Integrand size = 24, antiderivative size = 256 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{7/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{7/2}}{5 e (b d-a e) (d+e x)^{5/2}}-\frac {2 (7 b B d-2 A b e-5 a B e) (a+b x)^{5/2}}{15 e^2 (b d-a e) (d+e x)^{3/2}}-\frac {2 b (7 b B d-2 A b e-5 a B e) (a+b x)^{3/2}}{3 e^3 (b d-a e) \sqrt {d+e x}}+\frac {b^2 (7 b B d-2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{e^4 (b d-a e)}-\frac {b^{3/2} (7 b B d-2 A b e-5 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{9/2}} \]
-2/5*(-A*e+B*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)/(e*x+d)^(5/2)-2/15*(-2*A*b*e-5* B*a*e+7*B*b*d)*(b*x+a)^(5/2)/e^2/(-a*e+b*d)/(e*x+d)^(3/2)-b^(3/2)*(-2*A*b* e-5*B*a*e+7*B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/e^ (9/2)-2/3*b*(-2*A*b*e-5*B*a*e+7*B*b*d)*(b*x+a)^(3/2)/e^3/(-a*e+b*d)/(e*x+d )^(1/2)+b^2*(-2*A*b*e-5*B*a*e+7*B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/e^4/(-a *e+b*d)
Time = 0.51 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{7/2}} \, dx=-\frac {\sqrt {a+b x} \left (2 a^2 e^2 (2 B d+3 A e+5 B e x)+2 a b e \left (A e (5 d+11 e x)+B \left (20 d^2+49 d e x+35 e^2 x^2\right )\right )+b^2 \left (2 A e \left (15 d^2+35 d e x+23 e^2 x^2\right )-B \left (105 d^3+245 d^2 e x+161 d e^2 x^2+15 e^3 x^3\right )\right )\right )}{15 e^4 (d+e x)^{5/2}}+\frac {b^{3/2} (-7 b B d+2 A b e+5 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{e^{9/2}} \]
-1/15*(Sqrt[a + b*x]*(2*a^2*e^2*(2*B*d + 3*A*e + 5*B*e*x) + 2*a*b*e*(A*e*( 5*d + 11*e*x) + B*(20*d^2 + 49*d*e*x + 35*e^2*x^2)) + b^2*(2*A*e*(15*d^2 + 35*d*e*x + 23*e^2*x^2) - B*(105*d^3 + 245*d^2*e*x + 161*d*e^2*x^2 + 15*e^ 3*x^3))))/(e^4*(d + e*x)^(5/2)) + (b^(3/2)*(-7*b*B*d + 2*A*b*e + 5*a*B*e)* ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x])])/e^(9/2)
Time = 0.28 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.84, 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, 60, 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.
| \(\displaystyle \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-5 a B e-2 A b e+7 b B d) \int \frac {(a+b x)^{5/2}}{(d+e x)^{5/2}}dx}{5 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {(-5 a B e-2 A b e+7 b B d) \left (\frac {5 b \int \frac {(a+b x)^{3/2}}{(d+e x)^{3/2}}dx}{3 e}-\frac {2 (a+b x)^{5/2}}{3 e (d+e x)^{3/2}}\right )}{5 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {(-5 a B e-2 A b e+7 b B d) \left (\frac {5 b \left (\frac {3 b \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}}dx}{e}-\frac {2 (a+b x)^{3/2}}{e \sqrt {d+e x}}\right )}{3 e}-\frac {2 (a+b x)^{5/2}}{3 e (d+e x)^{3/2}}\right )}{5 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-5 a B e-2 A b e+7 b B d) \left (\frac {5 b \left (\frac {3 b \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{2 e}\right )}{e}-\frac {2 (a+b x)^{3/2}}{e \sqrt {d+e x}}\right )}{3 e}-\frac {2 (a+b x)^{5/2}}{3 e (d+e x)^{3/2}}\right )}{5 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {(-5 a B e-2 A b e+7 b B d) \left (\frac {5 b \left (\frac {3 b \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}}{e}\right )}{e}-\frac {2 (a+b x)^{3/2}}{e \sqrt {d+e x}}\right )}{3 e}-\frac {2 (a+b x)^{5/2}}{3 e (d+e x)^{3/2}}\right )}{5 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(-5 a B e-2 A b e+7 b B d) \left (\frac {5 b \left (\frac {3 b \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{3/2}}\right )}{e}-\frac {2 (a+b x)^{3/2}}{e \sqrt {d+e x}}\right )}{3 e}-\frac {2 (a+b x)^{5/2}}{3 e (d+e x)^{3/2}}\right )}{5 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}\) |
(-2*(B*d - A*e)*(a + b*x)^(7/2))/(5*e*(b*d - a*e)*(d + e*x)^(5/2)) + ((7*b *B*d - 2*A*b*e - 5*a*B*e)*((-2*(a + b*x)^(5/2))/(3*e*(d + e*x)^(3/2)) + (5 *b*((-2*(a + b*x)^(3/2))/(e*Sqrt[d + e*x]) + (3*b*((Sqrt[a + b*x]*Sqrt[d + e*x])/e - ((b*d - a*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(Sqrt[b]*e^(3/2))))/e))/(3*e)))/(5*e*(b*d - a*e))
3.23.29.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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. \(1091\) vs. \(2(222)=444\).
Time = 1.09 (sec) , antiderivative size = 1092, normalized size of antiderivative = 4.27
1/30*(-196*B*a*b*d*e^2*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-60*A*b^2*d^2* e*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+90*A*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^3*d^2*e^2*x+30*B*b^2*e^3*x^ 3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-140*A*b^2*d*e^2*x*(b*e)^(1/2)*((b*x+ a)*(e*x+d))^(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^3*d^4-20*A*a*b*d*e^2*(b*e)^(1/2)*((b*x+a)*(e*x +d))^(1/2)+30*A*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^3*d^3*e+90*A*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^3*d*e^3*x^2-80*B*a*b*d^2*e*((b*x+a)*( e*x+d))^(1/2)*(b*e)^(1/2)-12*A*a^2*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2) +210*B*b^2*d^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+30*A*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^3*e^4*x^3+225* 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^2*d*e^3*x^2+225*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^2*d^2*e^2*x-140*B*a*b*e^3*x^2*((b*x+a)*(e* x+d))^(1/2)*(b*e)^(1/2)+322*B*b^2*d*e^2*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^ (1/2)-44*A*a*b*e^3*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+490*B*b^2*d^2*e*x *((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-315*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^3*d^3*e*x+75*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^2*d...
Time = 3.24 (sec) , antiderivative size = 835, normalized size of antiderivative = 3.26 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{7/2}} \, dx=\left [-\frac {15 \, {\left (7 \, B b^{2} d^{4} - {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{3} e + {\left (7 \, B b^{2} d e^{3} - {\left (5 \, B a b + 2 \, A b^{2}\right )} e^{4}\right )} x^{3} + 3 \, {\left (7 \, B b^{2} d^{2} e^{2} - {\left (5 \, B a b + 2 \, A b^{2}\right )} d e^{3}\right )} x^{2} + 3 \, {\left (7 \, B b^{2} d^{3} e - {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{2} 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 (15 \, B b^{2} e^{3} x^{3} + 105 \, B b^{2} d^{3} - 6 \, A a^{2} e^{3} - 10 \, {\left (4 \, B a b + 3 \, A b^{2}\right )} d^{2} e - 2 \, {\left (2 \, B a^{2} + 5 \, A a b\right )} d e^{2} + {\left (161 \, B b^{2} d e^{2} - 2 \, {\left (35 \, B a b + 23 \, A b^{2}\right )} e^{3}\right )} x^{2} + {\left (245 \, B b^{2} d^{2} e - 14 \, {\left (7 \, B a b + 5 \, A b^{2}\right )} d e^{2} - 2 \, {\left (5 \, B a^{2} + 11 \, A a b\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{60 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}}, \frac {15 \, {\left (7 \, B b^{2} d^{4} - {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{3} e + {\left (7 \, B b^{2} d e^{3} - {\left (5 \, B a b + 2 \, A b^{2}\right )} e^{4}\right )} x^{3} + 3 \, {\left (7 \, B b^{2} d^{2} e^{2} - {\left (5 \, B a b + 2 \, A b^{2}\right )} d e^{3}\right )} x^{2} + 3 \, {\left (7 \, B b^{2} d^{3} e - {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{2} 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 (15 \, B b^{2} e^{3} x^{3} + 105 \, B b^{2} d^{3} - 6 \, A a^{2} e^{3} - 10 \, {\left (4 \, B a b + 3 \, A b^{2}\right )} d^{2} e - 2 \, {\left (2 \, B a^{2} + 5 \, A a b\right )} d e^{2} + {\left (161 \, B b^{2} d e^{2} - 2 \, {\left (35 \, B a b + 23 \, A b^{2}\right )} e^{3}\right )} x^{2} + {\left (245 \, B b^{2} d^{2} e - 14 \, {\left (7 \, B a b + 5 \, A b^{2}\right )} d e^{2} - 2 \, {\left (5 \, B a^{2} + 11 \, A a b\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{30 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}}\right ] \]
[-1/60*(15*(7*B*b^2*d^4 - (5*B*a*b + 2*A*b^2)*d^3*e + (7*B*b^2*d*e^3 - (5* B*a*b + 2*A*b^2)*e^4)*x^3 + 3*(7*B*b^2*d^2*e^2 - (5*B*a*b + 2*A*b^2)*d*e^3 )*x^2 + 3*(7*B*b^2*d^3*e - (5*B*a*b + 2*A*b^2)*d^2*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*(1 5*B*b^2*e^3*x^3 + 105*B*b^2*d^3 - 6*A*a^2*e^3 - 10*(4*B*a*b + 3*A*b^2)*d^2 *e - 2*(2*B*a^2 + 5*A*a*b)*d*e^2 + (161*B*b^2*d*e^2 - 2*(35*B*a*b + 23*A*b ^2)*e^3)*x^2 + (245*B*b^2*d^2*e - 14*(7*B*a*b + 5*A*b^2)*d*e^2 - 2*(5*B*a^ 2 + 11*A*a*b)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(e^7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4), 1/30*(15*(7*B*b^2*d^4 - (5*B*a*b + 2*A*b^2)*d^3* e + (7*B*b^2*d*e^3 - (5*B*a*b + 2*A*b^2)*e^4)*x^3 + 3*(7*B*b^2*d^2*e^2 - ( 5*B*a*b + 2*A*b^2)*d*e^3)*x^2 + 3*(7*B*b^2*d^3*e - (5*B*a*b + 2*A*b^2)*d^2 *e^2)*x)*sqrt(-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*(15*B*b^2*e ^3*x^3 + 105*B*b^2*d^3 - 6*A*a^2*e^3 - 10*(4*B*a*b + 3*A*b^2)*d^2*e - 2*(2 *B*a^2 + 5*A*a*b)*d*e^2 + (161*B*b^2*d*e^2 - 2*(35*B*a*b + 23*A*b^2)*e^3)* x^2 + (245*B*b^2*d^2*e - 14*(7*B*a*b + 5*A*b^2)*d*e^2 - 2*(5*B*a^2 + 11*A* a*b)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(e^7*x^3 + 3*d*e^6*x^2 + 3*d^2*e ^5*x + d^3*e^4)]
\[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{7/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{7/2}} \, dx=\text {Exception raised: ValueError} \]
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. 707 vs. \(2 (222) = 444\).
Time = 0.48 (sec) , antiderivative size = 707, normalized size of antiderivative = 2.76 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{7/2}} \, dx=\frac {{\left ({\left ({\left (b x + a\right )} {\left (\frac {15 \, {\left (B b^{9} d^{2} e^{6} {\left | b \right |} - 2 \, B a b^{8} d e^{7} {\left | b \right |} + B a^{2} b^{7} e^{8} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{6} d^{2} e^{7} - 2 \, a b^{5} d e^{8} + a^{2} b^{4} e^{9}} + \frac {23 \, {\left (7 \, B b^{10} d^{3} e^{5} {\left | b \right |} - 19 \, B a b^{9} d^{2} e^{6} {\left | b \right |} - 2 \, A b^{10} d^{2} e^{6} {\left | b \right |} + 17 \, B a^{2} b^{8} d e^{7} {\left | b \right |} + 4 \, A a b^{9} d e^{7} {\left | b \right |} - 5 \, B a^{3} b^{7} e^{8} {\left | b \right |} - 2 \, A a^{2} b^{8} e^{8} {\left | b \right |}\right )}}{b^{6} d^{2} e^{7} - 2 \, a b^{5} d e^{8} + a^{2} b^{4} e^{9}}\right )} + \frac {35 \, {\left (7 \, B b^{11} d^{4} e^{4} {\left | b \right |} - 26 \, B a b^{10} d^{3} e^{5} {\left | b \right |} - 2 \, A b^{11} d^{3} e^{5} {\left | b \right |} + 36 \, B a^{2} b^{9} d^{2} e^{6} {\left | b \right |} + 6 \, A a b^{10} d^{2} e^{6} {\left | b \right |} - 22 \, B a^{3} b^{8} d e^{7} {\left | b \right |} - 6 \, A a^{2} b^{9} d e^{7} {\left | b \right |} + 5 \, B a^{4} b^{7} e^{8} {\left | b \right |} + 2 \, A a^{3} b^{8} e^{8} {\left | b \right |}\right )}}{b^{6} d^{2} e^{7} - 2 \, a b^{5} d e^{8} + a^{2} b^{4} e^{9}}\right )} {\left (b x + a\right )} + \frac {15 \, {\left (7 \, B b^{12} d^{5} e^{3} {\left | b \right |} - 33 \, B a b^{11} d^{4} e^{4} {\left | b \right |} - 2 \, A b^{12} d^{4} e^{4} {\left | b \right |} + 62 \, B a^{2} b^{10} d^{3} e^{5} {\left | b \right |} + 8 \, A a b^{11} d^{3} e^{5} {\left | b \right |} - 58 \, B a^{3} b^{9} d^{2} e^{6} {\left | b \right |} - 12 \, A a^{2} b^{10} d^{2} e^{6} {\left | b \right |} + 27 \, B a^{4} b^{8} d e^{7} {\left | b \right |} + 8 \, A a^{3} b^{9} d e^{7} {\left | b \right |} - 5 \, B a^{5} b^{7} e^{8} {\left | b \right |} - 2 \, A a^{4} b^{8} e^{8} {\left | b \right |}\right )}}{b^{6} d^{2} e^{7} - 2 \, a b^{5} d e^{8} + a^{2} b^{4} e^{9}}\right )} \sqrt {b x + a}}{15 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {5}{2}}} + \frac {{\left (7 \, B b^{2} d {\left | b \right |} - 5 \, B a b e {\left | b \right |} - 2 \, A b^{2} e {\left | b \right |}\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}} \]
1/15*(((b*x + a)*(15*(B*b^9*d^2*e^6*abs(b) - 2*B*a*b^8*d*e^7*abs(b) + B*a^ 2*b^7*e^8*abs(b))*(b*x + a)/(b^6*d^2*e^7 - 2*a*b^5*d*e^8 + a^2*b^4*e^9) + 23*(7*B*b^10*d^3*e^5*abs(b) - 19*B*a*b^9*d^2*e^6*abs(b) - 2*A*b^10*d^2*e^6 *abs(b) + 17*B*a^2*b^8*d*e^7*abs(b) + 4*A*a*b^9*d*e^7*abs(b) - 5*B*a^3*b^7 *e^8*abs(b) - 2*A*a^2*b^8*e^8*abs(b))/(b^6*d^2*e^7 - 2*a*b^5*d*e^8 + a^2*b ^4*e^9)) + 35*(7*B*b^11*d^4*e^4*abs(b) - 26*B*a*b^10*d^3*e^5*abs(b) - 2*A* b^11*d^3*e^5*abs(b) + 36*B*a^2*b^9*d^2*e^6*abs(b) + 6*A*a*b^10*d^2*e^6*abs (b) - 22*B*a^3*b^8*d*e^7*abs(b) - 6*A*a^2*b^9*d*e^7*abs(b) + 5*B*a^4*b^7*e ^8*abs(b) + 2*A*a^3*b^8*e^8*abs(b))/(b^6*d^2*e^7 - 2*a*b^5*d*e^8 + a^2*b^4 *e^9))*(b*x + a) + 15*(7*B*b^12*d^5*e^3*abs(b) - 33*B*a*b^11*d^4*e^4*abs(b ) - 2*A*b^12*d^4*e^4*abs(b) + 62*B*a^2*b^10*d^3*e^5*abs(b) + 8*A*a*b^11*d^ 3*e^5*abs(b) - 58*B*a^3*b^9*d^2*e^6*abs(b) - 12*A*a^2*b^10*d^2*e^6*abs(b) + 27*B*a^4*b^8*d*e^7*abs(b) + 8*A*a^3*b^9*d*e^7*abs(b) - 5*B*a^5*b^7*e^8*a bs(b) - 2*A*a^4*b^8*e^8*abs(b))/(b^6*d^2*e^7 - 2*a*b^5*d*e^8 + a^2*b^4*e^9 ))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(5/2) + (7*B*b^2*d*abs(b) - 5*B*a*b*e*abs(b) - 2*A*b^2*e*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)
Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{7/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \]