Integrand size = 24, antiderivative size = 255 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{13/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{3/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {4 b (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{231 e (b d-a e)^3 (d+e x)^{7/2}}+\frac {16 b^2 (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{1155 e (b d-a e)^4 (d+e x)^{5/2}}+\frac {32 b^3 (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{3465 e (b d-a e)^5 (d+e x)^{3/2}} \]
-2/11*(-A*e+B*d)*(b*x+a)^(3/2)/e/(-a*e+b*d)/(e*x+d)^(11/2)+2/99*(8*A*b*e-1 1*B*a*e+3*B*b*d)*(b*x+a)^(3/2)/e/(-a*e+b*d)^2/(e*x+d)^(9/2)+4/231*b*(8*A*b *e-11*B*a*e+3*B*b*d)*(b*x+a)^(3/2)/e/(-a*e+b*d)^3/(e*x+d)^(7/2)+16/1155*b^ 2*(8*A*b*e-11*B*a*e+3*B*b*d)*(b*x+a)^(3/2)/e/(-a*e+b*d)^4/(e*x+d)^(5/2)+32 /3465*b^3*(8*A*b*e-11*B*a*e+3*B*b*d)*(b*x+a)^(3/2)/e/(-a*e+b*d)^5/(e*x+d)^ (3/2)
Time = 0.44 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{13/2}} \, dx=\frac {2 (a+b x)^{3/2} \left (-315 B d e^3 (a+b x)^4+315 A e^4 (a+b x)^4+1155 b B d e^2 (a+b x)^3 (d+e x)-1540 A b e^3 (a+b x)^3 (d+e x)+385 a B e^3 (a+b x)^3 (d+e x)-1485 b^2 B d e (a+b x)^2 (d+e x)^2+2970 A b^2 e^2 (a+b x)^2 (d+e x)^2-1485 a b B e^2 (a+b x)^2 (d+e x)^2+693 b^3 B d (a+b x) (d+e x)^3-2772 A b^3 e (a+b x) (d+e x)^3+2079 a b^2 B e (a+b x) (d+e x)^3+1155 A b^4 (d+e x)^4-1155 a b^3 B (d+e x)^4\right )}{3465 (b d-a e)^5 (d+e x)^{11/2}} \]
(2*(a + b*x)^(3/2)*(-315*B*d*e^3*(a + b*x)^4 + 315*A*e^4*(a + b*x)^4 + 115 5*b*B*d*e^2*(a + b*x)^3*(d + e*x) - 1540*A*b*e^3*(a + b*x)^3*(d + e*x) + 3 85*a*B*e^3*(a + b*x)^3*(d + e*x) - 1485*b^2*B*d*e*(a + b*x)^2*(d + e*x)^2 + 2970*A*b^2*e^2*(a + b*x)^2*(d + e*x)^2 - 1485*a*b*B*e^2*(a + b*x)^2*(d + e*x)^2 + 693*b^3*B*d*(a + b*x)*(d + e*x)^3 - 2772*A*b^3*e*(a + b*x)*(d + e*x)^3 + 2079*a*b^2*B*e*(a + b*x)*(d + e*x)^3 + 1155*A*b^4*(d + e*x)^4 - 1 155*a*b^3*B*(d + e*x)^4))/(3465*(b*d - a*e)^5*(d + e*x)^(11/2))
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.
| \(\displaystyle \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{13/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-11 a B e+8 A b e+3 b B d) \int \frac {\sqrt {a+b x}}{(d+e x)^{11/2}}dx}{11 e (b d-a e)}-\frac {2 (a+b x)^{3/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-11 a B e+8 A b e+3 b B d) \left (\frac {2 b \int \frac {\sqrt {a+b x}}{(d+e x)^{9/2}}dx}{3 (b d-a e)}+\frac {2 (a+b x)^{3/2}}{9 (d+e x)^{9/2} (b d-a e)}\right )}{11 e (b d-a e)}-\frac {2 (a+b x)^{3/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-11 a B e+8 A b e+3 b B d) \left (\frac {2 b \left (\frac {4 b \int \frac {\sqrt {a+b x}}{(d+e x)^{7/2}}dx}{7 (b d-a e)}+\frac {2 (a+b x)^{3/2}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{3 (b d-a e)}+\frac {2 (a+b x)^{3/2}}{9 (d+e x)^{9/2} (b d-a e)}\right )}{11 e (b d-a e)}-\frac {2 (a+b x)^{3/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-11 a B e+8 A b e+3 b B d) \left (\frac {2 b \left (\frac {4 b \left (\frac {2 b \int \frac {\sqrt {a+b x}}{(d+e x)^{5/2}}dx}{5 (b d-a e)}+\frac {2 (a+b x)^{3/2}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{7 (b d-a e)}+\frac {2 (a+b x)^{3/2}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{3 (b d-a e)}+\frac {2 (a+b x)^{3/2}}{9 (d+e x)^{9/2} (b d-a e)}\right )}{11 e (b d-a e)}-\frac {2 (a+b x)^{3/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\left (\frac {2 (a+b x)^{3/2}}{9 (d+e x)^{9/2} (b d-a e)}+\frac {2 b \left (\frac {2 (a+b x)^{3/2}}{7 (d+e x)^{7/2} (b d-a e)}+\frac {4 b \left (\frac {4 b (a+b x)^{3/2}}{15 (d+e x)^{3/2} (b d-a e)^2}+\frac {2 (a+b x)^{3/2}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{7 (b d-a e)}\right )}{3 (b d-a e)}\right ) (-11 a B e+8 A b e+3 b B d)}{11 e (b d-a e)}-\frac {2 (a+b x)^{3/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)}\) |
(-2*(B*d - A*e)*(a + b*x)^(3/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + ((3 *b*B*d + 8*A*b*e - 11*a*B*e)*((2*(a + b*x)^(3/2))/(9*(b*d - a*e)*(d + e*x) ^(9/2)) + (2*b*((2*(a + b*x)^(3/2))/(7*(b*d - a*e)*(d + e*x)^(7/2)) + (4*b *((2*(a + b*x)^(3/2))/(5*(b*d - a*e)*(d + e*x)^(5/2)) + (4*b*(a + b*x)^(3/ 2))/(15*(b*d - a*e)^2*(d + e*x)^(3/2))))/(7*(b*d - a*e))))/(3*(b*d - a*e)) ))/(11*e*(b*d - a*e))
3.23.9.3.1 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 = 449, normalized size of antiderivative = 1.76
| method | result | size |
| default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (128 A \,b^{4} e^{4} x^{4}-176 B a \,b^{3} e^{4} x^{4}+48 B \,b^{4} d \,e^{3} x^{4}-192 A a \,b^{3} e^{4} x^{3}+704 A \,b^{4} d \,e^{3} x^{3}+264 B \,a^{2} b^{2} e^{4} x^{3}-1040 B a \,b^{3} d \,e^{3} x^{3}+264 B \,b^{4} d^{2} e^{2} x^{3}+240 A \,a^{2} b^{2} e^{4} x^{2}-1056 A a \,b^{3} d \,e^{3} x^{2}+1584 A \,b^{4} d^{2} e^{2} x^{2}-330 B \,a^{3} b \,e^{4} x^{2}+1542 B \,a^{2} b^{2} d \,e^{3} x^{2}-2574 B a \,b^{3} d^{2} e^{2} x^{2}+594 B \,b^{4} d^{3} e \,x^{2}-280 A \,a^{3} b \,e^{4} x +1320 A \,a^{2} b^{2} d \,e^{3} x -2376 A a \,b^{3} d^{2} e^{2} x +1848 A \,b^{4} d^{3} e x +385 B \,a^{4} e^{4} x -1920 B \,a^{3} b d \,e^{3} x +3762 B \,a^{2} b^{2} d^{2} e^{2} x -3432 B a \,b^{3} d^{3} e x +693 B \,b^{4} d^{4} x +315 A \,a^{4} e^{4}-1540 A \,a^{3} b d \,e^{3}+2970 A \,a^{2} b^{2} d^{2} e^{2}-2772 A a \,b^{3} d^{3} e +1155 A \,b^{4} d^{4}+70 B \,a^{4} d \,e^{3}-330 B \,a^{3} b \,d^{2} e^{2}+594 B \,a^{2} b^{2} d^{3} e -462 B a \,b^{3} d^{4}\right )}{3465 \left (e x +d \right )^{\frac {11}{2}} \left (a e -b d \right )^{5}}\) | \(449\) |
| gosper | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (128 A \,b^{4} e^{4} x^{4}-176 B a \,b^{3} e^{4} x^{4}+48 B \,b^{4} d \,e^{3} x^{4}-192 A a \,b^{3} e^{4} x^{3}+704 A \,b^{4} d \,e^{3} x^{3}+264 B \,a^{2} b^{2} e^{4} x^{3}-1040 B a \,b^{3} d \,e^{3} x^{3}+264 B \,b^{4} d^{2} e^{2} x^{3}+240 A \,a^{2} b^{2} e^{4} x^{2}-1056 A a \,b^{3} d \,e^{3} x^{2}+1584 A \,b^{4} d^{2} e^{2} x^{2}-330 B \,a^{3} b \,e^{4} x^{2}+1542 B \,a^{2} b^{2} d \,e^{3} x^{2}-2574 B a \,b^{3} d^{2} e^{2} x^{2}+594 B \,b^{4} d^{3} e \,x^{2}-280 A \,a^{3} b \,e^{4} x +1320 A \,a^{2} b^{2} d \,e^{3} x -2376 A a \,b^{3} d^{2} e^{2} x +1848 A \,b^{4} d^{3} e x +385 B \,a^{4} e^{4} x -1920 B \,a^{3} b d \,e^{3} x +3762 B \,a^{2} b^{2} d^{2} e^{2} x -3432 B a \,b^{3} d^{3} e x +693 B \,b^{4} d^{4} x +315 A \,a^{4} e^{4}-1540 A \,a^{3} b d \,e^{3}+2970 A \,a^{2} b^{2} d^{2} e^{2}-2772 A a \,b^{3} d^{3} e +1155 A \,b^{4} d^{4}+70 B \,a^{4} d \,e^{3}-330 B \,a^{3} b \,d^{2} e^{2}+594 B \,a^{2} b^{2} d^{3} e -462 B a \,b^{3} d^{4}\right )}{3465 \left (e x +d \right )^{\frac {11}{2}} \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}\) | \(505\) |
-2/3465*(b*x+a)^(3/2)/(e*x+d)^(11/2)*(128*A*b^4*e^4*x^4-176*B*a*b^3*e^4*x^ 4+48*B*b^4*d*e^3*x^4-192*A*a*b^3*e^4*x^3+704*A*b^4*d*e^3*x^3+264*B*a^2*b^2 *e^4*x^3-1040*B*a*b^3*d*e^3*x^3+264*B*b^4*d^2*e^2*x^3+240*A*a^2*b^2*e^4*x^ 2-1056*A*a*b^3*d*e^3*x^2+1584*A*b^4*d^2*e^2*x^2-330*B*a^3*b*e^4*x^2+1542*B *a^2*b^2*d*e^3*x^2-2574*B*a*b^3*d^2*e^2*x^2+594*B*b^4*d^3*e*x^2-280*A*a^3* b*e^4*x+1320*A*a^2*b^2*d*e^3*x-2376*A*a*b^3*d^2*e^2*x+1848*A*b^4*d^3*e*x+3 85*B*a^4*e^4*x-1920*B*a^3*b*d*e^3*x+3762*B*a^2*b^2*d^2*e^2*x-3432*B*a*b^3* d^3*e*x+693*B*b^4*d^4*x+315*A*a^4*e^4-1540*A*a^3*b*d*e^3+2970*A*a^2*b^2*d^ 2*e^2-2772*A*a*b^3*d^3*e+1155*A*b^4*d^4+70*B*a^4*d*e^3-330*B*a^3*b*d^2*e^2 +594*B*a^2*b^2*d^3*e-462*B*a*b^3*d^4)/(a*e-b*d)^5
Leaf count of result is larger than twice the leaf count of optimal. 1045 vs. \(2 (225) = 450\).
Time = 61.05 (sec) , antiderivative size = 1045, normalized size of antiderivative = 4.10 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{13/2}} \, dx=\frac {2 \, {\left (315 \, A a^{5} e^{4} + 16 \, {\left (3 \, B b^{5} d e^{3} - {\left (11 \, B a b^{4} - 8 \, A b^{5}\right )} e^{4}\right )} x^{5} - 231 \, {\left (2 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} d^{4} + 198 \, {\left (3 \, B a^{3} b^{2} - 14 \, A a^{2} b^{3}\right )} d^{3} e - 330 \, {\left (B a^{4} b - 9 \, A a^{3} b^{2}\right )} d^{2} e^{2} + 70 \, {\left (B a^{5} - 22 \, A a^{4} b\right )} d e^{3} + 8 \, {\left (33 \, B b^{5} d^{2} e^{2} - 4 \, {\left (31 \, B a b^{4} - 22 \, A b^{5}\right )} d e^{3} + {\left (11 \, B a^{2} b^{3} - 8 \, A a b^{4}\right )} e^{4}\right )} x^{4} + 2 \, {\left (297 \, B b^{5} d^{3} e - 33 \, {\left (35 \, B a b^{4} - 24 \, A b^{5}\right )} d^{2} e^{2} + {\left (251 \, B a^{2} b^{3} - 176 \, A a b^{4}\right )} d e^{3} - 3 \, {\left (11 \, B a^{3} b^{2} - 8 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{3} + {\left (693 \, B b^{5} d^{4} - 66 \, {\left (43 \, B a b^{4} - 28 \, A b^{5}\right )} d^{3} e + 396 \, {\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{2} e^{2} - 6 \, {\left (63 \, B a^{3} b^{2} - 44 \, A a^{2} b^{3}\right )} d e^{3} + 5 \, {\left (11 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} e^{4}\right )} x^{2} + {\left (231 \, {\left (B a b^{4} + 5 \, A b^{5}\right )} d^{4} - 66 \, {\left (43 \, B a^{2} b^{3} + 14 \, A a b^{4}\right )} d^{3} e + 66 \, {\left (52 \, B a^{3} b^{2} + 9 \, A a^{2} b^{3}\right )} d^{2} e^{2} - 10 \, {\left (185 \, B a^{4} b + 22 \, A a^{3} b^{2}\right )} d e^{3} + 35 \, {\left (11 \, B a^{5} + A a^{4} b\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{3465 \, {\left (b^{5} d^{11} - 5 \, a b^{4} d^{10} e + 10 \, a^{2} b^{3} d^{9} e^{2} - 10 \, a^{3} b^{2} d^{8} e^{3} + 5 \, a^{4} b d^{7} e^{4} - a^{5} d^{6} e^{5} + {\left (b^{5} d^{5} e^{6} - 5 \, a b^{4} d^{4} e^{7} + 10 \, a^{2} b^{3} d^{3} e^{8} - 10 \, a^{3} b^{2} d^{2} e^{9} + 5 \, a^{4} b d e^{10} - a^{5} e^{11}\right )} x^{6} + 6 \, {\left (b^{5} d^{6} e^{5} - 5 \, a b^{4} d^{5} e^{6} + 10 \, a^{2} b^{3} d^{4} e^{7} - 10 \, a^{3} b^{2} d^{3} e^{8} + 5 \, a^{4} b d^{2} e^{9} - a^{5} d e^{10}\right )} x^{5} + 15 \, {\left (b^{5} d^{7} e^{4} - 5 \, a b^{4} d^{6} e^{5} + 10 \, a^{2} b^{3} d^{5} e^{6} - 10 \, a^{3} b^{2} d^{4} e^{7} + 5 \, a^{4} b d^{3} e^{8} - a^{5} d^{2} e^{9}\right )} x^{4} + 20 \, {\left (b^{5} d^{8} e^{3} - 5 \, a b^{4} d^{7} e^{4} + 10 \, a^{2} b^{3} d^{6} e^{5} - 10 \, a^{3} b^{2} d^{5} e^{6} + 5 \, a^{4} b d^{4} e^{7} - a^{5} d^{3} e^{8}\right )} x^{3} + 15 \, {\left (b^{5} d^{9} e^{2} - 5 \, a b^{4} d^{8} e^{3} + 10 \, a^{2} b^{3} d^{7} e^{4} - 10 \, a^{3} b^{2} d^{6} e^{5} + 5 \, a^{4} b d^{5} e^{6} - a^{5} d^{4} e^{7}\right )} x^{2} + 6 \, {\left (b^{5} d^{10} e - 5 \, a b^{4} d^{9} e^{2} + 10 \, a^{2} b^{3} d^{8} e^{3} - 10 \, a^{3} b^{2} d^{7} e^{4} + 5 \, a^{4} b d^{6} e^{5} - a^{5} d^{5} e^{6}\right )} x\right )}} \]
2/3465*(315*A*a^5*e^4 + 16*(3*B*b^5*d*e^3 - (11*B*a*b^4 - 8*A*b^5)*e^4)*x^ 5 - 231*(2*B*a^2*b^3 - 5*A*a*b^4)*d^4 + 198*(3*B*a^3*b^2 - 14*A*a^2*b^3)*d ^3*e - 330*(B*a^4*b - 9*A*a^3*b^2)*d^2*e^2 + 70*(B*a^5 - 22*A*a^4*b)*d*e^3 + 8*(33*B*b^5*d^2*e^2 - 4*(31*B*a*b^4 - 22*A*b^5)*d*e^3 + (11*B*a^2*b^3 - 8*A*a*b^4)*e^4)*x^4 + 2*(297*B*b^5*d^3*e - 33*(35*B*a*b^4 - 24*A*b^5)*d^2 *e^2 + (251*B*a^2*b^3 - 176*A*a*b^4)*d*e^3 - 3*(11*B*a^3*b^2 - 8*A*a^2*b^3 )*e^4)*x^3 + (693*B*b^5*d^4 - 66*(43*B*a*b^4 - 28*A*b^5)*d^3*e + 396*(3*B* a^2*b^3 - 2*A*a*b^4)*d^2*e^2 - 6*(63*B*a^3*b^2 - 44*A*a^2*b^3)*d*e^3 + 5*( 11*B*a^4*b - 8*A*a^3*b^2)*e^4)*x^2 + (231*(B*a*b^4 + 5*A*b^5)*d^4 - 66*(43 *B*a^2*b^3 + 14*A*a*b^4)*d^3*e + 66*(52*B*a^3*b^2 + 9*A*a^2*b^3)*d^2*e^2 - 10*(185*B*a^4*b + 22*A*a^3*b^2)*d*e^3 + 35*(11*B*a^5 + A*a^4*b)*e^4)*x)*s qrt(b*x + a)*sqrt(e*x + d)/(b^5*d^11 - 5*a*b^4*d^10*e + 10*a^2*b^3*d^9*e^2 - 10*a^3*b^2*d^8*e^3 + 5*a^4*b*d^7*e^4 - a^5*d^6*e^5 + (b^5*d^5*e^6 - 5*a *b^4*d^4*e^7 + 10*a^2*b^3*d^3*e^8 - 10*a^3*b^2*d^2*e^9 + 5*a^4*b*d*e^10 - a^5*e^11)*x^6 + 6*(b^5*d^6*e^5 - 5*a*b^4*d^5*e^6 + 10*a^2*b^3*d^4*e^7 - 10 *a^3*b^2*d^3*e^8 + 5*a^4*b*d^2*e^9 - a^5*d*e^10)*x^5 + 15*(b^5*d^7*e^4 - 5 *a*b^4*d^6*e^5 + 10*a^2*b^3*d^5*e^6 - 10*a^3*b^2*d^4*e^7 + 5*a^4*b*d^3*e^8 - a^5*d^2*e^9)*x^4 + 20*(b^5*d^8*e^3 - 5*a*b^4*d^7*e^4 + 10*a^2*b^3*d^6*e ^5 - 10*a^3*b^2*d^5*e^6 + 5*a^4*b*d^4*e^7 - a^5*d^3*e^8)*x^3 + 15*(b^5*d^9 *e^2 - 5*a*b^4*d^8*e^3 + 10*a^2*b^3*d^7*e^4 - 10*a^3*b^2*d^6*e^5 + 5*a^...
\[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{13/2}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {a + b x}}{\left (d + e x\right )^{\frac {13}{2}}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{13/2}} \, dx=\text {Exception raised: ValueError} \]
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. 937 vs. \(2 (225) = 450\).
Time = 0.61 (sec) , antiderivative size = 937, normalized size of antiderivative = 3.67 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{13/2}} \, dx=\frac {2 \, {\left ({\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (3 \, B b^{12} d e^{8} {\left | b \right |} - 11 \, B a b^{11} e^{9} {\left | b \right |} + 8 \, 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 (3 \, B b^{13} d^{2} e^{7} {\left | b \right |} - 14 \, B a b^{12} d e^{8} {\left | b \right |} + 8 \, A b^{13} d e^{8} {\left | b \right |} + 11 \, B a^{2} b^{11} e^{9} {\left | b \right |} - 8 \, A a 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 (3 \, B b^{14} d^{3} e^{6} {\left | b \right |} - 17 \, B a b^{13} d^{2} e^{7} {\left | b \right |} + 8 \, A b^{14} d^{2} e^{7} {\left | b \right |} + 25 \, B a^{2} b^{12} d e^{8} {\left | b \right |} - 16 \, A a b^{13} d e^{8} {\left | b \right |} - 11 \, B a^{3} b^{11} e^{9} {\left | b \right |} + 8 \, 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 )} {\left (b x + a\right )} + \frac {231 \, {\left (3 \, B b^{15} d^{4} e^{5} {\left | b \right |} - 20 \, B a b^{14} d^{3} e^{6} {\left | b \right |} + 8 \, A b^{15} d^{3} e^{6} {\left | b \right |} + 42 \, B a^{2} b^{13} d^{2} e^{7} {\left | b \right |} - 24 \, A a b^{14} d^{2} e^{7} {\left | b \right |} - 36 \, B a^{3} b^{12} d e^{8} {\left | b \right |} + 24 \, A a^{2} b^{13} d e^{8} {\left | b \right |} + 11 \, B a^{4} b^{11} e^{9} {\left | b \right |} - 8 \, 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 {1155 \, {\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 {3}{2}}}{3465 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {11}{2}}} \]
2/3465*((2*(4*(b*x + a)*(2*(3*B*b^12*d*e^8*abs(b) - 11*B*a*b^11*e^9*abs(b) + 8*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*(3 *B*b^13*d^2*e^7*abs(b) - 14*B*a*b^12*d*e^8*abs(b) + 8*A*b^13*d*e^8*abs(b) + 11*B*a^2*b^11*e^9*abs(b) - 8*A*a*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*(3*B*b^14*d^3*e^6*abs(b) - 17*B*a*b^13*d^2*e^7*abs(b) + 8 *A*b^14*d^2*e^7*abs(b) + 25*B*a^2*b^12*d*e^8*abs(b) - 16*A*a*b^13*d*e^8*ab s(b) - 11*B*a^3*b^11*e^9*abs(b) + 8*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))*(b*x + a) + 231*(3*B*b^15*d^4*e^5*abs(b) - 20*B*a*b^14* d^3*e^6*abs(b) + 8*A*b^15*d^3*e^6*abs(b) + 42*B*a^2*b^13*d^2*e^7*abs(b) - 24*A*a*b^14*d^2*e^7*abs(b) - 36*B*a^3*b^12*d*e^8*abs(b) + 24*A*a^2*b^13*d* e^8*abs(b) + 11*B*a^4*b^11*e^9*abs(b) - 8*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) - 1155*(B*a*b^15*d^4*e^5*abs(b) - A*b^1 6*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^1 2*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 -...
Time = 2.99 (sec) , antiderivative size = 585, normalized size of antiderivative = 2.29 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{13/2}} \, dx=-\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (140\,B\,a^5\,d\,e^3+630\,A\,a^5\,e^4-660\,B\,a^4\,b\,d^2\,e^2-3080\,A\,a^4\,b\,d\,e^3+1188\,B\,a^3\,b^2\,d^3\,e+5940\,A\,a^3\,b^2\,d^2\,e^2-924\,B\,a^2\,b^3\,d^4-5544\,A\,a^2\,b^3\,d^3\,e+2310\,A\,a\,b^4\,d^4\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^5}+\frac {x\,\sqrt {a+b\,x}\,\left (770\,B\,a^5\,e^4-3700\,B\,a^4\,b\,d\,e^3+70\,A\,a^4\,b\,e^4+6864\,B\,a^3\,b^2\,d^2\,e^2-440\,A\,a^3\,b^2\,d\,e^3-5676\,B\,a^2\,b^3\,d^3\,e+1188\,A\,a^2\,b^3\,d^2\,e^2+462\,B\,a\,b^4\,d^4-1848\,A\,a\,b^4\,d^3\,e+2310\,A\,b^5\,d^4\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^5}+\frac {32\,b^4\,x^5\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-11\,B\,a\,e+3\,B\,b\,d\right )}{3465\,e^3\,{\left (a\,e-b\,d\right )}^5}-\frac {16\,b^3\,x^4\,\left (a\,e-11\,b\,d\right )\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-11\,B\,a\,e+3\,B\,b\,d\right )}{3465\,e^4\,{\left (a\,e-b\,d\right )}^5}+\frac {4\,b^2\,x^3\,\sqrt {a+b\,x}\,\left (3\,a^2\,e^2-22\,a\,b\,d\,e+99\,b^2\,d^2\right )\,\left (8\,A\,b\,e-11\,B\,a\,e+3\,B\,b\,d\right )}{3465\,e^5\,{\left (a\,e-b\,d\right )}^5}-\frac {2\,b\,x^2\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-11\,B\,a\,e+3\,B\,b\,d\right )\,\left (5\,a^3\,e^3-33\,a^2\,b\,d\,e^2+99\,a\,b^2\,d^2\,e-231\,b^3\,d^3\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^5}\right )}{x^6+\frac {d^6}{e^6}+\frac {6\,d\,x^5}{e}+\frac {6\,d^5\,x}{e^5}+\frac {15\,d^2\,x^4}{e^2}+\frac {20\,d^3\,x^3}{e^3}+\frac {15\,d^4\,x^2}{e^4}} \]
-((d + e*x)^(1/2)*(((a + b*x)^(1/2)*(630*A*a^5*e^4 + 2310*A*a*b^4*d^4 + 14 0*B*a^5*d*e^3 - 924*B*a^2*b^3*d^4 - 5544*A*a^2*b^3*d^3*e + 1188*B*a^3*b^2* d^3*e - 660*B*a^4*b*d^2*e^2 + 5940*A*a^3*b^2*d^2*e^2 - 3080*A*a^4*b*d*e^3) )/(3465*e^6*(a*e - b*d)^5) + (x*(a + b*x)^(1/2)*(2310*A*b^5*d^4 + 770*B*a^ 5*e^4 + 70*A*a^4*b*e^4 + 462*B*a*b^4*d^4 - 440*A*a^3*b^2*d*e^3 - 5676*B*a^ 2*b^3*d^3*e + 1188*A*a^2*b^3*d^2*e^2 + 6864*B*a^3*b^2*d^2*e^2 - 1848*A*a*b ^4*d^3*e - 3700*B*a^4*b*d*e^3))/(3465*e^6*(a*e - b*d)^5) + (32*b^4*x^5*(a + b*x)^(1/2)*(8*A*b*e - 11*B*a*e + 3*B*b*d))/(3465*e^3*(a*e - b*d)^5) - (1 6*b^3*x^4*(a*e - 11*b*d)*(a + b*x)^(1/2)*(8*A*b*e - 11*B*a*e + 3*B*b*d))/( 3465*e^4*(a*e - b*d)^5) + (4*b^2*x^3*(a + b*x)^(1/2)*(3*a^2*e^2 + 99*b^2*d ^2 - 22*a*b*d*e)*(8*A*b*e - 11*B*a*e + 3*B*b*d))/(3465*e^5*(a*e - b*d)^5) - (2*b*x^2*(a + b*x)^(1/2)*(8*A*b*e - 11*B*a*e + 3*B*b*d)*(5*a^3*e^3 - 231 *b^3*d^3 + 99*a*b^2*d^2*e - 33*a^2*b*d*e^2))/(3465*e^6*(a*e - b*d)^5)))/(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)