\(\int (a+b x) (d+e x)^5 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\) [129]

Optimal result

Integrand size = 33, antiderivative size = 254 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {(b d-a e)^4 (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^5 (a+b x)}-\frac {4 b (b d-a e)^3 (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}+\frac {3 b^2 (b d-a e)^2 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x)}-\frac {4 b^3 (b d-a e) (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac {b^4 (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^5 (a+b x)} \] Output:

1/6*(-a*e+b*d)^4*(e*x+d)^6*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-4/7*b*(-a*e+b*d)^ 
3*(e*x+d)^7*((b*x+a)^2)^(1/2)/e^5/(b*x+a)+3/4*b^2*(-a*e+b*d)^2*(e*x+d)^8*( 
(b*x+a)^2)^(1/2)/e^5/(b*x+a)-4/9*b^3*(-a*e+b*d)*(e*x+d)^9*((b*x+a)^2)^(1/2 
)/e^5/(b*x+a)+1/10*b^4*(e*x+d)^10*((b*x+a)^2)^(1/2)/e^5/(b*x+a)
 

Mathematica [A] (verified)

Time = 1.12 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.27 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (210 a^4 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+120 a^3 b x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+45 a^2 b^2 x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+10 a b^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )+b^4 x^4 \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )\right )}{1260 (a+b x)} \] Input:

Integrate[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
 

Output:

(x*Sqrt[(a + b*x)^2]*(210*a^4*(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) + 120*a^3*b*x*(21*d^5 + 70*d^4*e*x + 10 
5*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 45*a^2*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 + 21*e^5*x^5) + 10*a*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) + b^4*x^4*(252*d^5 + 1050*d^ 
4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5) 
))/(1260*(a + b*x))
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.58, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 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.

\(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (d+e x)^5 \, dx\)

\(\Big \downarrow \) 1187

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^3 (a+b x)^4 (d+e x)^5dx}{b^3 (a+b x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^4 (d+e x)^5dx}{a+b x}\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^4 (d+e x)^9}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^8}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^7}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^6}{e^4}+\frac {(a e-b d)^4 (d+e x)^5}{e^4}\right )dx}{a+b x}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {4 b^3 (d+e x)^9 (b d-a e)}{9 e^5}+\frac {3 b^2 (d+e x)^8 (b d-a e)^2}{4 e^5}-\frac {4 b (d+e x)^7 (b d-a e)^3}{7 e^5}+\frac {(d+e x)^6 (b d-a e)^4}{6 e^5}+\frac {b^4 (d+e x)^{10}}{10 e^5}\right )}{a+b x}\)

Input:

Int[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
 

Output:

(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((b*d - a*e)^4*(d + e*x)^6)/(6*e^5) - (4*b 
*(b*d - a*e)^3*(d + e*x)^7)/(7*e^5) + (3*b^2*(b*d - a*e)^2*(d + e*x)^8)/(4 
*e^5) - (4*b^3*(b*d - a*e)*(d + e*x)^9)/(9*e^5) + (b^4*(d + e*x)^10)/(10*e 
^5)))/(a + b*x)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 49

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]
 

rule 1187

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(413\) vs. \(2(189)=378\).

Time = 1.55 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.63

method	result	size
gosper	\(\frac {x \left (126 b^{4} e^{5} x^{9}+560 x^{8} a \,b^{3} e^{5}+700 x^{8} b^{4} d \,e^{4}+945 x^{7} a^{2} b^{2} e^{5}+3150 x^{7} a \,b^{3} d \,e^{4}+1575 x^{7} d^{2} e^{3} b^{4}+720 x^{6} a^{3} b \,e^{5}+5400 x^{6} a^{2} b^{2} d \,e^{4}+7200 x^{6} a \,b^{3} d^{2} e^{3}+1800 x^{6} b^{4} d^{3} e^{2}+210 x^{5} a^{4} e^{5}+4200 x^{5} a^{3} b d \,e^{4}+12600 x^{5} a^{2} b^{2} d^{2} e^{3}+8400 x^{5} a \,b^{3} d^{3} e^{2}+1050 x^{5} b^{4} d^{4} e +1260 x^{4} a^{4} d \,e^{4}+10080 x^{4} a^{3} b \,d^{2} e^{3}+15120 x^{4} a^{2} b^{2} d^{3} e^{2}+5040 x^{4} a \,b^{3} d^{4} e +252 x^{4} b^{4} d^{5}+3150 x^{3} a^{4} d^{2} e^{3}+12600 x^{3} a^{3} b \,d^{3} e^{2}+9450 x^{3} a^{2} b^{2} d^{4} e +1260 x^{3} a \,b^{3} d^{5}+4200 x^{2} a^{4} d^{3} e^{2}+8400 x^{2} a^{3} b \,d^{4} e +2520 x^{2} a^{2} b^{2} d^{5}+3150 x \,a^{4} d^{4} e +2520 x \,a^{3} b \,d^{5}+1260 a^{4} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{1260 \left (b x +a \right )^{3}}\)	\(414\)
default	\(\frac {x \left (126 b^{4} e^{5} x^{9}+560 x^{8} a \,b^{3} e^{5}+700 x^{8} b^{4} d \,e^{4}+945 x^{7} a^{2} b^{2} e^{5}+3150 x^{7} a \,b^{3} d \,e^{4}+1575 x^{7} d^{2} e^{3} b^{4}+720 x^{6} a^{3} b \,e^{5}+5400 x^{6} a^{2} b^{2} d \,e^{4}+7200 x^{6} a \,b^{3} d^{2} e^{3}+1800 x^{6} b^{4} d^{3} e^{2}+210 x^{5} a^{4} e^{5}+4200 x^{5} a^{3} b d \,e^{4}+12600 x^{5} a^{2} b^{2} d^{2} e^{3}+8400 x^{5} a \,b^{3} d^{3} e^{2}+1050 x^{5} b^{4} d^{4} e +1260 x^{4} a^{4} d \,e^{4}+10080 x^{4} a^{3} b \,d^{2} e^{3}+15120 x^{4} a^{2} b^{2} d^{3} e^{2}+5040 x^{4} a \,b^{3} d^{4} e +252 x^{4} b^{4} d^{5}+3150 x^{3} a^{4} d^{2} e^{3}+12600 x^{3} a^{3} b \,d^{3} e^{2}+9450 x^{3} a^{2} b^{2} d^{4} e +1260 x^{3} a \,b^{3} d^{5}+4200 x^{2} a^{4} d^{3} e^{2}+8400 x^{2} a^{3} b \,d^{4} e +2520 x^{2} a^{2} b^{2} d^{5}+3150 x \,a^{4} d^{4} e +2520 x \,a^{3} b \,d^{5}+1260 a^{4} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{1260 \left (b x +a \right )^{3}}\)	\(414\)
orering	\(\frac {x \left (126 b^{4} e^{5} x^{9}+560 x^{8} a \,b^{3} e^{5}+700 x^{8} b^{4} d \,e^{4}+945 x^{7} a^{2} b^{2} e^{5}+3150 x^{7} a \,b^{3} d \,e^{4}+1575 x^{7} d^{2} e^{3} b^{4}+720 x^{6} a^{3} b \,e^{5}+5400 x^{6} a^{2} b^{2} d \,e^{4}+7200 x^{6} a \,b^{3} d^{2} e^{3}+1800 x^{6} b^{4} d^{3} e^{2}+210 x^{5} a^{4} e^{5}+4200 x^{5} a^{3} b d \,e^{4}+12600 x^{5} a^{2} b^{2} d^{2} e^{3}+8400 x^{5} a \,b^{3} d^{3} e^{2}+1050 x^{5} b^{4} d^{4} e +1260 x^{4} a^{4} d \,e^{4}+10080 x^{4} a^{3} b \,d^{2} e^{3}+15120 x^{4} a^{2} b^{2} d^{3} e^{2}+5040 x^{4} a \,b^{3} d^{4} e +252 x^{4} b^{4} d^{5}+3150 x^{3} a^{4} d^{2} e^{3}+12600 x^{3} a^{3} b \,d^{3} e^{2}+9450 x^{3} a^{2} b^{2} d^{4} e +1260 x^{3} a \,b^{3} d^{5}+4200 x^{2} a^{4} d^{3} e^{2}+8400 x^{2} a^{3} b \,d^{4} e +2520 x^{2} a^{2} b^{2} d^{5}+3150 x \,a^{4} d^{4} e +2520 x \,a^{3} b \,d^{5}+1260 a^{4} d^{5}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}{1260 \left (b x +a \right )^{3}}\)	\(423\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{4} e^{5} x^{10}}{10 b x +10 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a \,b^{3} e^{5}+5 b^{4} d \,e^{4}\right ) x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 a^{2} b^{2} e^{5}+20 a \,b^{3} d \,e^{4}+10 d^{2} e^{3} b^{4}\right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a^{3} b \,e^{5}+30 a^{2} b^{2} d \,e^{4}+40 a \,b^{3} d^{2} e^{3}+10 b^{4} d^{3} e^{2}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{4} e^{5}+20 a^{3} b d \,e^{4}+60 a^{2} b^{2} d^{2} e^{3}+40 a \,b^{3} d^{3} e^{2}+5 b^{4} d^{4} e \right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a^{4} d \,e^{4}+40 a^{3} b \,d^{2} e^{3}+60 a^{2} b^{2} d^{3} e^{2}+20 a \,b^{3} d^{4} e +b^{4} d^{5}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{4} d^{2} e^{3}+40 a^{3} b \,d^{3} e^{2}+30 a^{2} b^{2} d^{4} e +4 a \,b^{3} d^{5}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{4} d^{3} e^{2}+20 a^{3} b \,d^{4} e +6 a^{2} b^{2} d^{5}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a^{4} d^{4} e +4 a^{3} b \,d^{5}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, d^{5} a^{4} x}{b x +a}\)	\(521\)

Input:

int((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/1260*x*(126*b^4*e^5*x^9+560*a*b^3*e^5*x^8+700*b^4*d*e^4*x^8+945*a^2*b^2* 
e^5*x^7+3150*a*b^3*d*e^4*x^7+1575*b^4*d^2*e^3*x^7+720*a^3*b*e^5*x^6+5400*a 
^2*b^2*d*e^4*x^6+7200*a*b^3*d^2*e^3*x^6+1800*b^4*d^3*e^2*x^6+210*a^4*e^5*x 
^5+4200*a^3*b*d*e^4*x^5+12600*a^2*b^2*d^2*e^3*x^5+8400*a*b^3*d^3*e^2*x^5+1 
050*b^4*d^4*e*x^5+1260*a^4*d*e^4*x^4+10080*a^3*b*d^2*e^3*x^4+15120*a^2*b^2 
*d^3*e^2*x^4+5040*a*b^3*d^4*e*x^4+252*b^4*d^5*x^4+3150*a^4*d^2*e^3*x^3+126 
00*a^3*b*d^3*e^2*x^3+9450*a^2*b^2*d^4*e*x^3+1260*a*b^3*d^5*x^3+4200*a^4*d^ 
3*e^2*x^2+8400*a^3*b*d^4*e*x^2+2520*a^2*b^2*d^5*x^2+3150*a^4*d^4*e*x+2520* 
a^3*b*d^5*x+1260*a^4*d^5)*((b*x+a)^2)^(3/2)/(b*x+a)^3
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.42 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{10} \, b^{4} e^{5} x^{10} + a^{4} d^{5} x + \frac {1}{9} \, {\left (5 \, b^{4} d e^{4} + 4 \, a b^{3} e^{5}\right )} x^{9} + \frac {1}{4} \, {\left (5 \, b^{4} d^{2} e^{3} + 10 \, a b^{3} d e^{4} + 3 \, a^{2} b^{2} e^{5}\right )} x^{8} + \frac {2}{7} \, {\left (5 \, b^{4} d^{3} e^{2} + 20 \, a b^{3} d^{2} e^{3} + 15 \, a^{2} b^{2} d e^{4} + 2 \, a^{3} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (5 \, b^{4} d^{4} e + 40 \, a b^{3} d^{3} e^{2} + 60 \, a^{2} b^{2} d^{2} e^{3} + 20 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{5} + 20 \, a b^{3} d^{4} e + 60 \, a^{2} b^{2} d^{3} e^{2} + 40 \, a^{3} b d^{2} e^{3} + 5 \, a^{4} d e^{4}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, a b^{3} d^{5} + 15 \, a^{2} b^{2} d^{4} e + 20 \, a^{3} b d^{3} e^{2} + 5 \, a^{4} d^{2} e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} d^{5} + 10 \, a^{3} b d^{4} e + 5 \, a^{4} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{5} + 5 \, a^{4} d^{4} e\right )} x^{2} \] Input:

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fric 
as")
 

Output:

1/10*b^4*e^5*x^10 + a^4*d^5*x + 1/9*(5*b^4*d*e^4 + 4*a*b^3*e^5)*x^9 + 1/4* 
(5*b^4*d^2*e^3 + 10*a*b^3*d*e^4 + 3*a^2*b^2*e^5)*x^8 + 2/7*(5*b^4*d^3*e^2 
+ 20*a*b^3*d^2*e^3 + 15*a^2*b^2*d*e^4 + 2*a^3*b*e^5)*x^7 + 1/6*(5*b^4*d^4* 
e + 40*a*b^3*d^3*e^2 + 60*a^2*b^2*d^2*e^3 + 20*a^3*b*d*e^4 + a^4*e^5)*x^6 
+ 1/5*(b^4*d^5 + 20*a*b^3*d^4*e + 60*a^2*b^2*d^3*e^2 + 40*a^3*b*d^2*e^3 + 
5*a^4*d*e^4)*x^5 + 1/2*(2*a*b^3*d^5 + 15*a^2*b^2*d^4*e + 20*a^3*b*d^3*e^2 
+ 5*a^4*d^2*e^3)*x^4 + 2/3*(3*a^2*b^2*d^5 + 10*a^3*b*d^4*e + 5*a^4*d^3*e^2 
)*x^3 + 1/2*(4*a^3*b*d^5 + 5*a^4*d^4*e)*x^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 17473 vs. \(2 (184) = 368\).

Time = 1.83 (sec) , antiderivative size = 17473, normalized size of antiderivative = 68.79 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
 

Output:

Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**3*e**5*x**9/10 + x**8*(31* 
a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + x**7*(91*a**2*b**3*e**5/10 + 25 
*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) + 10*b**5* 
d**2*e**3)/(8*b**2) + x**6*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a* 
*2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15 
*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5* 
b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b**2 
) + x**5*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 100*a**2*b**3*d**2*e**3 - 
7*a**2*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 
+ 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b**2) + 50*a*b**4*d**3*e**2 
 - 13*a*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8*a**2*(31*a*b**4*e**5/ 
10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 15*a*(91*a**2*b**3*e* 
*5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b) 
+ 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b) + 5*b**5*d**4*e)/(6* 
b**2) + x**4*(a**5*e**5 + 25*a**4*b*d*e**4 + 100*a**3*b**2*d**2*e**3 + 100 
*a**2*b**3*d**3*e**2 - 6*a**2*(10*a**3*b**2*e**5 + 50*a**2*b**3*d*e**4 - 8 
*a**2*(31*a*b**4*e**5/10 + 5*b**5*d*e**4)/(9*b**2) + 50*a*b**4*d**2*e**3 - 
 15*a*(91*a**2*b**3*e**5/10 + 25*a*b**4*d*e**4 - 17*a*(31*a*b**4*e**5/10 + 
 5*b**5*d*e**4)/(9*b) + 10*b**5*d**2*e**3)/(8*b) + 10*b**5*d**3*e**2)/(7*b 
**2) + 25*a*b**4*d**4*e - 11*a*(5*a**4*b*e**5 + 50*a**3*b**2*d*e**4 + 1...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1323 vs. \(2 (189) = 378\).

Time = 0.04 (sec) , antiderivative size = 1323, normalized size of antiderivative = 5.21 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxi 
ma")
 

Output:

1/10*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*e^5*x^5/b - 1/6*(b^2*x^2 + 2*a*b*x + 
a^2)^(5/2)*a*e^5*x^4/b^2 + 5/24*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*e^5*x^ 
3/b^3 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*d^5*x + 1/4*(b^2*x^2 + 2*a*b 
*x + a^2)^(3/2)*a^6*e^5*x/b^5 - 13/56*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3* 
e^5*x^2/b^4 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*d^5/b + 1/4*(b^2*x^2 
 + 2*a*b*x + a^2)^(3/2)*a^7*e^5/b^6 + 41/168*(b^2*x^2 + 2*a*b*x + a^2)^(5/ 
2)*a^4*e^5*x/b^5 - 209/840*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5*e^5/b^6 + 1 
/9*(5*b*d*e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^4/b^2 - 13/72*(5* 
b*d*e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x^3/b^3 + 5/8*(2*b*d^2* 
e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^3/b^2 - 1/4*(5*b*d*e^4 + 
a*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^5*x/b^5 + 5/4*(2*b*d^2*e^3 + a*d* 
e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4*x/b^4 - 5/2*(b*d^3*e^2 + a*d^2*e^ 
3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*x/b^3 + 5/4*(b*d^4*e + 2*a*d^3*e^2) 
*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*x/b^2 - 1/4*(b*d^5 + 5*a*d^4*e)*(b^2* 
x^2 + 2*a*b*x + a^2)^(3/2)*a*x/b + 37/168*(5*b*d*e^4 + a*e^5)*(b^2*x^2 + 2 
*a*b*x + a^2)^(5/2)*a^2*x^2/b^4 - 55/56*(2*b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 
 2*a*b*x + a^2)^(5/2)*a*x^2/b^3 + 10/7*(b*d^3*e^2 + a*d^2*e^3)*(b^2*x^2 + 
2*a*b*x + a^2)^(5/2)*x^2/b^2 - 1/4*(5*b*d*e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x 
+ a^2)^(3/2)*a^6/b^6 + 5/4*(2*b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^ 
2)^(3/2)*a^5/b^5 - 5/2*(b*d^3*e^2 + a*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 657 vs. \(2 (189) = 378\).

Time = 0.80 (sec) , antiderivative size = 657, normalized size of antiderivative = 2.59 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac 
")
 

Output:

1/10*b^4*e^5*x^10*sgn(b*x + a) + 5/9*b^4*d*e^4*x^9*sgn(b*x + a) + 4/9*a*b^ 
3*e^5*x^9*sgn(b*x + a) + 5/4*b^4*d^2*e^3*x^8*sgn(b*x + a) + 5/2*a*b^3*d*e^ 
4*x^8*sgn(b*x + a) + 3/4*a^2*b^2*e^5*x^8*sgn(b*x + a) + 10/7*b^4*d^3*e^2*x 
^7*sgn(b*x + a) + 40/7*a*b^3*d^2*e^3*x^7*sgn(b*x + a) + 30/7*a^2*b^2*d*e^4 
*x^7*sgn(b*x + a) + 4/7*a^3*b*e^5*x^7*sgn(b*x + a) + 5/6*b^4*d^4*e*x^6*sgn 
(b*x + a) + 20/3*a*b^3*d^3*e^2*x^6*sgn(b*x + a) + 10*a^2*b^2*d^2*e^3*x^6*s 
gn(b*x + a) + 10/3*a^3*b*d*e^4*x^6*sgn(b*x + a) + 1/6*a^4*e^5*x^6*sgn(b*x 
+ a) + 1/5*b^4*d^5*x^5*sgn(b*x + a) + 4*a*b^3*d^4*e*x^5*sgn(b*x + a) + 12* 
a^2*b^2*d^3*e^2*x^5*sgn(b*x + a) + 8*a^3*b*d^2*e^3*x^5*sgn(b*x + a) + a^4* 
d*e^4*x^5*sgn(b*x + a) + a*b^3*d^5*x^4*sgn(b*x + a) + 15/2*a^2*b^2*d^4*e*x 
^4*sgn(b*x + a) + 10*a^3*b*d^3*e^2*x^4*sgn(b*x + a) + 5/2*a^4*d^2*e^3*x^4* 
sgn(b*x + a) + 2*a^2*b^2*d^5*x^3*sgn(b*x + a) + 20/3*a^3*b*d^4*e*x^3*sgn(b 
*x + a) + 10/3*a^4*d^3*e^2*x^3*sgn(b*x + a) + 2*a^3*b*d^5*x^2*sgn(b*x + a) 
 + 5/2*a^4*d^4*e*x^2*sgn(b*x + a) + a^4*d^5*x*sgn(b*x + a) + 1/1260*(252*a 
^5*b^5*d^5 - 210*a^6*b^4*d^4*e + 120*a^7*b^3*d^3*e^2 - 45*a^8*b^2*d^2*e^3 
+ 10*a^9*b*d*e^4 - a^10*e^5)*sgn(b*x + a)/b^6
 

Mupad [F(-1)]

Timed out. \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^5\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \] Input:

int((a + b*x)*(d + e*x)^5*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)
 

Output:

int((a + b*x)*(d + e*x)^5*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.56 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {x \left (126 b^{4} e^{5} x^{9}+560 a \,b^{3} e^{5} x^{8}+700 b^{4} d \,e^{4} x^{8}+945 a^{2} b^{2} e^{5} x^{7}+3150 a \,b^{3} d \,e^{4} x^{7}+1575 b^{4} d^{2} e^{3} x^{7}+720 a^{3} b \,e^{5} x^{6}+5400 a^{2} b^{2} d \,e^{4} x^{6}+7200 a \,b^{3} d^{2} e^{3} x^{6}+1800 b^{4} d^{3} e^{2} x^{6}+210 a^{4} e^{5} x^{5}+4200 a^{3} b d \,e^{4} x^{5}+12600 a^{2} b^{2} d^{2} e^{3} x^{5}+8400 a \,b^{3} d^{3} e^{2} x^{5}+1050 b^{4} d^{4} e \,x^{5}+1260 a^{4} d \,e^{4} x^{4}+10080 a^{3} b \,d^{2} e^{3} x^{4}+15120 a^{2} b^{2} d^{3} e^{2} x^{4}+5040 a \,b^{3} d^{4} e \,x^{4}+252 b^{4} d^{5} x^{4}+3150 a^{4} d^{2} e^{3} x^{3}+12600 a^{3} b \,d^{3} e^{2} x^{3}+9450 a^{2} b^{2} d^{4} e \,x^{3}+1260 a \,b^{3} d^{5} x^{3}+4200 a^{4} d^{3} e^{2} x^{2}+8400 a^{3} b \,d^{4} e \,x^{2}+2520 a^{2} b^{2} d^{5} x^{2}+3150 a^{4} d^{4} e x +2520 a^{3} b \,d^{5} x +1260 a^{4} d^{5}\right )}{1260} \] Input:

int((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
 

Output:

(x*(1260*a**4*d**5 + 3150*a**4*d**4*e*x + 4200*a**4*d**3*e**2*x**2 + 3150* 
a**4*d**2*e**3*x**3 + 1260*a**4*d*e**4*x**4 + 210*a**4*e**5*x**5 + 2520*a* 
*3*b*d**5*x + 8400*a**3*b*d**4*e*x**2 + 12600*a**3*b*d**3*e**2*x**3 + 1008 
0*a**3*b*d**2*e**3*x**4 + 4200*a**3*b*d*e**4*x**5 + 720*a**3*b*e**5*x**6 + 
 2520*a**2*b**2*d**5*x**2 + 9450*a**2*b**2*d**4*e*x**3 + 15120*a**2*b**2*d 
**3*e**2*x**4 + 12600*a**2*b**2*d**2*e**3*x**5 + 5400*a**2*b**2*d*e**4*x** 
6 + 945*a**2*b**2*e**5*x**7 + 1260*a*b**3*d**5*x**3 + 5040*a*b**3*d**4*e*x 
**4 + 8400*a*b**3*d**3*e**2*x**5 + 7200*a*b**3*d**2*e**3*x**6 + 3150*a*b** 
3*d*e**4*x**7 + 560*a*b**3*e**5*x**8 + 252*b**4*d**5*x**4 + 1050*b**4*d**4 
*e*x**5 + 1800*b**4*d**3*e**2*x**6 + 1575*b**4*d**2*e**3*x**7 + 700*b**4*d 
*e**4*x**8 + 126*b**4*e**5*x**9))/1260