Integrand size = 22, antiderivative size = 173 \[ \int (a+b x)^3 (A+B x) (d+e x)^{5/2} \, dx=\frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{7/2}}{7 e^5}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{9/2}}{9 e^5}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{11/2}}{11 e^5}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{13/2}}{13 e^5}+\frac {2 b^3 B (d+e x)^{15/2}}{15 e^5} \] Output:
2/7*(-a*e+b*d)^3*(-A*e+B*d)*(e*x+d)^(7/2)/e^5-2/9*(-a*e+b*d)^2*(-3*A*b*e-B *a*e+4*B*b*d)*(e*x+d)^(9/2)/e^5+6/11*b*(-a*e+b*d)*(-A*b*e-B*a*e+2*B*b*d)*( e*x+d)^(11/2)/e^5-2/13*b^2*(-A*b*e-3*B*a*e+4*B*b*d)*(e*x+d)^(13/2)/e^5+2/1 5*b^3*B*(e*x+d)^(15/2)/e^5
Time = 0.16 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.32 \[ \int (a+b x)^3 (A+B x) (d+e x)^{5/2} \, dx=\frac {2 (d+e x)^{7/2} \left (715 a^3 e^3 (-2 B d+9 A e+7 B e x)+195 a^2 b e^2 \left (11 A e (-2 d+7 e x)+B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-15 a b^2 e \left (-13 A e \left (8 d^2-28 d e x+63 e^2 x^2\right )+3 B \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )\right )+b^3 \left (15 A e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+B \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )\right )}{45045 e^5} \] Input:
Integrate[(a + b*x)^3*(A + B*x)*(d + e*x)^(5/2),x]
Output:
(2*(d + e*x)^(7/2)*(715*a^3*e^3*(-2*B*d + 9*A*e + 7*B*e*x) + 195*a^2*b*e^2 *(11*A*e*(-2*d + 7*e*x) + B*(8*d^2 - 28*d*e*x + 63*e^2*x^2)) - 15*a*b^2*e* (-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)) + b^3*(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))))/(45045*e^5)
Time = 0.36 (sec) , antiderivative size = 173, 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.
| \(\displaystyle \int (a+b x)^3 (A+B x) (d+e x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b^2 (d+e x)^{11/2} (3 a B e+A b e-4 b B d)}{e^4}-\frac {3 b (d+e x)^{9/2} (b d-a e) (a B e+A b e-2 b B d)}{e^4}+\frac {(d+e x)^{7/2} (a e-b d)^2 (a B e+3 A b e-4 b B d)}{e^4}+\frac {(d+e x)^{5/2} (a e-b d)^3 (A e-B d)}{e^4}+\frac {b^3 B (d+e x)^{13/2}}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^2 (d+e x)^{13/2} (-3 a B e-A b e+4 b B d)}{13 e^5}+\frac {6 b (d+e x)^{11/2} (b d-a e) (-a B e-A b e+2 b B d)}{11 e^5}-\frac {2 (d+e x)^{9/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{9 e^5}+\frac {2 (d+e x)^{7/2} (b d-a e)^3 (B d-A e)}{7 e^5}+\frac {2 b^3 B (d+e x)^{15/2}}{15 e^5}\) |
Input:
Int[(a + b*x)^3*(A + B*x)*(d + e*x)^(5/2),x]
Output:
(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^5) - (2*(b*d - a*e)^2*( 4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)^(9/2))/(9*e^5) + (6*b*(b*d - a*e)*(2* b*B*d - A*b*e - a*B*e)*(d + e*x)^(11/2))/(11*e^5) - (2*b^2*(4*b*B*d - A*b* e - 3*a*B*e)*(d + e*x)^(13/2))/(13*e^5) + (2*b^3*B*(d + e*x)^(15/2))/(15*e ^5)
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 = 0.75 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.99
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (3 \left (a e -d b \right ) b^{2} B +b^{3} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (3 \left (a e -d b \right )^{2} b B +3 \left (a e -d b \right ) b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -d b \right )^{3} B +3 \left (a e -d b \right )^{2} b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a e -d b \right )^{3} \left (A e -B d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{5}}\) | \(171\) |
| default | \(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (3 \left (a e -d b \right ) b^{2} B +b^{3} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (3 \left (a e -d b \right )^{2} b B +3 \left (a e -d b \right ) b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -d b \right )^{3} B +3 \left (a e -d b \right )^{2} b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a e -d b \right )^{3} \left (A e -B d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{5}}\) | \(171\) |
| pseudoelliptic | \(\frac {2 \left (\left (\left (\frac {7}{15} B \,x^{4}+\frac {7}{13} A \,x^{3}\right ) b^{3}+\frac {21 a \left (\frac {11 B x}{13}+A \right ) x^{2} b^{2}}{11}+\frac {7 a^{2} x \left (\frac {9 B x}{11}+A \right ) b}{3}+a^{3} \left (\frac {7 B x}{9}+A \right )\right ) e^{4}-\frac {2 \left (\frac {63 \left (\frac {44 B x}{45}+A \right ) x^{2} b^{3}}{143}+\frac {14 a \left (\frac {27 B x}{26}+A \right ) x \,b^{2}}{11}+a^{2} \left (\frac {14 B x}{11}+A \right ) b +\frac {a^{3} B}{3}\right ) d \,e^{3}}{3}+\frac {8 b \,d^{2} \left (\frac {7 \left (\frac {6 B x}{5}+A \right ) x \,b^{2}}{13}+a \left (\frac {21 B x}{13}+A \right ) b +a^{2} B \right ) e^{2}}{33}-\frac {16 \left (\left (\frac {28 B x}{15}+A \right ) b +3 B a \right ) b^{2} d^{3} e}{429}+\frac {128 b^{3} B \,d^{4}}{6435}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7 e^{5}}\) | \(192\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 B \,x^{4} b^{3} e^{4}+3465 A \,x^{3} b^{3} e^{4}+10395 B \,x^{3} a \,b^{2} e^{4}-1848 B \,x^{3} b^{3} d \,e^{3}+12285 A \,x^{2} a \,b^{2} e^{4}-1890 A \,x^{2} b^{3} d \,e^{3}+12285 B \,x^{2} a^{2} b \,e^{4}-5670 B \,x^{2} a \,b^{2} d \,e^{3}+1008 B \,x^{2} b^{3} d^{2} e^{2}+15015 A x \,a^{2} b \,e^{4}-5460 A x a \,b^{2} d \,e^{3}+840 A x \,b^{3} d^{2} e^{2}+5005 B x \,a^{3} e^{4}-5460 B x \,a^{2} b d \,e^{3}+2520 B x a \,b^{2} d^{2} e^{2}-448 B x \,b^{3} d^{3} e +6435 a^{3} A \,e^{4}-4290 A \,a^{2} b d \,e^{3}+1560 A a \,b^{2} d^{2} e^{2}-240 A \,b^{3} d^{3} e -1430 B \,a^{3} d \,e^{3}+1560 B \,a^{2} b \,d^{2} e^{2}-720 B a \,b^{2} d^{3} e +128 b^{3} B \,d^{4}\right )}{45045 e^{5}}\) | \(301\) |
| orering | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 B \,x^{4} b^{3} e^{4}+3465 A \,x^{3} b^{3} e^{4}+10395 B \,x^{3} a \,b^{2} e^{4}-1848 B \,x^{3} b^{3} d \,e^{3}+12285 A \,x^{2} a \,b^{2} e^{4}-1890 A \,x^{2} b^{3} d \,e^{3}+12285 B \,x^{2} a^{2} b \,e^{4}-5670 B \,x^{2} a \,b^{2} d \,e^{3}+1008 B \,x^{2} b^{3} d^{2} e^{2}+15015 A x \,a^{2} b \,e^{4}-5460 A x a \,b^{2} d \,e^{3}+840 A x \,b^{3} d^{2} e^{2}+5005 B x \,a^{3} e^{4}-5460 B x \,a^{2} b d \,e^{3}+2520 B x a \,b^{2} d^{2} e^{2}-448 B x \,b^{3} d^{3} e +6435 a^{3} A \,e^{4}-4290 A \,a^{2} b d \,e^{3}+1560 A a \,b^{2} d^{2} e^{2}-240 A \,b^{3} d^{3} e -1430 B \,a^{3} d \,e^{3}+1560 B \,a^{2} b \,d^{2} e^{2}-720 B a \,b^{2} d^{3} e +128 b^{3} B \,d^{4}\right )}{45045 e^{5}}\) | \(301\) |
| trager | \(\frac {2 \left (3003 B \,b^{3} e^{7} x^{7}+3465 A \,b^{3} e^{7} x^{6}+10395 B a \,b^{2} e^{7} x^{6}+7161 B \,b^{3} d \,e^{6} x^{6}+12285 A a \,b^{2} e^{7} x^{5}+8505 A \,b^{3} d \,e^{6} x^{5}+12285 B \,a^{2} b \,e^{7} x^{5}+25515 B a \,b^{2} d \,e^{6} x^{5}+4473 B \,b^{3} d^{2} e^{5} x^{5}+15015 A \,a^{2} b \,e^{7} x^{4}+31395 A a \,b^{2} d \,e^{6} x^{4}+5565 A \,b^{3} d^{2} e^{5} x^{4}+5005 B \,a^{3} e^{7} x^{4}+31395 B \,a^{2} b d \,e^{6} x^{4}+16695 B a \,b^{2} d^{2} e^{5} x^{4}+35 B \,b^{3} d^{3} e^{4} x^{4}+6435 A \,a^{3} e^{7} x^{3}+40755 A \,a^{2} b d \,e^{6} x^{3}+22035 A a \,b^{2} d^{2} e^{5} x^{3}+75 A \,b^{3} d^{3} e^{4} x^{3}+13585 B \,a^{3} d \,e^{6} x^{3}+22035 B \,a^{2} b \,d^{2} e^{5} x^{3}+225 B a \,b^{2} d^{3} e^{4} x^{3}-40 B \,b^{3} d^{4} e^{3} x^{3}+19305 A \,a^{3} d \,e^{6} x^{2}+32175 A \,a^{2} b \,d^{2} e^{5} x^{2}+585 A a \,b^{2} d^{3} e^{4} x^{2}-90 A \,b^{3} d^{4} e^{3} x^{2}+10725 B \,a^{3} d^{2} e^{5} x^{2}+585 B \,a^{2} b \,d^{3} e^{4} x^{2}-270 B a \,b^{2} d^{4} e^{3} x^{2}+48 B \,b^{3} d^{5} e^{2} x^{2}+19305 A \,a^{3} d^{2} e^{5} x +2145 A \,a^{2} b \,d^{3} e^{4} x -780 A a \,b^{2} d^{4} e^{3} x +120 A \,b^{3} d^{5} e^{2} x +715 B \,a^{3} d^{3} e^{4} x -780 B \,a^{2} b \,d^{4} e^{3} x +360 B a \,b^{2} d^{5} e^{2} x -64 B \,b^{3} d^{6} e x +6435 A \,a^{3} d^{3} e^{4}-4290 A \,a^{2} b \,d^{4} e^{3}+1560 A a \,b^{2} d^{5} e^{2}-240 A \,b^{3} d^{6} e -1430 B \,a^{3} d^{4} e^{3}+1560 B \,a^{2} b \,d^{5} e^{2}-720 B a \,b^{2} d^{6} e +128 B \,b^{3} d^{7}\right ) \sqrt {e x +d}}{45045 e^{5}}\) | \(669\) |
| risch | \(\frac {2 \left (3003 B \,b^{3} e^{7} x^{7}+3465 A \,b^{3} e^{7} x^{6}+10395 B a \,b^{2} e^{7} x^{6}+7161 B \,b^{3} d \,e^{6} x^{6}+12285 A a \,b^{2} e^{7} x^{5}+8505 A \,b^{3} d \,e^{6} x^{5}+12285 B \,a^{2} b \,e^{7} x^{5}+25515 B a \,b^{2} d \,e^{6} x^{5}+4473 B \,b^{3} d^{2} e^{5} x^{5}+15015 A \,a^{2} b \,e^{7} x^{4}+31395 A a \,b^{2} d \,e^{6} x^{4}+5565 A \,b^{3} d^{2} e^{5} x^{4}+5005 B \,a^{3} e^{7} x^{4}+31395 B \,a^{2} b d \,e^{6} x^{4}+16695 B a \,b^{2} d^{2} e^{5} x^{4}+35 B \,b^{3} d^{3} e^{4} x^{4}+6435 A \,a^{3} e^{7} x^{3}+40755 A \,a^{2} b d \,e^{6} x^{3}+22035 A a \,b^{2} d^{2} e^{5} x^{3}+75 A \,b^{3} d^{3} e^{4} x^{3}+13585 B \,a^{3} d \,e^{6} x^{3}+22035 B \,a^{2} b \,d^{2} e^{5} x^{3}+225 B a \,b^{2} d^{3} e^{4} x^{3}-40 B \,b^{3} d^{4} e^{3} x^{3}+19305 A \,a^{3} d \,e^{6} x^{2}+32175 A \,a^{2} b \,d^{2} e^{5} x^{2}+585 A a \,b^{2} d^{3} e^{4} x^{2}-90 A \,b^{3} d^{4} e^{3} x^{2}+10725 B \,a^{3} d^{2} e^{5} x^{2}+585 B \,a^{2} b \,d^{3} e^{4} x^{2}-270 B a \,b^{2} d^{4} e^{3} x^{2}+48 B \,b^{3} d^{5} e^{2} x^{2}+19305 A \,a^{3} d^{2} e^{5} x +2145 A \,a^{2} b \,d^{3} e^{4} x -780 A a \,b^{2} d^{4} e^{3} x +120 A \,b^{3} d^{5} e^{2} x +715 B \,a^{3} d^{3} e^{4} x -780 B \,a^{2} b \,d^{4} e^{3} x +360 B a \,b^{2} d^{5} e^{2} x -64 B \,b^{3} d^{6} e x +6435 A \,a^{3} d^{3} e^{4}-4290 A \,a^{2} b \,d^{4} e^{3}+1560 A a \,b^{2} d^{5} e^{2}-240 A \,b^{3} d^{6} e -1430 B \,a^{3} d^{4} e^{3}+1560 B \,a^{2} b \,d^{5} e^{2}-720 B a \,b^{2} d^{6} e +128 B \,b^{3} d^{7}\right ) \sqrt {e x +d}}{45045 e^{5}}\) | \(669\) |
Input:
int((b*x+a)^3*(B*x+A)*(e*x+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/e^5*(1/15*b^3*B*(e*x+d)^(15/2)+1/13*(3*(a*e-b*d)*b^2*B+b^3*(A*e-B*d))*(e *x+d)^(13/2)+1/11*(3*(a*e-b*d)^2*b*B+3*(a*e-b*d)*b^2*(A*e-B*d))*(e*x+d)^(1 1/2)+1/9*((a*e-b*d)^3*B+3*(a*e-b*d)^2*b*(A*e-B*d))*(e*x+d)^(9/2)+1/7*(a*e- b*d)^3*(A*e-B*d)*(e*x+d)^(7/2))
Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (153) = 306\).
Time = 0.08 (sec) , antiderivative size = 539, normalized size of antiderivative = 3.12 \[ \int (a+b x)^3 (A+B x) (d+e x)^{5/2} \, dx=\frac {2 \, {\left (3003 \, B b^{3} e^{7} x^{7} + 128 \, B b^{3} d^{7} + 6435 \, A a^{3} d^{3} e^{4} - 240 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{6} e + 1560 \, {\left (B a^{2} b + A a b^{2}\right )} d^{5} e^{2} - 1430 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e^{3} + 231 \, {\left (31 \, B b^{3} d e^{6} + 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{7}\right )} x^{6} + 63 \, {\left (71 \, B b^{3} d^{2} e^{5} + 135 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{6} + 195 \, {\left (B a^{2} b + A a b^{2}\right )} e^{7}\right )} x^{5} + 35 \, {\left (B b^{3} d^{3} e^{4} + 159 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{5} + 897 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{6} + 143 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x^{4} - 5 \, {\left (8 \, B b^{3} d^{4} e^{3} - 1287 \, A a^{3} e^{7} - 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{4} - 4407 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{5} - 2717 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{5} e^{2} + 6435 \, A a^{3} d e^{6} - 30 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{3} + 195 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{4} + 3575 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{5}\right )} x^{2} - {\left (64 \, B b^{3} d^{6} e - 19305 \, A a^{3} d^{2} e^{5} - 120 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e^{2} + 780 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{3} - 715 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{4}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \] Input:
integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(5/2),x, algorithm="fricas")
Output:
2/45045*(3003*B*b^3*e^7*x^7 + 128*B*b^3*d^7 + 6435*A*a^3*d^3*e^4 - 240*(3* B*a*b^2 + A*b^3)*d^6*e + 1560*(B*a^2*b + A*a*b^2)*d^5*e^2 - 1430*(B*a^3 + 3*A*a^2*b)*d^4*e^3 + 231*(31*B*b^3*d*e^6 + 15*(3*B*a*b^2 + A*b^3)*e^7)*x^6 + 63*(71*B*b^3*d^2*e^5 + 135*(3*B*a*b^2 + A*b^3)*d*e^6 + 195*(B*a^2*b + A *a*b^2)*e^7)*x^5 + 35*(B*b^3*d^3*e^4 + 159*(3*B*a*b^2 + A*b^3)*d^2*e^5 + 8 97*(B*a^2*b + A*a*b^2)*d*e^6 + 143*(B*a^3 + 3*A*a^2*b)*e^7)*x^4 - 5*(8*B*b ^3*d^4*e^3 - 1287*A*a^3*e^7 - 15*(3*B*a*b^2 + A*b^3)*d^3*e^4 - 4407*(B*a^2 *b + A*a*b^2)*d^2*e^5 - 2717*(B*a^3 + 3*A*a^2*b)*d*e^6)*x^3 + 3*(16*B*b^3* d^5*e^2 + 6435*A*a^3*d*e^6 - 30*(3*B*a*b^2 + A*b^3)*d^4*e^3 + 195*(B*a^2*b + A*a*b^2)*d^3*e^4 + 3575*(B*a^3 + 3*A*a^2*b)*d^2*e^5)*x^2 - (64*B*b^3*d^ 6*e - 19305*A*a^3*d^2*e^5 - 120*(3*B*a*b^2 + A*b^3)*d^5*e^2 + 780*(B*a^2*b + A*a*b^2)*d^4*e^3 - 715*(B*a^3 + 3*A*a^2*b)*d^3*e^4)*x)*sqrt(e*x + d)/e^ 5
Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (170) = 340\).
Time = 1.28 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.45 \[ \int (a+b x)^3 (A+B x) (d+e x)^{5/2} \, dx=\begin {cases} \frac {2 \left (\frac {B b^{3} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{4}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \left (A b^{3} e + 3 B a b^{2} e - 4 B b^{3} d\right )}{13 e^{4}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (3 A a b^{2} e^{2} - 3 A b^{3} d e + 3 B a^{2} b e^{2} - 9 B a b^{2} d e + 6 B b^{3} d^{2}\right )}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (3 A a^{2} b e^{3} - 6 A a b^{2} d e^{2} + 3 A b^{3} d^{2} e + B a^{3} e^{3} - 6 B a^{2} b d e^{2} + 9 B a b^{2} d^{2} e - 4 B b^{3} d^{3}\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A a^{3} e^{4} - 3 A a^{2} b d e^{3} + 3 A a b^{2} d^{2} e^{2} - A b^{3} d^{3} e - B a^{3} d e^{3} + 3 B a^{2} b d^{2} e^{2} - 3 B a b^{2} d^{3} e + B b^{3} d^{4}\right )}{7 e^{4}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (A a^{3} x + \frac {B b^{3} x^{5}}{5} + \frac {x^{4} \left (A b^{3} + 3 B a b^{2}\right )}{4} + \frac {x^{3} \cdot \left (3 A a b^{2} + 3 B a^{2} b\right )}{3} + \frac {x^{2} \cdot \left (3 A a^{2} b + B a^{3}\right )}{2}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**3*(B*x+A)*(e*x+d)**(5/2),x)
Output:
Piecewise((2*(B*b**3*(d + e*x)**(15/2)/(15*e**4) + (d + e*x)**(13/2)*(A*b* *3*e + 3*B*a*b**2*e - 4*B*b**3*d)/(13*e**4) + (d + e*x)**(11/2)*(3*A*a*b** 2*e**2 - 3*A*b**3*d*e + 3*B*a**2*b*e**2 - 9*B*a*b**2*d*e + 6*B*b**3*d**2)/ (11*e**4) + (d + e*x)**(9/2)*(3*A*a**2*b*e**3 - 6*A*a*b**2*d*e**2 + 3*A*b* *3*d**2*e + B*a**3*e**3 - 6*B*a**2*b*d*e**2 + 9*B*a*b**2*d**2*e - 4*B*b**3 *d**3)/(9*e**4) + (d + e*x)**(7/2)*(A*a**3*e**4 - 3*A*a**2*b*d*e**3 + 3*A* a*b**2*d**2*e**2 - A*b**3*d**3*e - B*a**3*d*e**3 + 3*B*a**2*b*d**2*e**2 - 3*B*a*b**2*d**3*e + B*b**3*d**4)/(7*e**4))/e, Ne(e, 0)), (d**(5/2)*(A*a**3 *x + B*b**3*x**5/5 + x**4*(A*b**3 + 3*B*a*b**2)/4 + x**3*(3*A*a*b**2 + 3*B *a**2*b)/3 + x**2*(3*A*a**2*b + B*a**3)/2), True))
Time = 0.03 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.53 \[ \int (a+b x)^3 (A+B x) (d+e x)^{5/2} \, dx=\frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} B b^{3} - 3465 \, {\left (4 \, B b^{3} d - {\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 12285 \, {\left (2 \, B b^{3} d^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e + {\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 5005 \, {\left (4 \, B b^{3} d^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 6435 \, {\left (B b^{3} d^{4} + A a^{3} e^{4} - {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{45045 \, e^{5}} \] Input:
integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(5/2),x, algorithm="maxima")
Output:
2/45045*(3003*(e*x + d)^(15/2)*B*b^3 - 3465*(4*B*b^3*d - (3*B*a*b^2 + A*b^ 3)*e)*(e*x + d)^(13/2) + 12285*(2*B*b^3*d^2 - (3*B*a*b^2 + A*b^3)*d*e + (B *a^2*b + A*a*b^2)*e^2)*(e*x + d)^(11/2) - 5005*(4*B*b^3*d^3 - 3*(3*B*a*b^2 + A*b^3)*d^2*e + 6*(B*a^2*b + A*a*b^2)*d*e^2 - (B*a^3 + 3*A*a^2*b)*e^3)*( e*x + d)^(9/2) + 6435*(B*b^3*d^4 + A*a^3*e^4 - (3*B*a*b^2 + A*b^3)*d^3*e + 3*(B*a^2*b + A*a*b^2)*d^2*e^2 - (B*a^3 + 3*A*a^2*b)*d*e^3)*(e*x + d)^(7/2 ))/e^5
Leaf count of result is larger than twice the leaf count of optimal. 1947 vs. \(2 (153) = 306\).
Time = 0.14 (sec) , antiderivative size = 1947, normalized size of antiderivative = 11.25 \[ \int (a+b x)^3 (A+B x) (d+e x)^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(5/2),x, algorithm="giac")
Output:
2/45045*(45045*sqrt(e*x + d)*A*a^3*d^3 + 45045*((e*x + d)^(3/2) - 3*sqrt(e *x + d)*d)*A*a^3*d^2 + 15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*B*a^3*d ^3/e + 45045*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^2*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^3*d + 9009*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*a ^2*b*d^3/e^2 + 9009*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e* x + d)*d^2)*A*a*b^2*d^3/e^2 + 9009*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2) *d + 15*sqrt(e*x + d)*d^2)*B*a^3*d^2/e + 27027*(3*(e*x + d)^(5/2) - 10*(e* x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a^2*b*d^2/e + 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)*A*a^3 + 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*b^2*d^3/e^3 + 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)*A *b^3*d^3/e^3 + 11583*(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*d^2/e^2 + 11583*(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^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^3*d/e + 11583*(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*d/e + 143*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 3...
Time = 0.05 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.89 \[ \int (a+b x)^3 (A+B x) (d+e x)^{5/2} \, dx=\frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,A\,b^3\,e-8\,B\,b^3\,d+6\,B\,a\,b^2\,e\right )}{13\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}\,\left (3\,A\,b\,e+B\,a\,e-4\,B\,b\,d\right )}{9\,e^5}+\frac {2\,B\,b^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^5}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {6\,b\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{11\,e^5} \] Input:
int((A + B*x)*(a + b*x)^3*(d + e*x)^(5/2),x)
Output:
((d + e*x)^(13/2)*(2*A*b^3*e - 8*B*b^3*d + 6*B*a*b^2*e))/(13*e^5) + (2*(a* e - b*d)^2*(d + e*x)^(9/2)*(3*A*b*e + B*a*e - 4*B*b*d))/(9*e^5) + (2*B*b^3 *(d + e*x)^(15/2))/(15*e^5) + (2*(A*e - B*d)*(a*e - b*d)^3*(d + e*x)^(7/2) )/(7*e^5) + (6*b*(a*e - b*d)*(d + e*x)^(11/2)*(A*b*e + B*a*e - 2*B*b*d))/( 11*e^5)
Time = 0.16 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.34 \[ \int (a+b x)^3 (A+B x) (d+e x)^{5/2} \, dx=\frac {2 \sqrt {e x +d}\, \left (3003 b^{4} e^{7} x^{7}+13860 a \,b^{3} e^{7} x^{6}+7161 b^{4} d \,e^{6} x^{6}+24570 a^{2} b^{2} e^{7} x^{5}+34020 a \,b^{3} d \,e^{6} x^{5}+4473 b^{4} d^{2} e^{5} x^{5}+20020 a^{3} b \,e^{7} x^{4}+62790 a^{2} b^{2} d \,e^{6} x^{4}+22260 a \,b^{3} d^{2} e^{5} x^{4}+35 b^{4} d^{3} e^{4} x^{4}+6435 a^{4} e^{7} x^{3}+54340 a^{3} b d \,e^{6} x^{3}+44070 a^{2} b^{2} d^{2} e^{5} x^{3}+300 a \,b^{3} d^{3} e^{4} x^{3}-40 b^{4} d^{4} e^{3} x^{3}+19305 a^{4} d \,e^{6} x^{2}+42900 a^{3} b \,d^{2} e^{5} x^{2}+1170 a^{2} b^{2} d^{3} e^{4} x^{2}-360 a \,b^{3} d^{4} e^{3} x^{2}+48 b^{4} d^{5} e^{2} x^{2}+19305 a^{4} d^{2} e^{5} x +2860 a^{3} b \,d^{3} e^{4} x -1560 a^{2} b^{2} d^{4} e^{3} x +480 a \,b^{3} d^{5} e^{2} x -64 b^{4} d^{6} e x +6435 a^{4} d^{3} e^{4}-5720 a^{3} b \,d^{4} e^{3}+3120 a^{2} b^{2} d^{5} e^{2}-960 a \,b^{3} d^{6} e +128 b^{4} d^{7}\right )}{45045 e^{5}} \] Input:
int((b*x+a)^3*(B*x+A)*(e*x+d)^(5/2),x)
Output:
(2*sqrt(d + e*x)*(6435*a**4*d**3*e**4 + 19305*a**4*d**2*e**5*x + 19305*a** 4*d*e**6*x**2 + 6435*a**4*e**7*x**3 - 5720*a**3*b*d**4*e**3 + 2860*a**3*b* d**3*e**4*x + 42900*a**3*b*d**2*e**5*x**2 + 54340*a**3*b*d*e**6*x**3 + 200 20*a**3*b*e**7*x**4 + 3120*a**2*b**2*d**5*e**2 - 1560*a**2*b**2*d**4*e**3* x + 1170*a**2*b**2*d**3*e**4*x**2 + 44070*a**2*b**2*d**2*e**5*x**3 + 62790 *a**2*b**2*d*e**6*x**4 + 24570*a**2*b**2*e**7*x**5 - 960*a*b**3*d**6*e + 4 80*a*b**3*d**5*e**2*x - 360*a*b**3*d**4*e**3*x**2 + 300*a*b**3*d**3*e**4*x **3 + 22260*a*b**3*d**2*e**5*x**4 + 34020*a*b**3*d*e**6*x**5 + 13860*a*b** 3*e**7*x**6 + 128*b**4*d**7 - 64*b**4*d**6*e*x + 48*b**4*d**5*e**2*x**2 - 40*b**4*d**4*e**3*x**3 + 35*b**4*d**3*e**4*x**4 + 4473*b**4*d**2*e**5*x**5 + 7161*b**4*d*e**6*x**6 + 3003*b**4*e**7*x**7))/(45045*e**5)