Integrand size = 33, antiderivative size = 214 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 (b d-a e)^4 (B d-A e)}{e^6 \sqrt {d+e x}}+\frac {2 (b d-a e)^3 (5 b B d-4 A b e-a B e) \sqrt {d+e x}}{e^6}-\frac {4 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) (d+e x)^{3/2}}{3 e^6}+\frac {4 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^{5/2}}{5 e^6}-\frac {2 b^3 (5 b B d-A b e-4 a B e) (d+e x)^{7/2}}{7 e^6}+\frac {2 b^4 B (d+e x)^{9/2}}{9 e^6} \]
-4/3*b*(-a*e+b*d)^2*(-3*A*b*e-2*B*a*e+5*B*b*d)*(e*x+d)^(3/2)/e^6+4/5*b^2*( -a*e+b*d)*(-2*A*b*e-3*B*a*e+5*B*b*d)*(e*x+d)^(5/2)/e^6-2/7*b^3*(-A*b*e-4*B *a*e+5*B*b*d)*(e*x+d)^(7/2)/e^6+2/9*b^4*B*(e*x+d)^(9/2)/e^6+2*(-a*e+b*d)^4 *(-A*e+B*d)/e^6/(e*x+d)^(1/2)+2*(-a*e+b*d)^3*(-4*A*b*e-B*a*e+5*B*b*d)*(e*x +d)^(1/2)/e^6
Time = 0.26 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.57 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {630 a^4 e^4 (2 B d-A e+B e x)+840 a^3 b e^3 \left (3 A e (2 d+e x)+B \left (-8 d^2-4 d e x+e^2 x^2\right )\right )+252 a^2 b^2 e^2 \left (5 A e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 B \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )-72 a b^3 e \left (-7 A e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+B \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )\right )+2 b^4 \left (9 A e \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+5 B \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )\right )}{315 e^6 \sqrt {d+e x}} \]
(630*a^4*e^4*(2*B*d - A*e + B*e*x) + 840*a^3*b*e^3*(3*A*e*(2*d + e*x) + B* (-8*d^2 - 4*d*e*x + e^2*x^2)) + 252*a^2*b^2*e^2*(5*A*e*(-8*d^2 - 4*d*e*x + e^2*x^2) + 3*B*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)) - 72*a*b^3*e *(-7*A*e*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + B*(128*d^4 + 64*d^ 3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4)) + 2*b^4*(9*A*e*(-128*d^ 4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3*x^3 + 5*e^4*x^4) + 5*B*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5 )))/(315*e^6*Sqrt[d + e*x])
Time = 0.36 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1184, 27, 86, 2009}
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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2 (A+B x)}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^4 (a+b x)^4 (A+B x)}{(d+e x)^{3/2}}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^4 (A+B x)}{(d+e x)^{3/2}}dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b^3 (d+e x)^{5/2} (4 a B e+A b e-5 b B d)}{e^5}-\frac {2 b^2 (d+e x)^{3/2} (b d-a e) (3 a B e+2 A b e-5 b B d)}{e^5}+\frac {2 b \sqrt {d+e x} (b d-a e)^2 (2 a B e+3 A b e-5 b B d)}{e^5}+\frac {(a e-b d)^3 (a B e+4 A b e-5 b B d)}{e^5 \sqrt {d+e x}}+\frac {(a e-b d)^4 (A e-B d)}{e^5 (d+e x)^{3/2}}+\frac {b^4 B (d+e x)^{7/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^3 (d+e x)^{7/2} (-4 a B e-A b e+5 b B d)}{7 e^6}+\frac {4 b^2 (d+e x)^{5/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{5 e^6}-\frac {4 b (d+e x)^{3/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{3 e^6}+\frac {2 \sqrt {d+e x} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{e^6}+\frac {2 (b d-a e)^4 (B d-A e)}{e^6 \sqrt {d+e x}}+\frac {2 b^4 B (d+e x)^{9/2}}{9 e^6}\) |
(2*(b*d - a*e)^4*(B*d - A*e))/(e^6*Sqrt[d + e*x]) + (2*(b*d - a*e)^3*(5*b* B*d - 4*A*b*e - a*B*e)*Sqrt[d + e*x])/e^6 - (4*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e)*(d + e*x)^(3/2))/(3*e^6) + (4*b^2*(b*d - a*e)*(5*b*B*d - 2*A*b*e - 3*a*B*e)*(d + e*x)^(5/2))/(5*e^6) - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^(7/2))/(7*e^6) + (2*b^4*B*(d + e*x)^(9/2))/(9*e^6)
3.18.97.3.1 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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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 = 0.31 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.31
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\left (-\frac {x^{4} \left (\frac {7 B x}{9}+A \right ) b^{4}}{7}-\frac {4 x^{3} \left (\frac {5 B x}{7}+A \right ) a \,b^{3}}{5}-2 \left (\frac {3 B x}{5}+A \right ) x^{2} a^{2} b^{2}-4 \left (\frac {B x}{3}+A \right ) x \,a^{3} b +a^{4} \left (-B x +A \right )\right ) e^{5}-8 \left (-\frac {\left (\frac {25 B x}{36}+A \right ) x^{3} b^{4}}{35}-\frac {\left (\frac {4 B x}{7}+A \right ) x^{2} a \,b^{3}}{5}-x \,a^{2} \left (\frac {3 B x}{10}+A \right ) b^{2}+a^{3} \left (A -\frac {2 B x}{3}\right ) b +\frac {B \,a^{4}}{4}\right ) d \,e^{4}+16 b \left (-\frac {\left (\frac {5 B x}{9}+A \right ) x^{2} b^{3}}{35}-\frac {2 \left (\frac {2 B x}{7}+A \right ) x a \,b^{2}}{5}+a^{2} \left (-\frac {3 B x}{5}+A \right ) b +\frac {2 B \,a^{3}}{3}\right ) d^{2} e^{3}-\frac {64 b^{2} d^{3} \left (-\frac {x \left (\frac {5 B x}{18}+A \right ) b^{2}}{7}+a \left (-\frac {4 B x}{7}+A \right ) b +\frac {3 B \,a^{2}}{2}\right ) e^{2}}{5}+\frac {128 b^{3} d^{4} \left (\left (-\frac {5 B x}{9}+A \right ) b +4 B a \right ) e}{35}-\frac {256 b^{4} B \,d^{5}}{63}\right )}{\sqrt {e x +d}\, e^{6}}\) | \(280\) |
| risch | \(\frac {2 \left (35 B \,b^{4} x^{4} e^{4}+45 A \,b^{4} e^{4} x^{3}+180 B a \,b^{3} e^{4} x^{3}-85 B \,b^{4} d \,e^{3} x^{3}+252 A a \,b^{3} e^{4} x^{2}-117 A \,b^{4} d \,e^{3} x^{2}+378 B \,a^{2} b^{2} e^{4} x^{2}-468 B a \,b^{3} d \,e^{3} x^{2}+165 B \,b^{4} d^{2} e^{2} x^{2}+630 A \,a^{2} b^{2} e^{4} x -756 A a \,b^{3} d \,e^{3} x +261 A \,b^{4} d^{2} e^{2} x +420 B \,a^{3} b \,e^{4} x -1134 B \,a^{2} b^{2} d \,e^{3} x +1044 B a \,b^{3} d^{2} e^{2} x -325 B \,b^{4} d^{3} e x +1260 A \,a^{3} b \,e^{4}-3150 A \,a^{2} b^{2} d \,e^{3}+2772 A a \,b^{3} d^{2} e^{2}-837 A \,b^{4} d^{3} e +315 B \,a^{4} e^{4}-2100 B \,a^{3} b d \,e^{3}+4158 B \,a^{2} b^{2} d^{2} e^{2}-3348 B a \,b^{3} d^{3} e +965 b^{4} B \,d^{4}\right ) \sqrt {e x +d}}{315 e^{6}}-\frac {2 \left (A \,a^{4} e^{5}-4 A \,a^{3} b d \,e^{4}+6 A \,a^{2} b^{2} d^{2} e^{3}-4 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e -B \,a^{4} d \,e^{4}+4 B \,a^{3} b \,d^{2} e^{3}-6 B \,a^{2} b^{2} d^{3} e^{2}+4 B a \,b^{3} d^{4} e -b^{4} B \,d^{5}\right )}{e^{6} \sqrt {e x +d}}\) | \(451\) |
| gosper | \(-\frac {2 \left (-35 B \,x^{5} b^{4} e^{5}-45 A \,b^{4} e^{5} x^{4}-180 B \,x^{4} a \,b^{3} e^{5}+50 B \,x^{4} b^{4} d \,e^{4}-252 A \,x^{3} a \,b^{3} e^{5}+72 A \,x^{3} b^{4} d \,e^{4}-378 B \,x^{3} a^{2} b^{2} e^{5}+288 B \,x^{3} a \,b^{3} d \,e^{4}-80 B \,x^{3} b^{4} d^{2} e^{3}-630 A \,x^{2} a^{2} b^{2} e^{5}+504 A \,x^{2} a \,b^{3} d \,e^{4}-144 A \,x^{2} b^{4} d^{2} e^{3}-420 B \,x^{2} a^{3} b \,e^{5}+756 B \,x^{2} a^{2} b^{2} d \,e^{4}-576 B \,x^{2} a \,b^{3} d^{2} e^{3}+160 B \,x^{2} b^{4} d^{3} e^{2}-1260 A x \,a^{3} b \,e^{5}+2520 A x \,a^{2} b^{2} d \,e^{4}-2016 A x a \,b^{3} d^{2} e^{3}+576 A x \,b^{4} d^{3} e^{2}-315 B x \,a^{4} e^{5}+1680 B x \,a^{3} b d \,e^{4}-3024 B x \,a^{2} b^{2} d^{2} e^{3}+2304 B x a \,b^{3} d^{3} e^{2}-640 B x \,b^{4} d^{4} e +315 A \,a^{4} e^{5}-2520 A \,a^{3} b d \,e^{4}+5040 A \,a^{2} b^{2} d^{2} e^{3}-4032 A a \,b^{3} d^{3} e^{2}+1152 A \,b^{4} d^{4} e -630 B \,a^{4} d \,e^{4}+3360 B \,a^{3} b \,d^{2} e^{3}-6048 B \,a^{2} b^{2} d^{3} e^{2}+4608 B a \,b^{3} d^{4} e -1280 b^{4} B \,d^{5}\right )}{315 \sqrt {e x +d}\, e^{6}}\) | \(469\) |
| trager | \(-\frac {2 \left (-35 B \,x^{5} b^{4} e^{5}-45 A \,b^{4} e^{5} x^{4}-180 B \,x^{4} a \,b^{3} e^{5}+50 B \,x^{4} b^{4} d \,e^{4}-252 A \,x^{3} a \,b^{3} e^{5}+72 A \,x^{3} b^{4} d \,e^{4}-378 B \,x^{3} a^{2} b^{2} e^{5}+288 B \,x^{3} a \,b^{3} d \,e^{4}-80 B \,x^{3} b^{4} d^{2} e^{3}-630 A \,x^{2} a^{2} b^{2} e^{5}+504 A \,x^{2} a \,b^{3} d \,e^{4}-144 A \,x^{2} b^{4} d^{2} e^{3}-420 B \,x^{2} a^{3} b \,e^{5}+756 B \,x^{2} a^{2} b^{2} d \,e^{4}-576 B \,x^{2} a \,b^{3} d^{2} e^{3}+160 B \,x^{2} b^{4} d^{3} e^{2}-1260 A x \,a^{3} b \,e^{5}+2520 A x \,a^{2} b^{2} d \,e^{4}-2016 A x a \,b^{3} d^{2} e^{3}+576 A x \,b^{4} d^{3} e^{2}-315 B x \,a^{4} e^{5}+1680 B x \,a^{3} b d \,e^{4}-3024 B x \,a^{2} b^{2} d^{2} e^{3}+2304 B x a \,b^{3} d^{3} e^{2}-640 B x \,b^{4} d^{4} e +315 A \,a^{4} e^{5}-2520 A \,a^{3} b d \,e^{4}+5040 A \,a^{2} b^{2} d^{2} e^{3}-4032 A a \,b^{3} d^{3} e^{2}+1152 A \,b^{4} d^{4} e -630 B \,a^{4} d \,e^{4}+3360 B \,a^{3} b \,d^{2} e^{3}-6048 B \,a^{2} b^{2} d^{3} e^{2}+4608 B a \,b^{3} d^{4} e -1280 b^{4} B \,d^{5}\right )}{315 \sqrt {e x +d}\, e^{6}}\) | \(469\) |
| derivativedivides | \(\frac {\frac {2 b^{4} B \left (e x +d \right )^{\frac {9}{2}}}{9}-16 B \,a^{3} b d \,e^{3} \sqrt {e x +d}+36 B \,a^{2} b^{2} d^{2} e^{2} \sqrt {e x +d}+24 A a \,b^{3} d^{2} e^{2} \sqrt {e x +d}-12 B \,a^{2} b^{2} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {32 B a \,b^{3} d e \left (e x +d \right )^{\frac {5}{2}}}{5}-24 A \,a^{2} b^{2} d \,e^{3} \sqrt {e x +d}-32 B a \,b^{3} d^{3} e \sqrt {e x +d}+16 B a \,b^{3} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-8 A a \,b^{3} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {2 \left (A \,a^{4} e^{5}-4 A \,a^{3} b d \,e^{4}+6 A \,a^{2} b^{2} d^{2} e^{3}-4 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e -B \,a^{4} d \,e^{4}+4 B \,a^{3} b \,d^{2} e^{3}-6 B \,a^{2} b^{2} d^{3} e^{2}+4 B a \,b^{3} d^{4} e -b^{4} B \,d^{5}\right )}{\sqrt {e x +d}}-\frac {10 B \,b^{4} d \left (e x +d \right )^{\frac {7}{2}}}{7}+4 B \,b^{4} d^{2} \left (e x +d \right )^{\frac {5}{2}}-\frac {20 B \,b^{4} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 B \,a^{4} e^{4} \sqrt {e x +d}+10 B \,b^{4} d^{4} \sqrt {e x +d}+\frac {2 A \,b^{4} e \left (e x +d \right )^{\frac {7}{2}}}{7}-8 A \,b^{4} d^{3} e \sqrt {e x +d}+8 A \,a^{3} b \,e^{4} \sqrt {e x +d}+\frac {8 B a \,b^{3} e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {8 A a \,b^{3} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 A \,b^{4} d e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {12 B \,a^{2} b^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+4 A \,a^{2} b^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}+4 A \,b^{4} d^{2} e \left (e x +d \right )^{\frac {3}{2}}+\frac {8 B \,a^{3} b \,e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{6}}\) | \(558\) |
| default | \(\frac {\frac {2 b^{4} B \left (e x +d \right )^{\frac {9}{2}}}{9}-16 B \,a^{3} b d \,e^{3} \sqrt {e x +d}+36 B \,a^{2} b^{2} d^{2} e^{2} \sqrt {e x +d}+24 A a \,b^{3} d^{2} e^{2} \sqrt {e x +d}-12 B \,a^{2} b^{2} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {32 B a \,b^{3} d e \left (e x +d \right )^{\frac {5}{2}}}{5}-24 A \,a^{2} b^{2} d \,e^{3} \sqrt {e x +d}-32 B a \,b^{3} d^{3} e \sqrt {e x +d}+16 B a \,b^{3} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-8 A a \,b^{3} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {2 \left (A \,a^{4} e^{5}-4 A \,a^{3} b d \,e^{4}+6 A \,a^{2} b^{2} d^{2} e^{3}-4 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e -B \,a^{4} d \,e^{4}+4 B \,a^{3} b \,d^{2} e^{3}-6 B \,a^{2} b^{2} d^{3} e^{2}+4 B a \,b^{3} d^{4} e -b^{4} B \,d^{5}\right )}{\sqrt {e x +d}}-\frac {10 B \,b^{4} d \left (e x +d \right )^{\frac {7}{2}}}{7}+4 B \,b^{4} d^{2} \left (e x +d \right )^{\frac {5}{2}}-\frac {20 B \,b^{4} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 B \,a^{4} e^{4} \sqrt {e x +d}+10 B \,b^{4} d^{4} \sqrt {e x +d}+\frac {2 A \,b^{4} e \left (e x +d \right )^{\frac {7}{2}}}{7}-8 A \,b^{4} d^{3} e \sqrt {e x +d}+8 A \,a^{3} b \,e^{4} \sqrt {e x +d}+\frac {8 B a \,b^{3} e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {8 A a \,b^{3} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 A \,b^{4} d e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {12 B \,a^{2} b^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+4 A \,a^{2} b^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}+4 A \,b^{4} d^{2} e \left (e x +d \right )^{\frac {3}{2}}+\frac {8 B \,a^{3} b \,e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{6}}\) | \(558\) |
-2*((-1/7*x^4*(7/9*B*x+A)*b^4-4/5*x^3*(5/7*B*x+A)*a*b^3-2*(3/5*B*x+A)*x^2* a^2*b^2-4*(1/3*B*x+A)*x*a^3*b+a^4*(-B*x+A))*e^5-8*(-1/35*(25/36*B*x+A)*x^3 *b^4-1/5*(4/7*B*x+A)*x^2*a*b^3-x*a^2*(3/10*B*x+A)*b^2+a^3*(A-2/3*B*x)*b+1/ 4*B*a^4)*d*e^4+16*b*(-1/35*(5/9*B*x+A)*x^2*b^3-2/5*(2/7*B*x+A)*x*a*b^2+a^2 *(-3/5*B*x+A)*b+2/3*B*a^3)*d^2*e^3-64/5*b^2*d^3*(-1/7*x*(5/18*B*x+A)*b^2+a *(-4/7*B*x+A)*b+3/2*B*a^2)*e^2+128/35*b^3*d^4*((-5/9*B*x+A)*b+4*B*a)*e-256 /63*b^4*B*d^5)/(e*x+d)^(1/2)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (194) = 388\).
Time = 0.31 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.95 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (35 \, B b^{4} e^{5} x^{5} + 1280 \, B b^{4} d^{5} - 315 \, A a^{4} e^{5} - 1152 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2016 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 1680 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 630 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 5 \, {\left (10 \, B b^{4} d e^{4} - 9 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 2 \, {\left (40 \, B b^{4} d^{2} e^{3} - 36 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 63 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} - 2 \, {\left (80 \, B b^{4} d^{3} e^{2} - 72 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 126 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} - 105 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + {\left (640 \, B b^{4} d^{4} e - 576 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 1008 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 840 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 315 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x\right )} \sqrt {e x + d}}{315 \, {\left (e^{7} x + d e^{6}\right )}} \]
2/315*(35*B*b^4*e^5*x^5 + 1280*B*b^4*d^5 - 315*A*a^4*e^5 - 1152*(4*B*a*b^3 + A*b^4)*d^4*e + 2016*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 1680*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + 630*(B*a^4 + 4*A*a^3*b)*d*e^4 - 5*(10*B*b^4*d*e^ 4 - 9*(4*B*a*b^3 + A*b^4)*e^5)*x^4 + 2*(40*B*b^4*d^2*e^3 - 36*(4*B*a*b^3 + A*b^4)*d*e^4 + 63*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 - 2*(80*B*b^4*d^3*e^ 2 - 72*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 126*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 - 105*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + (640*B*b^4*d^4*e - 576*(4*B*a*b^ 3 + A*b^4)*d^3*e^2 + 1008*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 - 840*(2*B*a^3 *b + 3*A*a^2*b^2)*d*e^4 + 315*(B*a^4 + 4*A*a^3*b)*e^5)*x)*sqrt(e*x + d)/(e ^7*x + d*e^6)
Leaf count of result is larger than twice the leaf count of optimal. 507 vs. \(2 (218) = 436\).
Time = 21.61 (sec) , antiderivative size = 507, normalized size of antiderivative = 2.37 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {B b^{4} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A b^{4} e + 4 B a b^{3} e - 5 B b^{4} d\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (4 A a b^{3} e^{2} - 4 A b^{4} d e + 6 B a^{2} b^{2} e^{2} - 16 B a b^{3} d e + 10 B b^{4} d^{2}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (6 A a^{2} b^{2} e^{3} - 12 A a b^{3} d e^{2} + 6 A b^{4} d^{2} e + 4 B a^{3} b e^{3} - 18 B a^{2} b^{2} d e^{2} + 24 B a b^{3} d^{2} e - 10 B b^{4} d^{3}\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (4 A a^{3} b e^{4} - 12 A a^{2} b^{2} d e^{3} + 12 A a b^{3} d^{2} e^{2} - 4 A b^{4} d^{3} e + B a^{4} e^{4} - 8 B a^{3} b d e^{3} + 18 B a^{2} b^{2} d^{2} e^{2} - 16 B a b^{3} d^{3} e + 5 B b^{4} d^{4}\right )}{e^{5}} + \frac {\left (- A e + B d\right ) \left (a e - b d\right )^{4}}{e^{5} \sqrt {d + e x}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {A a^{4} x + \frac {B b^{4} x^{6}}{6} + \frac {x^{5} \left (A b^{4} + 4 B a b^{3}\right )}{5} + \frac {x^{4} \cdot \left (4 A a b^{3} + 6 B a^{2} b^{2}\right )}{4} + \frac {x^{3} \cdot \left (6 A a^{2} b^{2} + 4 B a^{3} b\right )}{3} + \frac {x^{2} \cdot \left (4 A a^{3} b + B a^{4}\right )}{2}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(B*b**4*(d + e*x)**(9/2)/(9*e**5) + (d + e*x)**(7/2)*(A*b**4* e + 4*B*a*b**3*e - 5*B*b**4*d)/(7*e**5) + (d + e*x)**(5/2)*(4*A*a*b**3*e** 2 - 4*A*b**4*d*e + 6*B*a**2*b**2*e**2 - 16*B*a*b**3*d*e + 10*B*b**4*d**2)/ (5*e**5) + (d + e*x)**(3/2)*(6*A*a**2*b**2*e**3 - 12*A*a*b**3*d*e**2 + 6*A *b**4*d**2*e + 4*B*a**3*b*e**3 - 18*B*a**2*b**2*d*e**2 + 24*B*a*b**3*d**2* e - 10*B*b**4*d**3)/(3*e**5) + sqrt(d + e*x)*(4*A*a**3*b*e**4 - 12*A*a**2* b**2*d*e**3 + 12*A*a*b**3*d**2*e**2 - 4*A*b**4*d**3*e + B*a**4*e**4 - 8*B* a**3*b*d*e**3 + 18*B*a**2*b**2*d**2*e**2 - 16*B*a*b**3*d**3*e + 5*B*b**4*d **4)/e**5 + (-A*e + B*d)*(a*e - b*d)**4/(e**5*sqrt(d + e*x)))/e, Ne(e, 0)) , ((A*a**4*x + B*b**4*x**6/6 + x**5*(A*b**4 + 4*B*a*b**3)/5 + x**4*(4*A*a* b**3 + 6*B*a**2*b**2)/4 + x**3*(6*A*a**2*b**2 + 4*B*a**3*b)/3 + x**2*(4*A* a**3*b + B*a**4)/2)/d**(3/2), True))
Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (194) = 388\).
Time = 0.20 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.95 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {35 \, {\left (e x + d\right )}^{\frac {9}{2}} B b^{4} - 45 \, {\left (5 \, B b^{4} d - {\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 126 \, {\left (5 \, B b^{4} d^{2} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 210 \, {\left (5 \, B b^{4} d^{3} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} - {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 315 \, {\left (5 \, B b^{4} d^{4} - 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )} \sqrt {e x + d}}{e^{5}} + \frac {315 \, {\left (B b^{4} d^{5} - A a^{4} e^{5} - {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}\right )}}{\sqrt {e x + d} e^{5}}\right )}}{315 \, e} \]
2/315*((35*(e*x + d)^(9/2)*B*b^4 - 45*(5*B*b^4*d - (4*B*a*b^3 + A*b^4)*e)* (e*x + d)^(7/2) + 126*(5*B*b^4*d^2 - 2*(4*B*a*b^3 + A*b^4)*d*e + (3*B*a^2* b^2 + 2*A*a*b^3)*e^2)*(e*x + d)^(5/2) - 210*(5*B*b^4*d^3 - 3*(4*B*a*b^3 + A*b^4)*d^2*e + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^2 - (2*B*a^3*b + 3*A*a^2*b^ 2)*e^3)*(e*x + d)^(3/2) + 315*(5*B*b^4*d^4 - 4*(4*B*a*b^3 + A*b^4)*d^3*e + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^2 - 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^3 + (B*a^4 + 4*A*a^3*b)*e^4)*sqrt(e*x + d))/e^5 + 315*(B*b^4*d^5 - A*a^4*e^5 - (4*B*a*b^3 + A*b^4)*d^4*e + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 2*(2*B *a^3*b + 3*A*a^2*b^2)*d^2*e^3 + (B*a^4 + 4*A*a^3*b)*d*e^4)/(sqrt(e*x + d)* e^5))/e
Leaf count of result is larger than twice the leaf count of optimal. 593 vs. \(2 (194) = 388\).
Time = 0.31 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.77 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (B b^{4} d^{5} - 4 \, B a b^{3} d^{4} e - A b^{4} d^{4} e + 6 \, B a^{2} b^{2} d^{3} e^{2} + 4 \, A a b^{3} d^{3} e^{2} - 4 \, B a^{3} b d^{2} e^{3} - 6 \, A a^{2} b^{2} d^{2} e^{3} + B a^{4} d e^{4} + 4 \, A a^{3} b d e^{4} - A a^{4} e^{5}\right )}}{\sqrt {e x + d} e^{6}} + \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} B b^{4} e^{48} - 225 \, {\left (e x + d\right )}^{\frac {7}{2}} B b^{4} d e^{48} + 630 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{4} d^{2} e^{48} - 1050 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{4} d^{3} e^{48} + 1575 \, \sqrt {e x + d} B b^{4} d^{4} e^{48} + 180 \, {\left (e x + d\right )}^{\frac {7}{2}} B a b^{3} e^{49} + 45 \, {\left (e x + d\right )}^{\frac {7}{2}} A b^{4} e^{49} - 1008 \, {\left (e x + d\right )}^{\frac {5}{2}} B a b^{3} d e^{49} - 252 \, {\left (e x + d\right )}^{\frac {5}{2}} A b^{4} d e^{49} + 2520 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{3} d^{2} e^{49} + 630 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{4} d^{2} e^{49} - 5040 \, \sqrt {e x + d} B a b^{3} d^{3} e^{49} - 1260 \, \sqrt {e x + d} A b^{4} d^{3} e^{49} + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{2} b^{2} e^{50} + 252 \, {\left (e x + d\right )}^{\frac {5}{2}} A a b^{3} e^{50} - 1890 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} b^{2} d e^{50} - 1260 \, {\left (e x + d\right )}^{\frac {3}{2}} A a b^{3} d e^{50} + 5670 \, \sqrt {e x + d} B a^{2} b^{2} d^{2} e^{50} + 3780 \, \sqrt {e x + d} A a b^{3} d^{2} e^{50} + 420 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{3} b e^{51} + 630 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{2} b^{2} e^{51} - 2520 \, \sqrt {e x + d} B a^{3} b d e^{51} - 3780 \, \sqrt {e x + d} A a^{2} b^{2} d e^{51} + 315 \, \sqrt {e x + d} B a^{4} e^{52} + 1260 \, \sqrt {e x + d} A a^{3} b e^{52}\right )}}{315 \, e^{54}} \]
2*(B*b^4*d^5 - 4*B*a*b^3*d^4*e - A*b^4*d^4*e + 6*B*a^2*b^2*d^3*e^2 + 4*A*a *b^3*d^3*e^2 - 4*B*a^3*b*d^2*e^3 - 6*A*a^2*b^2*d^2*e^3 + B*a^4*d*e^4 + 4*A *a^3*b*d*e^4 - A*a^4*e^5)/(sqrt(e*x + d)*e^6) + 2/315*(35*(e*x + d)^(9/2)* B*b^4*e^48 - 225*(e*x + d)^(7/2)*B*b^4*d*e^48 + 630*(e*x + d)^(5/2)*B*b^4* d^2*e^48 - 1050*(e*x + d)^(3/2)*B*b^4*d^3*e^48 + 1575*sqrt(e*x + d)*B*b^4* d^4*e^48 + 180*(e*x + d)^(7/2)*B*a*b^3*e^49 + 45*(e*x + d)^(7/2)*A*b^4*e^4 9 - 1008*(e*x + d)^(5/2)*B*a*b^3*d*e^49 - 252*(e*x + d)^(5/2)*A*b^4*d*e^49 + 2520*(e*x + d)^(3/2)*B*a*b^3*d^2*e^49 + 630*(e*x + d)^(3/2)*A*b^4*d^2*e ^49 - 5040*sqrt(e*x + d)*B*a*b^3*d^3*e^49 - 1260*sqrt(e*x + d)*A*b^4*d^3*e ^49 + 378*(e*x + d)^(5/2)*B*a^2*b^2*e^50 + 252*(e*x + d)^(5/2)*A*a*b^3*e^5 0 - 1890*(e*x + d)^(3/2)*B*a^2*b^2*d*e^50 - 1260*(e*x + d)^(3/2)*A*a*b^3*d *e^50 + 5670*sqrt(e*x + d)*B*a^2*b^2*d^2*e^50 + 3780*sqrt(e*x + d)*A*a*b^3 *d^2*e^50 + 420*(e*x + d)^(3/2)*B*a^3*b*e^51 + 630*(e*x + d)^(3/2)*A*a^2*b ^2*e^51 - 2520*sqrt(e*x + d)*B*a^3*b*d*e^51 - 3780*sqrt(e*x + d)*A*a^2*b^2 *d*e^51 + 315*sqrt(e*x + d)*B*a^4*e^52 + 1260*sqrt(e*x + d)*A*a^3*b*e^52)/ e^54
Time = 10.87 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.38 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b^4\,e-10\,B\,b^4\,d+8\,B\,a\,b^3\,e\right )}{7\,e^6}-\frac {-2\,B\,a^4\,d\,e^4+2\,A\,a^4\,e^5+8\,B\,a^3\,b\,d^2\,e^3-8\,A\,a^3\,b\,d\,e^4-12\,B\,a^2\,b^2\,d^3\,e^2+12\,A\,a^2\,b^2\,d^2\,e^3+8\,B\,a\,b^3\,d^4\,e-8\,A\,a\,b^3\,d^3\,e^2-2\,B\,b^4\,d^5+2\,A\,b^4\,d^4\,e}{e^6\,\sqrt {d+e\,x}}+\frac {2\,{\left (a\,e-b\,d\right )}^3\,\sqrt {d+e\,x}\,\left (4\,A\,b\,e+B\,a\,e-5\,B\,b\,d\right )}{e^6}+\frac {2\,B\,b^4\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}+\frac {4\,b\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}\,\left (3\,A\,b\,e+2\,B\,a\,e-5\,B\,b\,d\right )}{3\,e^6}+\frac {4\,b^2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,b\,e+3\,B\,a\,e-5\,B\,b\,d\right )}{5\,e^6} \]
((d + e*x)^(7/2)*(2*A*b^4*e - 10*B*b^4*d + 8*B*a*b^3*e))/(7*e^6) - (2*A*a^ 4*e^5 - 2*B*b^4*d^5 + 2*A*b^4*d^4*e - 2*B*a^4*d*e^4 - 8*A*a*b^3*d^3*e^2 + 8*B*a^3*b*d^2*e^3 + 12*A*a^2*b^2*d^2*e^3 - 12*B*a^2*b^2*d^3*e^2 - 8*A*a^3* b*d*e^4 + 8*B*a*b^3*d^4*e)/(e^6*(d + e*x)^(1/2)) + (2*(a*e - b*d)^3*(d + e *x)^(1/2)*(4*A*b*e + B*a*e - 5*B*b*d))/e^6 + (2*B*b^4*(d + e*x)^(9/2))/(9* e^6) + (4*b*(a*e - b*d)^2*(d + e*x)^(3/2)*(3*A*b*e + 2*B*a*e - 5*B*b*d))/( 3*e^6) + (4*b^2*(a*e - b*d)*(d + e*x)^(5/2)*(2*A*b*e + 3*B*a*e - 5*B*b*d)) /(5*e^6)