Integrand size = 24, antiderivative size = 167 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{7/2}}{7 e (b d-a e) (d+e x)^{7/2}}-\frac {2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}}-\frac {2 b^2 B \sqrt {a+b x}}{e^4 \sqrt {d+e x}}+\frac {2 b^{5/2} B \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{9/2}} \]
-2/7*(-A*e+B*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)/(e*x+d)^(7/2)-2/5*B*(b*x+a)^(5/ 2)/e^2/(e*x+d)^(5/2)-2/3*b*B*(b*x+a)^(3/2)/e^3/(e*x+d)^(3/2)+2*b^(5/2)*B*a rctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/e^(9/2)-2*b^2*B*(b*x+a )^(1/2)/e^4/(e*x+d)^(1/2)
Time = 0.51 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.29 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx=-\frac {2 \sqrt {a+b x} \left (-15 B d e^3 (a+b x)^3+15 A e^4 (a+b x)^3-21 b B d e^2 (a+b x)^2 (d+e x)+21 a B e^3 (a+b x)^2 (d+e x)-35 b^2 B d e (a+b x) (d+e x)^2+35 a b B e^2 (a+b x) (d+e x)^2-105 b^3 B d (d+e x)^3+105 a b^2 B e (d+e x)^3\right )}{105 e^4 (-b d+a e) (d+e x)^{7/2}}+\frac {2 b^{5/2} B \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{e^{9/2}} \]
(-2*Sqrt[a + b*x]*(-15*B*d*e^3*(a + b*x)^3 + 15*A*e^4*(a + b*x)^3 - 21*b*B *d*e^2*(a + b*x)^2*(d + e*x) + 21*a*B*e^3*(a + b*x)^2*(d + e*x) - 35*b^2*B *d*e*(a + b*x)*(d + e*x)^2 + 35*a*b*B*e^2*(a + b*x)*(d + e*x)^2 - 105*b^3* B*d*(d + e*x)^3 + 105*a*b^2*B*e*(d + e*x)^3))/(105*e^4*(-(b*d) + a*e)*(d + e*x)^(7/2)) + (2*b^(5/2)*B*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[ a + b*x])])/e^(9/2)
Time = 0.24 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {87, 57, 57, 57, 66, 221}
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.
| \(\displaystyle \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {B \int \frac {(a+b x)^{5/2}}{(d+e x)^{7/2}}dx}{e}-\frac {2 (a+b x)^{7/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {B \left (\frac {b \int \frac {(a+b x)^{3/2}}{(d+e x)^{5/2}}dx}{e}-\frac {2 (a+b x)^{5/2}}{5 e (d+e x)^{5/2}}\right )}{e}-\frac {2 (a+b x)^{7/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {B \left (\frac {b \left (\frac {b \int \frac {\sqrt {a+b x}}{(d+e x)^{3/2}}dx}{e}-\frac {2 (a+b x)^{3/2}}{3 e (d+e x)^{3/2}}\right )}{e}-\frac {2 (a+b x)^{5/2}}{5 e (d+e x)^{5/2}}\right )}{e}-\frac {2 (a+b x)^{7/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {B \left (\frac {b \left (\frac {b \left (\frac {b \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{e}-\frac {2 \sqrt {a+b x}}{e \sqrt {d+e x}}\right )}{e}-\frac {2 (a+b x)^{3/2}}{3 e (d+e x)^{3/2}}\right )}{e}-\frac {2 (a+b x)^{5/2}}{5 e (d+e x)^{5/2}}\right )}{e}-\frac {2 (a+b x)^{7/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {B \left (\frac {b \left (\frac {b \left (\frac {2 b \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}}{e}-\frac {2 \sqrt {a+b x}}{e \sqrt {d+e x}}\right )}{e}-\frac {2 (a+b x)^{3/2}}{3 e (d+e x)^{3/2}}\right )}{e}-\frac {2 (a+b x)^{5/2}}{5 e (d+e x)^{5/2}}\right )}{e}-\frac {2 (a+b x)^{7/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {B \left (\frac {b \left (\frac {b \left (\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{3/2}}-\frac {2 \sqrt {a+b x}}{e \sqrt {d+e x}}\right )}{e}-\frac {2 (a+b x)^{3/2}}{3 e (d+e x)^{3/2}}\right )}{e}-\frac {2 (a+b x)^{5/2}}{5 e (d+e x)^{5/2}}\right )}{e}-\frac {2 (a+b x)^{7/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}\) |
(-2*(B*d - A*e)*(a + b*x)^(7/2))/(7*e*(b*d - a*e)*(d + e*x)^(7/2)) + (B*(( -2*(a + b*x)^(5/2))/(5*e*(d + e*x)^(5/2)) + (b*((-2*(a + b*x)^(3/2))/(3*e* (d + e*x)^(3/2)) + (b*((-2*Sqrt[a + b*x])/(e*Sqrt[d + e*x]) + (2*Sqrt[b]*A rcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/e^(3/2)))/e))/e)) /e
3.23.30.3.1 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/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c , d, m, n, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
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]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Leaf count of result is larger than twice the leaf count of optimal. \(1088\) vs. \(2(133)=266\).
Time = 3.57 (sec) , antiderivative size = 1089, normalized size of antiderivative = 6.52
-1/105*(b*x+a)^(1/2)*(568*B*a*b^2*d*e^3*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^ (1/2)+92*B*a^2*b*d*e^3*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+476*B*a*b^2*d ^2*e^2*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+322*B*a*b^2*e^4*x^3*((b*x+a)* (e*x+d))^(1/2)*(b*e)^(1/2)-352*B*b^3*d*e^3*x^3*((b*x+a)*(e*x+d))^(1/2)*(b* e)^(1/2)+90*A*a*b^2*e^4*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+154*B*a^2* b*e^4*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-812*B*b^3*d^2*e^2*x^2*((b*x+ a)*(e*x+d))^(1/2)*(b*e)^(1/2)+90*A*a^2*b*e^4*x*((b*x+a)*(e*x+d))^(1/2)*(b* e)^(1/2)-700*B*b^3*d^3*e*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+28*B*a^2*b* d^2*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+105*B*ln(1/2*(2*b*e*x+2*((b*x+ a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^4*d^5+140*B*a*b^2*d^ 3*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-420*B*ln(1/2*(2*b*e*x+2*((b*x+a)*( e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^3*d*e^4*x^3-630*B*ln(1 /2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a* b^3*d^2*e^3*x^2-420*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2 )+a*e+b*d)/(b*e)^(1/2))*a*b^3*d^3*e^2*x+105*B*ln(1/2*(2*b*e*x+2*((b*x+a)*( e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^4*d*e^4*x^4+420*B*ln(1/2 *(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^4* d^2*e^3*x^3+630*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a* e+b*d)/(b*e)^(1/2))*b^4*d^3*e^2*x^2+420*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+ d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^4*d^4*e*x-105*B*ln(1/2*(2...
Leaf count of result is larger than twice the leaf count of optimal. 514 vs. \(2 (133) = 266\).
Time = 8.04 (sec) , antiderivative size = 1053, normalized size of antiderivative = 6.31 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx=\left [\frac {105 \, {\left (B b^{3} d^{5} - B a b^{2} d^{4} e + {\left (B b^{3} d e^{4} - B a b^{2} e^{5}\right )} x^{4} + 4 \, {\left (B b^{3} d^{2} e^{3} - B a b^{2} d e^{4}\right )} x^{3} + 6 \, {\left (B b^{3} d^{3} e^{2} - B a b^{2} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (B b^{3} d^{4} e - B a b^{2} d^{3} e^{2}\right )} x\right )} \sqrt {\frac {b}{e}} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e^{2} x + b d e + a e^{2}\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {\frac {b}{e}} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \, {\left (105 \, B b^{3} d^{4} - 70 \, B a b^{2} d^{3} e - 14 \, B a^{2} b d^{2} e^{2} - 6 \, B a^{3} d e^{3} - 15 \, A a^{3} e^{4} + {\left (176 \, B b^{3} d e^{3} - {\left (161 \, B a b^{2} + 15 \, A b^{3}\right )} e^{4}\right )} x^{3} + {\left (406 \, B b^{3} d^{2} e^{2} - 284 \, B a b^{2} d e^{3} - {\left (77 \, B a^{2} b + 45 \, A a b^{2}\right )} e^{4}\right )} x^{2} + {\left (350 \, B b^{3} d^{3} e - 238 \, B a b^{2} d^{2} e^{2} - 46 \, B a^{2} b d e^{3} - 3 \, {\left (7 \, B a^{3} + 15 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{210 \, {\left (b d^{5} e^{4} - a d^{4} e^{5} + {\left (b d e^{8} - a e^{9}\right )} x^{4} + 4 \, {\left (b d^{2} e^{7} - a d e^{8}\right )} x^{3} + 6 \, {\left (b d^{3} e^{6} - a d^{2} e^{7}\right )} x^{2} + 4 \, {\left (b d^{4} e^{5} - a d^{3} e^{6}\right )} x\right )}}, -\frac {105 \, {\left (B b^{3} d^{5} - B a b^{2} d^{4} e + {\left (B b^{3} d e^{4} - B a b^{2} e^{5}\right )} x^{4} + 4 \, {\left (B b^{3} d^{2} e^{3} - B a b^{2} d e^{4}\right )} x^{3} + 6 \, {\left (B b^{3} d^{3} e^{2} - B a b^{2} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (B b^{3} d^{4} e - B a b^{2} d^{3} e^{2}\right )} x\right )} \sqrt {-\frac {b}{e}} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {-\frac {b}{e}}}{2 \, {\left (b^{2} e x^{2} + a b d + {\left (b^{2} d + a b e\right )} x\right )}}\right ) + 2 \, {\left (105 \, B b^{3} d^{4} - 70 \, B a b^{2} d^{3} e - 14 \, B a^{2} b d^{2} e^{2} - 6 \, B a^{3} d e^{3} - 15 \, A a^{3} e^{4} + {\left (176 \, B b^{3} d e^{3} - {\left (161 \, B a b^{2} + 15 \, A b^{3}\right )} e^{4}\right )} x^{3} + {\left (406 \, B b^{3} d^{2} e^{2} - 284 \, B a b^{2} d e^{3} - {\left (77 \, B a^{2} b + 45 \, A a b^{2}\right )} e^{4}\right )} x^{2} + {\left (350 \, B b^{3} d^{3} e - 238 \, B a b^{2} d^{2} e^{2} - 46 \, B a^{2} b d e^{3} - 3 \, {\left (7 \, B a^{3} + 15 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{105 \, {\left (b d^{5} e^{4} - a d^{4} e^{5} + {\left (b d e^{8} - a e^{9}\right )} x^{4} + 4 \, {\left (b d^{2} e^{7} - a d e^{8}\right )} x^{3} + 6 \, {\left (b d^{3} e^{6} - a d^{2} e^{7}\right )} x^{2} + 4 \, {\left (b d^{4} e^{5} - a d^{3} e^{6}\right )} x\right )}}\right ] \]
[1/210*(105*(B*b^3*d^5 - B*a*b^2*d^4*e + (B*b^3*d*e^4 - B*a*b^2*e^5)*x^4 + 4*(B*b^3*d^2*e^3 - B*a*b^2*d*e^4)*x^3 + 6*(B*b^3*d^3*e^2 - B*a*b^2*d^2*e^ 3)*x^2 + 4*(B*b^3*d^4*e - B*a*b^2*d^3*e^2)*x)*sqrt(b/e)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 + 4*(2*b*e^2*x + b*d*e + a*e^2)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt(b/e) + 8*(b^2*d*e + a*b*e^2)*x) - 4*(105*B*b^3*d^4 - 70*B*a*b^2*d^3*e - 14*B*a^2*b*d^2*e^2 - 6*B*a^3*d*e^3 - 15*A*a^3*e^4 + ( 176*B*b^3*d*e^3 - (161*B*a*b^2 + 15*A*b^3)*e^4)*x^3 + (406*B*b^3*d^2*e^2 - 284*B*a*b^2*d*e^3 - (77*B*a^2*b + 45*A*a*b^2)*e^4)*x^2 + (350*B*b^3*d^3*e - 238*B*a*b^2*d^2*e^2 - 46*B*a^2*b*d*e^3 - 3*(7*B*a^3 + 15*A*a^2*b)*e^4)* x)*sqrt(b*x + a)*sqrt(e*x + d))/(b*d^5*e^4 - a*d^4*e^5 + (b*d*e^8 - a*e^9) *x^4 + 4*(b*d^2*e^7 - a*d*e^8)*x^3 + 6*(b*d^3*e^6 - a*d^2*e^7)*x^2 + 4*(b* d^4*e^5 - a*d^3*e^6)*x), -1/105*(105*(B*b^3*d^5 - B*a*b^2*d^4*e + (B*b^3*d *e^4 - B*a*b^2*e^5)*x^4 + 4*(B*b^3*d^2*e^3 - B*a*b^2*d*e^4)*x^3 + 6*(B*b^3 *d^3*e^2 - B*a*b^2*d^2*e^3)*x^2 + 4*(B*b^3*d^4*e - B*a*b^2*d^3*e^2)*x)*sqr t(-b/e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt( -b/e)/(b^2*e*x^2 + a*b*d + (b^2*d + a*b*e)*x)) + 2*(105*B*b^3*d^4 - 70*B*a *b^2*d^3*e - 14*B*a^2*b*d^2*e^2 - 6*B*a^3*d*e^3 - 15*A*a^3*e^4 + (176*B*b^ 3*d*e^3 - (161*B*a*b^2 + 15*A*b^3)*e^4)*x^3 + (406*B*b^3*d^2*e^2 - 284*B*a *b^2*d*e^3 - (77*B*a^2*b + 45*A*a*b^2)*e^4)*x^2 + (350*B*b^3*d^3*e - 238*B *a*b^2*d^2*e^2 - 46*B*a^2*b*d*e^3 - 3*(7*B*a^3 + 15*A*a^2*b)*e^4)*x)*sq...
\[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx=\text {Exception raised: ValueError} \]
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. 678 vs. \(2 (133) = 266\).
Time = 0.60 (sec) , antiderivative size = 678, normalized size of antiderivative = 4.06 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx=-\frac {2 \, B b^{2} {\left | b \right |} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e} e^{4}} - \frac {2 \, {\left ({\left ({\left (b x + a\right )} {\left (\frac {{\left (176 \, B b^{10} d^{3} e^{6} {\left | b \right |} - 513 \, B a b^{9} d^{2} e^{7} {\left | b \right |} - 15 \, A b^{10} d^{2} e^{7} {\left | b \right |} + 498 \, B a^{2} b^{8} d e^{8} {\left | b \right |} + 30 \, A a b^{9} d e^{8} {\left | b \right |} - 161 \, B a^{3} b^{7} e^{9} {\left | b \right |} - 15 \, A a^{2} b^{8} e^{9} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}} + \frac {406 \, {\left (B b^{11} d^{4} e^{5} {\left | b \right |} - 4 \, B a b^{10} d^{3} e^{6} {\left | b \right |} + 6 \, B a^{2} b^{9} d^{2} e^{7} {\left | b \right |} - 4 \, B a^{3} b^{8} d e^{8} {\left | b \right |} + B a^{4} b^{7} e^{9} {\left | b \right |}\right )}}{b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}}\right )} + \frac {350 \, {\left (B b^{12} d^{5} e^{4} {\left | b \right |} - 5 \, B a b^{11} d^{4} e^{5} {\left | b \right |} + 10 \, B a^{2} b^{10} d^{3} e^{6} {\left | b \right |} - 10 \, B a^{3} b^{9} d^{2} e^{7} {\left | b \right |} + 5 \, B a^{4} b^{8} d e^{8} {\left | b \right |} - B a^{5} b^{7} e^{9} {\left | b \right |}\right )}}{b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}}\right )} {\left (b x + a\right )} + \frac {105 \, {\left (B b^{13} d^{6} e^{3} {\left | b \right |} - 6 \, B a b^{12} d^{5} e^{4} {\left | b \right |} + 15 \, B a^{2} b^{11} d^{4} e^{5} {\left | b \right |} - 20 \, B a^{3} b^{10} d^{3} e^{6} {\left | b \right |} + 15 \, B a^{4} b^{9} d^{2} e^{7} {\left | b \right |} - 6 \, B a^{5} b^{8} d e^{8} {\left | b \right |} + B a^{6} b^{7} e^{9} {\left | b \right |}\right )}}{b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}}\right )} \sqrt {b x + a}}{105 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {7}{2}}} \]
-2*B*b^2*abs(b)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)* b*e - a*b*e)))/(sqrt(b*e)*e^4) - 2/105*(((b*x + a)*((176*B*b^10*d^3*e^6*ab s(b) - 513*B*a*b^9*d^2*e^7*abs(b) - 15*A*b^10*d^2*e^7*abs(b) + 498*B*a^2*b ^8*d*e^8*abs(b) + 30*A*a*b^9*d*e^8*abs(b) - 161*B*a^3*b^7*e^9*abs(b) - 15* A*a^2*b^8*e^9*abs(b))*(b*x + a)/(b^5*d^3*e^7 - 3*a*b^4*d^2*e^8 + 3*a^2*b^3 *d*e^9 - a^3*b^2*e^10) + 406*(B*b^11*d^4*e^5*abs(b) - 4*B*a*b^10*d^3*e^6*a bs(b) + 6*B*a^2*b^9*d^2*e^7*abs(b) - 4*B*a^3*b^8*d*e^8*abs(b) + B*a^4*b^7* e^9*abs(b))/(b^5*d^3*e^7 - 3*a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 - a^3*b^2*e^1 0)) + 350*(B*b^12*d^5*e^4*abs(b) - 5*B*a*b^11*d^4*e^5*abs(b) + 10*B*a^2*b^ 10*d^3*e^6*abs(b) - 10*B*a^3*b^9*d^2*e^7*abs(b) + 5*B*a^4*b^8*d*e^8*abs(b) - B*a^5*b^7*e^9*abs(b))/(b^5*d^3*e^7 - 3*a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 - a^3*b^2*e^10))*(b*x + a) + 105*(B*b^13*d^6*e^3*abs(b) - 6*B*a*b^12*d^5*e ^4*abs(b) + 15*B*a^2*b^11*d^4*e^5*abs(b) - 20*B*a^3*b^10*d^3*e^6*abs(b) + 15*B*a^4*b^9*d^2*e^7*abs(b) - 6*B*a^5*b^8*d*e^8*abs(b) + B*a^6*b^7*e^9*abs (b))/(b^5*d^3*e^7 - 3*a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 - a^3*b^2*e^10))*sqr t(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(7/2)
Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]