Integrand size = 24, antiderivative size = 201 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{5/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {8 b (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{693 e (b d-a e)^3 (d+e x)^{7/2}}+\frac {16 b^2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{3465 e (b d-a e)^4 (d+e x)^{5/2}} \]
-2/11*(-A*e+B*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)/(e*x+d)^(11/2)+2/99*(6*A*b*e-1 1*B*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^2/(e*x+d)^(9/2)+8/693*b*(6*A*b *e-11*B*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^3/(e*x+d)^(7/2)+16/3465*b^ 2*(6*A*b*e-11*B*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^4/(e*x+d)^(5/2)
Time = 0.28 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx=\frac {2 (a+b x)^{5/2} \left (315 B d e^2 (a+b x)^3-315 A e^3 (a+b x)^3-770 b B d e (a+b x)^2 (d+e x)+1155 A b e^2 (a+b x)^2 (d+e x)-385 a B e^2 (a+b x)^2 (d+e x)+495 b^2 B d (a+b x) (d+e x)^2-1485 A b^2 e (a+b x) (d+e x)^2+990 a b B e (a+b x) (d+e x)^2+693 A b^3 (d+e x)^3-693 a b^2 B (d+e x)^3\right )}{3465 (b d-a e)^4 (d+e x)^{11/2}} \]
(2*(a + b*x)^(5/2)*(315*B*d*e^2*(a + b*x)^3 - 315*A*e^3*(a + b*x)^3 - 770* b*B*d*e*(a + b*x)^2*(d + e*x) + 1155*A*b*e^2*(a + b*x)^2*(d + e*x) - 385*a *B*e^2*(a + b*x)^2*(d + e*x) + 495*b^2*B*d*(a + b*x)*(d + e*x)^2 - 1485*A* b^2*e*(a + b*x)*(d + e*x)^2 + 990*a*b*B*e*(a + b*x)*(d + e*x)^2 + 693*A*b^ 3*(d + e*x)^3 - 693*a*b^2*B*(d + e*x)^3))/(3465*(b*d - a*e)^4*(d + e*x)^(1 1/2))
Time = 0.26 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {87, 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.
| \(\displaystyle \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-11 a B e+6 A b e+5 b B d) \int \frac {(a+b x)^{3/2}}{(d+e x)^{11/2}}dx}{11 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-11 a B e+6 A b e+5 b B d) \left (\frac {4 b \int \frac {(a+b x)^{3/2}}{(d+e x)^{9/2}}dx}{9 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{9 (d+e x)^{9/2} (b d-a e)}\right )}{11 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-11 a B e+6 A b e+5 b B d) \left (\frac {4 b \left (\frac {2 b \int \frac {(a+b x)^{3/2}}{(d+e x)^{7/2}}dx}{7 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{9 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{9 (d+e x)^{9/2} (b d-a e)}\right )}{11 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\left (\frac {2 (a+b x)^{5/2}}{9 (d+e x)^{9/2} (b d-a e)}+\frac {4 b \left (\frac {4 b (a+b x)^{5/2}}{35 (d+e x)^{5/2} (b d-a e)^2}+\frac {2 (a+b x)^{5/2}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{9 (b d-a e)}\right ) (-11 a B e+6 A b e+5 b B d)}{11 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)}\) |
(-2*(B*d - A*e)*(a + b*x)^(5/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + ((5 *b*B*d + 6*A*b*e - 11*a*B*e)*((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*e*(b*d - a*e))
3.23.20.3.1 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]))))
Time = 1.08 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.60
| method | result | size |
| gosper | \(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-48 A \,b^{3} e^{3} x^{3}+88 B a \,b^{2} e^{3} x^{3}-40 B \,b^{3} d \,e^{2} x^{3}+120 A a \,b^{2} e^{3} x^{2}-264 A \,b^{3} d \,e^{2} x^{2}-220 B \,a^{2} b \,e^{3} x^{2}+584 B a \,b^{2} d \,e^{2} x^{2}-220 B \,b^{3} d^{2} e \,x^{2}-210 A \,a^{2} b \,e^{3} x +660 A a \,b^{2} d \,e^{2} x -594 A \,b^{3} d^{2} e x +385 B \,a^{3} e^{3} x -1385 B \,a^{2} b d \,e^{2} x +1639 B a \,b^{2} d^{2} e x -495 b^{3} B \,d^{3} x +315 a^{3} A \,e^{3}-1155 A \,a^{2} b d \,e^{2}+1485 A a \,b^{2} d^{2} e -693 A \,b^{3} d^{3}+70 B \,a^{3} d \,e^{2}-220 B \,a^{2} b \,d^{2} e +198 B a \,b^{2} d^{3}\right )}{3465 \left (e x +d \right )^{\frac {11}{2}} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\) | \(322\) |
| default | \(-\frac {2 \left (-48 A \,b^{4} e^{3} x^{4}+88 B a \,b^{3} e^{3} x^{4}-40 B \,b^{4} d \,e^{2} x^{4}+72 A a \,b^{3} e^{3} x^{3}-264 A \,b^{4} d \,e^{2} x^{3}-132 B \,a^{2} b^{2} e^{3} x^{3}+544 B a \,b^{3} d \,e^{2} x^{3}-220 B \,b^{4} d^{2} e \,x^{3}-90 A \,a^{2} b^{2} e^{3} x^{2}+396 A a \,b^{3} d \,e^{2} x^{2}-594 A \,b^{4} d^{2} e \,x^{2}+165 B \,a^{3} b \,e^{3} x^{2}-801 B \,a^{2} b^{2} d \,e^{2} x^{2}+1419 B a \,b^{3} d^{2} e \,x^{2}-495 B \,b^{4} d^{3} x^{2}+105 A \,a^{3} b \,e^{3} x -495 A \,a^{2} b^{2} d \,e^{2} x +891 A a \,b^{3} d^{2} e x -693 A \,b^{4} d^{3} x +385 B \,a^{4} e^{3} x -1315 B \,a^{3} b d \,e^{2} x +1419 B \,a^{2} b^{2} d^{2} e x -297 B a \,b^{3} d^{3} x +315 A \,a^{4} e^{3}-1155 A \,a^{3} b d \,e^{2}+1485 A \,a^{2} b^{2} d^{2} e -693 A a \,b^{3} d^{3}+70 B \,a^{4} d \,e^{2}-220 B \,a^{3} b \,d^{2} e +198 B \,a^{2} b^{2} d^{3}\right ) \left (b x +a \right )^{\frac {3}{2}}}{3465 \left (e x +d \right )^{\frac {11}{2}} \left (a e -b d \right )^{4}}\) | \(401\) |
-2/3465*(b*x+a)^(5/2)*(-48*A*b^3*e^3*x^3+88*B*a*b^2*e^3*x^3-40*B*b^3*d*e^2 *x^3+120*A*a*b^2*e^3*x^2-264*A*b^3*d*e^2*x^2-220*B*a^2*b*e^3*x^2+584*B*a*b ^2*d*e^2*x^2-220*B*b^3*d^2*e*x^2-210*A*a^2*b*e^3*x+660*A*a*b^2*d*e^2*x-594 *A*b^3*d^2*e*x+385*B*a^3*e^3*x-1385*B*a^2*b*d*e^2*x+1639*B*a*b^2*d^2*e*x-4 95*B*b^3*d^3*x+315*A*a^3*e^3-1155*A*a^2*b*d*e^2+1485*A*a*b^2*d^2*e-693*A*b ^3*d^3+70*B*a^3*d*e^2-220*B*a^2*b*d^2*e+198*B*a*b^2*d^3)/(e*x+d)^(11/2)/(a ^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)
Leaf count of result is larger than twice the leaf count of optimal. 880 vs. \(2 (177) = 354\).
Time = 62.91 (sec) , antiderivative size = 880, normalized size of antiderivative = 4.38 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx=-\frac {2 \, {\left (315 \, A a^{5} e^{3} - 8 \, {\left (5 \, B b^{5} d e^{2} - {\left (11 \, B a b^{4} - 6 \, A b^{5}\right )} e^{3}\right )} x^{5} - 4 \, {\left (55 \, B b^{5} d^{2} e - 6 \, {\left (21 \, B a b^{4} - 11 \, A b^{5}\right )} d e^{2} + {\left (11 \, B a^{2} b^{3} - 6 \, A a b^{4}\right )} e^{3}\right )} x^{4} + 99 \, {\left (2 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} d^{3} - 55 \, {\left (4 \, B a^{4} b - 27 \, A a^{3} b^{2}\right )} d^{2} e + 35 \, {\left (2 \, B a^{5} - 33 \, A a^{4} b\right )} d e^{2} - {\left (495 \, B b^{5} d^{3} - 11 \, {\left (109 \, B a b^{4} - 54 \, A b^{5}\right )} d^{2} e + {\left (257 \, B a^{2} b^{3} - 132 \, A a b^{4}\right )} d e^{2} - 3 \, {\left (11 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} e^{3}\right )} x^{3} - {\left (99 \, {\left (8 \, B a b^{4} + 7 \, A b^{5}\right )} d^{3} - 33 \, {\left (86 \, B a^{2} b^{3} + 9 \, A a b^{4}\right )} d^{2} e + {\left (2116 \, B a^{3} b^{2} + 99 \, A a^{2} b^{3}\right )} d e^{2} - 5 \, {\left (110 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} e^{3}\right )} x^{2} - {\left (99 \, {\left (B a^{2} b^{3} + 14 \, A a b^{4}\right )} d^{3} - 11 \, {\left (109 \, B a^{3} b^{2} + 216 \, A a^{2} b^{3}\right )} d^{2} e + 15 \, {\left (83 \, B a^{4} b + 110 \, A a^{3} b^{2}\right )} d e^{2} - 35 \, {\left (11 \, B a^{5} + 12 \, A a^{4} b\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{3465 \, {\left (b^{4} d^{10} - 4 \, a b^{3} d^{9} e + 6 \, a^{2} b^{2} d^{8} e^{2} - 4 \, a^{3} b d^{7} e^{3} + a^{4} d^{6} e^{4} + {\left (b^{4} d^{4} e^{6} - 4 \, a b^{3} d^{3} e^{7} + 6 \, a^{2} b^{2} d^{2} e^{8} - 4 \, a^{3} b d e^{9} + a^{4} e^{10}\right )} x^{6} + 6 \, {\left (b^{4} d^{5} e^{5} - 4 \, a b^{3} d^{4} e^{6} + 6 \, a^{2} b^{2} d^{3} e^{7} - 4 \, a^{3} b d^{2} e^{8} + a^{4} d e^{9}\right )} x^{5} + 15 \, {\left (b^{4} d^{6} e^{4} - 4 \, a b^{3} d^{5} e^{5} + 6 \, a^{2} b^{2} d^{4} e^{6} - 4 \, a^{3} b d^{3} e^{7} + a^{4} d^{2} e^{8}\right )} x^{4} + 20 \, {\left (b^{4} d^{7} e^{3} - 4 \, a b^{3} d^{6} e^{4} + 6 \, a^{2} b^{2} d^{5} e^{5} - 4 \, a^{3} b d^{4} e^{6} + a^{4} d^{3} e^{7}\right )} x^{3} + 15 \, {\left (b^{4} d^{8} e^{2} - 4 \, a b^{3} d^{7} e^{3} + 6 \, a^{2} b^{2} d^{6} e^{4} - 4 \, a^{3} b d^{5} e^{5} + a^{4} d^{4} e^{6}\right )} x^{2} + 6 \, {\left (b^{4} d^{9} e - 4 \, a b^{3} d^{8} e^{2} + 6 \, a^{2} b^{2} d^{7} e^{3} - 4 \, a^{3} b d^{6} e^{4} + a^{4} d^{5} e^{5}\right )} x\right )}} \]
-2/3465*(315*A*a^5*e^3 - 8*(5*B*b^5*d*e^2 - (11*B*a*b^4 - 6*A*b^5)*e^3)*x^ 5 - 4*(55*B*b^5*d^2*e - 6*(21*B*a*b^4 - 11*A*b^5)*d*e^2 + (11*B*a^2*b^3 - 6*A*a*b^4)*e^3)*x^4 + 99*(2*B*a^3*b^2 - 7*A*a^2*b^3)*d^3 - 55*(4*B*a^4*b - 27*A*a^3*b^2)*d^2*e + 35*(2*B*a^5 - 33*A*a^4*b)*d*e^2 - (495*B*b^5*d^3 - 11*(109*B*a*b^4 - 54*A*b^5)*d^2*e + (257*B*a^2*b^3 - 132*A*a*b^4)*d*e^2 - 3*(11*B*a^3*b^2 - 6*A*a^2*b^3)*e^3)*x^3 - (99*(8*B*a*b^4 + 7*A*b^5)*d^3 - 33*(86*B*a^2*b^3 + 9*A*a*b^4)*d^2*e + (2116*B*a^3*b^2 + 99*A*a^2*b^3)*d*e^ 2 - 5*(110*B*a^4*b + 3*A*a^3*b^2)*e^3)*x^2 - (99*(B*a^2*b^3 + 14*A*a*b^4)* d^3 - 11*(109*B*a^3*b^2 + 216*A*a^2*b^3)*d^2*e + 15*(83*B*a^4*b + 110*A*a^ 3*b^2)*d*e^2 - 35*(11*B*a^5 + 12*A*a^4*b)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^4*d^10 - 4*a*b^3*d^9*e + 6*a^2*b^2*d^8*e^2 - 4*a^3*b*d^7*e^3 + a^4* d^6*e^4 + (b^4*d^4*e^6 - 4*a*b^3*d^3*e^7 + 6*a^2*b^2*d^2*e^8 - 4*a^3*b*d*e ^9 + a^4*e^10)*x^6 + 6*(b^4*d^5*e^5 - 4*a*b^3*d^4*e^6 + 6*a^2*b^2*d^3*e^7 - 4*a^3*b*d^2*e^8 + a^4*d*e^9)*x^5 + 15*(b^4*d^6*e^4 - 4*a*b^3*d^5*e^5 + 6 *a^2*b^2*d^4*e^6 - 4*a^3*b*d^3*e^7 + a^4*d^2*e^8)*x^4 + 20*(b^4*d^7*e^3 - 4*a*b^3*d^6*e^4 + 6*a^2*b^2*d^5*e^5 - 4*a^3*b*d^4*e^6 + a^4*d^3*e^7)*x^3 + 15*(b^4*d^8*e^2 - 4*a*b^3*d^7*e^3 + 6*a^2*b^2*d^6*e^4 - 4*a^3*b*d^5*e^5 + a^4*d^4*e^6)*x^2 + 6*(b^4*d^9*e - 4*a*b^3*d^8*e^2 + 6*a^2*b^2*d^7*e^3 - 4 *a^3*b*d^6*e^4 + a^4*d^5*e^5)*x)
\[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {13}{2}}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx=\text {Exception raised: ValueError} \]
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. 815 vs. \(2 (177) = 354\).
Time = 0.68 (sec) , antiderivative size = 815, normalized size of antiderivative = 4.05 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx=\frac {2 \, {\left ({\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (5 \, B b^{13} d^{2} e^{7} {\left | b \right |} - 16 \, B a b^{12} d e^{8} {\left | b \right |} + 6 \, A b^{13} d e^{8} {\left | b \right |} + 11 \, B a^{2} b^{11} e^{9} {\left | b \right |} - 6 \, A a b^{12} e^{9} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}} + \frac {11 \, {\left (5 \, B b^{14} d^{3} e^{6} {\left | b \right |} - 21 \, B a b^{13} d^{2} e^{7} {\left | b \right |} + 6 \, A b^{14} d^{2} e^{7} {\left | b \right |} + 27 \, B a^{2} b^{12} d e^{8} {\left | b \right |} - 12 \, A a b^{13} d e^{8} {\left | b \right |} - 11 \, B a^{3} b^{11} e^{9} {\left | b \right |} + 6 \, A a^{2} b^{12} e^{9} {\left | b \right |}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} + \frac {99 \, {\left (5 \, B b^{15} d^{4} e^{5} {\left | b \right |} - 26 \, B a b^{14} d^{3} e^{6} {\left | b \right |} + 6 \, A b^{15} d^{3} e^{6} {\left | b \right |} + 48 \, B a^{2} b^{13} d^{2} e^{7} {\left | b \right |} - 18 \, A a b^{14} d^{2} e^{7} {\left | b \right |} - 38 \, B a^{3} b^{12} d e^{8} {\left | b \right |} + 18 \, A a^{2} b^{13} d e^{8} {\left | b \right |} + 11 \, B a^{4} b^{11} e^{9} {\left | b \right |} - 6 \, A a^{3} b^{12} e^{9} {\left | b \right |}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} {\left (b x + a\right )} - \frac {693 \, {\left (B a b^{15} d^{4} e^{5} {\left | b \right |} - A b^{16} d^{4} e^{5} {\left | b \right |} - 4 \, B a^{2} b^{14} d^{3} e^{6} {\left | b \right |} + 4 \, A a b^{15} d^{3} e^{6} {\left | b \right |} + 6 \, B a^{3} b^{13} d^{2} e^{7} {\left | b \right |} - 6 \, A a^{2} b^{14} d^{2} e^{7} {\left | b \right |} - 4 \, B a^{4} b^{12} d e^{8} {\left | b \right |} + 4 \, A a^{3} b^{13} d e^{8} {\left | b \right |} + B a^{5} b^{11} e^{9} {\left | b \right |} - A a^{4} b^{12} e^{9} {\left | b \right |}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} {\left (b x + a\right )}^{\frac {5}{2}}}{3465 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {11}{2}}} \]
2/3465*((4*(b*x + a)*(2*(5*B*b^13*d^2*e^7*abs(b) - 16*B*a*b^12*d*e^8*abs(b ) + 6*A*b^13*d*e^8*abs(b) + 11*B*a^2*b^11*e^9*abs(b) - 6*A*a*b^12*e^9*abs( b))*(b*x + a)/(b^7*d^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^3 *b^4*d^2*e^8 + 5*a^4*b^3*d*e^9 - a^5*b^2*e^10) + 11*(5*B*b^14*d^3*e^6*abs( b) - 21*B*a*b^13*d^2*e^7*abs(b) + 6*A*b^14*d^2*e^7*abs(b) + 27*B*a^2*b^12* d*e^8*abs(b) - 12*A*a*b^13*d*e^8*abs(b) - 11*B*a^3*b^11*e^9*abs(b) + 6*A*a ^2*b^12*e^9*abs(b))/(b^7*d^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^3*b^4*d^2*e^8 + 5*a^4*b^3*d*e^9 - a^5*b^2*e^10)) + 99*(5*B*b^15*d^4*e ^5*abs(b) - 26*B*a*b^14*d^3*e^6*abs(b) + 6*A*b^15*d^3*e^6*abs(b) + 48*B*a^ 2*b^13*d^2*e^7*abs(b) - 18*A*a*b^14*d^2*e^7*abs(b) - 38*B*a^3*b^12*d*e^8*a bs(b) + 18*A*a^2*b^13*d*e^8*abs(b) + 11*B*a^4*b^11*e^9*abs(b) - 6*A*a^3*b^ 12*e^9*abs(b))/(b^7*d^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^ 3*b^4*d^2*e^8 + 5*a^4*b^3*d*e^9 - a^5*b^2*e^10))*(b*x + a) - 693*(B*a*b^15 *d^4*e^5*abs(b) - A*b^16*d^4*e^5*abs(b) - 4*B*a^2*b^14*d^3*e^6*abs(b) + 4* A*a*b^15*d^3*e^6*abs(b) + 6*B*a^3*b^13*d^2*e^7*abs(b) - 6*A*a^2*b^14*d^2*e ^7*abs(b) - 4*B*a^4*b^12*d*e^8*abs(b) + 4*A*a^3*b^13*d*e^8*abs(b) + B*a^5* b^11*e^9*abs(b) - A*a^4*b^12*e^9*abs(b))/(b^7*d^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^3*b^4*d^2*e^8 + 5*a^4*b^3*d*e^9 - a^5*b^2*e^10)) *(b*x + a)^(5/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(11/2)
Time = 3.07 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.84 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx=-\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (140\,B\,a^5\,d\,e^2+630\,A\,a^5\,e^3-440\,B\,a^4\,b\,d^2\,e-2310\,A\,a^4\,b\,d\,e^2+396\,B\,a^3\,b^2\,d^3+2970\,A\,a^3\,b^2\,d^2\,e-1386\,A\,a^2\,b^3\,d^3\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^4}+\frac {x\,\sqrt {a+b\,x}\,\left (770\,B\,a^5\,e^3-2490\,B\,a^4\,b\,d\,e^2+840\,A\,a^4\,b\,e^3+2398\,B\,a^3\,b^2\,d^2\,e-3300\,A\,a^3\,b^2\,d\,e^2-198\,B\,a^2\,b^3\,d^3+4752\,A\,a^2\,b^3\,d^2\,e-2772\,A\,a\,b^4\,d^3\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^4}-\frac {x^2\,\sqrt {a+b\,x}\,\left (-1100\,B\,a^4\,b\,e^3+4232\,B\,a^3\,b^2\,d\,e^2-30\,A\,a^3\,b^2\,e^3-5676\,B\,a^2\,b^3\,d^2\,e+198\,A\,a^2\,b^3\,d\,e^2+1584\,B\,a\,b^4\,d^3-594\,A\,a\,b^4\,d^2\,e+1386\,A\,b^5\,d^3\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^4}-\frac {16\,b^4\,x^5\,\sqrt {a+b\,x}\,\left (6\,A\,b\,e-11\,B\,a\,e+5\,B\,b\,d\right )}{3465\,e^4\,{\left (a\,e-b\,d\right )}^4}+\frac {8\,b^3\,x^4\,\left (a\,e-11\,b\,d\right )\,\sqrt {a+b\,x}\,\left (6\,A\,b\,e-11\,B\,a\,e+5\,B\,b\,d\right )}{3465\,e^5\,{\left (a\,e-b\,d\right )}^4}-\frac {2\,b^2\,x^3\,\sqrt {a+b\,x}\,\left (3\,a^2\,e^2-22\,a\,b\,d\,e+99\,b^2\,d^2\right )\,\left (6\,A\,b\,e-11\,B\,a\,e+5\,B\,b\,d\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^4}\right )}{x^6+\frac {d^6}{e^6}+\frac {6\,d\,x^5}{e}+\frac {6\,d^5\,x}{e^5}+\frac {15\,d^2\,x^4}{e^2}+\frac {20\,d^3\,x^3}{e^3}+\frac {15\,d^4\,x^2}{e^4}} \]
-((d + e*x)^(1/2)*(((a + b*x)^(1/2)*(630*A*a^5*e^3 + 140*B*a^5*d*e^2 - 138 6*A*a^2*b^3*d^3 + 396*B*a^3*b^2*d^3 + 2970*A*a^3*b^2*d^2*e - 2310*A*a^4*b* d*e^2 - 440*B*a^4*b*d^2*e))/(3465*e^6*(a*e - b*d)^4) + (x*(a + b*x)^(1/2)* (770*B*a^5*e^3 - 2772*A*a*b^4*d^3 + 840*A*a^4*b*e^3 - 198*B*a^2*b^3*d^3 + 4752*A*a^2*b^3*d^2*e - 3300*A*a^3*b^2*d*e^2 + 2398*B*a^3*b^2*d^2*e - 2490* B*a^4*b*d*e^2))/(3465*e^6*(a*e - b*d)^4) - (x^2*(a + b*x)^(1/2)*(1386*A*b^ 5*d^3 + 1584*B*a*b^4*d^3 - 1100*B*a^4*b*e^3 - 30*A*a^3*b^2*e^3 + 198*A*a^2 *b^3*d*e^2 - 5676*B*a^2*b^3*d^2*e + 4232*B*a^3*b^2*d*e^2 - 594*A*a*b^4*d^2 *e))/(3465*e^6*(a*e - b*d)^4) - (16*b^4*x^5*(a + b*x)^(1/2)*(6*A*b*e - 11* B*a*e + 5*B*b*d))/(3465*e^4*(a*e - b*d)^4) + (8*b^3*x^4*(a*e - 11*b*d)*(a + b*x)^(1/2)*(6*A*b*e - 11*B*a*e + 5*B*b*d))/(3465*e^5*(a*e - b*d)^4) - (2 *b^2*x^3*(a + b*x)^(1/2)*(3*a^2*e^2 + 99*b^2*d^2 - 22*a*b*d*e)*(6*A*b*e - 11*B*a*e + 5*B*b*d))/(3465*e^6*(a*e - b*d)^4)))/(x^6 + d^6/e^6 + (6*d*x^5) /e + (6*d^5*x)/e^5 + (15*d^2*x^4)/e^2 + (20*d^3*x^3)/e^3 + (15*d^4*x^2)/e^ 4)