Integrand size = 24, antiderivative size = 249 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx=\frac {5 (b d-a e) (b B d+6 A b e-7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^4}+\frac {5 (b B d+6 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^3}+\frac {(b B d+6 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^2 (b d-a e)}-\frac {2 (A b-a B) (d+e x)^{7/2}}{b (b d-a e) \sqrt {a+b x}}+\frac {5 (b d-a e)^2 (b B d+6 A b e-7 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{9/2} \sqrt {e}} \]
5/8*(-a*e+b*d)^2*(6*A*b*e-7*B*a*e+B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^( 1/2)/(e*x+d)^(1/2))/b^(9/2)/e^(1/2)-2*(A*b-B*a)*(e*x+d)^(7/2)/b/(-a*e+b*d) /(b*x+a)^(1/2)+5/12*(6*A*b*e-7*B*a*e+B*b*d)*(e*x+d)^(3/2)*(b*x+a)^(1/2)/b^ 3+1/3*(6*A*b*e-7*B*a*e+B*b*d)*(e*x+d)^(5/2)*(b*x+a)^(1/2)/b^2/(-a*e+b*d)+5 /8*(-a*e+b*d)*(6*A*b*e-7*B*a*e+B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b^4
Time = 0.69 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.88 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (-6 A b \left (15 a^2 e^2+5 a b e (-5 d+e x)+b^2 \left (8 d^2-9 d e x-2 e^2 x^2\right )\right )+B \left (105 a^3 e^2+5 a^2 b e (-38 d+7 e x)+a b^2 \left (81 d^2-68 d e x-14 e^2 x^2\right )+b^3 x \left (33 d^2+26 d e x+8 e^2 x^2\right )\right )\right )}{24 b^4 \sqrt {a+b x}}+\frac {5 (b d-a e)^2 (b B d+6 A b e-7 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{8 b^{9/2} \sqrt {e}} \]
(Sqrt[d + e*x]*(-6*A*b*(15*a^2*e^2 + 5*a*b*e*(-5*d + e*x) + b^2*(8*d^2 - 9 *d*e*x - 2*e^2*x^2)) + B*(105*a^3*e^2 + 5*a^2*b*e*(-38*d + 7*e*x) + a*b^2* (81*d^2 - 68*d*e*x - 14*e^2*x^2) + b^3*x*(33*d^2 + 26*d*e*x + 8*e^2*x^2))) )/(24*b^4*Sqrt[a + b*x]) + (5*(b*d - a*e)^2*(b*B*d + 6*A*b*e - 7*a*B*e)*Ar cTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x])])/(8*b^(9/2)*Sqrt[e] )
Time = 0.30 (sec) , antiderivative size = 225, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-7 a B e+6 A b e+b B d) \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x}}dx}{b (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{b \sqrt {a+b x} (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-7 a B e+6 A b e+b B d) \left (\frac {5 (b d-a e) \int \frac {(d+e x)^{3/2}}{\sqrt {a+b x}}dx}{6 b}+\frac {\sqrt {a+b x} (d+e x)^{5/2}}{3 b}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{b \sqrt {a+b x} (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-7 a B e+6 A b e+b B d) \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x}}dx}{4 b}+\frac {\sqrt {a+b x} (d+e x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (d+e x)^{5/2}}{3 b}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{b \sqrt {a+b x} (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-7 a B e+6 A b e+b B d) \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{2 b}+\frac {\sqrt {a+b x} \sqrt {d+e x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (d+e x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (d+e x)^{5/2}}{3 b}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{b \sqrt {a+b x} (b d-a e)}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {(-7 a B e+6 A b e+b B d) \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}}{b}+\frac {\sqrt {a+b x} \sqrt {d+e x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (d+e x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (d+e x)^{5/2}}{3 b}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{b \sqrt {a+b x} (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(-7 a B e+6 A b e+b B d) \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{3/2} \sqrt {e}}+\frac {\sqrt {a+b x} \sqrt {d+e x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (d+e x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (d+e x)^{5/2}}{3 b}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{b \sqrt {a+b x} (b d-a e)}\) |
(-2*(A*b - a*B)*(d + e*x)^(7/2))/(b*(b*d - a*e)*Sqrt[a + b*x]) + ((b*B*d + 6*A*b*e - 7*a*B*e)*((Sqrt[a + b*x]*(d + e*x)^(5/2))/(3*b) + (5*(b*d - a*e )*((Sqrt[a + b*x]*(d + e*x)^(3/2))/(2*b) + (3*(b*d - a*e)*((Sqrt[a + b*x]* Sqrt[d + e*x])/b + ((b*d - a*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*S qrt[d + e*x])])/(b^(3/2)*Sqrt[e])))/(4*b)))/(6*b)))/(b*(b*d - a*e))
3.23.45.3.1 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. \(2(213)=426\).
Time = 1.10 (sec) , antiderivative size = 1184, normalized size of antiderivative = 4.76
1/48*(e*x+d)^(1/2)*(162*B*a*b^2*d^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(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^2*e^3*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^4*d^2*e*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))*a^3*b*e^3*x-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^2*b^2* d*e^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*b^3*d^2*e+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^3*b*d*e^2-136*B*a*b^2*d*e*x*((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))*a^4*e^3+16*B*b^3*e^2*x^3*((b*x+a)*(e*x+d) )^(1/2)*(b*e)^(1/2)+24*A*b^3*e^2*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+6 6*B*b^3*d^2*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-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^2*d^2*e-18 0*A*a^2*b*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-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*b^3*d^2*e*x-96 *A*b^3*d^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+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))*b^4*d^3*x-28*B*a*b^2 *e^2*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(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^3*b*e^3+210*B*a^3...
Leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (213) = 426\).
Time = 0.61 (sec) , antiderivative size = 872, normalized size of antiderivative = 3.50 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx=\left [-\frac {15 \, {\left (B a b^{3} d^{3} - 3 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{2} e + 3 \, {\left (5 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} d e^{2} - {\left (7 \, B a^{4} - 6 \, A a^{3} b\right )} e^{3} + {\left (B b^{4} d^{3} - 3 \, {\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} d^{2} e + 3 \, {\left (5 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} d e^{2} - {\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} e^{3}\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^{4} e^{3} x^{3} + 3 \, {\left (27 \, B a b^{3} - 16 \, A b^{4}\right )} d^{2} e - 10 \, {\left (19 \, B a^{2} b^{2} - 15 \, A a b^{3}\right )} d e^{2} + 15 \, {\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} e^{3} + 2 \, {\left (13 \, B b^{4} d e^{2} - {\left (7 \, B a b^{3} - 6 \, A b^{4}\right )} e^{3}\right )} x^{2} + {\left (33 \, B b^{4} d^{2} e - 2 \, {\left (34 \, B a b^{3} - 27 \, A b^{4}\right )} d e^{2} + 5 \, {\left (7 \, B a^{2} b^{2} - 6 \, A a b^{3}\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{96 \, {\left (b^{6} e x + a b^{5} e\right )}}, -\frac {15 \, {\left (B a b^{3} d^{3} - 3 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{2} e + 3 \, {\left (5 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} d e^{2} - {\left (7 \, B a^{4} - 6 \, A a^{3} b\right )} e^{3} + {\left (B b^{4} d^{3} - 3 \, {\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} d^{2} e + 3 \, {\left (5 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} d e^{2} - {\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} e^{3}\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^{4} e^{3} x^{3} + 3 \, {\left (27 \, B a b^{3} - 16 \, A b^{4}\right )} d^{2} e - 10 \, {\left (19 \, B a^{2} b^{2} - 15 \, A a b^{3}\right )} d e^{2} + 15 \, {\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} e^{3} + 2 \, {\left (13 \, B b^{4} d e^{2} - {\left (7 \, B a b^{3} - 6 \, A b^{4}\right )} e^{3}\right )} x^{2} + {\left (33 \, B b^{4} d^{2} e - 2 \, {\left (34 \, B a b^{3} - 27 \, A b^{4}\right )} d e^{2} + 5 \, {\left (7 \, B a^{2} b^{2} - 6 \, A a b^{3}\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{48 \, {\left (b^{6} e x + a b^{5} e\right )}}\right ] \]
[-1/96*(15*(B*a*b^3*d^3 - 3*(3*B*a^2*b^2 - 2*A*a*b^3)*d^2*e + 3*(5*B*a^3*b - 4*A*a^2*b^2)*d*e^2 - (7*B*a^4 - 6*A*a^3*b)*e^3 + (B*b^4*d^3 - 3*(3*B*a* b^3 - 2*A*b^4)*d^2*e + 3*(5*B*a^2*b^2 - 4*A*a*b^3)*d*e^2 - (7*B*a^3*b - 6* A*a^2*b^2)*e^3)*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^4*e^3*x^3 + 3*(27*B*a*b^3 - 16*A*b^4)*d^2 *e - 10*(19*B*a^2*b^2 - 15*A*a*b^3)*d*e^2 + 15*(7*B*a^3*b - 6*A*a^2*b^2)*e ^3 + 2*(13*B*b^4*d*e^2 - (7*B*a*b^3 - 6*A*b^4)*e^3)*x^2 + (33*B*b^4*d^2*e - 2*(34*B*a*b^3 - 27*A*b^4)*d*e^2 + 5*(7*B*a^2*b^2 - 6*A*a*b^3)*e^3)*x)*sq rt(b*x + a)*sqrt(e*x + d))/(b^6*e*x + a*b^5*e), -1/48*(15*(B*a*b^3*d^3 - 3 *(3*B*a^2*b^2 - 2*A*a*b^3)*d^2*e + 3*(5*B*a^3*b - 4*A*a^2*b^2)*d*e^2 - (7* B*a^4 - 6*A*a^3*b)*e^3 + (B*b^4*d^3 - 3*(3*B*a*b^3 - 2*A*b^4)*d^2*e + 3*(5 *B*a^2*b^2 - 4*A*a*b^3)*d*e^2 - (7*B*a^3*b - 6*A*a^2*b^2)*e^3)*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^4*e^3*x^3 + 3 *(27*B*a*b^3 - 16*A*b^4)*d^2*e - 10*(19*B*a^2*b^2 - 15*A*a*b^3)*d*e^2 + 15 *(7*B*a^3*b - 6*A*a^2*b^2)*e^3 + 2*(13*B*b^4*d*e^2 - (7*B*a*b^3 - 6*A*b^4) *e^3)*x^2 + (33*B*b^4*d^2*e - 2*(34*B*a*b^3 - 27*A*b^4)*d*e^2 + 5*(7*B*a^2 *b^2 - 6*A*a*b^3)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^6*e*x + a*b^5*e) ]
\[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (213) = 426\).
Time = 0.59 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.92 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx=\frac {1}{24} \, \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} B e^{2} {\left | b \right |}}{b^{6}} + \frac {13 \, B b^{18} d e^{5} {\left | b \right |} - 19 \, B a b^{17} e^{6} {\left | b \right |} + 6 \, A b^{18} e^{6} {\left | b \right |}}{b^{23} e^{4}}\right )} + \frac {3 \, {\left (11 \, B b^{19} d^{2} e^{4} {\left | b \right |} - 40 \, B a b^{18} d e^{5} {\left | b \right |} + 18 \, A b^{19} d e^{5} {\left | b \right |} + 29 \, B a^{2} b^{17} e^{6} {\left | b \right |} - 18 \, A a b^{18} e^{6} {\left | b \right |}\right )}}{b^{23} e^{4}}\right )} - \frac {5 \, {\left (B b^{3} d^{3} {\left | b \right |} - 9 \, B a b^{2} d^{2} e {\left | b \right |} + 6 \, A b^{3} d^{2} e {\left | b \right |} + 15 \, B a^{2} b d e^{2} {\left | b \right |} - 12 \, A a b^{2} d e^{2} {\left | b \right |} - 7 \, B a^{3} e^{3} {\left | b \right |} + 6 \, A a^{2} b e^{3} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}{16 \, \sqrt {b e} b^{5}} + \frac {4 \, {\left (B a b^{3} d^{3} e {\left | b \right |} - A b^{4} d^{3} e {\left | b \right |} - 3 \, B a^{2} b^{2} d^{2} e^{2} {\left | b \right |} + 3 \, A a b^{3} d^{2} e^{2} {\left | b \right |} + 3 \, B a^{3} b d e^{3} {\left | b \right |} - 3 \, A a^{2} b^{2} d e^{3} {\left | b \right |} - B a^{4} e^{4} {\left | b \right |} + A a^{3} b e^{4} {\left | b \right |}\right )}}{{\left (b^{2} d - a b e - {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )} \sqrt {b e} b^{4}} \]
1/24*sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b* x + a)*B*e^2*abs(b)/b^6 + (13*B*b^18*d*e^5*abs(b) - 19*B*a*b^17*e^6*abs(b) + 6*A*b^18*e^6*abs(b))/(b^23*e^4)) + 3*(11*B*b^19*d^2*e^4*abs(b) - 40*B*a *b^18*d*e^5*abs(b) + 18*A*b^19*d*e^5*abs(b) + 29*B*a^2*b^17*e^6*abs(b) - 1 8*A*a*b^18*e^6*abs(b))/(b^23*e^4)) - 5/16*(B*b^3*d^3*abs(b) - 9*B*a*b^2*d^ 2*e*abs(b) + 6*A*b^3*d^2*e*abs(b) + 15*B*a^2*b*d*e^2*abs(b) - 12*A*a*b^2*d *e^2*abs(b) - 7*B*a^3*e^3*abs(b) + 6*A*a^2*b*e^3*abs(b))*log((sqrt(b*e)*sq rt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2)/(sqrt(b*e)*b^5) + 4* (B*a*b^3*d^3*e*abs(b) - A*b^4*d^3*e*abs(b) - 3*B*a^2*b^2*d^2*e^2*abs(b) + 3*A*a*b^3*d^2*e^2*abs(b) + 3*B*a^3*b*d*e^3*abs(b) - 3*A*a^2*b^2*d*e^3*abs( b) - B*a^4*e^4*abs(b) + A*a^3*b*e^4*abs(b))/((b^2*d - a*b*e - (sqrt(b*e)*s qrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2)*sqrt(b*e)*b^4)
Timed out. \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{5/2}}{{\left (a+b\,x\right )}^{3/2}} \,d x \]