Integrand size = 24, antiderivative size = 304 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{17/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{5/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac {2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{39 e (b d-a e)^2 (d+e x)^{13/2}}+\frac {16 b (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{11/2}}+\frac {32 b^2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{1287 e (b d-a e)^4 (d+e x)^{9/2}}+\frac {128 b^3 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{9009 e (b d-a e)^5 (d+e x)^{7/2}}+\frac {256 b^4 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{45045 e (b d-a e)^6 (d+e x)^{5/2}} \]
-2/15*(-A*e+B*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)/(e*x+d)^(15/2)+2/39*(2*A*b*e-3 *B*a*e+B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^2/(e*x+d)^(13/2)+16/429*b*(2*A*b* e-3*B*a*e+B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^3/(e*x+d)^(11/2)+32/1287*b^2*( 2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^4/(e*x+d)^(9/2)+128/9009 *b^3*(2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^5/(e*x+d)^(7/2)+25 6/45045*b^4*(2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^6/(e*x+d)^( 5/2)
Time = 1.10 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{17/2}} \, dx=\frac {2 (a+b x)^{5/2} \left (3003 B d e^4 (a+b x)^5-3003 A e^5 (a+b x)^5-13860 b B d e^3 (a+b x)^4 (d+e x)+17325 A b e^4 (a+b x)^4 (d+e x)-3465 a B e^4 (a+b x)^4 (d+e x)+24570 b^2 B d e^2 (a+b x)^3 (d+e x)^2-40950 A b^2 e^3 (a+b x)^3 (d+e x)^2+16380 a b B e^3 (a+b x)^3 (d+e x)^2-20020 b^3 B d e (a+b x)^2 (d+e x)^3+50050 A b^3 e^2 (a+b x)^2 (d+e x)^3-30030 a b^2 B e^2 (a+b x)^2 (d+e x)^3+6435 b^4 B d (a+b x) (d+e x)^4-32175 A b^4 e (a+b x) (d+e x)^4+25740 a b^3 B e (a+b x) (d+e x)^4+9009 A b^5 (d+e x)^5-9009 a b^4 B (d+e x)^5\right )}{45045 (b d-a e)^6 (d+e x)^{15/2}} \]
(2*(a + b*x)^(5/2)*(3003*B*d*e^4*(a + b*x)^5 - 3003*A*e^5*(a + b*x)^5 - 13 860*b*B*d*e^3*(a + b*x)^4*(d + e*x) + 17325*A*b*e^4*(a + b*x)^4*(d + e*x) - 3465*a*B*e^4*(a + b*x)^4*(d + e*x) + 24570*b^2*B*d*e^2*(a + b*x)^3*(d + e*x)^2 - 40950*A*b^2*e^3*(a + b*x)^3*(d + e*x)^2 + 16380*a*b*B*e^3*(a + b* x)^3*(d + e*x)^2 - 20020*b^3*B*d*e*(a + b*x)^2*(d + e*x)^3 + 50050*A*b^3*e ^2*(a + b*x)^2*(d + e*x)^3 - 30030*a*b^2*B*e^2*(a + b*x)^2*(d + e*x)^3 + 6 435*b^4*B*d*(a + b*x)*(d + e*x)^4 - 32175*A*b^4*e*(a + b*x)*(d + e*x)^4 + 25740*a*b^3*B*e*(a + b*x)*(d + e*x)^4 + 9009*A*b^5*(d + e*x)^5 - 9009*a*b^ 4*B*(d + e*x)^5))/(45045*(b*d - a*e)^6*(d + e*x)^(15/2))
Time = 0.28 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {87, 55, 55, 55, 55, 48}
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.
| \(\displaystyle \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{17/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \int \frac {(a+b x)^{3/2}}{(d+e x)^{15/2}}dx}{3 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {8 b \int \frac {(a+b x)^{3/2}}{(d+e x)^{13/2}}dx}{13 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{13 (d+e x)^{13/2} (b d-a e)}\right )}{3 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {8 b \left (\frac {6 b \int \frac {(a+b x)^{3/2}}{(d+e x)^{11/2}}dx}{11 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{11 (d+e x)^{11/2} (b d-a e)}\right )}{13 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{13 (d+e x)^{13/2} (b d-a e)}\right )}{3 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {8 b \left (\frac {6 b \left (\frac {4 b \int \frac {(a+b x)^{3/2}}{(d+e x)^{9/2}}dx}{9 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{9 (d+e x)^{9/2} (b d-a e)}\right )}{11 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{11 (d+e x)^{11/2} (b d-a e)}\right )}{13 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{13 (d+e x)^{13/2} (b d-a e)}\right )}{3 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {8 b \left (\frac {6 b \left (\frac {4 b \left (\frac {2 b \int \frac {(a+b x)^{3/2}}{(d+e x)^{7/2}}dx}{7 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{9 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{9 (d+e x)^{9/2} (b d-a e)}\right )}{11 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{11 (d+e x)^{11/2} (b d-a e)}\right )}{13 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{13 (d+e x)^{13/2} (b d-a e)}\right )}{3 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\left (\frac {2 (a+b x)^{5/2}}{13 (d+e x)^{13/2} (b d-a e)}+\frac {8 b \left (\frac {2 (a+b x)^{5/2}}{11 (d+e x)^{11/2} (b d-a e)}+\frac {6 b \left (\frac {2 (a+b x)^{5/2}}{9 (d+e x)^{9/2} (b d-a e)}+\frac {4 b \left (\frac {4 b (a+b x)^{5/2}}{35 (d+e x)^{5/2} (b d-a e)^2}+\frac {2 (a+b x)^{5/2}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{9 (b d-a e)}\right )}{11 (b d-a e)}\right )}{13 (b d-a e)}\right ) (-3 a B e+2 A b e+b B d)}{3 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)}\) |
(-2*(B*d - A*e)*(a + b*x)^(5/2))/(15*e*(b*d - a*e)*(d + e*x)^(15/2)) + ((b *B*d + 2*A*b*e - 3*a*B*e)*((2*(a + b*x)^(5/2))/(13*(b*d - a*e)*(d + e*x)^( 13/2)) + (8*b*((2*(a + b*x)^(5/2))/(11*(b*d - a*e)*(d + e*x)^(11/2)) + (6* b*((2*(a + b*x)^(5/2))/(9*(b*d - a*e)*(d + e*x)^(9/2)) + (4*b*((2*(a + b*x )^(5/2))/(7*(b*d - a*e)*(d + e*x)^(7/2)) + (4*b*(a + b*x)^(5/2))/(35*(b*d - a*e)^2*(d + e*x)^(5/2))))/(9*(b*d - a*e))))/(11*(b*d - a*e))))/(13*(b*d - a*e))))/(3*e*(b*d - a*e))
3.23.22.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Leaf count of result is larger than twice the leaf count of optimal. \(721\) vs. \(2(268)=536\).
Time = 1.15 (sec) , antiderivative size = 722, normalized size of antiderivative = 2.38
| method | result | size |
| gosper | \(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-256 A \,b^{5} e^{5} x^{5}+384 B a \,b^{4} e^{5} x^{5}-128 B \,b^{5} d \,e^{4} x^{5}+640 A a \,b^{4} e^{5} x^{4}-1920 A \,b^{5} d \,e^{4} x^{4}-960 B \,a^{2} b^{3} e^{5} x^{4}+3200 B a \,b^{4} d \,e^{4} x^{4}-960 B \,b^{5} d^{2} e^{3} x^{4}-1120 A \,a^{2} b^{3} e^{5} x^{3}+4800 A a \,b^{4} d \,e^{4} x^{3}-6240 A \,b^{5} d^{2} e^{3} x^{3}+1680 B \,a^{3} b^{2} e^{5} x^{3}-7760 B \,a^{2} b^{3} d \,e^{4} x^{3}+11760 B a \,b^{4} d^{2} e^{3} x^{3}-3120 B \,b^{5} d^{3} e^{2} x^{3}+1680 A \,a^{3} b^{2} e^{5} x^{2}-8400 A \,a^{2} b^{3} d \,e^{4} x^{2}+15600 A a \,b^{4} d^{2} e^{3} x^{2}-11440 A \,b^{5} d^{3} e^{2} x^{2}-2520 B \,a^{4} b \,e^{5} x^{2}+13440 B \,a^{3} b^{2} d \,e^{4} x^{2}-27600 B \,a^{2} b^{3} d^{2} e^{3} x^{2}+24960 B a \,b^{4} d^{3} e^{2} x^{2}-5720 B \,b^{5} d^{4} e \,x^{2}-2310 A \,a^{4} b \,e^{5} x +12600 A \,a^{3} b^{2} d \,e^{4} x -27300 A \,a^{2} b^{3} d^{2} e^{3} x +28600 A a \,b^{4} d^{3} e^{2} x -12870 A \,b^{5} d^{4} e x +3465 B \,a^{5} e^{5} x -20055 B \,a^{4} b d \,e^{4} x +47250 B \,a^{3} b^{2} d^{2} e^{3} x -56550 B \,a^{2} b^{3} d^{3} e^{2} x +33605 B a \,b^{4} d^{4} e x -6435 B \,b^{5} d^{5} x +3003 A \,a^{5} e^{5}-17325 A \,a^{4} b d \,e^{4}+40950 A \,a^{3} b^{2} d^{2} e^{3}-50050 A \,a^{2} b^{3} d^{3} e^{2}+32175 A a \,b^{4} d^{4} e -9009 A \,b^{5} d^{5}+462 B \,a^{5} d \,e^{4}-2520 B \,a^{4} b \,d^{2} e^{3}+5460 B \,a^{3} b^{2} d^{3} e^{2}-5720 B \,a^{2} b^{3} d^{4} e +2574 B a \,b^{4} d^{5}\right )}{45045 \left (e x +d \right )^{\frac {15}{2}} \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}\) | \(722\) |
| default | \(-\frac {2 \left (-256 A \,b^{6} e^{5} x^{6}+384 B a \,b^{5} e^{5} x^{6}-128 B \,b^{6} d \,e^{4} x^{6}+384 A a \,b^{5} e^{5} x^{5}-1920 A \,b^{6} d \,e^{4} x^{5}-576 B \,a^{2} b^{4} e^{5} x^{5}+3072 B a \,b^{5} d \,e^{4} x^{5}-960 B \,b^{6} d^{2} e^{3} x^{5}-480 A \,a^{2} b^{4} e^{5} x^{4}+2880 A a \,b^{5} d \,e^{4} x^{4}-6240 A \,b^{6} d^{2} e^{3} x^{4}+720 B \,a^{3} b^{3} e^{5} x^{4}-4560 B \,a^{2} b^{4} d \,e^{4} x^{4}+10800 B a \,b^{5} d^{2} e^{3} x^{4}-3120 B \,b^{6} d^{3} e^{2} x^{4}+560 A \,a^{3} b^{3} e^{5} x^{3}-3600 A \,a^{2} b^{4} d \,e^{4} x^{3}+9360 A a \,b^{5} d^{2} e^{3} x^{3}-11440 A \,b^{6} d^{3} e^{2} x^{3}-840 B \,a^{4} b^{2} e^{5} x^{3}+5680 B \,a^{3} b^{3} d \,e^{4} x^{3}-15840 B \,a^{2} b^{4} d^{2} e^{3} x^{3}+21840 B a \,b^{5} d^{3} e^{2} x^{3}-5720 B \,b^{6} d^{4} e \,x^{3}-630 A \,a^{4} b^{2} e^{5} x^{2}+4200 A \,a^{3} b^{3} d \,e^{4} x^{2}-11700 A \,a^{2} b^{4} d^{2} e^{3} x^{2}+17160 A a \,b^{5} d^{3} e^{2} x^{2}-12870 A \,b^{6} d^{4} e \,x^{2}+945 B \,a^{5} b \,e^{5} x^{2}-6615 B \,a^{4} b^{2} d \,e^{4} x^{2}+19650 B \,a^{3} b^{3} d^{2} e^{3} x^{2}-31590 B \,a^{2} b^{4} d^{3} e^{2} x^{2}+27885 B a \,b^{5} d^{4} e \,x^{2}-6435 B \,b^{6} d^{5} x^{2}+693 A \,a^{5} b \,e^{5} x -4725 A \,a^{4} b^{2} d \,e^{4} x +13650 A \,a^{3} b^{3} d^{2} e^{3} x -21450 A \,a^{2} b^{4} d^{3} e^{2} x +19305 A a \,b^{5} d^{4} e x -9009 A \,b^{6} d^{5} x +3465 B \,a^{6} e^{5} x -19593 B \,a^{5} b d \,e^{4} x +44730 B \,a^{4} b^{2} d^{2} e^{3} x -51090 B \,a^{3} b^{3} d^{3} e^{2} x +27885 B \,a^{2} b^{4} d^{4} e x -3861 B a \,b^{5} d^{5} x +3003 A \,a^{6} e^{5}-17325 A \,a^{5} b d \,e^{4}+40950 A \,a^{4} b^{2} d^{2} e^{3}-50050 A \,a^{3} b^{3} d^{3} e^{2}+32175 A \,a^{2} b^{4} d^{4} e -9009 A a \,b^{5} d^{5}+462 B \,a^{6} d \,e^{4}-2520 B \,a^{5} b \,d^{2} e^{3}+5460 B \,a^{4} b^{2} d^{3} e^{2}-5720 B \,a^{3} b^{3} d^{4} e +2574 B \,a^{2} b^{4} d^{5}\right ) \left (b x +a \right )^{\frac {3}{2}}}{45045 \left (e x +d \right )^{\frac {15}{2}} \left (a e -b d \right )^{6}}\) | \(845\) |
-2/45045*(b*x+a)^(5/2)*(-256*A*b^5*e^5*x^5+384*B*a*b^4*e^5*x^5-128*B*b^5*d *e^4*x^5+640*A*a*b^4*e^5*x^4-1920*A*b^5*d*e^4*x^4-960*B*a^2*b^3*e^5*x^4+32 00*B*a*b^4*d*e^4*x^4-960*B*b^5*d^2*e^3*x^4-1120*A*a^2*b^3*e^5*x^3+4800*A*a *b^4*d*e^4*x^3-6240*A*b^5*d^2*e^3*x^3+1680*B*a^3*b^2*e^5*x^3-7760*B*a^2*b^ 3*d*e^4*x^3+11760*B*a*b^4*d^2*e^3*x^3-3120*B*b^5*d^3*e^2*x^3+1680*A*a^3*b^ 2*e^5*x^2-8400*A*a^2*b^3*d*e^4*x^2+15600*A*a*b^4*d^2*e^3*x^2-11440*A*b^5*d ^3*e^2*x^2-2520*B*a^4*b*e^5*x^2+13440*B*a^3*b^2*d*e^4*x^2-27600*B*a^2*b^3* d^2*e^3*x^2+24960*B*a*b^4*d^3*e^2*x^2-5720*B*b^5*d^4*e*x^2-2310*A*a^4*b*e^ 5*x+12600*A*a^3*b^2*d*e^4*x-27300*A*a^2*b^3*d^2*e^3*x+28600*A*a*b^4*d^3*e^ 2*x-12870*A*b^5*d^4*e*x+3465*B*a^5*e^5*x-20055*B*a^4*b*d*e^4*x+47250*B*a^3 *b^2*d^2*e^3*x-56550*B*a^2*b^3*d^3*e^2*x+33605*B*a*b^4*d^4*e*x-6435*B*b^5* d^5*x+3003*A*a^5*e^5-17325*A*a^4*b*d*e^4+40950*A*a^3*b^2*d^2*e^3-50050*A*a ^2*b^3*d^3*e^2+32175*A*a*b^4*d^4*e-9009*A*b^5*d^5+462*B*a^5*d*e^4-2520*B*a ^4*b*d^2*e^3+5460*B*a^3*b^2*d^3*e^2-5720*B*a^2*b^3*d^4*e+2574*B*a*b^4*d^5) /(e*x+d)^(15/2)/(a^6*e^6-6*a^5*b*d*e^5+15*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e ^3+15*a^2*b^4*d^4*e^2-6*a*b^5*d^5*e+b^6*d^6)
Leaf count of result is larger than twice the leaf count of optimal. 1674 vs. \(2 (268) = 536\).
Time = 239.57 (sec) , antiderivative size = 1674, normalized size of antiderivative = 5.51 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{17/2}} \, dx=\text {Too large to display} \]
-2/45045*(3003*A*a^7*e^5 - 128*(B*b^7*d*e^4 - (3*B*a*b^6 - 2*A*b^7)*e^5)*x ^7 - 64*(15*B*b^7*d^2*e^3 - 2*(23*B*a*b^6 - 15*A*b^7)*d*e^4 + (3*B*a^2*b^5 - 2*A*a*b^6)*e^5)*x^6 + 1287*(2*B*a^3*b^4 - 7*A*a^2*b^5)*d^5 - 715*(8*B*a ^4*b^3 - 45*A*a^3*b^4)*d^4*e + 910*(6*B*a^5*b^2 - 55*A*a^4*b^3)*d^3*e^2 - 630*(4*B*a^6*b - 65*A*a^5*b^2)*d^2*e^3 + 231*(2*B*a^7 - 75*A*a^6*b)*d*e^4 - 48*(65*B*b^7*d^3*e^2 - 5*(41*B*a*b^6 - 26*A*b^7)*d^2*e^3 + (31*B*a^2*b^5 - 20*A*a*b^6)*d*e^4 - (3*B*a^3*b^4 - 2*A*a^2*b^5)*e^5)*x^5 - 40*(143*B*b^ 7*d^4*e - 26*(18*B*a*b^6 - 11*A*b^7)*d^3*e^2 + 6*(21*B*a^2*b^5 - 13*A*a*b^ 6)*d^2*e^3 - 2*(14*B*a^3*b^4 - 9*A*a^2*b^5)*d*e^4 + (3*B*a^4*b^3 - 2*A*a^3 *b^4)*e^5)*x^4 - 5*(1287*B*b^7*d^5 - 143*(31*B*a*b^6 - 18*A*b^7)*d^4*e + 2 6*(75*B*a^2*b^5 - 44*A*a*b^6)*d^3*e^2 - 6*(127*B*a^3*b^4 - 78*A*a^2*b^5)*d ^2*e^3 + (187*B*a^4*b^3 - 120*A*a^3*b^4)*d*e^4 - 7*(3*B*a^5*b^2 - 2*A*a^4* b^3)*e^5)*x^3 - 3*(429*(8*B*a*b^6 + 7*A*b^7)*d^5 - 715*(26*B*a^2*b^5 + 3*A *a*b^6)*d^4*e + 130*(212*B*a^3*b^4 + 11*A*a^2*b^5)*d^3*e^2 - 10*(2146*B*a^ 4*b^3 + 65*A*a^3*b^4)*d^2*e^3 + 7*(1248*B*a^5*b^2 + 25*A*a^4*b^3)*d*e^4 - 21*(70*B*a^6*b + A*a^5*b^2)*e^5)*x^2 - (1287*(B*a^2*b^5 + 14*A*a*b^6)*d^5 - 715*(31*B*a^3*b^4 + 72*A*a^2*b^5)*d^4*e + 130*(351*B*a^4*b^3 + 550*A*a^3 *b^4)*d^3*e^2 - 210*(201*B*a^5*b^2 + 260*A*a^4*b^3)*d^2*e^3 + 21*(911*B*a^ 6*b + 1050*A*a^5*b^2)*d*e^4 - 231*(15*B*a^7 + 16*A*a^6*b)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^6*d^14 - 6*a*b^5*d^13*e + 15*a^2*b^4*d^12*e^2 - ...
Timed out. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{17/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{17/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*(a*e-b*d)>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 1588 vs. \(2 (268) = 536\).
Time = 1.12 (sec) , antiderivative size = 1588, normalized size of antiderivative = 5.22 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{17/2}} \, dx=\text {Too large to display} \]
2/45045*((8*(2*(4*(b*x + a)*(2*(B*b^17*d^2*e^11*abs(b) - 4*B*a*b^16*d*e^12 *abs(b) + 2*A*b^17*d*e^12*abs(b) + 3*B*a^2*b^15*e^13*abs(b) - 2*A*a*b^16*e ^13*abs(b))*(b*x + a)/(b^9*d^7*e^7 - 7*a*b^8*d^6*e^8 + 21*a^2*b^7*d^5*e^9 - 35*a^3*b^6*d^4*e^10 + 35*a^4*b^5*d^3*e^11 - 21*a^5*b^4*d^2*e^12 + 7*a^6* b^3*d*e^13 - a^7*b^2*e^14) + 15*(B*b^18*d^3*e^10*abs(b) - 5*B*a*b^17*d^2*e ^11*abs(b) + 2*A*b^18*d^2*e^11*abs(b) + 7*B*a^2*b^16*d*e^12*abs(b) - 4*A*a *b^17*d*e^12*abs(b) - 3*B*a^3*b^15*e^13*abs(b) + 2*A*a^2*b^16*e^13*abs(b)) /(b^9*d^7*e^7 - 7*a*b^8*d^6*e^8 + 21*a^2*b^7*d^5*e^9 - 35*a^3*b^6*d^4*e^10 + 35*a^4*b^5*d^3*e^11 - 21*a^5*b^4*d^2*e^12 + 7*a^6*b^3*d*e^13 - a^7*b^2* e^14)) + 195*(B*b^19*d^4*e^9*abs(b) - 6*B*a*b^18*d^3*e^10*abs(b) + 2*A*b^1 9*d^3*e^10*abs(b) + 12*B*a^2*b^17*d^2*e^11*abs(b) - 6*A*a*b^18*d^2*e^11*ab s(b) - 10*B*a^3*b^16*d*e^12*abs(b) + 6*A*a^2*b^17*d*e^12*abs(b) + 3*B*a^4* b^15*e^13*abs(b) - 2*A*a^3*b^16*e^13*abs(b))/(b^9*d^7*e^7 - 7*a*b^8*d^6*e^ 8 + 21*a^2*b^7*d^5*e^9 - 35*a^3*b^6*d^4*e^10 + 35*a^4*b^5*d^3*e^11 - 21*a^ 5*b^4*d^2*e^12 + 7*a^6*b^3*d*e^13 - a^7*b^2*e^14))*(b*x + a) + 715*(B*b^20 *d^5*e^8*abs(b) - 7*B*a*b^19*d^4*e^9*abs(b) + 2*A*b^20*d^4*e^9*abs(b) + 18 *B*a^2*b^18*d^3*e^10*abs(b) - 8*A*a*b^19*d^3*e^10*abs(b) - 22*B*a^3*b^17*d ^2*e^11*abs(b) + 12*A*a^2*b^18*d^2*e^11*abs(b) + 13*B*a^4*b^16*d*e^12*abs( b) - 8*A*a^3*b^17*d*e^12*abs(b) - 3*B*a^5*b^15*e^13*abs(b) + 2*A*a^4*b^16* e^13*abs(b))/(b^9*d^7*e^7 - 7*a*b^8*d^6*e^8 + 21*a^2*b^7*d^5*e^9 - 35*a...
Time = 4.58 (sec) , antiderivative size = 941, normalized size of antiderivative = 3.10 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{17/2}} \, dx=-\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (924\,B\,a^7\,d\,e^4+6006\,A\,a^7\,e^5-5040\,B\,a^6\,b\,d^2\,e^3-34650\,A\,a^6\,b\,d\,e^4+10920\,B\,a^5\,b^2\,d^3\,e^2+81900\,A\,a^5\,b^2\,d^2\,e^3-11440\,B\,a^4\,b^3\,d^4\,e-100100\,A\,a^4\,b^3\,d^3\,e^2+5148\,B\,a^3\,b^4\,d^5+64350\,A\,a^3\,b^4\,d^4\,e-18018\,A\,a^2\,b^5\,d^5\right )}{45045\,e^8\,{\left (a\,e-b\,d\right )}^6}-\frac {x^2\,\sqrt {a+b\,x}\,\left (-8820\,B\,a^6\,b\,e^5+52416\,B\,a^5\,b^2\,d\,e^4-126\,A\,a^5\,b^2\,e^5-128760\,B\,a^4\,b^3\,d^2\,e^3+1050\,A\,a^4\,b^3\,d\,e^4+165360\,B\,a^3\,b^4\,d^3\,e^2-3900\,A\,a^3\,b^4\,d^2\,e^3-111540\,B\,a^2\,b^5\,d^4\,e+8580\,A\,a^2\,b^5\,d^3\,e^2+20592\,B\,a\,b^6\,d^5-12870\,A\,a\,b^6\,d^4\,e+18018\,A\,b^7\,d^5\right )}{45045\,e^8\,{\left (a\,e-b\,d\right )}^6}+\frac {x\,\sqrt {a+b\,x}\,\left (6930\,B\,a^7\,e^5-38262\,B\,a^6\,b\,d\,e^4+7392\,A\,a^6\,b\,e^5+84420\,B\,a^5\,b^2\,d^2\,e^3-44100\,A\,a^5\,b^2\,d\,e^4-91260\,B\,a^4\,b^3\,d^3\,e^2+109200\,A\,a^4\,b^3\,d^2\,e^3+44330\,B\,a^3\,b^4\,d^4\,e-143000\,A\,a^3\,b^4\,d^3\,e^2-2574\,B\,a^2\,b^5\,d^5+102960\,A\,a^2\,b^5\,d^4\,e-36036\,A\,a\,b^6\,d^5\right )}{45045\,e^8\,{\left (a\,e-b\,d\right )}^6}-\frac {256\,b^6\,x^7\,\sqrt {a+b\,x}\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )}{45045\,e^4\,{\left (a\,e-b\,d\right )}^6}+\frac {16\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )\,\left (a^3\,e^3-9\,a^2\,b\,d\,e^2+39\,a\,b^2\,d^2\,e-143\,b^3\,d^3\right )}{9009\,e^7\,{\left (a\,e-b\,d\right )}^6}+\frac {128\,b^5\,x^6\,\left (a\,e-15\,b\,d\right )\,\sqrt {a+b\,x}\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )}{45045\,e^5\,{\left (a\,e-b\,d\right )}^6}-\frac {2\,b^2\,x^3\,\sqrt {a+b\,x}\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )\,\left (7\,a^4\,e^4-60\,a^3\,b\,d\,e^3+234\,a^2\,b^2\,d^2\,e^2-572\,a\,b^3\,d^3\,e+1287\,b^4\,d^4\right )}{9009\,e^8\,{\left (a\,e-b\,d\right )}^6}-\frac {32\,b^4\,x^5\,\sqrt {a+b\,x}\,\left (a^2\,e^2-10\,a\,b\,d\,e+65\,b^2\,d^2\right )\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )}{15015\,e^6\,{\left (a\,e-b\,d\right )}^6}\right )}{x^8+\frac {d^8}{e^8}+\frac {8\,d\,x^7}{e}+\frac {8\,d^7\,x}{e^7}+\frac {28\,d^2\,x^6}{e^2}+\frac {56\,d^3\,x^5}{e^3}+\frac {70\,d^4\,x^4}{e^4}+\frac {56\,d^5\,x^3}{e^5}+\frac {28\,d^6\,x^2}{e^6}} \]
-((d + e*x)^(1/2)*(((a + b*x)^(1/2)*(6006*A*a^7*e^5 + 924*B*a^7*d*e^4 - 18 018*A*a^2*b^5*d^5 + 5148*B*a^3*b^4*d^5 + 64350*A*a^3*b^4*d^4*e - 11440*B*a ^4*b^3*d^4*e - 5040*B*a^6*b*d^2*e^3 - 100100*A*a^4*b^3*d^3*e^2 + 81900*A*a ^5*b^2*d^2*e^3 + 10920*B*a^5*b^2*d^3*e^2 - 34650*A*a^6*b*d*e^4))/(45045*e^ 8*(a*e - b*d)^6) - (x^2*(a + b*x)^(1/2)*(18018*A*b^7*d^5 + 20592*B*a*b^6*d ^5 - 8820*B*a^6*b*e^5 - 126*A*a^5*b^2*e^5 + 1050*A*a^4*b^3*d*e^4 - 111540* B*a^2*b^5*d^4*e + 52416*B*a^5*b^2*d*e^4 + 8580*A*a^2*b^5*d^3*e^2 - 3900*A* a^3*b^4*d^2*e^3 + 165360*B*a^3*b^4*d^3*e^2 - 128760*B*a^4*b^3*d^2*e^3 - 12 870*A*a*b^6*d^4*e))/(45045*e^8*(a*e - b*d)^6) + (x*(a + b*x)^(1/2)*(6930*B *a^7*e^5 - 36036*A*a*b^6*d^5 + 7392*A*a^6*b*e^5 - 2574*B*a^2*b^5*d^5 + 102 960*A*a^2*b^5*d^4*e - 44100*A*a^5*b^2*d*e^4 + 44330*B*a^3*b^4*d^4*e - 1430 00*A*a^3*b^4*d^3*e^2 + 109200*A*a^4*b^3*d^2*e^3 - 91260*B*a^4*b^3*d^3*e^2 + 84420*B*a^5*b^2*d^2*e^3 - 38262*B*a^6*b*d*e^4))/(45045*e^8*(a*e - b*d)^6 ) - (256*b^6*x^7*(a + b*x)^(1/2)*(2*A*b*e - 3*B*a*e + B*b*d))/(45045*e^4*( a*e - b*d)^6) + (16*b^3*x^4*(a + b*x)^(1/2)*(2*A*b*e - 3*B*a*e + B*b*d)*(a ^3*e^3 - 143*b^3*d^3 + 39*a*b^2*d^2*e - 9*a^2*b*d*e^2))/(9009*e^7*(a*e - b *d)^6) + (128*b^5*x^6*(a*e - 15*b*d)*(a + b*x)^(1/2)*(2*A*b*e - 3*B*a*e + B*b*d))/(45045*e^5*(a*e - b*d)^6) - (2*b^2*x^3*(a + b*x)^(1/2)*(2*A*b*e - 3*B*a*e + B*b*d)*(7*a^4*e^4 + 1287*b^4*d^4 + 234*a^2*b^2*d^2*e^2 - 572*a*b ^3*d^3*e - 60*a^3*b*d*e^3))/(9009*e^8*(a*e - b*d)^6) - (32*b^4*x^5*(a +...