{7} + 510 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{8}\right )} x^{6} + 63 \, {\left (5 \, B b^{4} d^{3} e^{5} + 1207 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{6} + 4590 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{7} + 2210 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{8}\right )} x^{5} - 35 \, {\left (10 \, B b^{4} d^{4} e^{4} - 17 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{5} - 5406 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{6} - 10166 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{7} - 2431 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{8}\right )} x^{4} + 5 \, {\left (80 \, B b^{4} d^{5} e^{3} + 21879 \, A a^{4} e^{8} - 136 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e^{4} + 510 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{5} + 49946 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{6} + 46189 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{7}\right )} x^{3} - 3 \, {\left (160 \, B b^{4} d^{6} e^{2} - 109395 \, A a^{4} d e^{7} - 272 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{5} e^{3} + 1020 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{4} e^{4} - 2210 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{3} e^{5} - 60775 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2} e^{6}\right )} x^{2} + {\left (640 \, B b^{4} d^{7} e + 328185 \, A a^{4} d^{2} e^{6} - 1088 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{6} e^{2} + 4080 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{5} e^{3} - 8840 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{4} e^{4} + 12155 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{3} e^{5}\right )} x\right )} \sqrt {e x + d}}{765765 \, e^{6}} \]
2/765765*(45045*B*b^4*e^8*x^8 - 1280*B*b^4*d^8 + 109395*A*a^4*d^3*e^5 + 21 76*(4*B*a*b^3 + A*b^4)*d^7*e - 8160*(3*B*a^2*b^2 + 2*A*a*b^3)*d^6*e^2 + 17 680*(2*B*a^3*b + 3*A*a^2*b^2)*d^5*e^3 - 24310*(B*a^4 + 4*A*a^3*b)*d^4*e^4 + 3003*(35*B*b^4*d*e^7 + 17*(4*B*a*b^3 + A*b^4)*e^8)*x^7 + 231*(275*B*b^4* d^2*e^6 + 527*(4*B*a*b^3 + A*b^4)*d*e^7 + 510*(3*B*a^2*b^2 + 2*A*a*b^3)*e^ 8)*x^6 + 63*(5*B*b^4*d^3*e^5 + 1207*(4*B*a*b^3 + A*b^4)*d^2*e^6 + 4590*(3* B*a^2*b^2 + 2*A*a*b^3)*d*e^7 + 2210*(2*B*a^3*b + 3*A*a^2*b^2)*e^8)*x^5 - 3 5*(10*B*b^4*d^4*e^4 - 17*(4*B*a*b^3 + A*b^4)*d^3*e^5 - 5406*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^6 - 10166*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^7 - 2431*(B*a^4 + 4*A*a^3*b)*e^8)*x^4 + 5*(80*B*b^4*d^5*e^3 + 21879*A*a^4*e^8 - 136*(4*B*a *b^3 + A*b^4)*d^4*e^4 + 510*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^5 + 49946*(2*B *a^3*b + 3*A*a^2*b^2)*d^2*e^6 + 46189*(B*a^4 + 4*A*a^3*b)*d*e^7)*x^3 - 3*( 160*B*b^4*d^6*e^2 - 109395*A*a^4*d*e^7 - 272*(4*B*a*b^3 + A*b^4)*d^5*e^3 + 1020*(3*B*a^2*b^2 + 2*A*a*b^3)*d^4*e^4 - 2210*(2*B*a^3*b + 3*A*a^2*b^2)*d ^3*e^5 - 60775*(B*a^4 + 4*A*a^3*b)*d^2*e^6)*x^2 + (640*B*b^4*d^7*e + 32818 5*A*a^4*d^2*e^6 - 1088*(4*B*a*b^3 + A*b^4)*d^6*e^2 + 4080*(3*B*a^2*b^2 + 2 *A*a*b^3)*d^5*e^3 - 8840*(2*B*a^3*b + 3*A*a^2*b^2)*d^4*e^4 + 12155*(B*a^4 + 4*A*a^3*b)*d^3*e^5)*x)*sqrt(e*x + d)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (221) = 442\).
Time = 1.84 (sec) , antiderivative size = 624, normalized size of antiderivative = 2.86 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {B b^{4} \left (d + e x\right )^{\frac {17}{2}}}{17 e^{5}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \left (A b^{4} e + 4 B a b^{3} e - 5 B b^{4} d\right )}{15 e^{5}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (4 A a b^{3} e^{2} - 4 A b^{4} d e + 6 B a^{2} b^{2} e^{2} - 16 B a b^{3} d e + 10 B b^{4} d^{2}\right )}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (6 A a^{2} b^{2} e^{3} - 12 A a b^{3} d e^{2} + 6 A b^{4} d^{2} e + 4 B a^{3} b e^{3} - 18 B a^{2} b^{2} d e^{2} + 24 B a b^{3} d^{2} e - 10 B b^{4} d^{3}\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (4 A a^{3} b e^{4} - 12 A a^{2} b^{2} d e^{3} + 12 A a b^{3} d^{2} e^{2} - 4 A b^{4} d^{3} e + B a^{4} e^{4} - 8 B a^{3} b d e^{3} + 18 B a^{2} b^{2} d^{2} e^{2} - 16 B a b^{3} d^{3} e + 5 B b^{4} d^{4}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A a^{4} e^{5} - 4 A a^{3} b d e^{4} + 6 A a^{2} b^{2} d^{2} e^{3} - 4 A a b^{3} d^{3} e^{2} + A b^{4} d^{4} e - B a^{4} d e^{4} + 4 B a^{3} b d^{2} e^{3} - 6 B a^{2} b^{2} d^{3} e^{2} + 4 B a b^{3} d^{4} e - B b^{4} d^{5}\right )}{7 e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (A a^{4} x + \frac {B b^{4} x^{6}}{6} + \frac {x^{5} \left (A b^{4} + 4 B a b^{3}\right )}{5} + \frac {x^{4} \cdot \left (4 A a b^{3} + 6 B a^{2} b^{2}\right )}{4} + \frac {x^{3} \cdot \left (6 A a^{2} b^{2} + 4 B a^{3} b\right )}{3} + \frac {x^{2} \cdot \left (4 A a^{3} b + B a^{4}\right )}{2}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(B*b**4*(d + e*x)**(17/2)/(17*e**5) + (d + e*x)**(15/2)*(A*b* *4*e + 4*B*a*b**3*e - 5*B*b**4*d)/(15*e**5) + (d + e*x)**(13/2)*(4*A*a*b** 3*e**2 - 4*A*b**4*d*e + 6*B*a**2*b**2*e**2 - 16*B*a*b**3*d*e + 10*B*b**4*d **2)/(13*e**5) + (d + e*x)**(11/2)*(6*A*a**2*b**2*e**3 - 12*A*a*b**3*d*e** 2 + 6*A*b**4*d**2*e + 4*B*a**3*b*e**3 - 18*B*a**2*b**2*d*e**2 + 24*B*a*b** 3*d**2*e - 10*B*b**4*d**3)/(11*e**5) + (d + e*x)**(9/2)*(4*A*a**3*b*e**4 - 12*A*a**2*b**2*d*e**3 + 12*A*a*b**3*d**2*e**2 - 4*A*b**4*d**3*e + B*a**4* e**4 - 8*B*a**3*b*d*e**3 + 18*B*a**2*b**2*d**2*e**2 - 16*B*a*b**3*d**3*e + 5*B*b**4*d**4)/(9*e**5) + (d + e*x)**(7/2)*(A*a**4*e**5 - 4*A*a**3*b*d*e* *4 + 6*A*a**2*b**2*d**2*e**3 - 4*A*a*b**3*d**3*e**2 + A*b**4*d**4*e - B*a* *4*d*e**4 + 4*B*a**3*b*d**2*e**3 - 6*B*a**2*b**2*d**3*e**2 + 4*B*a*b**3*d* *4*e - B*b**4*d**5)/(7*e**5))/e, Ne(e, 0)), (d**(5/2)*(A*a**4*x + B*b**4*x **6/6 + x**5*(A*b**4 + 4*B*a*b**3)/5 + x**4*(4*A*a*b**3 + 6*B*a**2*b**2)/4 + x**3*(6*A*a**2*b**2 + 4*B*a**3*b)/3 + x**2*(4*A*a**3*b + B*a**4)/2), Tr ue))
Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (194) = 388\).
Time = 0.19 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.88 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (45045 \, {\left (e x + d\right )}^{\frac {17}{2}} B b^{4} - 51051 \, {\left (5 \, B b^{4} d - {\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 117810 \, {\left (5 \, B b^{4} d^{2} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 139230 \, {\left (5 \, B b^{4} d^{3} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} - {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 85085 \, {\left (5 \, B b^{4} d^{4} - 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 109395 \, {\left (B b^{4} d^{5} - A a^{4} e^{5} - {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{765765 \, e^{6}} \]
2/765765*(45045*(e*x + d)^(17/2)*B*b^4 - 51051*(5*B*b^4*d - (4*B*a*b^3 + A *b^4)*e)*(e*x + d)^(15/2) + 117810*(5*B*b^4*d^2 - 2*(4*B*a*b^3 + A*b^4)*d* e + (3*B*a^2*b^2 + 2*A*a*b^3)*e^2)*(e*x + d)^(13/2) - 139230*(5*B*b^4*d^3 - 3*(4*B*a*b^3 + A*b^4)*d^2*e + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^2 - (2*B*a ^3*b + 3*A*a^2*b^2)*e^3)*(e*x + d)^(11/2) + 85085*(5*B*b^4*d^4 - 4*(4*B*a* b^3 + A*b^4)*d^3*e + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^2 - 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^3 + (B*a^4 + 4*A*a^3*b)*e^4)*(e*x + d)^(9/2) - 109395*(B* b^4*d^5 - A*a^4*e^5 - (4*B*a*b^3 + A*b^4)*d^4*e + 2*(3*B*a^2*b^2 + 2*A*a*b ^3)*d^3*e^2 - 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + (B*a^4 + 4*A*a^3*b)*d* e^4)*(e*x + d)^(7/2))/e^6
Leaf count of result is larger than twice the leaf count of optimal. 2697 vs. \(2 (194) = 388\).
Time = 0.31 (sec) , antiderivative size = 2697, normalized size of antiderivative = 12.37 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \]
2/765765*(765765*sqrt(e*x + d)*A*a^4*d^3 + 765765*((e*x + d)^(3/2) - 3*sqr t(e*x + d)*d)*A*a^4*d^2 + 255255*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*B*a ^4*d^3/e + 1021020*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^3*b*d^3/e + 1 53153*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A* a^4*d + 204204*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d )*d^2)*B*a^3*b*d^3/e^2 + 306306*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a^2*b^2*d^3/e^2 + 153153*(3*(e*x + d)^(5/2) - 10 *(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*a^4*d^2/e + 612612*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a^3*b*d^2/e + 21 879*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 3 5*sqrt(e*x + d)*d^3)*A*a^4 + 131274*(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*b^2*d^3/e^3 + 8 7516*(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^3*d^3/e^3 + 262548*(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^3*b*d^ 2/e^2 + 393822*(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*b^2*d^2/e^2 + 65637*(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^4*d/e + 262548*(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^3*b*d/e + 9724*(35*(e*x + d)^(...
Time = 10.86 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.90 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {{\left (d+e\,x\right )}^{15/2}\,\left (2\,A\,b^4\,e-10\,B\,b^4\,d+8\,B\,a\,b^3\,e\right )}{15\,e^6}+\frac {2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}\,\left (4\,A\,b\,e+B\,a\,e-5\,B\,b\,d\right )}{9\,e^6}+\frac {2\,B\,b^4\,{\left (d+e\,x\right )}^{17/2}}{17\,e^6}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}+\frac {4\,b\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{11/2}\,\left (3\,A\,b\,e+2\,B\,a\,e-5\,B\,b\,d\right )}{11\,e^6}+\frac {4\,b^2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{13/2}\,\left (2\,A\,b\,e+3\,B\,a\,e-5\,B\,b\,d\right )}{13\,e^6} \]
((d + e*x)^(15/2)*(2*A*b^4*e - 10*B*b^4*d + 8*B*a*b^3*e))/(15*e^6) + (2*(a *e - b*d)^3*(d + e*x)^(9/2)*(4*A*b*e + B*a*e - 5*B*b*d))/(9*e^6) + (2*B*b^ 4*(d + e*x)^(17/2))/(17*e^6) + (2*(A*e - B*d)*(a*e - b*d)^4*(d + e*x)^(7/2 ))/(7*e^6) + (4*b*(a*e - b*d)^2*(d + e*x)^(11/2)*(3*A*b*e + 2*B*a*e - 5*B* b*d))/(11*e^6) + (4*b^2*(a*e - b*d)*(d + e*x)^(13/2)*(2*A*b*e + 3*B*a*e - 5*B*b*d))/(13*e^6)