Integrand size = 33, antiderivative size = 324 \[ \int (A+B x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(A b-a B) (b d-a e)^4 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^6}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^6}+\frac {e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^6}+\frac {2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^6}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{10 b^6}+\frac {B e^4 (a+b x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{11 b^6} \]
1/6*(A*b-B*a)*(-a*e+b*d)^4*(b*x+a)^5*((b*x+a)^2)^(1/2)/b^6+1/7*(-a*e+b*d)^ 3*(4*A*b*e-5*B*a*e+B*b*d)*(b*x+a)^6*((b*x+a)^2)^(1/2)/b^6+1/4*e*(-a*e+b*d) ^2*(3*A*b*e-5*B*a*e+2*B*b*d)*(b*x+a)^7*((b*x+a)^2)^(1/2)/b^6+2/9*e^2*(-a*e +b*d)*(2*A*b*e-5*B*a*e+3*B*b*d)*(b*x+a)^8*((b*x+a)^2)^(1/2)/b^6+1/10*e^3*( A*b*e-5*B*a*e+4*B*b*d)*(b*x+a)^9*((b*x+a)^2)^(1/2)/b^6+1/11*B*e^4*(b*x+a)^ 10*((b*x+a)^2)^(1/2)/b^6
Time = 1.16 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.89 \[ \int (A+B x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (462 a^5 \left (6 A \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+B x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )\right )+330 a^4 b x \left (7 A \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+2 B x \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )\right )+165 a^3 b^2 x^2 \left (8 A \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+3 B x \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )\right )+55 a^2 b^3 x^3 \left (9 A \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )+4 B x \left (126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right )\right )+55 a b^4 x^4 \left (2 A \left (126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right )+B x \left (210 d^4+720 d^3 e x+945 d^2 e^2 x^2+560 d e^3 x^3+126 e^4 x^4\right )\right )+b^5 x^5 \left (11 A \left (210 d^4+720 d^3 e x+945 d^2 e^2 x^2+560 d e^3 x^3+126 e^4 x^4\right )+6 B x \left (330 d^4+1155 d^3 e x+1540 d^2 e^2 x^2+924 d e^3 x^3+210 e^4 x^4\right )\right )\right )}{13860 (a+b x)} \]
(x*Sqrt[(a + b*x)^2]*(462*a^5*(6*A*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + B*x*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d* e^3*x^3 + 5*e^4*x^4)) + 330*a^4*b*x*(7*A*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2 *x^2 + 24*d*e^3*x^3 + 5*e^4*x^4) + 2*B*x*(35*d^4 + 105*d^3*e*x + 126*d^2*e ^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4)) + 165*a^3*b^2*x^2*(8*A*(35*d^4 + 105* d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4) + 3*B*x*(70*d^4 + 2 24*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^4)) + 55*a^2*b^3*x ^3*(9*A*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x ^4) + 4*B*x*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70* e^4*x^4)) + 55*a*b^4*x^4*(2*A*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 3 15*d*e^3*x^3 + 70*e^4*x^4) + B*x*(210*d^4 + 720*d^3*e*x + 945*d^2*e^2*x^2 + 560*d*e^3*x^3 + 126*e^4*x^4)) + b^5*x^5*(11*A*(210*d^4 + 720*d^3*e*x + 9 45*d^2*e^2*x^2 + 560*d*e^3*x^3 + 126*e^4*x^4) + 6*B*x*(330*d^4 + 1155*d^3* e*x + 1540*d^2*e^2*x^2 + 924*d*e^3*x^3 + 210*e^4*x^4))))/(13860*(a + b*x))
Time = 0.75 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.72, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x) (d+e x)^4 \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 (a+b x)^5 (A+B x) (d+e x)^4dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^5 (A+B x) (d+e x)^4dx}{a+b x}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {B e^4 (a+b x)^{10}}{b^5}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x)^9}{b^5}+\frac {2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)^8}{b^5}+\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) (a+b x)^7}{b^5}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e) (a+b x)^6}{b^5}+\frac {(A b-a B) (b d-a e)^4 (a+b x)^5}{b^5}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {e^3 (a+b x)^{10} (-5 a B e+A b e+4 b B d)}{10 b^6}+\frac {2 e^2 (a+b x)^9 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{9 b^6}+\frac {e (a+b x)^8 (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{4 b^6}+\frac {(a+b x)^7 (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{7 b^6}+\frac {(a+b x)^6 (A b-a B) (b d-a e)^4}{6 b^6}+\frac {B e^4 (a+b x)^{11}}{11 b^6}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((A*b - a*B)*(b*d - a*e)^4*(a + b*x)^6)/(6 *b^6) + ((b*d - a*e)^3*(b*B*d + 4*A*b*e - 5*a*B*e)*(a + b*x)^7)/(7*b^6) + (e*(b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5*a*B*e)*(a + b*x)^8)/(4*b^6) + (2*e ^2*(b*d - a*e)*(3*b*B*d + 2*A*b*e - 5*a*B*e)*(a + b*x)^9)/(9*b^6) + (e^3*( 4*b*B*d + A*b*e - 5*a*B*e)*(a + b*x)^10)/(10*b^6) + (B*e^4*(a + b*x)^11)/( 11*b^6)))/(a + b*x)
3.18.40.3.1 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[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[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, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(860\) vs. \(2(246)=492\).
Time = 0.62 (sec) , antiderivative size = 861, normalized size of antiderivative = 2.66
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{5} B \,e^{4} x^{11}}{11 b x +11 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (A \,e^{4}+4 B \,e^{3} d \right ) b^{5}+5 B \,e^{4} b^{4} a \right ) x^{10}}{10 b x +10 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (4 A \,e^{3} d +6 B \,d^{2} e^{2}\right ) b^{5}+5 \left (A \,e^{4}+4 B \,e^{3} d \right ) b^{4} a +10 B \,e^{4} a^{2} b^{3}\right ) x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) b^{5}+5 \left (4 A \,e^{3} d +6 B \,d^{2} e^{2}\right ) b^{4} a +10 \left (A \,e^{4}+4 B \,e^{3} d \right ) a^{2} b^{3}+10 B \,e^{4} a^{3} b^{2}\right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (4 A \,d^{3} e +B \,d^{4}\right ) b^{5}+5 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) b^{4} a +10 \left (4 A \,e^{3} d +6 B \,d^{2} e^{2}\right ) a^{2} b^{3}+10 \left (A \,e^{4}+4 B \,e^{3} d \right ) a^{3} b^{2}+5 B \,e^{4} a^{4} b \right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A \,b^{5} d^{4}+5 \left (4 A \,d^{3} e +B \,d^{4}\right ) b^{4} a +10 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a^{2} b^{3}+10 \left (4 A \,e^{3} d +6 B \,d^{2} e^{2}\right ) a^{3} b^{2}+5 \left (A \,e^{4}+4 B \,e^{3} d \right ) a^{4} b +B \,a^{5} e^{4}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A \,d^{4} b^{4} a +10 \left (4 A \,d^{3} e +B \,d^{4}\right ) a^{2} b^{3}+10 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a^{3} b^{2}+5 \left (4 A \,e^{3} d +6 B \,d^{2} e^{2}\right ) a^{4} b +\left (A \,e^{4}+4 B \,e^{3} d \right ) a^{5}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 A \,d^{4} a^{2} b^{3}+10 \left (4 A \,d^{3} e +B \,d^{4}\right ) a^{3} b^{2}+5 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a^{4} b +\left (4 A \,e^{3} d +6 B \,d^{2} e^{2}\right ) a^{5}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 A \,d^{4} a^{3} b^{2}+5 \left (4 A \,d^{3} e +B \,d^{4}\right ) a^{4} b +\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a^{5}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A \,d^{4} a^{4} b +\left (4 A \,d^{3} e +B \,d^{4}\right ) a^{5}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, A \,d^{4} a^{5} x}{b x +a}\) | \(861\) |
| gosper | \(\frac {x \left (1260 b^{5} B \,e^{4} x^{10}+1386 x^{9} A \,b^{5} e^{4}+6930 x^{9} B \,e^{4} b^{4} a +5544 x^{9} B \,b^{5} d \,e^{3}+7700 x^{8} A a \,b^{4} e^{4}+6160 x^{8} A \,b^{5} d \,e^{3}+15400 x^{8} B \,e^{4} a^{2} b^{3}+30800 x^{8} B a \,b^{4} d \,e^{3}+9240 x^{8} B \,b^{5} d^{2} e^{2}+17325 x^{7} A \,a^{2} b^{3} e^{4}+34650 x^{7} A a \,b^{4} d \,e^{3}+10395 x^{7} A \,b^{5} d^{2} e^{2}+17325 x^{7} B \,e^{4} a^{3} b^{2}+69300 x^{7} B \,a^{2} b^{3} d \,e^{3}+51975 x^{7} B a \,b^{4} d^{2} e^{2}+6930 x^{7} B \,b^{5} d^{3} e +19800 x^{6} A \,a^{3} b^{2} e^{4}+79200 x^{6} A \,a^{2} b^{3} d \,e^{3}+59400 x^{6} A a \,b^{4} d^{2} e^{2}+7920 x^{6} A \,b^{5} d^{3} e +9900 x^{6} B \,e^{4} a^{4} b +79200 x^{6} B \,a^{3} b^{2} d \,e^{3}+118800 x^{6} B \,a^{2} b^{3} d^{2} e^{2}+39600 x^{6} B a \,b^{4} d^{3} e +1980 x^{6} B \,b^{5} d^{4}+11550 x^{5} A \,a^{4} b \,e^{4}+92400 x^{5} A \,a^{3} b^{2} d \,e^{3}+138600 x^{5} A \,a^{2} b^{3} d^{2} e^{2}+46200 x^{5} A \,b^{4} d^{3} e a +2310 x^{5} A \,b^{5} d^{4}+2310 x^{5} B \,a^{5} e^{4}+46200 x^{5} B \,a^{4} b d \,e^{3}+138600 x^{5} B \,a^{3} b^{2} d^{2} e^{2}+92400 x^{5} B \,a^{2} b^{3} d^{3} e +11550 x^{5} B \,b^{4} d^{4} a +2772 x^{4} A \,a^{5} e^{4}+55440 x^{4} A \,a^{4} b d \,e^{3}+166320 x^{4} A \,a^{3} b^{2} d^{2} e^{2}+110880 x^{4} A \,a^{2} b^{3} d^{3} e +13860 x^{4} A \,d^{4} b^{4} a +11088 x^{4} B \,a^{5} d \,e^{3}+83160 x^{4} B \,a^{4} b \,d^{2} e^{2}+110880 x^{4} B \,a^{3} b^{2} d^{3} e +27720 x^{4} B \,a^{2} b^{3} d^{4}+13860 x^{3} A \,a^{5} d \,e^{3}+103950 x^{3} A \,a^{4} b \,d^{2} e^{2}+138600 x^{3} A \,a^{3} b^{2} d^{3} e +34650 x^{3} A \,d^{4} a^{2} b^{3}+20790 x^{3} B \,a^{5} d^{2} e^{2}+69300 x^{3} B \,a^{4} b \,d^{3} e +34650 x^{3} B \,a^{3} b^{2} d^{4}+27720 x^{2} A \,a^{5} d^{2} e^{2}+92400 x^{2} A \,a^{4} b \,d^{3} e +46200 x^{2} A \,d^{4} a^{3} b^{2}+18480 x^{2} B \,a^{5} d^{3} e +23100 x^{2} B \,a^{4} b \,d^{4}+27720 x A \,a^{5} d^{3} e +34650 x A \,d^{4} a^{4} b +6930 x B \,a^{5} d^{4}+13860 A \,d^{4} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{13860 \left (b x +a \right )^{5}}\) | \(872\) |
| default | \(\frac {x \left (1260 b^{5} B \,e^{4} x^{10}+1386 x^{9} A \,b^{5} e^{4}+6930 x^{9} B \,e^{4} b^{4} a +5544 x^{9} B \,b^{5} d \,e^{3}+7700 x^{8} A a \,b^{4} e^{4}+6160 x^{8} A \,b^{5} d \,e^{3}+15400 x^{8} B \,e^{4} a^{2} b^{3}+30800 x^{8} B a \,b^{4} d \,e^{3}+9240 x^{8} B \,b^{5} d^{2} e^{2}+17325 x^{7} A \,a^{2} b^{3} e^{4}+34650 x^{7} A a \,b^{4} d \,e^{3}+10395 x^{7} A \,b^{5} d^{2} e^{2}+17325 x^{7} B \,e^{4} a^{3} b^{2}+69300 x^{7} B \,a^{2} b^{3} d \,e^{3}+51975 x^{7} B a \,b^{4} d^{2} e^{2}+6930 x^{7} B \,b^{5} d^{3} e +19800 x^{6} A \,a^{3} b^{2} e^{4}+79200 x^{6} A \,a^{2} b^{3} d \,e^{3}+59400 x^{6} A a \,b^{4} d^{2} e^{2}+7920 x^{6} A \,b^{5} d^{3} e +9900 x^{6} B \,e^{4} a^{4} b +79200 x^{6} B \,a^{3} b^{2} d \,e^{3}+118800 x^{6} B \,a^{2} b^{3} d^{2} e^{2}+39600 x^{6} B a \,b^{4} d^{3} e +1980 x^{6} B \,b^{5} d^{4}+11550 x^{5} A \,a^{4} b \,e^{4}+92400 x^{5} A \,a^{3} b^{2} d \,e^{3}+138600 x^{5} A \,a^{2} b^{3} d^{2} e^{2}+46200 x^{5} A \,b^{4} d^{3} e a +2310 x^{5} A \,b^{5} d^{4}+2310 x^{5} B \,a^{5} e^{4}+46200 x^{5} B \,a^{4} b d \,e^{3}+138600 x^{5} B \,a^{3} b^{2} d^{2} e^{2}+92400 x^{5} B \,a^{2} b^{3} d^{3} e +11550 x^{5} B \,b^{4} d^{4} a +2772 x^{4} A \,a^{5} e^{4}+55440 x^{4} A \,a^{4} b d \,e^{3}+166320 x^{4} A \,a^{3} b^{2} d^{2} e^{2}+110880 x^{4} A \,a^{2} b^{3} d^{3} e +13860 x^{4} A \,d^{4} b^{4} a +11088 x^{4} B \,a^{5} d \,e^{3}+83160 x^{4} B \,a^{4} b \,d^{2} e^{2}+110880 x^{4} B \,a^{3} b^{2} d^{3} e +27720 x^{4} B \,a^{2} b^{3} d^{4}+13860 x^{3} A \,a^{5} d \,e^{3}+103950 x^{3} A \,a^{4} b \,d^{2} e^{2}+138600 x^{3} A \,a^{3} b^{2} d^{3} e +34650 x^{3} A \,d^{4} a^{2} b^{3}+20790 x^{3} B \,a^{5} d^{2} e^{2}+69300 x^{3} B \,a^{4} b \,d^{3} e +34650 x^{3} B \,a^{3} b^{2} d^{4}+27720 x^{2} A \,a^{5} d^{2} e^{2}+92400 x^{2} A \,a^{4} b \,d^{3} e +46200 x^{2} A \,d^{4} a^{3} b^{2}+18480 x^{2} B \,a^{5} d^{3} e +23100 x^{2} B \,a^{4} b \,d^{4}+27720 x A \,a^{5} d^{3} e +34650 x A \,d^{4} a^{4} b +6930 x B \,a^{5} d^{4}+13860 A \,d^{4} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{13860 \left (b x +a \right )^{5}}\) | \(872\) |
1/11*((b*x+a)^2)^(1/2)/(b*x+a)*b^5*B*e^4*x^11+1/10*((b*x+a)^2)^(1/2)/(b*x+ a)*((A*e^4+4*B*d*e^3)*b^5+5*B*e^4*b^4*a)*x^10+1/9*((b*x+a)^2)^(1/2)/(b*x+a )*((4*A*d*e^3+6*B*d^2*e^2)*b^5+5*(A*e^4+4*B*d*e^3)*b^4*a+10*B*e^4*a^2*b^3) *x^9+1/8*((b*x+a)^2)^(1/2)/(b*x+a)*((6*A*d^2*e^2+4*B*d^3*e)*b^5+5*(4*A*d*e ^3+6*B*d^2*e^2)*b^4*a+10*(A*e^4+4*B*d*e^3)*a^2*b^3+10*B*e^4*a^3*b^2)*x^8+1 /7*((b*x+a)^2)^(1/2)/(b*x+a)*((4*A*d^3*e+B*d^4)*b^5+5*(6*A*d^2*e^2+4*B*d^3 *e)*b^4*a+10*(4*A*d*e^3+6*B*d^2*e^2)*a^2*b^3+10*(A*e^4+4*B*d*e^3)*a^3*b^2+ 5*B*e^4*a^4*b)*x^7+1/6*((b*x+a)^2)^(1/2)/(b*x+a)*(A*b^5*d^4+5*(4*A*d^3*e+B *d^4)*b^4*a+10*(6*A*d^2*e^2+4*B*d^3*e)*a^2*b^3+10*(4*A*d*e^3+6*B*d^2*e^2)* a^3*b^2+5*(A*e^4+4*B*d*e^3)*a^4*b+B*a^5*e^4)*x^6+1/5*((b*x+a)^2)^(1/2)/(b* x+a)*(5*A*d^4*b^4*a+10*(4*A*d^3*e+B*d^4)*a^2*b^3+10*(6*A*d^2*e^2+4*B*d^3*e )*a^3*b^2+5*(4*A*d*e^3+6*B*d^2*e^2)*a^4*b+(A*e^4+4*B*d*e^3)*a^5)*x^5+1/4*( (b*x+a)^2)^(1/2)/(b*x+a)*(10*A*d^4*a^2*b^3+10*(4*A*d^3*e+B*d^4)*a^3*b^2+5* (6*A*d^2*e^2+4*B*d^3*e)*a^4*b+(4*A*d*e^3+6*B*d^2*e^2)*a^5)*x^4+1/3*((b*x+a )^2)^(1/2)/(b*x+a)*(10*A*d^4*a^3*b^2+5*(4*A*d^3*e+B*d^4)*a^4*b+(6*A*d^2*e^ 2+4*B*d^3*e)*a^5)*x^3+1/2*((b*x+a)^2)^(1/2)/(b*x+a)*(5*A*d^4*a^4*b+(4*A*d^ 3*e+B*d^4)*a^5)*x^2+((b*x+a)^2)^(1/2)/(b*x+a)*A*d^4*a^5*x
Leaf count of result is larger than twice the leaf count of optimal. 677 vs. \(2 (246) = 492\).
Time = 0.29 (sec) , antiderivative size = 677, normalized size of antiderivative = 2.09 \[ \int (A+B x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{11} \, B b^{5} e^{4} x^{11} + A a^{5} d^{4} x + \frac {1}{10} \, {\left (4 \, B b^{5} d e^{3} + {\left (5 \, B a b^{4} + A b^{5}\right )} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (6 \, B b^{5} d^{2} e^{2} + 4 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{3} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{4}\right )} x^{9} + \frac {1}{4} \, {\left (2 \, B b^{5} d^{3} e + 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{2} + 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{3} + 5 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (B b^{5} d^{4} + 4 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{2} + 40 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{3} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left ({\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} + 20 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e + 60 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{2} + 20 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{3} + {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (A a^{5} e^{4} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} + 40 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e + 30 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{2} + 4 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{3}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, A a^{5} d e^{3} + 5 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{4} + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{3} e + 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{5} d^{2} e^{2} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{4} + 4 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{5} d^{3} e + {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{4}\right )} x^{2} \]
1/11*B*b^5*e^4*x^11 + A*a^5*d^4*x + 1/10*(4*B*b^5*d*e^3 + (5*B*a*b^4 + A*b ^5)*e^4)*x^10 + 1/9*(6*B*b^5*d^2*e^2 + 4*(5*B*a*b^4 + A*b^5)*d*e^3 + 5*(2* B*a^2*b^3 + A*a*b^4)*e^4)*x^9 + 1/4*(2*B*b^5*d^3*e + 3*(5*B*a*b^4 + A*b^5) *d^2*e^2 + 10*(2*B*a^2*b^3 + A*a*b^4)*d*e^3 + 5*(B*a^3*b^2 + A*a^2*b^3)*e^ 4)*x^8 + 1/7*(B*b^5*d^4 + 4*(5*B*a*b^4 + A*b^5)*d^3*e + 30*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^2 + 40*(B*a^3*b^2 + A*a^2*b^3)*d*e^3 + 5*(B*a^4*b + 2*A*a^3 *b^2)*e^4)*x^7 + 1/6*((5*B*a*b^4 + A*b^5)*d^4 + 20*(2*B*a^2*b^3 + A*a*b^4) *d^3*e + 60*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^2 + 20*(B*a^4*b + 2*A*a^3*b^2)*d *e^3 + (B*a^5 + 5*A*a^4*b)*e^4)*x^6 + 1/5*(A*a^5*e^4 + 5*(2*B*a^2*b^3 + A* a*b^4)*d^4 + 40*(B*a^3*b^2 + A*a^2*b^3)*d^3*e + 30*(B*a^4*b + 2*A*a^3*b^2) *d^2*e^2 + 4*(B*a^5 + 5*A*a^4*b)*d*e^3)*x^5 + 1/2*(2*A*a^5*d*e^3 + 5*(B*a^ 3*b^2 + A*a^2*b^3)*d^4 + 10*(B*a^4*b + 2*A*a^3*b^2)*d^3*e + 3*(B*a^5 + 5*A *a^4*b)*d^2*e^2)*x^4 + 1/3*(6*A*a^5*d^2*e^2 + 5*(B*a^4*b + 2*A*a^3*b^2)*d^ 4 + 4*(B*a^5 + 5*A*a^4*b)*d^3*e)*x^3 + 1/2*(4*A*a^5*d^3*e + (B*a^5 + 5*A*a ^4*b)*d^4)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 48020 vs. \(2 (265) = 530\).
Time = 1.86 (sec) , antiderivative size = 48020, normalized size of antiderivative = 148.21 \[ \int (A+B x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(B*b**4*e**4*x**10/11 + x**9*( A*b**6*e**4 + 45*B*a*b**5*e**4/11 + 4*B*b**6*d*e**3)/(10*b**2) + x**8*(6*A *a*b**5*e**4 + 4*A*b**6*d*e**3 + 155*B*a**2*b**4*e**4/11 + 24*B*a*b**5*d*e **3 + 6*B*b**6*d**2*e**2 - 19*a*(A*b**6*e**4 + 45*B*a*b**5*e**4/11 + 4*B*b **6*d*e**3)/(10*b))/(9*b**2) + x**7*(15*A*a**2*b**4*e**4 + 24*A*a*b**5*d*e **3 + 6*A*b**6*d**2*e**2 + 20*B*a**3*b**3*e**4 + 60*B*a**2*b**4*d*e**3 + 3 6*B*a*b**5*d**2*e**2 + 4*B*b**6*d**3*e - 9*a**2*(A*b**6*e**4 + 45*B*a*b**5 *e**4/11 + 4*B*b**6*d*e**3)/(10*b**2) - 17*a*(6*A*a*b**5*e**4 + 4*A*b**6*d *e**3 + 155*B*a**2*b**4*e**4/11 + 24*B*a*b**5*d*e**3 + 6*B*b**6*d**2*e**2 - 19*a*(A*b**6*e**4 + 45*B*a*b**5*e**4/11 + 4*B*b**6*d*e**3)/(10*b))/(9*b) )/(8*b**2) + x**6*(20*A*a**3*b**3*e**4 + 60*A*a**2*b**4*d*e**3 + 36*A*a*b* *5*d**2*e**2 + 4*A*b**6*d**3*e + 15*B*a**4*b**2*e**4 + 80*B*a**3*b**3*d*e* *3 + 90*B*a**2*b**4*d**2*e**2 + 24*B*a*b**5*d**3*e + B*b**6*d**4 - 8*a**2* (6*A*a*b**5*e**4 + 4*A*b**6*d*e**3 + 155*B*a**2*b**4*e**4/11 + 24*B*a*b**5 *d*e**3 + 6*B*b**6*d**2*e**2 - 19*a*(A*b**6*e**4 + 45*B*a*b**5*e**4/11 + 4 *B*b**6*d*e**3)/(10*b))/(9*b**2) - 15*a*(15*A*a**2*b**4*e**4 + 24*A*a*b**5 *d*e**3 + 6*A*b**6*d**2*e**2 + 20*B*a**3*b**3*e**4 + 60*B*a**2*b**4*d*e**3 + 36*B*a*b**5*d**2*e**2 + 4*B*b**6*d**3*e - 9*a**2*(A*b**6*e**4 + 45*B*a* b**5*e**4/11 + 4*B*b**6*d*e**3)/(10*b**2) - 17*a*(6*A*a*b**5*e**4 + 4*A*b* *6*d*e**3 + 155*B*a**2*b**4*e**4/11 + 24*B*a*b**5*d*e**3 + 6*B*b**6*d**...
Leaf count of result is larger than twice the leaf count of optimal. 1004 vs. \(2 (246) = 492\).
Time = 0.21 (sec) , antiderivative size = 1004, normalized size of antiderivative = 3.10 \[ \int (A+B x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
1/11*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*e^4*x^4/b^2 - 3/22*(b^2*x^2 + 2*a*b *x + a^2)^(7/2)*B*a*e^4*x^3/b^3 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*d^ 4*x - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^5*e^4*x/b^5 + 31/198*(b^2*x^ 2 + 2*a*b*x + a^2)^(7/2)*B*a^2*e^4*x^2/b^4 + 1/6*(b^2*x^2 + 2*a*b*x + a^2) ^(5/2)*A*a*d^4/b - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^6*e^4/b^6 - 65/ 396*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^3*e^4*x/b^5 + 461/2772*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^4*e^4/b^6 + 1/10*(4*B*d*e^3 + A*e^4)*(b^2*x^2 + 2 *a*b*x + a^2)^(7/2)*x^3/b^2 + 1/6*(4*B*d*e^3 + A*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*x/b^4 - 1/3*(3*B*d^2*e^2 + 2*A*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*x/b^3 + 1/3*(2*B*d^3*e + 3*A*d^2*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x/b^2 - 1/6*(B*d^4 + 4*A*d^3*e)*(b^2*x^2 + 2*a*b*x + a^2)^ (5/2)*a*x/b - 13/90*(4*B*d*e^3 + A*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a* x^2/b^3 + 2/9*(3*B*d^2*e^2 + 2*A*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^ 2/b^2 + 1/6*(4*B*d*e^3 + A*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5/b^5 - 1/3*(3*B*d^2*e^2 + 2*A*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4/b^4 + 1/ 3*(2*B*d^3*e + 3*A*d^2*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3/b^3 - 1/6* (B*d^4 + 4*A*d^3*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2/b^2 + 29/180*(4*B* d*e^3 + A*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x/b^4 - 11/36*(3*B*d^2* e^2 + 2*A*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x/b^3 + 1/4*(2*B*d^3*e + 3*A*d^2*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x/b^2 - 209/1260*(4*B*d*...
Leaf count of result is larger than twice the leaf count of optimal. 1351 vs. \(2 (246) = 492\).
Time = 0.30 (sec) , antiderivative size = 1351, normalized size of antiderivative = 4.17 \[ \int (A+B x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
1/11*B*b^5*e^4*x^11*sgn(b*x + a) + 2/5*B*b^5*d*e^3*x^10*sgn(b*x + a) + 1/2 *B*a*b^4*e^4*x^10*sgn(b*x + a) + 1/10*A*b^5*e^4*x^10*sgn(b*x + a) + 2/3*B* b^5*d^2*e^2*x^9*sgn(b*x + a) + 20/9*B*a*b^4*d*e^3*x^9*sgn(b*x + a) + 4/9*A *b^5*d*e^3*x^9*sgn(b*x + a) + 10/9*B*a^2*b^3*e^4*x^9*sgn(b*x + a) + 5/9*A* a*b^4*e^4*x^9*sgn(b*x + a) + 1/2*B*b^5*d^3*e*x^8*sgn(b*x + a) + 15/4*B*a*b ^4*d^2*e^2*x^8*sgn(b*x + a) + 3/4*A*b^5*d^2*e^2*x^8*sgn(b*x + a) + 5*B*a^2 *b^3*d*e^3*x^8*sgn(b*x + a) + 5/2*A*a*b^4*d*e^3*x^8*sgn(b*x + a) + 5/4*B*a ^3*b^2*e^4*x^8*sgn(b*x + a) + 5/4*A*a^2*b^3*e^4*x^8*sgn(b*x + a) + 1/7*B*b ^5*d^4*x^7*sgn(b*x + a) + 20/7*B*a*b^4*d^3*e*x^7*sgn(b*x + a) + 4/7*A*b^5* d^3*e*x^7*sgn(b*x + a) + 60/7*B*a^2*b^3*d^2*e^2*x^7*sgn(b*x + a) + 30/7*A* a*b^4*d^2*e^2*x^7*sgn(b*x + a) + 40/7*B*a^3*b^2*d*e^3*x^7*sgn(b*x + a) + 4 0/7*A*a^2*b^3*d*e^3*x^7*sgn(b*x + a) + 5/7*B*a^4*b*e^4*x^7*sgn(b*x + a) + 10/7*A*a^3*b^2*e^4*x^7*sgn(b*x + a) + 5/6*B*a*b^4*d^4*x^6*sgn(b*x + a) + 1 /6*A*b^5*d^4*x^6*sgn(b*x + a) + 20/3*B*a^2*b^3*d^3*e*x^6*sgn(b*x + a) + 10 /3*A*a*b^4*d^3*e*x^6*sgn(b*x + a) + 10*B*a^3*b^2*d^2*e^2*x^6*sgn(b*x + a) + 10*A*a^2*b^3*d^2*e^2*x^6*sgn(b*x + a) + 10/3*B*a^4*b*d*e^3*x^6*sgn(b*x + a) + 20/3*A*a^3*b^2*d*e^3*x^6*sgn(b*x + a) + 1/6*B*a^5*e^4*x^6*sgn(b*x + a) + 5/6*A*a^4*b*e^4*x^6*sgn(b*x + a) + 2*B*a^2*b^3*d^4*x^5*sgn(b*x + a) + A*a*b^4*d^4*x^5*sgn(b*x + a) + 8*B*a^3*b^2*d^3*e*x^5*sgn(b*x + a) + 8*A*a ^2*b^3*d^3*e*x^5*sgn(b*x + a) + 6*B*a^4*b*d^2*e^2*x^5*sgn(b*x + a) + 12...
Timed out. \[ \int (A+B x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^4\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]