Integrand size = 24, antiderivative size = 255 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{15/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{5/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {4 b (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {16 b^2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac {32 b^3 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{15015 e (b d-a e)^5 (d+e x)^{5/2}} \]
-2/13*(-A*e+B*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)/(e*x+d)^(13/2)+2/143*(8*A*b*e- 13*B*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^2/(e*x+d)^(11/2)+4/429*b*(8*A *b*e-13*B*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^3/(e*x+d)^(9/2)+16/3003* b^2*(8*A*b*e-13*B*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^4/(e*x+d)^(7/2)+ 32/15015*b^3*(8*A*b*e-13*B*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^5/(e*x+ d)^(5/2)
Time = 0.43 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{15/2}} \, dx=\frac {2 (a+b x)^{5/2} \left (-1155 B d e^3 (a+b x)^4+1155 A e^4 (a+b x)^4+4095 b B d e^2 (a+b x)^3 (d+e x)-5460 A b e^3 (a+b x)^3 (d+e x)+1365 a B e^3 (a+b x)^3 (d+e x)-5005 b^2 B d e (a+b x)^2 (d+e x)^2+10010 A b^2 e^2 (a+b x)^2 (d+e x)^2-5005 a b B e^2 (a+b x)^2 (d+e x)^2+2145 b^3 B d (a+b x) (d+e x)^3-8580 A b^3 e (a+b x) (d+e x)^3+6435 a b^2 B e (a+b x) (d+e x)^3+3003 A b^4 (d+e x)^4-3003 a b^3 B (d+e x)^4\right )}{15015 (b d-a e)^5 (d+e x)^{13/2}} \]
(2*(a + b*x)^(5/2)*(-1155*B*d*e^3*(a + b*x)^4 + 1155*A*e^4*(a + b*x)^4 + 4 095*b*B*d*e^2*(a + b*x)^3*(d + e*x) - 5460*A*b*e^3*(a + b*x)^3*(d + e*x) + 1365*a*B*e^3*(a + b*x)^3*(d + e*x) - 5005*b^2*B*d*e*(a + b*x)^2*(d + e*x) ^2 + 10010*A*b^2*e^2*(a + b*x)^2*(d + e*x)^2 - 5005*a*b*B*e^2*(a + b*x)^2* (d + e*x)^2 + 2145*b^3*B*d*(a + b*x)*(d + e*x)^3 - 8580*A*b^3*e*(a + b*x)* (d + e*x)^3 + 6435*a*b^2*B*e*(a + b*x)*(d + e*x)^3 + 3003*A*b^4*(d + e*x)^ 4 - 3003*a*b^3*B*(d + e*x)^4))/(15015*(b*d - a*e)^5*(d + e*x)^(13/2))
Time = 0.27 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {87, 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.
| \(\displaystyle \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{15/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-13 a B e+8 A b e+5 b B d) \int \frac {(a+b x)^{3/2}}{(d+e x)^{13/2}}dx}{13 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-13 a B e+8 A b e+5 b B d) \left (\frac {6 b \int \frac {(a+b x)^{3/2}}{(d+e x)^{11/2}}dx}{11 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{11 (d+e x)^{11/2} (b d-a e)}\right )}{13 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-13 a B e+8 A b e+5 b B d) \left (\frac {6 b \left (\frac {4 b \int \frac {(a+b x)^{3/2}}{(d+e x)^{9/2}}dx}{9 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{9 (d+e x)^{9/2} (b d-a e)}\right )}{11 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{11 (d+e x)^{11/2} (b d-a e)}\right )}{13 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-13 a B e+8 A b e+5 b B d) \left (\frac {6 b \left (\frac {4 b \left (\frac {2 b \int \frac {(a+b x)^{3/2}}{(d+e x)^{7/2}}dx}{7 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{9 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{9 (d+e x)^{9/2} (b d-a e)}\right )}{11 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{11 (d+e x)^{11/2} (b d-a e)}\right )}{13 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\left (\frac {2 (a+b x)^{5/2}}{11 (d+e x)^{11/2} (b d-a e)}+\frac {6 b \left (\frac {2 (a+b x)^{5/2}}{9 (d+e x)^{9/2} (b d-a e)}+\frac {4 b \left (\frac {4 b (a+b x)^{5/2}}{35 (d+e x)^{5/2} (b d-a e)^2}+\frac {2 (a+b x)^{5/2}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{9 (b d-a e)}\right )}{11 (b d-a e)}\right ) (-13 a B e+8 A b e+5 b B d)}{13 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)}\) |
(-2*(B*d - A*e)*(a + b*x)^(5/2))/(13*e*(b*d - a*e)*(d + e*x)^(13/2)) + ((5 *b*B*d + 8*A*b*e - 13*a*B*e)*((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*e*(b*d - a*e))
3.23.21.3.1 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. \(504\) vs. \(2(225)=450\).
Time = 1.09 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.98
| method | result | size |
| gosper | \(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (128 A \,b^{4} e^{4} x^{4}-208 B a \,b^{3} e^{4} x^{4}+80 B \,b^{4} d \,e^{3} x^{4}-320 A a \,b^{3} e^{4} x^{3}+832 A \,b^{4} d \,e^{3} x^{3}+520 B \,a^{2} b^{2} e^{4} x^{3}-1552 B a \,b^{3} d \,e^{3} x^{3}+520 B \,b^{4} d^{2} e^{2} x^{3}+560 A \,a^{2} b^{2} e^{4} x^{2}-2080 A a \,b^{3} d \,e^{3} x^{2}+2288 A \,b^{4} d^{2} e^{2} x^{2}-910 B \,a^{3} b \,e^{4} x^{2}+3730 B \,a^{2} b^{2} d \,e^{3} x^{2}-5018 B a \,b^{3} d^{2} e^{2} x^{2}+1430 B \,b^{4} d^{3} e \,x^{2}-840 A \,a^{3} b \,e^{4} x +3640 A \,a^{2} b^{2} d \,e^{3} x -5720 A a \,b^{3} d^{2} e^{2} x +3432 A \,b^{4} d^{3} e x +1365 B \,a^{4} e^{4} x -6440 B \,a^{3} b d \,e^{3} x +11570 B \,a^{2} b^{2} d^{2} e^{2} x -9152 B a \,b^{3} d^{3} e x +2145 B \,b^{4} d^{4} x +1155 A \,a^{4} e^{4}-5460 A \,a^{3} b d \,e^{3}+10010 A \,a^{2} b^{2} d^{2} e^{2}-8580 A a \,b^{3} d^{3} e +3003 A \,b^{4} d^{4}+210 B \,a^{4} d \,e^{3}-910 B \,a^{3} b \,d^{2} e^{2}+1430 B \,a^{2} b^{2} d^{3} e -858 B a \,b^{3} d^{4}\right )}{15015 \left (e x +d \right )^{\frac {13}{2}} \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}\) | \(505\) |
| default | \(-\frac {2 \left (128 A \,b^{5} e^{4} x^{5}-208 B a \,b^{4} e^{4} x^{5}+80 B \,b^{5} d \,e^{3} x^{5}-192 A a \,b^{4} e^{4} x^{4}+832 A \,b^{5} d \,e^{3} x^{4}+312 B \,a^{2} b^{3} e^{4} x^{4}-1472 B a \,b^{4} d \,e^{3} x^{4}+520 B \,b^{5} d^{2} e^{2} x^{4}+240 A \,a^{2} b^{3} e^{4} x^{3}-1248 A a \,b^{4} d \,e^{3} x^{3}+2288 A \,b^{5} d^{2} e^{2} x^{3}-390 B \,a^{3} b^{2} e^{4} x^{3}+2178 B \,a^{2} b^{3} d \,e^{3} x^{3}-4498 B a \,b^{4} d^{2} e^{2} x^{3}+1430 B \,b^{5} d^{3} e \,x^{3}-280 A \,a^{3} b^{2} e^{4} x^{2}+1560 A \,a^{2} b^{3} d \,e^{3} x^{2}-3432 A a \,b^{4} d^{2} e^{2} x^{2}+3432 A \,b^{5} d^{3} e \,x^{2}+455 B \,a^{4} b \,e^{4} x^{2}-2710 B \,a^{3} b^{2} d \,e^{3} x^{2}+6552 B \,a^{2} b^{3} d^{2} e^{2} x^{2}-7722 B a \,b^{4} d^{3} e \,x^{2}+2145 B \,b^{5} d^{4} x^{2}+315 A \,a^{4} b \,e^{4} x -1820 A \,a^{3} b^{2} d \,e^{3} x +4290 A \,a^{2} b^{3} d^{2} e^{2} x -5148 A a \,b^{4} d^{3} e x +3003 A \,b^{5} d^{4} x +1365 B \,a^{5} e^{4} x -6230 B \,a^{4} b d \,e^{3} x +10660 B \,a^{3} b^{2} d^{2} e^{2} x -7722 B \,a^{2} b^{3} d^{3} e x +1287 B a \,b^{4} d^{4} x +1155 A \,a^{5} e^{4}-5460 A \,a^{4} b d \,e^{3}+10010 A \,a^{3} b^{2} d^{2} e^{2}-8580 A \,a^{2} b^{3} d^{3} e +3003 A a \,b^{4} d^{4}+210 B \,a^{5} d \,e^{3}-910 B \,a^{4} b \,d^{2} e^{2}+1430 B \,a^{3} b^{2} d^{3} e -858 B \,a^{2} b^{3} d^{4}\right ) \left (b x +a \right )^{\frac {3}{2}}}{15015 \left (e x +d \right )^{\frac {13}{2}} \left (a e -b d \right )^{5}}\) | \(605\) |
-2/15015*(b*x+a)^(5/2)*(128*A*b^4*e^4*x^4-208*B*a*b^3*e^4*x^4+80*B*b^4*d*e ^3*x^4-320*A*a*b^3*e^4*x^3+832*A*b^4*d*e^3*x^3+520*B*a^2*b^2*e^4*x^3-1552* B*a*b^3*d*e^3*x^3+520*B*b^4*d^2*e^2*x^3+560*A*a^2*b^2*e^4*x^2-2080*A*a*b^3 *d*e^3*x^2+2288*A*b^4*d^2*e^2*x^2-910*B*a^3*b*e^4*x^2+3730*B*a^2*b^2*d*e^3 *x^2-5018*B*a*b^3*d^2*e^2*x^2+1430*B*b^4*d^3*e*x^2-840*A*a^3*b*e^4*x+3640* A*a^2*b^2*d*e^3*x-5720*A*a*b^3*d^2*e^2*x+3432*A*b^4*d^3*e*x+1365*B*a^4*e^4 *x-6440*B*a^3*b*d*e^3*x+11570*B*a^2*b^2*d^2*e^2*x-9152*B*a*b^3*d^3*e*x+214 5*B*b^4*d^4*x+1155*A*a^4*e^4-5460*A*a^3*b*d*e^3+10010*A*a^2*b^2*d^2*e^2-85 80*A*a*b^3*d^3*e+3003*A*b^4*d^4+210*B*a^4*d*e^3-910*B*a^3*b*d^2*e^2+1430*B *a^2*b^2*d^3*e-858*B*a*b^3*d^4)/(e*x+d)^(13/2)/(a^5*e^5-5*a^4*b*d*e^4+10*a ^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)
Leaf count of result is larger than twice the leaf count of optimal. 1252 vs. \(2 (225) = 450\).
Time = 127.88 (sec) , antiderivative size = 1252, normalized size of antiderivative = 4.91 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{15/2}} \, dx=\text {Too large to display} \]
2/15015*(1155*A*a^6*e^4 + 16*(5*B*b^6*d*e^3 - (13*B*a*b^5 - 8*A*b^6)*e^4)* x^6 + 8*(65*B*b^6*d^2*e^2 - 2*(87*B*a*b^5 - 52*A*b^6)*d*e^3 + (13*B*a^2*b^ 4 - 8*A*a*b^5)*e^4)*x^5 - 429*(2*B*a^3*b^3 - 7*A*a^2*b^4)*d^4 + 1430*(B*a^ 4*b^2 - 6*A*a^3*b^3)*d^3*e - 910*(B*a^5*b - 11*A*a^4*b^2)*d^2*e^2 + 210*(B *a^6 - 26*A*a^5*b)*d*e^3 + 2*(715*B*b^6*d^3*e - 13*(153*B*a*b^5 - 88*A*b^6 )*d^2*e^2 + (353*B*a^2*b^4 - 208*A*a*b^5)*d*e^3 - 3*(13*B*a^3*b^3 - 8*A*a^ 2*b^4)*e^4)*x^4 + (2145*B*b^6*d^4 - 572*(11*B*a*b^5 - 6*A*b^6)*d^3*e + 26* (79*B*a^2*b^4 - 44*A*a*b^5)*d^2*e^2 - 4*(133*B*a^3*b^3 - 78*A*a^2*b^4)*d*e ^3 + 5*(13*B*a^4*b^2 - 8*A*a^3*b^3)*e^4)*x^3 + (429*(8*B*a*b^5 + 7*A*b^6)* d^4 - 1716*(9*B*a^2*b^4 + A*a*b^5)*d^3*e + 26*(662*B*a^3*b^3 + 33*A*a^2*b^ 4)*d^2*e^2 - 20*(447*B*a^4*b^2 + 13*A*a^3*b^3)*d*e^3 + 35*(52*B*a^5*b + A* a^4*b^2)*e^4)*x^2 + (429*(B*a^2*b^4 + 14*A*a*b^5)*d^4 - 572*(11*B*a^3*b^3 + 24*A*a^2*b^4)*d^3*e + 650*(15*B*a^4*b^2 + 22*A*a^3*b^3)*d^2*e^2 - 140*(4 3*B*a^5*b + 52*A*a^4*b^2)*d*e^3 + 105*(13*B*a^6 + 14*A*a^5*b)*e^4)*x)*sqrt (b*x + a)*sqrt(e*x + d)/(b^5*d^12 - 5*a*b^4*d^11*e + 10*a^2*b^3*d^10*e^2 - 10*a^3*b^2*d^9*e^3 + 5*a^4*b*d^8*e^4 - a^5*d^7*e^5 + (b^5*d^5*e^7 - 5*a*b ^4*d^4*e^8 + 10*a^2*b^3*d^3*e^9 - 10*a^3*b^2*d^2*e^10 + 5*a^4*b*d*e^11 - a ^5*e^12)*x^7 + 7*(b^5*d^6*e^6 - 5*a*b^4*d^5*e^7 + 10*a^2*b^3*d^4*e^8 - 10* a^3*b^2*d^3*e^9 + 5*a^4*b*d^2*e^10 - a^5*d*e^11)*x^6 + 21*(b^5*d^7*e^5 - 5 *a*b^4*d^6*e^6 + 10*a^2*b^3*d^5*e^7 - 10*a^3*b^2*d^4*e^8 + 5*a^4*b*d^3*...
Timed out. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{15/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{15/2}} \, dx=\text {Exception raised: ValueError} \]
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. 1168 vs. \(2 (225) = 450\).
Time = 0.84 (sec) , antiderivative size = 1168, normalized size of antiderivative = 4.58 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{15/2}} \, dx=\text {Too large to display} \]
2/15015*((2*(4*(b*x + a)*(2*(5*B*b^15*d^2*e^9*abs(b) - 18*B*a*b^14*d*e^10* abs(b) + 8*A*b^15*d*e^10*abs(b) + 13*B*a^2*b^13*e^11*abs(b) - 8*A*a*b^14*e ^11*abs(b))*(b*x + a)/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 20*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 6*a^5*b^3*d*e^11 + a^6*b^2*e^ 12) + 13*(5*B*b^16*d^3*e^8*abs(b) - 23*B*a*b^15*d^2*e^9*abs(b) + 8*A*b^16* d^2*e^9*abs(b) + 31*B*a^2*b^14*d*e^10*abs(b) - 16*A*a*b^15*d*e^10*abs(b) - 13*B*a^3*b^13*e^11*abs(b) + 8*A*a^2*b^14*e^11*abs(b))/(b^8*d^6*e^6 - 6*a* b^7*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 20*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d^2*e^1 0 - 6*a^5*b^3*d*e^11 + a^6*b^2*e^12)) + 143*(5*B*b^17*d^4*e^7*abs(b) - 28* B*a*b^16*d^3*e^8*abs(b) + 8*A*b^17*d^3*e^8*abs(b) + 54*B*a^2*b^15*d^2*e^9* abs(b) - 24*A*a*b^16*d^2*e^9*abs(b) - 44*B*a^3*b^14*d*e^10*abs(b) + 24*A*a ^2*b^15*d*e^10*abs(b) + 13*B*a^4*b^13*e^11*abs(b) - 8*A*a^3*b^14*e^11*abs( b))/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 20*a^3*b^5*d^3*e ^9 + 15*a^4*b^4*d^2*e^10 - 6*a^5*b^3*d*e^11 + a^6*b^2*e^12))*(b*x + a) + 4 29*(5*B*b^18*d^5*e^6*abs(b) - 33*B*a*b^17*d^4*e^7*abs(b) + 8*A*b^18*d^4*e^ 7*abs(b) + 82*B*a^2*b^16*d^3*e^8*abs(b) - 32*A*a*b^17*d^3*e^8*abs(b) - 98* B*a^3*b^15*d^2*e^9*abs(b) + 48*A*a^2*b^16*d^2*e^9*abs(b) + 57*B*a^4*b^14*d *e^10*abs(b) - 32*A*a^3*b^15*d*e^10*abs(b) - 13*B*a^5*b^13*e^11*abs(b) + 8 *A*a^4*b^14*e^11*abs(b))/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b^6*d^4*e ^8 - 20*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 6*a^5*b^3*d*e^11 + a^6*...
Time = 3.46 (sec) , antiderivative size = 752, normalized size of antiderivative = 2.95 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{15/2}} \, dx=-\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (420\,B\,a^6\,d\,e^3+2310\,A\,a^6\,e^4-1820\,B\,a^5\,b\,d^2\,e^2-10920\,A\,a^5\,b\,d\,e^3+2860\,B\,a^4\,b^2\,d^3\,e+20020\,A\,a^4\,b^2\,d^2\,e^2-1716\,B\,a^3\,b^3\,d^4-17160\,A\,a^3\,b^3\,d^3\,e+6006\,A\,a^2\,b^4\,d^4\right )}{15015\,e^7\,{\left (a\,e-b\,d\right )}^5}+\frac {x\,\sqrt {a+b\,x}\,\left (2730\,B\,a^6\,e^4-12040\,B\,a^5\,b\,d\,e^3+2940\,A\,a^5\,b\,e^4+19500\,B\,a^4\,b^2\,d^2\,e^2-14560\,A\,a^4\,b^2\,d\,e^3-12584\,B\,a^3\,b^3\,d^3\,e+28600\,A\,a^3\,b^3\,d^2\,e^2+858\,B\,a^2\,b^4\,d^4-27456\,A\,a^2\,b^4\,d^3\,e+12012\,A\,a\,b^5\,d^4\right )}{15015\,e^7\,{\left (a\,e-b\,d\right )}^5}+\frac {x^2\,\sqrt {a+b\,x}\,\left (3640\,B\,a^5\,b\,e^4-17880\,B\,a^4\,b^2\,d\,e^3+70\,A\,a^4\,b^2\,e^4+34424\,B\,a^3\,b^3\,d^2\,e^2-520\,A\,a^3\,b^3\,d\,e^3-30888\,B\,a^2\,b^4\,d^3\,e+1716\,A\,a^2\,b^4\,d^2\,e^2+6864\,B\,a\,b^5\,d^4-3432\,A\,a\,b^5\,d^3\,e+6006\,A\,b^6\,d^4\right )}{15015\,e^7\,{\left (a\,e-b\,d\right )}^5}+\frac {32\,b^5\,x^6\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-13\,B\,a\,e+5\,B\,b\,d\right )}{15015\,e^4\,{\left (a\,e-b\,d\right )}^5}-\frac {2\,b^2\,x^3\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-13\,B\,a\,e+5\,B\,b\,d\right )\,\left (5\,a^3\,e^3-39\,a^2\,b\,d\,e^2+143\,a\,b^2\,d^2\,e-429\,b^3\,d^3\right )}{15015\,e^7\,{\left (a\,e-b\,d\right )}^5}-\frac {16\,b^4\,x^5\,\left (a\,e-13\,b\,d\right )\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-13\,B\,a\,e+5\,B\,b\,d\right )}{15015\,e^5\,{\left (a\,e-b\,d\right )}^5}+\frac {4\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (3\,a^2\,e^2-26\,a\,b\,d\,e+143\,b^2\,d^2\right )\,\left (8\,A\,b\,e-13\,B\,a\,e+5\,B\,b\,d\right )}{15015\,e^6\,{\left (a\,e-b\,d\right )}^5}\right )}{x^7+\frac {d^7}{e^7}+\frac {7\,d\,x^6}{e}+\frac {7\,d^6\,x}{e^6}+\frac {21\,d^2\,x^5}{e^2}+\frac {35\,d^3\,x^4}{e^3}+\frac {35\,d^4\,x^3}{e^4}+\frac {21\,d^5\,x^2}{e^5}} \]
-((d + e*x)^(1/2)*(((a + b*x)^(1/2)*(2310*A*a^6*e^4 + 420*B*a^6*d*e^3 + 60 06*A*a^2*b^4*d^4 - 1716*B*a^3*b^3*d^4 - 17160*A*a^3*b^3*d^3*e + 2860*B*a^4 *b^2*d^3*e - 1820*B*a^5*b*d^2*e^2 + 20020*A*a^4*b^2*d^2*e^2 - 10920*A*a^5* b*d*e^3))/(15015*e^7*(a*e - b*d)^5) + (x*(a + b*x)^(1/2)*(2730*B*a^6*e^4 + 12012*A*a*b^5*d^4 + 2940*A*a^5*b*e^4 + 858*B*a^2*b^4*d^4 - 27456*A*a^2*b^ 4*d^3*e - 14560*A*a^4*b^2*d*e^3 - 12584*B*a^3*b^3*d^3*e + 28600*A*a^3*b^3* d^2*e^2 + 19500*B*a^4*b^2*d^2*e^2 - 12040*B*a^5*b*d*e^3))/(15015*e^7*(a*e - b*d)^5) + (x^2*(a + b*x)^(1/2)*(6006*A*b^6*d^4 + 6864*B*a*b^5*d^4 + 3640 *B*a^5*b*e^4 + 70*A*a^4*b^2*e^4 - 520*A*a^3*b^3*d*e^3 - 30888*B*a^2*b^4*d^ 3*e - 17880*B*a^4*b^2*d*e^3 + 1716*A*a^2*b^4*d^2*e^2 + 34424*B*a^3*b^3*d^2 *e^2 - 3432*A*a*b^5*d^3*e))/(15015*e^7*(a*e - b*d)^5) + (32*b^5*x^6*(a + b *x)^(1/2)*(8*A*b*e - 13*B*a*e + 5*B*b*d))/(15015*e^4*(a*e - b*d)^5) - (2*b ^2*x^3*(a + b*x)^(1/2)*(8*A*b*e - 13*B*a*e + 5*B*b*d)*(5*a^3*e^3 - 429*b^3 *d^3 + 143*a*b^2*d^2*e - 39*a^2*b*d*e^2))/(15015*e^7*(a*e - b*d)^5) - (16* b^4*x^5*(a*e - 13*b*d)*(a + b*x)^(1/2)*(8*A*b*e - 13*B*a*e + 5*B*b*d))/(15 015*e^5*(a*e - b*d)^5) + (4*b^3*x^4*(a + b*x)^(1/2)*(3*a^2*e^2 + 143*b^2*d ^2 - 26*a*b*d*e)*(8*A*b*e - 13*B*a*e + 5*B*b*d))/(15015*e^6*(a*e - b*d)^5) ))/(x^7 + d^7/e^7 + (7*d*x^6)/e + (7*d^6*x)/e^6 + (21*d^2*x^5)/e^2 + (35*d ^3*x^4)/e^3 + (35*d^4*x^3)/e^4 + (21*d^5*x^2)/e^5)