Integrand size = 24, antiderivative size = 253 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{3/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{7/2}}{e (b d-a e) \sqrt {d+e x}}+\frac {5 (b d-a e) (7 b B d-6 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 e^4}-\frac {5 (7 b B d-6 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{12 e^3}+\frac {(7 b B d-6 A b e-a B e) (a+b x)^{5/2} \sqrt {d+e x}}{3 e^2 (b d-a e)}-\frac {5 (b d-a e)^2 (7 b B d-6 A b e-a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 \sqrt {b} e^{9/2}} \]
-5/8*(-a*e+b*d)^2*(-6*A*b*e-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)/b^(1/2)-2*(-A*e+B*d)*(b*x+a)^(7/2)/e/(-a*e+b *d)/(e*x+d)^(1/2)-5/12*(-6*A*b*e-B*a*e+7*B*b*d)*(b*x+a)^(3/2)*(e*x+d)^(1/2 )/e^3+1/3*(-6*A*b*e-B*a*e+7*B*b*d)*(b*x+a)^(5/2)*(e*x+d)^(1/2)/e^2/(-a*e+b *d)+5/8*(-a*e+b*d)*(-6*A*b*e-B*a*e+7*B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/e^ 4
Time = 0.63 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {a+b x} \left (3 a^2 e^2 (27 B d-16 A e+11 B e x)+2 a b e \left (3 A e (25 d+9 e x)+B \left (-95 d^2-34 d e x+13 e^2 x^2\right )\right )+b^2 \left (6 A e \left (-15 d^2-5 d e x+2 e^2 x^2\right )+B \left (105 d^3+35 d^2 e x-14 d e^2 x^2+8 e^3 x^3\right )\right )\right )}{24 e^4 \sqrt {d+e x}}+\frac {5 (b d-a e)^2 (-7 b B d+6 A b e+a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 \sqrt {b} e^{9/2}} \]
(Sqrt[a + b*x]*(3*a^2*e^2*(27*B*d - 16*A*e + 11*B*e*x) + 2*a*b*e*(3*A*e*(2 5*d + 9*e*x) + B*(-95*d^2 - 34*d*e*x + 13*e^2*x^2)) + b^2*(6*A*e*(-15*d^2 - 5*d*e*x + 2*e^2*x^2) + B*(105*d^3 + 35*d^2*e*x - 14*d*e^2*x^2 + 8*e^3*x^ 3))))/(24*e^4*Sqrt[d + e*x]) + (5*(b*d - a*e)^2*(-7*b*B*d + 6*A*b*e + a*B* e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(8*Sqrt[b]*e^ (9/2))
Time = 0.28 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {87, 60, 60, 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.
| \(\displaystyle \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-a B e-6 A b e+7 b B d) \int \frac {(a+b x)^{5/2}}{\sqrt {d+e x}}dx}{e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{e \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-a B e-6 A b e+7 b B d) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}}dx}{6 e}\right )}{e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{e \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-a B e-6 A b e+7 b B d) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}}dx}{4 e}\right )}{6 e}\right )}{e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{e \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-a B e-6 A b e+7 b B d) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{2 e}\right )}{4 e}\right )}{6 e}\right )}{e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{e \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {(-a B e-6 A b e+7 b B d) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}}{e}\right )}{4 e}\right )}{6 e}\right )}{e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{e \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(-a B e-6 A b e+7 b B d) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{3/2}}\right )}{4 e}\right )}{6 e}\right )}{e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{e \sqrt {d+e x} (b d-a e)}\) |
(-2*(B*d - A*e)*(a + b*x)^(7/2))/(e*(b*d - a*e)*Sqrt[d + e*x]) + ((7*b*B*d - 6*A*b*e - a*B*e)*(((a + b*x)^(5/2)*Sqrt[d + e*x])/(3*e) - (5*(b*d - a*e )*(((a + b*x)^(3/2)*Sqrt[d + e*x])/(2*e) - (3*(b*d - a*e)*((Sqrt[a + b*x]* Sqrt[d + e*x])/e - ((b*d - a*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*S qrt[d + e*x])])/(Sqrt[b]*e^(3/2))))/(4*e)))/(6*e)))/(e*(b*d - a*e))
3.23.27.3.1 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 + 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. \(1183\) vs. \(2(217)=434\).
Time = 1.10 (sec) , antiderivative size = 1184, normalized size of antiderivative = 4.68
1/48*(b*x+a)^(1/2)*(-136*B*a*b*d*e^2*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2) -180*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))*a^2*b*e^4*x+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+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))*a^2*b*d*e^3-180*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))*a*b^2*d^2*e^2-135*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^2*b* d^2*e^2+16*B*b^2*e^3*x^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-60*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+300*A*a*b*d*e^2*(b *e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-180*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))*a*b^2*d*e^3*x-135*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^2*b*d*e^ 3*x+15*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^3*e^4*x+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^3*e+15*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^3*d*e^3-380*B*a*b*d^2*e*((b *x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-96*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)+225*B*ln(1/2*...
Time = 0.74 (sec) , antiderivative size = 858, normalized size of antiderivative = 3.39 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\left [-\frac {15 \, {\left (7 \, B b^{3} d^{4} - 3 \, {\left (5 \, B a b^{2} + 2 \, A b^{3}\right )} d^{3} e + 3 \, {\left (3 \, B a^{2} b + 4 \, A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 6 \, A a^{2} b\right )} d e^{3} + {\left (7 \, B b^{3} d^{3} e - 3 \, {\left (5 \, B a b^{2} + 2 \, A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (3 \, B a^{2} b + 4 \, A a b^{2}\right )} d e^{3} - {\left (B a^{3} + 6 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt {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 x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \, {\left (8 \, B b^{3} e^{4} x^{3} + 105 \, B b^{3} d^{3} e - 48 \, A a^{2} b e^{4} - 10 \, {\left (19 \, B a b^{2} + 9 \, A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (27 \, B a^{2} b + 50 \, A a b^{2}\right )} d e^{3} - 2 \, {\left (7 \, B b^{3} d e^{3} - {\left (13 \, B a b^{2} + 6 \, A b^{3}\right )} e^{4}\right )} x^{2} + {\left (35 \, B b^{3} d^{2} e^{2} - 2 \, {\left (34 \, B a b^{2} + 15 \, A b^{3}\right )} d e^{3} + 3 \, {\left (11 \, B a^{2} b + 18 \, A a b^{2}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{96 \, {\left (b e^{6} x + b d e^{5}\right )}}, \frac {15 \, {\left (7 \, B b^{3} d^{4} - 3 \, {\left (5 \, B a b^{2} + 2 \, A b^{3}\right )} d^{3} e + 3 \, {\left (3 \, B a^{2} b + 4 \, A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 6 \, A a^{2} b\right )} d e^{3} + {\left (7 \, B b^{3} d^{3} e - 3 \, {\left (5 \, B a b^{2} + 2 \, A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (3 \, B a^{2} b + 4 \, A a b^{2}\right )} d e^{3} - {\left (B a^{3} + 6 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) + 2 \, {\left (8 \, B b^{3} e^{4} x^{3} + 105 \, B b^{3} d^{3} e - 48 \, A a^{2} b e^{4} - 10 \, {\left (19 \, B a b^{2} + 9 \, A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (27 \, B a^{2} b + 50 \, A a b^{2}\right )} d e^{3} - 2 \, {\left (7 \, B b^{3} d e^{3} - {\left (13 \, B a b^{2} + 6 \, A b^{3}\right )} e^{4}\right )} x^{2} + {\left (35 \, B b^{3} d^{2} e^{2} - 2 \, {\left (34 \, B a b^{2} + 15 \, A b^{3}\right )} d e^{3} + 3 \, {\left (11 \, B a^{2} b + 18 \, A a b^{2}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{48 \, {\left (b e^{6} x + b d e^{5}\right )}}\right ] \]
[-1/96*(15*(7*B*b^3*d^4 - 3*(5*B*a*b^2 + 2*A*b^3)*d^3*e + 3*(3*B*a^2*b + 4 *A*a*b^2)*d^2*e^2 - (B*a^3 + 6*A*a^2*b)*d*e^3 + (7*B*b^3*d^3*e - 3*(5*B*a* b^2 + 2*A*b^3)*d^2*e^2 + 3*(3*B*a^2*b + 4*A*a*b^2)*d*e^3 - (B*a^3 + 6*A*a^ 2*b)*e^4)*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*x + b*d + a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d* e + a*b*e^2)*x) - 4*(8*B*b^3*e^4*x^3 + 105*B*b^3*d^3*e - 48*A*a^2*b*e^4 - 10*(19*B*a*b^2 + 9*A*b^3)*d^2*e^2 + 3*(27*B*a^2*b + 50*A*a*b^2)*d*e^3 - 2* (7*B*b^3*d*e^3 - (13*B*a*b^2 + 6*A*b^3)*e^4)*x^2 + (35*B*b^3*d^2*e^2 - 2*( 34*B*a*b^2 + 15*A*b^3)*d*e^3 + 3*(11*B*a^2*b + 18*A*a*b^2)*e^4)*x)*sqrt(b* x + a)*sqrt(e*x + d))/(b*e^6*x + b*d*e^5), 1/48*(15*(7*B*b^3*d^4 - 3*(5*B* a*b^2 + 2*A*b^3)*d^3*e + 3*(3*B*a^2*b + 4*A*a*b^2)*d^2*e^2 - (B*a^3 + 6*A* a^2*b)*d*e^3 + (7*B*b^3*d^3*e - 3*(5*B*a*b^2 + 2*A*b^3)*d^2*e^2 + 3*(3*B*a ^2*b + 4*A*a*b^2)*d*e^3 - (B*a^3 + 6*A*a^2*b)*e^4)*x)*sqrt(-b*e)*arctan(1/ 2*(2*b*e*x + b*d + a*e)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d)/(b^2*e^2*x^ 2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) + 2*(8*B*b^3*e^4*x^3 + 105*B*b^3*d^3 *e - 48*A*a^2*b*e^4 - 10*(19*B*a*b^2 + 9*A*b^3)*d^2*e^2 + 3*(27*B*a^2*b + 50*A*a*b^2)*d*e^3 - 2*(7*B*b^3*d*e^3 - (13*B*a*b^2 + 6*A*b^3)*e^4)*x^2 + ( 35*B*b^3*d^2*e^2 - 2*(34*B*a*b^2 + 15*A*b^3)*d*e^3 + 3*(11*B*a^2*b + 18*A* a*b^2)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b*e^6*x + b*d*e^5)]
\[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Time = 0.36 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.66 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\frac {{\left ({\left (b x + a\right )} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} B {\left | b \right |}}{b e} - \frac {7 \, B b^{2} d e^{5} {\left | b \right |} - B a b e^{6} {\left | b \right |} - 6 \, A b^{2} e^{6} {\left | b \right |}}{b^{2} e^{7}}\right )} + \frac {5 \, {\left (7 \, B b^{3} d^{2} e^{4} {\left | b \right |} - 8 \, B a b^{2} d e^{5} {\left | b \right |} - 6 \, A b^{3} d e^{5} {\left | b \right |} + B a^{2} b e^{6} {\left | b \right |} + 6 \, A a b^{2} e^{6} {\left | b \right |}\right )}}{b^{2} e^{7}}\right )} + \frac {15 \, {\left (7 \, B b^{4} d^{3} e^{3} {\left | b \right |} - 15 \, B a b^{3} d^{2} e^{4} {\left | b \right |} - 6 \, A b^{4} d^{2} e^{4} {\left | b \right |} + 9 \, B a^{2} b^{2} d e^{5} {\left | b \right |} + 12 \, A a b^{3} d e^{5} {\left | b \right |} - B a^{3} b e^{6} {\left | b \right |} - 6 \, A a^{2} b^{2} e^{6} {\left | b \right |}\right )}}{b^{2} e^{7}}\right )} \sqrt {b x + a}}{24 \, \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}} + \frac {5 \, {\left (7 \, B b^{3} d^{3} {\left | b \right |} - 15 \, B a b^{2} d^{2} e {\left | b \right |} - 6 \, A b^{3} d^{2} e {\left | b \right |} + 9 \, B a^{2} b d e^{2} {\left | b \right |} + 12 \, A a b^{2} d e^{2} {\left | b \right |} - B a^{3} e^{3} {\left | b \right |} - 6 \, A a^{2} b e^{3} {\left | b \right |}\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{8 \, \sqrt {b e} b e^{4}} \]
1/24*((b*x + a)*(2*(b*x + a)*(4*(b*x + a)*B*abs(b)/(b*e) - (7*B*b^2*d*e^5* abs(b) - B*a*b*e^6*abs(b) - 6*A*b^2*e^6*abs(b))/(b^2*e^7)) + 5*(7*B*b^3*d^ 2*e^4*abs(b) - 8*B*a*b^2*d*e^5*abs(b) - 6*A*b^3*d*e^5*abs(b) + B*a^2*b*e^6 *abs(b) + 6*A*a*b^2*e^6*abs(b))/(b^2*e^7)) + 15*(7*B*b^4*d^3*e^3*abs(b) - 15*B*a*b^3*d^2*e^4*abs(b) - 6*A*b^4*d^2*e^4*abs(b) + 9*B*a^2*b^2*d*e^5*abs (b) + 12*A*a*b^3*d*e^5*abs(b) - B*a^3*b*e^6*abs(b) - 6*A*a^2*b^2*e^6*abs(b ))/(b^2*e^7))*sqrt(b*x + a)/sqrt(b^2*d + (b*x + a)*b*e - a*b*e) + 5/8*(7*B *b^3*d^3*abs(b) - 15*B*a*b^2*d^2*e*abs(b) - 6*A*b^3*d^2*e*abs(b) + 9*B*a^2 *b*d*e^2*abs(b) + 12*A*a*b^2*d*e^2*abs(b) - B*a^3*e^3*abs(b) - 6*A*a^2*b*e ^3*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)*b*e^4)
Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \]