Integrand size = 24, antiderivative size = 198 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{11/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{3/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{21 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {8 b (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{105 e (b d-a e)^3 (d+e x)^{5/2}}+\frac {16 b^2 (b B d+2 A b e-3 a B e) (a+b x)^{3/2}}{315 e (b d-a e)^4 (d+e x)^{3/2}} \]
-2/9*(-A*e+B*d)*(b*x+a)^(3/2)/e/(-a*e+b*d)/(e*x+d)^(9/2)+2/21*(2*A*b*e-3*B *a*e+B*b*d)*(b*x+a)^(3/2)/e/(-a*e+b*d)^2/(e*x+d)^(7/2)+8/105*b*(2*A*b*e-3* B*a*e+B*b*d)*(b*x+a)^(3/2)/e/(-a*e+b*d)^3/(e*x+d)^(5/2)+16/315*b^2*(2*A*b* e-3*B*a*e+B*b*d)*(b*x+a)^(3/2)/e/(-a*e+b*d)^4/(e*x+d)^(3/2)
Time = 0.26 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{11/2}} \, dx=\frac {2 (a+b x)^{3/2} \left (35 B d e^2 (a+b x)^3-35 A e^3 (a+b x)^3-90 b B d e (a+b x)^2 (d+e x)+135 A b e^2 (a+b x)^2 (d+e x)-45 a B e^2 (a+b x)^2 (d+e x)+63 b^2 B d (a+b x) (d+e x)^2-189 A b^2 e (a+b x) (d+e x)^2+126 a b B e (a+b x) (d+e x)^2+105 A b^3 (d+e x)^3-105 a b^2 B (d+e x)^3\right )}{315 (b d-a e)^4 (d+e x)^{9/2}} \]
(2*(a + b*x)^(3/2)*(35*B*d*e^2*(a + b*x)^3 - 35*A*e^3*(a + b*x)^3 - 90*b*B *d*e*(a + b*x)^2*(d + e*x) + 135*A*b*e^2*(a + b*x)^2*(d + e*x) - 45*a*B*e^ 2*(a + b*x)^2*(d + e*x) + 63*b^2*B*d*(a + b*x)*(d + e*x)^2 - 189*A*b^2*e*( a + b*x)*(d + e*x)^2 + 126*a*b*B*e*(a + b*x)*(d + e*x)^2 + 105*A*b^3*(d + e*x)^3 - 105*a*b^2*B*(d + e*x)^3))/(315*(b*d - a*e)^4*(d + e*x)^(9/2))
Time = 0.24 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.96, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \int \frac {\sqrt {a+b x}}{(d+e x)^{9/2}}dx}{3 e (b d-a e)}-\frac {2 (a+b x)^{3/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {4 b \int \frac {\sqrt {a+b x}}{(d+e x)^{7/2}}dx}{7 (b d-a e)}+\frac {2 (a+b x)^{3/2}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{3 e (b d-a e)}-\frac {2 (a+b x)^{3/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {4 b \left (\frac {2 b \int \frac {\sqrt {a+b x}}{(d+e x)^{5/2}}dx}{5 (b d-a e)}+\frac {2 (a+b x)^{3/2}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{7 (b d-a e)}+\frac {2 (a+b x)^{3/2}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{3 e (b d-a e)}-\frac {2 (a+b x)^{3/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\left (\frac {2 (a+b x)^{3/2}}{7 (d+e x)^{7/2} (b d-a e)}+\frac {4 b \left (\frac {4 b (a+b x)^{3/2}}{15 (d+e x)^{3/2} (b d-a e)^2}+\frac {2 (a+b x)^{3/2}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{7 (b d-a e)}\right ) (-3 a B e+2 A b e+b B d)}{3 e (b d-a e)}-\frac {2 (a+b x)^{3/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)}\) |
(-2*(B*d - A*e)*(a + b*x)^(3/2))/(9*e*(b*d - a*e)*(d + e*x)^(9/2)) + ((b*B *d + 2*A*b*e - 3*a*B*e)*((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*e*( b*d - a*e))
3.23.8.3.1 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 = 3.52 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.42
| method | result | size |
| default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-16 A \,b^{3} e^{3} x^{3}+24 B a \,b^{2} e^{3} x^{3}-8 B \,b^{3} d \,e^{2} x^{3}+24 A a \,b^{2} e^{3} x^{2}-72 A \,b^{3} d \,e^{2} x^{2}-36 B \,a^{2} b \,e^{3} x^{2}+120 B a \,b^{2} d \,e^{2} x^{2}-36 B \,b^{3} d^{2} e \,x^{2}-30 A \,a^{2} b \,e^{3} x +108 A a \,b^{2} d \,e^{2} x -126 A \,b^{3} d^{2} e x +45 B \,a^{3} e^{3} x -177 B \,a^{2} b d \,e^{2} x +243 B a \,b^{2} d^{2} e x -63 b^{3} B \,d^{3} x +35 a^{3} A \,e^{3}-135 A \,a^{2} b d \,e^{2}+189 A a \,b^{2} d^{2} e -105 A \,b^{3} d^{3}+10 B \,a^{3} d \,e^{2}-36 B \,a^{2} b \,d^{2} e +42 B a \,b^{2} d^{3}\right )}{315 \left (e x +d \right )^{\frac {9}{2}} \left (a e -b d \right )^{4}}\) | \(281\) |
| gosper | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-16 A \,b^{3} e^{3} x^{3}+24 B a \,b^{2} e^{3} x^{3}-8 B \,b^{3} d \,e^{2} x^{3}+24 A a \,b^{2} e^{3} x^{2}-72 A \,b^{3} d \,e^{2} x^{2}-36 B \,a^{2} b \,e^{3} x^{2}+120 B a \,b^{2} d \,e^{2} x^{2}-36 B \,b^{3} d^{2} e \,x^{2}-30 A \,a^{2} b \,e^{3} x +108 A a \,b^{2} d \,e^{2} x -126 A \,b^{3} d^{2} e x +45 B \,a^{3} e^{3} x -177 B \,a^{2} b d \,e^{2} x +243 B a \,b^{2} d^{2} e x -63 b^{3} B \,d^{3} x +35 a^{3} A \,e^{3}-135 A \,a^{2} b d \,e^{2}+189 A a \,b^{2} d^{2} e -105 A \,b^{3} d^{3}+10 B \,a^{3} d \,e^{2}-36 B \,a^{2} b \,d^{2} e +42 B a \,b^{2} d^{3}\right )}{315 \left (e x +d \right )^{\frac {9}{2}} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\) | \(322\) |
-2/315*(b*x+a)^(3/2)/(e*x+d)^(9/2)*(-16*A*b^3*e^3*x^3+24*B*a*b^2*e^3*x^3-8 *B*b^3*d*e^2*x^3+24*A*a*b^2*e^3*x^2-72*A*b^3*d*e^2*x^2-36*B*a^2*b*e^3*x^2+ 120*B*a*b^2*d*e^2*x^2-36*B*b^3*d^2*e*x^2-30*A*a^2*b*e^3*x+108*A*a*b^2*d*e^ 2*x-126*A*b^3*d^2*e*x+45*B*a^3*e^3*x-177*B*a^2*b*d*e^2*x+243*B*a*b^2*d^2*e *x-63*B*b^3*d^3*x+35*A*a^3*e^3-135*A*a^2*b*d*e^2+189*A*a*b^2*d^2*e-105*A*b ^3*d^3+10*B*a^3*d*e^2-36*B*a^2*b*d^2*e+42*B*a*b^2*d^3)/(a*e-b*d)^4
Leaf count of result is larger than twice the leaf count of optimal. 710 vs. \(2 (174) = 348\).
Time = 26.76 (sec) , antiderivative size = 710, normalized size of antiderivative = 3.59 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{11/2}} \, dx=-\frac {2 \, {\left (35 \, A a^{4} e^{3} - 8 \, {\left (B b^{4} d e^{2} - {\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} e^{3}\right )} x^{4} + 21 \, {\left (2 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} d^{3} - 9 \, {\left (4 \, B a^{3} b - 21 \, A a^{2} b^{2}\right )} d^{2} e + 5 \, {\left (2 \, B a^{4} - 27 \, A a^{3} b\right )} d e^{2} - 4 \, {\left (9 \, B b^{4} d^{2} e - 2 \, {\left (14 \, B a b^{3} - 9 \, A b^{4}\right )} d e^{2} + {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} e^{3}\right )} x^{3} - 3 \, {\left (21 \, B b^{4} d^{3} - 3 \, {\left (23 \, B a b^{3} - 14 \, A b^{4}\right )} d^{2} e + {\left (19 \, B a^{2} b^{2} - 12 \, A a b^{3}\right )} d e^{2} - {\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} e^{3}\right )} x^{2} - {\left (21 \, {\left (B a b^{3} + 5 \, A b^{4}\right )} d^{3} - 9 \, {\left (23 \, B a^{2} b^{2} + 7 \, A a b^{3}\right )} d^{2} e + {\left (167 \, B a^{3} b + 27 \, A a^{2} b^{2}\right )} d e^{2} - 5 \, {\left (9 \, B a^{4} + A a^{3} b\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{315 \, {\left (b^{4} d^{9} - 4 \, a b^{3} d^{8} e + 6 \, a^{2} b^{2} d^{7} e^{2} - 4 \, a^{3} b d^{6} e^{3} + a^{4} d^{5} e^{4} + {\left (b^{4} d^{4} e^{5} - 4 \, a b^{3} d^{3} e^{6} + 6 \, a^{2} b^{2} d^{2} e^{7} - 4 \, a^{3} b d e^{8} + a^{4} e^{9}\right )} x^{5} + 5 \, {\left (b^{4} d^{5} e^{4} - 4 \, a b^{3} d^{4} e^{5} + 6 \, a^{2} b^{2} d^{3} e^{6} - 4 \, a^{3} b d^{2} e^{7} + a^{4} d e^{8}\right )} x^{4} + 10 \, {\left (b^{4} d^{6} e^{3} - 4 \, a b^{3} d^{5} e^{4} + 6 \, a^{2} b^{2} d^{4} e^{5} - 4 \, a^{3} b d^{3} e^{6} + a^{4} d^{2} e^{7}\right )} x^{3} + 10 \, {\left (b^{4} d^{7} e^{2} - 4 \, a b^{3} d^{6} e^{3} + 6 \, a^{2} b^{2} d^{5} e^{4} - 4 \, a^{3} b d^{4} e^{5} + a^{4} d^{3} e^{6}\right )} x^{2} + 5 \, {\left (b^{4} d^{8} e - 4 \, a b^{3} d^{7} e^{2} + 6 \, a^{2} b^{2} d^{6} e^{3} - 4 \, a^{3} b d^{5} e^{4} + a^{4} d^{4} e^{5}\right )} x\right )}} \]
-2/315*(35*A*a^4*e^3 - 8*(B*b^4*d*e^2 - (3*B*a*b^3 - 2*A*b^4)*e^3)*x^4 + 2 1*(2*B*a^2*b^2 - 5*A*a*b^3)*d^3 - 9*(4*B*a^3*b - 21*A*a^2*b^2)*d^2*e + 5*( 2*B*a^4 - 27*A*a^3*b)*d*e^2 - 4*(9*B*b^4*d^2*e - 2*(14*B*a*b^3 - 9*A*b^4)* d*e^2 + (3*B*a^2*b^2 - 2*A*a*b^3)*e^3)*x^3 - 3*(21*B*b^4*d^3 - 3*(23*B*a*b ^3 - 14*A*b^4)*d^2*e + (19*B*a^2*b^2 - 12*A*a*b^3)*d*e^2 - (3*B*a^3*b - 2* A*a^2*b^2)*e^3)*x^2 - (21*(B*a*b^3 + 5*A*b^4)*d^3 - 9*(23*B*a^2*b^2 + 7*A* a*b^3)*d^2*e + (167*B*a^3*b + 27*A*a^2*b^2)*d*e^2 - 5*(9*B*a^4 + A*a^3*b)* e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^4*d^9 - 4*a*b^3*d^8*e + 6*a^2*b^2*d ^7*e^2 - 4*a^3*b*d^6*e^3 + a^4*d^5*e^4 + (b^4*d^4*e^5 - 4*a*b^3*d^3*e^6 + 6*a^2*b^2*d^2*e^7 - 4*a^3*b*d*e^8 + a^4*e^9)*x^5 + 5*(b^4*d^5*e^4 - 4*a*b^ 3*d^4*e^5 + 6*a^2*b^2*d^3*e^6 - 4*a^3*b*d^2*e^7 + a^4*d*e^8)*x^4 + 10*(b^4 *d^6*e^3 - 4*a*b^3*d^5*e^4 + 6*a^2*b^2*d^4*e^5 - 4*a^3*b*d^3*e^6 + a^4*d^2 *e^7)*x^3 + 10*(b^4*d^7*e^2 - 4*a*b^3*d^6*e^3 + 6*a^2*b^2*d^5*e^4 - 4*a^3* b*d^4*e^5 + a^4*d^3*e^6)*x^2 + 5*(b^4*d^8*e - 4*a*b^3*d^7*e^2 + 6*a^2*b^2* d^6*e^3 - 4*a^3*b*d^5*e^4 + a^4*d^4*e^5)*x)
\[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{11/2}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {a + b x}}{\left (d + e x\right )^{\frac {11}{2}}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{11/2}} \, dx=\text {Exception raised: ValueError} \]
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. 620 vs. \(2 (174) = 348\).
Time = 0.54 (sec) , antiderivative size = 620, normalized size of antiderivative = 3.13 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{11/2}} \, dx=\frac {2 \, {\left ({\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (B b^{10} d e^{6} {\left | b \right |} - 3 \, B a b^{9} e^{7} {\left | b \right |} + 2 \, A b^{10} e^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}} + \frac {9 \, {\left (B b^{11} d^{2} e^{5} {\left | b \right |} - 4 \, B a b^{10} d e^{6} {\left | b \right |} + 2 \, A b^{11} d e^{6} {\left | b \right |} + 3 \, B a^{2} b^{9} e^{7} {\left | b \right |} - 2 \, A a b^{10} e^{7} {\left | b \right |}\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}}\right )} + \frac {63 \, {\left (B b^{12} d^{3} e^{4} {\left | b \right |} - 5 \, B a b^{11} d^{2} e^{5} {\left | b \right |} + 2 \, A b^{12} d^{2} e^{5} {\left | b \right |} + 7 \, B a^{2} b^{10} d e^{6} {\left | b \right |} - 4 \, A a b^{11} d e^{6} {\left | b \right |} - 3 \, B a^{3} b^{9} e^{7} {\left | b \right |} + 2 \, A a^{2} b^{10} e^{7} {\left | b \right |}\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}}\right )} {\left (b x + a\right )} - \frac {105 \, {\left (B a b^{12} d^{3} e^{4} {\left | b \right |} - A b^{13} d^{3} e^{4} {\left | b \right |} - 3 \, B a^{2} b^{11} d^{2} e^{5} {\left | b \right |} + 3 \, A a b^{12} d^{2} e^{5} {\left | b \right |} + 3 \, B a^{3} b^{10} d e^{6} {\left | b \right |} - 3 \, A a^{2} b^{11} d e^{6} {\left | b \right |} - B a^{4} b^{9} e^{7} {\left | b \right |} + A a^{3} b^{10} e^{7} {\left | b \right |}\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}}\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{315 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {9}{2}}} \]
2/315*((4*(b*x + a)*(2*(B*b^10*d*e^6*abs(b) - 3*B*a*b^9*e^7*abs(b) + 2*A*b ^10*e^7*abs(b))*(b*x + a)/(b^6*d^4*e^4 - 4*a*b^5*d^3*e^5 + 6*a^2*b^4*d^2*e ^6 - 4*a^3*b^3*d*e^7 + a^4*b^2*e^8) + 9*(B*b^11*d^2*e^5*abs(b) - 4*B*a*b^1 0*d*e^6*abs(b) + 2*A*b^11*d*e^6*abs(b) + 3*B*a^2*b^9*e^7*abs(b) - 2*A*a*b^ 10*e^7*abs(b))/(b^6*d^4*e^4 - 4*a*b^5*d^3*e^5 + 6*a^2*b^4*d^2*e^6 - 4*a^3* b^3*d*e^7 + a^4*b^2*e^8)) + 63*(B*b^12*d^3*e^4*abs(b) - 5*B*a*b^11*d^2*e^5 *abs(b) + 2*A*b^12*d^2*e^5*abs(b) + 7*B*a^2*b^10*d*e^6*abs(b) - 4*A*a*b^11 *d*e^6*abs(b) - 3*B*a^3*b^9*e^7*abs(b) + 2*A*a^2*b^10*e^7*abs(b))/(b^6*d^4 *e^4 - 4*a*b^5*d^3*e^5 + 6*a^2*b^4*d^2*e^6 - 4*a^3*b^3*d*e^7 + a^4*b^2*e^8 ))*(b*x + a) - 105*(B*a*b^12*d^3*e^4*abs(b) - A*b^13*d^3*e^4*abs(b) - 3*B* a^2*b^11*d^2*e^5*abs(b) + 3*A*a*b^12*d^2*e^5*abs(b) + 3*B*a^3*b^10*d*e^6*a bs(b) - 3*A*a^2*b^11*d*e^6*abs(b) - B*a^4*b^9*e^7*abs(b) + A*a^3*b^10*e^7* abs(b))/(b^6*d^4*e^4 - 4*a*b^5*d^3*e^5 + 6*a^2*b^4*d^2*e^6 - 4*a^3*b^3*d*e ^7 + a^4*b^2*e^8))*(b*x + a)^(3/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(9/2)
Time = 2.64 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.16 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{11/2}} \, dx=\frac {\sqrt {d+e\,x}\,\left (\frac {x\,\sqrt {a+b\,x}\,\left (-90\,B\,a^4\,e^3+334\,B\,a^3\,b\,d\,e^2-10\,A\,a^3\,b\,e^3-414\,B\,a^2\,b^2\,d^2\,e+54\,A\,a^2\,b^2\,d\,e^2+42\,B\,a\,b^3\,d^3-126\,A\,a\,b^3\,d^2\,e+210\,A\,b^4\,d^3\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^4}-\frac {\sqrt {a+b\,x}\,\left (20\,B\,a^4\,d\,e^2+70\,A\,a^4\,e^3-72\,B\,a^3\,b\,d^2\,e-270\,A\,a^3\,b\,d\,e^2+84\,B\,a^2\,b^2\,d^3+378\,A\,a^2\,b^2\,d^2\,e-210\,A\,a\,b^3\,d^3\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^4}+\frac {16\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )}{315\,e^3\,{\left (a\,e-b\,d\right )}^4}-\frac {8\,b^2\,x^3\,\left (a\,e-9\,b\,d\right )\,\sqrt {a+b\,x}\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )}{315\,e^4\,{\left (a\,e-b\,d\right )}^4}+\frac {2\,b\,x^2\,\sqrt {a+b\,x}\,\left (a^2\,e^2-6\,a\,b\,d\,e+21\,b^2\,d^2\right )\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )}{105\,e^5\,{\left (a\,e-b\,d\right )}^4}\right )}{x^5+\frac {d^5}{e^5}+\frac {5\,d\,x^4}{e}+\frac {5\,d^4\,x}{e^4}+\frac {10\,d^2\,x^3}{e^2}+\frac {10\,d^3\,x^2}{e^3}} \]
((d + e*x)^(1/2)*((x*(a + b*x)^(1/2)*(210*A*b^4*d^3 - 90*B*a^4*e^3 - 10*A* a^3*b*e^3 + 42*B*a*b^3*d^3 + 54*A*a^2*b^2*d*e^2 - 414*B*a^2*b^2*d^2*e - 12 6*A*a*b^3*d^2*e + 334*B*a^3*b*d*e^2))/(315*e^5*(a*e - b*d)^4) - ((a + b*x) ^(1/2)*(70*A*a^4*e^3 - 210*A*a*b^3*d^3 + 20*B*a^4*d*e^2 + 84*B*a^2*b^2*d^3 + 378*A*a^2*b^2*d^2*e - 270*A*a^3*b*d*e^2 - 72*B*a^3*b*d^2*e))/(315*e^5*( a*e - b*d)^4) + (16*b^3*x^4*(a + b*x)^(1/2)*(2*A*b*e - 3*B*a*e + B*b*d))/( 315*e^3*(a*e - b*d)^4) - (8*b^2*x^3*(a*e - 9*b*d)*(a + b*x)^(1/2)*(2*A*b*e - 3*B*a*e + B*b*d))/(315*e^4*(a*e - b*d)^4) + (2*b*x^2*(a + b*x)^(1/2)*(a ^2*e^2 + 21*b^2*d^2 - 6*a*b*d*e)*(2*A*b*e - 3*B*a*e + B*b*d))/(105*e^5*(a* e - b*d)^4)))/(x^5 + d^5/e^5 + (5*d*x^4)/e + (5*d^4*x)/e^4 + (10*d^2*x^3)/ e^2 + (10*d^3*x^2)/e^3)