Integrand size = 33, antiderivative size = 209 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {e (6 b B d-A b e-5 a B e) \sqrt {d+e x}}{8 b^3 (b d-a e) (a+b x)}-\frac {(6 b B d-A b e-5 a B e) (d+e x)^{3/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{5/2}}{3 b (b d-a e) (a+b x)^3}-\frac {e^2 (6 b B d-A b e-5 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{7/2} (b d-a e)^{3/2}} \]
-1/12*(-A*b*e-5*B*a*e+6*B*b*d)*(e*x+d)^(3/2)/b^2/(-a*e+b*d)/(b*x+a)^2-1/3* (A*b-B*a)*(e*x+d)^(5/2)/b/(-a*e+b*d)/(b*x+a)^3-1/8*e^2*(-A*b*e-5*B*a*e+6*B *b*d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(7/2)/(-a*e+b*d)^( 3/2)-1/8*e*(-A*b*e-5*B*a*e+6*B*b*d)*(e*x+d)^(1/2)/b^3/(-a*e+b*d)/(b*x+a)
Time = 1.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.01 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\sqrt {d+e x} \left (B \left (-15 a^3 e^2+8 a^2 b e (d-5 e x)+6 b^3 d x (2 d+5 e x)+a b^2 \left (4 d^2+22 d e x-33 e^2 x^2\right )\right )+A b \left (-3 a^2 e^2-2 a b e (d+4 e x)+b^2 \left (8 d^2+14 d e x+3 e^2 x^2\right )\right )\right )}{24 b^3 (-b d+a e) (a+b x)^3}+\frac {e^2 (-6 b B d+A b e+5 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{8 b^{7/2} (-b d+a e)^{3/2}} \]
(Sqrt[d + e*x]*(B*(-15*a^3*e^2 + 8*a^2*b*e*(d - 5*e*x) + 6*b^3*d*x*(2*d + 5*e*x) + a*b^2*(4*d^2 + 22*d*e*x - 33*e^2*x^2)) + A*b*(-3*a^2*e^2 - 2*a*b* e*(d + 4*e*x) + b^2*(8*d^2 + 14*d*e*x + 3*e^2*x^2))))/(24*b^3*(-(b*d) + a* e)*(a + b*x)^3) + (e^2*(-6*b*B*d + A*b*e + 5*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(8*b^(7/2)*(-(b*d) + a*e)^(3/2))
Time = 0.30 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.85, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {1184, 27, 87, 51, 51, 73, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^4 \int \frac {(A+B x) (d+e x)^{3/2}}{b^4 (a+b x)^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^4}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-5 a B e-A b e+6 b B d) \int \frac {(d+e x)^{3/2}}{(a+b x)^3}dx}{6 b (b d-a e)}-\frac {(d+e x)^{5/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(-5 a B e-A b e+6 b B d) \left (\frac {3 e \int \frac {\sqrt {d+e x}}{(a+b x)^2}dx}{4 b}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}\right )}{6 b (b d-a e)}-\frac {(d+e x)^{5/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(-5 a B e-A b e+6 b B d) \left (\frac {3 e \left (\frac {e \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{2 b}-\frac {\sqrt {d+e x}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}\right )}{6 b (b d-a e)}-\frac {(d+e x)^{5/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(-5 a B e-A b e+6 b B d) \left (\frac {3 e \left (\frac {\int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b}-\frac {\sqrt {d+e x}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}\right )}{6 b (b d-a e)}-\frac {(d+e x)^{5/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(-5 a B e-A b e+6 b B d) \left (\frac {3 e \left (-\frac {e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {b d-a e}}-\frac {\sqrt {d+e x}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}\right )}{6 b (b d-a e)}-\frac {(d+e x)^{5/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)}\) |
-1/3*((A*b - a*B)*(d + e*x)^(5/2))/(b*(b*d - a*e)*(a + b*x)^3) + ((6*b*B*d - A*b*e - 5*a*B*e)*(-1/2*(d + e*x)^(3/2)/(b*(a + b*x)^2) + (3*e*(-(Sqrt[d + e*x]/(b*(a + b*x))) - (e*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e ]])/(b^(3/2)*Sqrt[b*d - a*e])))/(4*b)))/(6*b*(b*d - a*e))
3.19.19.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.34 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.99
| method | result | size |
| derivativedivides | \(2 e^{2} \left (\frac {\frac {\left (A b e -11 B a e +10 B b d \right ) \left (e x +d \right )^{\frac {5}{2}}}{16 \left (a e -b d \right ) b}-\frac {\left (A b e +5 B a e -6 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{6 b^{2}}-\frac {\left (A a b \,e^{2}-A \,b^{2} d e +5 a^{2} B \,e^{2}-11 B a b d e +6 B \,b^{2} d^{2}\right ) \sqrt {e x +d}}{16 b^{3}}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {\left (A b e +5 B a e -6 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{16 \left (a e -b d \right ) b^{3} \sqrt {\left (a e -b d \right ) b}}\right )\) | \(207\) |
| default | \(2 e^{2} \left (\frac {\frac {\left (A b e -11 B a e +10 B b d \right ) \left (e x +d \right )^{\frac {5}{2}}}{16 \left (a e -b d \right ) b}-\frac {\left (A b e +5 B a e -6 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{6 b^{2}}-\frac {\left (A a b \,e^{2}-A \,b^{2} d e +5 a^{2} B \,e^{2}-11 B a b d e +6 B \,b^{2} d^{2}\right ) \sqrt {e x +d}}{16 b^{3}}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {\left (A b e +5 B a e -6 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{16 \left (a e -b d \right ) b^{3} \sqrt {\left (a e -b d \right ) b}}\right )\) | \(207\) |
| pseudoelliptic | \(\frac {-\sqrt {\left (a e -b d \right ) b}\, \left (\left (-A \,b^{3} x^{2}+\frac {8 \left (\frac {33 B x}{8}+A \right ) x a \,b^{2}}{3}+a^{2} \left (A +\frac {40 B x}{3}\right ) b +5 B \,a^{3}\right ) e^{2}+\frac {2 b d \left (\left (-15 B \,x^{2}-7 A x \right ) b^{2}+a \left (-11 B x +A \right ) b -4 B \,a^{2}\right ) e}{3}-\frac {8 b^{2} d^{2} \left (\left (\frac {3 B x}{2}+A \right ) b +\frac {B a}{2}\right )}{3}\right ) \sqrt {e x +d}+\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \left (\left (A b +5 B a \right ) e -6 B b d \right ) \left (b x +a \right )^{3} e^{2}}{8 \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right ) b^{3} \left (b x +a \right )^{3}}\) | \(208\) |
2*e^2*((1/16*(A*b*e-11*B*a*e+10*B*b*d)/(a*e-b*d)/b*(e*x+d)^(5/2)-1/6*(A*b* e+5*B*a*e-6*B*b*d)/b^2*(e*x+d)^(3/2)-1/16*(A*a*b*e^2-A*b^2*d*e+5*B*a^2*e^2 -11*B*a*b*d*e+6*B*b^2*d^2)/b^3*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^3+1/16*( A*b*e+5*B*a*e-6*B*b*d)/(a*e-b*d)/b^3/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^ (1/2)/((a*e-b*d)*b)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 551 vs. \(2 (185) = 370\).
Time = 0.39 (sec) , antiderivative size = 1115, normalized size of antiderivative = 5.33 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\left [\frac {3 \, {\left (6 \, B a^{3} b d e^{2} - {\left (5 \, B a^{4} + A a^{3} b\right )} e^{3} + {\left (6 \, B b^{4} d e^{2} - {\left (5 \, B a b^{3} + A b^{4}\right )} e^{3}\right )} x^{3} + 3 \, {\left (6 \, B a b^{3} d e^{2} - {\left (5 \, B a^{2} b^{2} + A a b^{3}\right )} e^{3}\right )} x^{2} + 3 \, {\left (6 \, B a^{2} b^{2} d e^{2} - {\left (5 \, B a^{3} b + A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (4 \, {\left (B a b^{4} + 2 \, A b^{5}\right )} d^{3} + 2 \, {\left (2 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} d^{2} e - {\left (23 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{2} + 3 \, {\left (5 \, B a^{4} b + A a^{3} b^{2}\right )} e^{3} + 3 \, {\left (10 \, B b^{5} d^{2} e - {\left (21 \, B a b^{4} - A b^{5}\right )} d e^{2} + {\left (11 \, B a^{2} b^{3} - A a b^{4}\right )} e^{3}\right )} x^{2} + 2 \, {\left (6 \, B b^{5} d^{3} + {\left (5 \, B a b^{4} + 7 \, A b^{5}\right )} d^{2} e - {\left (31 \, B a^{2} b^{3} + 11 \, A a b^{4}\right )} d e^{2} + 4 \, {\left (5 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{48 \, {\left (a^{3} b^{6} d^{2} - 2 \, a^{4} b^{5} d e + a^{5} b^{4} e^{2} + {\left (b^{9} d^{2} - 2 \, a b^{8} d e + a^{2} b^{7} e^{2}\right )} x^{3} + 3 \, {\left (a b^{8} d^{2} - 2 \, a^{2} b^{7} d e + a^{3} b^{6} e^{2}\right )} x^{2} + 3 \, {\left (a^{2} b^{7} d^{2} - 2 \, a^{3} b^{6} d e + a^{4} b^{5} e^{2}\right )} x\right )}}, \frac {3 \, {\left (6 \, B a^{3} b d e^{2} - {\left (5 \, B a^{4} + A a^{3} b\right )} e^{3} + {\left (6 \, B b^{4} d e^{2} - {\left (5 \, B a b^{3} + A b^{4}\right )} e^{3}\right )} x^{3} + 3 \, {\left (6 \, B a b^{3} d e^{2} - {\left (5 \, B a^{2} b^{2} + A a b^{3}\right )} e^{3}\right )} x^{2} + 3 \, {\left (6 \, B a^{2} b^{2} d e^{2} - {\left (5 \, B a^{3} b + A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (4 \, {\left (B a b^{4} + 2 \, A b^{5}\right )} d^{3} + 2 \, {\left (2 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} d^{2} e - {\left (23 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{2} + 3 \, {\left (5 \, B a^{4} b + A a^{3} b^{2}\right )} e^{3} + 3 \, {\left (10 \, B b^{5} d^{2} e - {\left (21 \, B a b^{4} - A b^{5}\right )} d e^{2} + {\left (11 \, B a^{2} b^{3} - A a b^{4}\right )} e^{3}\right )} x^{2} + 2 \, {\left (6 \, B b^{5} d^{3} + {\left (5 \, B a b^{4} + 7 \, A b^{5}\right )} d^{2} e - {\left (31 \, B a^{2} b^{3} + 11 \, A a b^{4}\right )} d e^{2} + 4 \, {\left (5 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (a^{3} b^{6} d^{2} - 2 \, a^{4} b^{5} d e + a^{5} b^{4} e^{2} + {\left (b^{9} d^{2} - 2 \, a b^{8} d e + a^{2} b^{7} e^{2}\right )} x^{3} + 3 \, {\left (a b^{8} d^{2} - 2 \, a^{2} b^{7} d e + a^{3} b^{6} e^{2}\right )} x^{2} + 3 \, {\left (a^{2} b^{7} d^{2} - 2 \, a^{3} b^{6} d e + a^{4} b^{5} e^{2}\right )} x\right )}}\right ] \]
[1/48*(3*(6*B*a^3*b*d*e^2 - (5*B*a^4 + A*a^3*b)*e^3 + (6*B*b^4*d*e^2 - (5* B*a*b^3 + A*b^4)*e^3)*x^3 + 3*(6*B*a*b^3*d*e^2 - (5*B*a^2*b^2 + A*a*b^3)*e ^3)*x^2 + 3*(6*B*a^2*b^2*d*e^2 - (5*B*a^3*b + A*a^2*b^2)*e^3)*x)*sqrt(b^2* d - a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d)) /(b*x + a)) - 2*(4*(B*a*b^4 + 2*A*b^5)*d^3 + 2*(2*B*a^2*b^3 - 5*A*a*b^4)*d ^2*e - (23*B*a^3*b^2 + A*a^2*b^3)*d*e^2 + 3*(5*B*a^4*b + A*a^3*b^2)*e^3 + 3*(10*B*b^5*d^2*e - (21*B*a*b^4 - A*b^5)*d*e^2 + (11*B*a^2*b^3 - A*a*b^4)* e^3)*x^2 + 2*(6*B*b^5*d^3 + (5*B*a*b^4 + 7*A*b^5)*d^2*e - (31*B*a^2*b^3 + 11*A*a*b^4)*d*e^2 + 4*(5*B*a^3*b^2 + A*a^2*b^3)*e^3)*x)*sqrt(e*x + d))/(a^ 3*b^6*d^2 - 2*a^4*b^5*d*e + a^5*b^4*e^2 + (b^9*d^2 - 2*a*b^8*d*e + a^2*b^7 *e^2)*x^3 + 3*(a*b^8*d^2 - 2*a^2*b^7*d*e + a^3*b^6*e^2)*x^2 + 3*(a^2*b^7*d ^2 - 2*a^3*b^6*d*e + a^4*b^5*e^2)*x), 1/24*(3*(6*B*a^3*b*d*e^2 - (5*B*a^4 + A*a^3*b)*e^3 + (6*B*b^4*d*e^2 - (5*B*a*b^3 + A*b^4)*e^3)*x^3 + 3*(6*B*a* b^3*d*e^2 - (5*B*a^2*b^2 + A*a*b^3)*e^3)*x^2 + 3*(6*B*a^2*b^2*d*e^2 - (5*B *a^3*b + A*a^2*b^2)*e^3)*x)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b* e)*sqrt(e*x + d)/(b*e*x + b*d)) - (4*(B*a*b^4 + 2*A*b^5)*d^3 + 2*(2*B*a^2* b^3 - 5*A*a*b^4)*d^2*e - (23*B*a^3*b^2 + A*a^2*b^3)*d*e^2 + 3*(5*B*a^4*b + A*a^3*b^2)*e^3 + 3*(10*B*b^5*d^2*e - (21*B*a*b^4 - A*b^5)*d*e^2 + (11*B*a ^2*b^3 - A*a*b^4)*e^3)*x^2 + 2*(6*B*b^5*d^3 + (5*B*a*b^4 + 7*A*b^5)*d^2*e - (31*B*a^2*b^3 + 11*A*a*b^4)*d*e^2 + 4*(5*B*a^3*b^2 + A*a^2*b^3)*e^3)*...
Timed out. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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. 377 vs. \(2 (185) = 370\).
Time = 0.29 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.80 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {{\left (6 \, B b d e^{2} - 5 \, B a e^{3} - A b e^{3}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, {\left (b^{4} d - a b^{3} e\right )} \sqrt {-b^{2} d + a b e}} - \frac {30 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{3} d e^{2} - 48 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{3} d^{2} e^{2} + 18 \, \sqrt {e x + d} B b^{3} d^{3} e^{2} - 33 \, {\left (e x + d\right )}^{\frac {5}{2}} B a b^{2} e^{3} + 3 \, {\left (e x + d\right )}^{\frac {5}{2}} A b^{3} e^{3} + 88 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{2} d e^{3} + 8 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{3} d e^{3} - 51 \, \sqrt {e x + d} B a b^{2} d^{2} e^{3} - 3 \, \sqrt {e x + d} A b^{3} d^{2} e^{3} - 40 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} b e^{4} - 8 \, {\left (e x + d\right )}^{\frac {3}{2}} A a b^{2} e^{4} + 48 \, \sqrt {e x + d} B a^{2} b d e^{4} + 6 \, \sqrt {e x + d} A a b^{2} d e^{4} - 15 \, \sqrt {e x + d} B a^{3} e^{5} - 3 \, \sqrt {e x + d} A a^{2} b e^{5}}{24 \, {\left (b^{4} d - a b^{3} e\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{3}} \]
1/8*(6*B*b*d*e^2 - 5*B*a*e^3 - A*b*e^3)*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^4*d - a*b^3*e)*sqrt(-b^2*d + a*b*e)) - 1/24*(30*(e*x + d)^( 5/2)*B*b^3*d*e^2 - 48*(e*x + d)^(3/2)*B*b^3*d^2*e^2 + 18*sqrt(e*x + d)*B*b ^3*d^3*e^2 - 33*(e*x + d)^(5/2)*B*a*b^2*e^3 + 3*(e*x + d)^(5/2)*A*b^3*e^3 + 88*(e*x + d)^(3/2)*B*a*b^2*d*e^3 + 8*(e*x + d)^(3/2)*A*b^3*d*e^3 - 51*sq rt(e*x + d)*B*a*b^2*d^2*e^3 - 3*sqrt(e*x + d)*A*b^3*d^2*e^3 - 40*(e*x + d) ^(3/2)*B*a^2*b*e^4 - 8*(e*x + d)^(3/2)*A*a*b^2*e^4 + 48*sqrt(e*x + d)*B*a^ 2*b*d*e^4 + 6*sqrt(e*x + d)*A*a*b^2*d*e^4 - 15*sqrt(e*x + d)*B*a^3*e^5 - 3 *sqrt(e*x + d)*A*a^2*b*e^5)/((b^4*d - a*b^3*e)*((e*x + d)*b - b*d + a*e)^3 )
Time = 10.86 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.56 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^2\,\sqrt {d+e\,x}\,\left (A\,b\,e+5\,B\,a\,e-6\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (A\,b\,e^3+5\,B\,a\,e^3-6\,B\,b\,d\,e^2\right )}\right )\,\left (A\,b\,e+5\,B\,a\,e-6\,B\,b\,d\right )}{8\,b^{7/2}\,{\left (a\,e-b\,d\right )}^{3/2}}-\frac {\frac {{\left (d+e\,x\right )}^{3/2}\,\left (A\,b\,e^3+5\,B\,a\,e^3-6\,B\,b\,d\,e^2\right )}{3\,b^2}+\frac {\left (a\,e-b\,d\right )\,\sqrt {d+e\,x}\,\left (A\,b\,e^3+5\,B\,a\,e^3-6\,B\,b\,d\,e^2\right )}{8\,b^3}-\frac {{\left (d+e\,x\right )}^{5/2}\,\left (A\,b\,e^3-11\,B\,a\,e^3+10\,B\,b\,d\,e^2\right )}{8\,b\,\left (a\,e-b\,d\right )}}{\left (d+e\,x\right )\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )+b^3\,{\left (d+e\,x\right )}^3-\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^2+a^3\,e^3-b^3\,d^3+3\,a\,b^2\,d^2\,e-3\,a^2\,b\,d\,e^2} \]
(e^2*atan((b^(1/2)*e^2*(d + e*x)^(1/2)*(A*b*e + 5*B*a*e - 6*B*b*d))/((a*e - b*d)^(1/2)*(A*b*e^3 + 5*B*a*e^3 - 6*B*b*d*e^2)))*(A*b*e + 5*B*a*e - 6*B* b*d))/(8*b^(7/2)*(a*e - b*d)^(3/2)) - (((d + e*x)^(3/2)*(A*b*e^3 + 5*B*a*e ^3 - 6*B*b*d*e^2))/(3*b^2) + ((a*e - b*d)*(d + e*x)^(1/2)*(A*b*e^3 + 5*B*a *e^3 - 6*B*b*d*e^2))/(8*b^3) - ((d + e*x)^(5/2)*(A*b*e^3 - 11*B*a*e^3 + 10 *B*b*d*e^2))/(8*b*(a*e - b*d)))/((d + e*x)*(3*b^3*d^2 + 3*a^2*b*e^2 - 6*a* b^2*d*e) + b^3*(d + e*x)^3 - (3*b^3*d - 3*a*b^2*e)*(d + e*x)^2 + a^3*e^3 - b^3*d^3 + 3*a*b^2*d^2*e - 3*a^2*b*d*e^2)