Integrand size = 28, antiderivative size = 200 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=-\frac {(b d-a e)^3 (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^4 (a+b x)}+\frac {3 b (b d-a e)^2 (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^4 (a+b x)}-\frac {3 b^2 (b d-a e) (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^4 (a+b x)}+\frac {b^3 (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^4 (a+b x)} \]
-1/6*(-a*e+b*d)^3*(e*x+d)^6*((b*x+a)^2)^(1/2)/e^4/(b*x+a)+3/7*b*(-a*e+b*d) ^2*(e*x+d)^7*((b*x+a)^2)^(1/2)/e^4/(b*x+a)-3/8*b^2*(-a*e+b*d)*(e*x+d)^8*(( b*x+a)^2)^(1/2)/e^4/(b*x+a)+1/9*b^3*(e*x+d)^9*((b*x+a)^2)^(1/2)/e^4/(b*x+a )
Time = 1.06 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.30 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (84 a^3 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+36 a^2 b x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+9 a b^2 x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+b^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )\right )}{504 (a+b x)} \]
(x*Sqrt[(a + b*x)^2]*(84*a^3*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2 *e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) + 36*a^2*b*x*(21*d^5 + 70*d^4*e*x + 105* d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 9*a*b^2*x^2*(56 *d^5 + 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 2 1*e^5*x^5) + b^3*x^3*(126*d^5 + 504*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^ 3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5)))/(504*(a + b*x))
Time = 0.37 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.60, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1102, 27, 49, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (d+e x)^5 \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^3 (a+b x)^3 (d+e x)^5dx}{b^3 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^3 (d+e x)^5dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^3 (d+e x)^8}{e^3}-\frac {3 b^2 (b d-a e) (d+e x)^7}{e^3}+\frac {3 b (b d-a e)^2 (d+e x)^6}{e^3}+\frac {(a e-b d)^3 (d+e x)^5}{e^3}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {3 b^2 (d+e x)^8 (b d-a e)}{8 e^4}+\frac {3 b (d+e x)^7 (b d-a e)^2}{7 e^4}-\frac {(d+e x)^6 (b d-a e)^3}{6 e^4}+\frac {b^3 (d+e x)^9}{9 e^4}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/6*((b*d - a*e)^3*(d + e*x)^6)/e^4 + (3* b*(b*d - a*e)^2*(d + e*x)^7)/(7*e^4) - (3*b^2*(b*d - a*e)*(d + e*x)^8)/(8* e^4) + (b^3*(d + e*x)^9)/(9*e^4)))/(a + b*x)
3.16.54.3.1 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] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(321\) vs. \(2(148)=296\).
Time = 2.48 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.61
| method | result | size |
| gosper | \(\frac {x \left (56 b^{3} e^{5} x^{8}+189 x^{7} a \,b^{2} e^{5}+315 x^{7} b^{3} d \,e^{4}+216 x^{6} a^{2} b \,e^{5}+1080 x^{6} a \,b^{2} d \,e^{4}+720 x^{6} b^{3} d^{2} e^{3}+84 x^{5} a^{3} e^{5}+1260 x^{5} a^{2} b d \,e^{4}+2520 x^{5} a \,b^{2} d^{2} e^{3}+840 x^{5} b^{3} d^{3} e^{2}+504 a^{3} d \,e^{4} x^{4}+3024 a^{2} b \,d^{2} e^{3} x^{4}+3024 a \,b^{2} d^{3} e^{2} x^{4}+504 b^{3} d^{4} e \,x^{4}+1260 x^{3} a^{3} d^{2} e^{3}+3780 x^{3} a^{2} b \,d^{3} e^{2}+1890 x^{3} a \,b^{2} d^{4} e +126 x^{3} b^{3} d^{5}+1680 x^{2} a^{3} d^{3} e^{2}+2520 x^{2} a^{2} b \,d^{4} e +504 x^{2} a \,b^{2} d^{5}+1260 x \,a^{3} d^{4} e +756 x \,a^{2} b \,d^{5}+504 a^{3} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{504 \left (b x +a \right )^{3}}\) | \(322\) |
| default | \(\frac {x \left (56 b^{3} e^{5} x^{8}+189 x^{7} a \,b^{2} e^{5}+315 x^{7} b^{3} d \,e^{4}+216 x^{6} a^{2} b \,e^{5}+1080 x^{6} a \,b^{2} d \,e^{4}+720 x^{6} b^{3} d^{2} e^{3}+84 x^{5} a^{3} e^{5}+1260 x^{5} a^{2} b d \,e^{4}+2520 x^{5} a \,b^{2} d^{2} e^{3}+840 x^{5} b^{3} d^{3} e^{2}+504 a^{3} d \,e^{4} x^{4}+3024 a^{2} b \,d^{2} e^{3} x^{4}+3024 a \,b^{2} d^{3} e^{2} x^{4}+504 b^{3} d^{4} e \,x^{4}+1260 x^{3} a^{3} d^{2} e^{3}+3780 x^{3} a^{2} b \,d^{3} e^{2}+1890 x^{3} a \,b^{2} d^{4} e +126 x^{3} b^{3} d^{5}+1680 x^{2} a^{3} d^{3} e^{2}+2520 x^{2} a^{2} b \,d^{4} e +504 x^{2} a \,b^{2} d^{5}+1260 x \,a^{3} d^{4} e +756 x \,a^{2} b \,d^{5}+504 a^{3} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{504 \left (b x +a \right )^{3}}\) | \(322\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{3} e^{5} x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 a \,b^{2} e^{5}+5 b^{3} d \,e^{4}\right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 a^{2} b \,e^{5}+15 a \,b^{2} d \,e^{4}+10 b^{3} d^{2} e^{3}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{3} e^{5}+15 a^{2} b d \,e^{4}+30 a \,b^{2} d^{2} e^{3}+10 b^{3} d^{3} e^{2}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 d \,e^{4} a^{3}+30 a^{2} b \,d^{2} e^{3}+30 a \,b^{2} d^{3} e^{2}+5 b^{3} d^{4} e \right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{3} d^{2} e^{3}+30 a^{2} b \,d^{3} e^{2}+15 a \,b^{2} d^{4} e +b^{3} d^{5}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{3} d^{3} e^{2}+15 a^{2} b \,d^{4} e +3 a \,b^{2} d^{5}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a^{3} d^{4} e +3 a^{2} b \,d^{5}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{3} d^{5} x}{b x +a}\) | \(425\) |
1/504*x*(56*b^3*e^5*x^8+189*a*b^2*e^5*x^7+315*b^3*d*e^4*x^7+216*a^2*b*e^5* x^6+1080*a*b^2*d*e^4*x^6+720*b^3*d^2*e^3*x^6+84*a^3*e^5*x^5+1260*a^2*b*d*e ^4*x^5+2520*a*b^2*d^2*e^3*x^5+840*b^3*d^3*e^2*x^5+504*a^3*d*e^4*x^4+3024*a ^2*b*d^2*e^3*x^4+3024*a*b^2*d^3*e^2*x^4+504*b^3*d^4*e*x^4+1260*a^3*d^2*e^3 *x^3+3780*a^2*b*d^3*e^2*x^3+1890*a*b^2*d^4*e*x^3+126*b^3*d^5*x^3+1680*a^3* d^3*e^2*x^2+2520*a^2*b*d^4*e*x^2+504*a*b^2*d^5*x^2+1260*a^3*d^4*e*x+756*a^ 2*b*d^5*x+504*a^3*d^5)*((b*x+a)^2)^(3/2)/(b*x+a)^3
Time = 0.28 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.38 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{9} \, b^{3} e^{5} x^{9} + a^{3} d^{5} x + \frac {1}{8} \, {\left (5 \, b^{3} d e^{4} + 3 \, a b^{2} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (10 \, b^{3} d^{2} e^{3} + 15 \, a b^{2} d e^{4} + 3 \, a^{2} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, b^{3} d^{3} e^{2} + 30 \, a b^{2} d^{2} e^{3} + 15 \, a^{2} b d e^{4} + a^{3} e^{5}\right )} x^{6} + {\left (b^{3} d^{4} e + 6 \, a b^{2} d^{3} e^{2} + 6 \, a^{2} b d^{2} e^{3} + a^{3} d e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} d^{5} + 15 \, a b^{2} d^{4} e + 30 \, a^{2} b d^{3} e^{2} + 10 \, a^{3} d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a b^{2} d^{5} + 15 \, a^{2} b d^{4} e + 10 \, a^{3} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{5} + 5 \, a^{3} d^{4} e\right )} x^{2} \]
1/9*b^3*e^5*x^9 + a^3*d^5*x + 1/8*(5*b^3*d*e^4 + 3*a*b^2*e^5)*x^8 + 1/7*(1 0*b^3*d^2*e^3 + 15*a*b^2*d*e^4 + 3*a^2*b*e^5)*x^7 + 1/6*(10*b^3*d^3*e^2 + 30*a*b^2*d^2*e^3 + 15*a^2*b*d*e^4 + a^3*e^5)*x^6 + (b^3*d^4*e + 6*a*b^2*d^ 3*e^2 + 6*a^2*b*d^2*e^3 + a^3*d*e^4)*x^5 + 1/4*(b^3*d^5 + 15*a*b^2*d^4*e + 30*a^2*b*d^3*e^2 + 10*a^3*d^2*e^3)*x^4 + 1/3*(3*a*b^2*d^5 + 15*a^2*b*d^4* e + 10*a^3*d^3*e^2)*x^3 + 1/2*(3*a^2*b*d^5 + 5*a^3*d^4*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 10178 vs. \(2 (143) = 286\).
Time = 1.16 (sec) , antiderivative size = 10178, normalized size of antiderivative = 50.89 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**2*e**5*x**8/9 + x**7*(19*a *b**3*e**5/9 + 5*b**4*d*e**4)/(8*b**2) + x**6*(46*a**2*b**2*e**5/9 + 20*a* b**3*d*e**4 - 15*a*(19*a*b**3*e**5/9 + 5*b**4*d*e**4)/(8*b) + 10*b**4*d**2 *e**3)/(7*b**2) + x**5*(4*a**3*b*e**5 + 30*a**2*b**2*d*e**4 - 7*a**2*(19*a *b**3*e**5/9 + 5*b**4*d*e**4)/(8*b**2) + 40*a*b**3*d**2*e**3 - 13*a*(46*a* *2*b**2*e**5/9 + 20*a*b**3*d*e**4 - 15*a*(19*a*b**3*e**5/9 + 5*b**4*d*e**4 )/(8*b) + 10*b**4*d**2*e**3)/(7*b) + 10*b**4*d**3*e**2)/(6*b**2) + x**4*(a **4*e**5 + 20*a**3*b*d*e**4 + 60*a**2*b**2*d**2*e**3 - 6*a**2*(46*a**2*b** 2*e**5/9 + 20*a*b**3*d*e**4 - 15*a*(19*a*b**3*e**5/9 + 5*b**4*d*e**4)/(8*b ) + 10*b**4*d**2*e**3)/(7*b**2) + 40*a*b**3*d**3*e**2 - 11*a*(4*a**3*b*e** 5 + 30*a**2*b**2*d*e**4 - 7*a**2*(19*a*b**3*e**5/9 + 5*b**4*d*e**4)/(8*b** 2) + 40*a*b**3*d**2*e**3 - 13*a*(46*a**2*b**2*e**5/9 + 20*a*b**3*d*e**4 - 15*a*(19*a*b**3*e**5/9 + 5*b**4*d*e**4)/(8*b) + 10*b**4*d**2*e**3)/(7*b) + 10*b**4*d**3*e**2)/(6*b) + 5*b**4*d**4*e)/(5*b**2) + x**3*(5*a**4*d*e**4 + 40*a**3*b*d**2*e**3 + 60*a**2*b**2*d**3*e**2 - 5*a**2*(4*a**3*b*e**5 + 3 0*a**2*b**2*d*e**4 - 7*a**2*(19*a*b**3*e**5/9 + 5*b**4*d*e**4)/(8*b**2) + 40*a*b**3*d**2*e**3 - 13*a*(46*a**2*b**2*e**5/9 + 20*a*b**3*d*e**4 - 15*a* (19*a*b**3*e**5/9 + 5*b**4*d*e**4)/(8*b) + 10*b**4*d**2*e**3)/(7*b) + 10*b **4*d**3*e**2)/(6*b**2) + 20*a*b**3*d**4*e - 9*a*(a**4*e**5 + 20*a**3*b*d* e**4 + 60*a**2*b**2*d**2*e**3 - 6*a**2*(46*a**2*b**2*e**5/9 + 20*a*b**3...
Leaf count of result is larger than twice the leaf count of optimal. 814 vs. \(2 (148) = 296\).
Time = 0.24 (sec) , antiderivative size = 814, normalized size of antiderivative = 4.07 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} e^{5} x^{4}}{9 \, b^{2}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d e^{4} x^{3}}{8 \, b^{2}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a e^{5} x^{3}}{72 \, b^{3}} + \frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} d^{5} x - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{4} e x}{4 \, b} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d^{3} e^{2} x}{2 \, b^{2}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} d^{2} e^{3} x}{2 \, b^{3}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4} d e^{4} x}{4 \, b^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{5} e^{5} x}{4 \, b^{5}} + \frac {10 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{2} e^{3} x^{2}}{7 \, b^{2}} - \frac {55 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d e^{4} x^{2}}{56 \, b^{3}} + \frac {37 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} e^{5} x^{2}}{168 \, b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{5}}{4 \, b} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d^{4} e}{4 \, b^{2}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} d^{3} e^{2}}{2 \, b^{3}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4} d^{2} e^{3}}{2 \, b^{4}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{5} d e^{4}}{4 \, b^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{6} e^{5}}{4 \, b^{6}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{3} e^{2} x}{3 \, b^{2}} - \frac {15 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{2} e^{3} x}{7 \, b^{3}} + \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d e^{4} x}{56 \, b^{4}} - \frac {121 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} e^{5} x}{504 \, b^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{4} e}{b^{2}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{3} e^{2}}{3 \, b^{3}} + \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{2} e^{3}}{7 \, b^{4}} - \frac {69 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} d e^{4}}{56 \, b^{5}} + \frac {125 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} e^{5}}{504 \, b^{6}} \]
1/9*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*e^5*x^4/b^2 + 5/8*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*d*e^4*x^3/b^2 - 13/72*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*e^5*x^ 3/b^3 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*d^5*x - 5/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*d^4*e*x/b + 5/2*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*d^3*e^ 2*x/b^2 - 5/2*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*d^2*e^3*x/b^3 + 5/4*(b^2 *x^2 + 2*a*b*x + a^2)^(3/2)*a^4*d*e^4*x/b^4 - 1/4*(b^2*x^2 + 2*a*b*x + a^2 )^(3/2)*a^5*e^5*x/b^5 + 10/7*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*d^2*e^3*x^2/b ^2 - 55/56*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d*e^4*x^2/b^3 + 37/168*(b^2*x ^2 + 2*a*b*x + a^2)^(5/2)*a^2*e^5*x^2/b^4 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^ (3/2)*a*d^5/b - 5/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*d^4*e/b^2 + 5/2*(b ^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*d^3*e^2/b^3 - 5/2*(b^2*x^2 + 2*a*b*x + a ^2)^(3/2)*a^4*d^2*e^3/b^4 + 5/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^5*d*e^4/ b^5 - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^6*e^5/b^6 + 5/3*(b^2*x^2 + 2*a *b*x + a^2)^(5/2)*d^3*e^2*x/b^2 - 15/7*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d ^2*e^3*x/b^3 + 65/56*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d*e^4*x/b^4 - 121 /504*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*e^5*x/b^5 + (b^2*x^2 + 2*a*b*x + a^2)^(5/2)*d^4*e/b^2 - 7/3*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^3*e^2/b^3 + 17/7*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d^2*e^3/b^4 - 69/56*(b^2*x^2 + 2 *a*b*x + a^2)^(5/2)*a^3*d*e^4/b^5 + 125/504*(b^2*x^2 + 2*a*b*x + a^2)^(5/2 )*a^4*e^5/b^6
Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (148) = 296\).
Time = 0.30 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.64 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{9} \, b^{3} e^{5} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{8} \, b^{3} d e^{4} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{8} \, a b^{2} e^{5} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, b^{3} d^{2} e^{3} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{7} \, a b^{2} d e^{4} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, a^{2} b e^{5} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, b^{3} d^{3} e^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + 5 \, a b^{2} d^{2} e^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{2} b d e^{4} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, a^{3} e^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + b^{3} d^{4} e x^{5} \mathrm {sgn}\left (b x + a\right ) + 6 \, a b^{2} d^{3} e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b d^{2} e^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + a^{3} d e^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, b^{3} d^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{4} \, a b^{2} d^{4} e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a^{2} b d^{3} e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{3} d^{2} e^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + a b^{2} d^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{2} b d^{4} e x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{3} d^{3} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{2} b d^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{3} d^{4} e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{3} d^{5} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (126 \, a^{4} b^{5} d^{5} - 126 \, a^{5} b^{4} d^{4} e + 84 \, a^{6} b^{3} d^{3} e^{2} - 36 \, a^{7} b^{2} d^{2} e^{3} + 9 \, a^{8} b d e^{4} - a^{9} e^{5}\right )} \mathrm {sgn}\left (b x + a\right )}{504 \, b^{6}} \]
1/9*b^3*e^5*x^9*sgn(b*x + a) + 5/8*b^3*d*e^4*x^8*sgn(b*x + a) + 3/8*a*b^2* e^5*x^8*sgn(b*x + a) + 10/7*b^3*d^2*e^3*x^7*sgn(b*x + a) + 15/7*a*b^2*d*e^ 4*x^7*sgn(b*x + a) + 3/7*a^2*b*e^5*x^7*sgn(b*x + a) + 5/3*b^3*d^3*e^2*x^6* sgn(b*x + a) + 5*a*b^2*d^2*e^3*x^6*sgn(b*x + a) + 5/2*a^2*b*d*e^4*x^6*sgn( b*x + a) + 1/6*a^3*e^5*x^6*sgn(b*x + a) + b^3*d^4*e*x^5*sgn(b*x + a) + 6*a *b^2*d^3*e^2*x^5*sgn(b*x + a) + 6*a^2*b*d^2*e^3*x^5*sgn(b*x + a) + a^3*d*e ^4*x^5*sgn(b*x + a) + 1/4*b^3*d^5*x^4*sgn(b*x + a) + 15/4*a*b^2*d^4*e*x^4* sgn(b*x + a) + 15/2*a^2*b*d^3*e^2*x^4*sgn(b*x + a) + 5/2*a^3*d^2*e^3*x^4*s gn(b*x + a) + a*b^2*d^5*x^3*sgn(b*x + a) + 5*a^2*b*d^4*e*x^3*sgn(b*x + a) + 10/3*a^3*d^3*e^2*x^3*sgn(b*x + a) + 3/2*a^2*b*d^5*x^2*sgn(b*x + a) + 5/2 *a^3*d^4*e*x^2*sgn(b*x + a) + a^3*d^5*x*sgn(b*x + a) + 1/504*(126*a^4*b^5* d^5 - 126*a^5*b^4*d^4*e + 84*a^6*b^3*d^3*e^2 - 36*a^7*b^2*d^2*e^3 + 9*a^8* b*d*e^4 - a^9*e^5)*sgn(b*x + a)/b^6
Timed out. \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int {\left (d+e\,x\right )}^5\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]