\(\int (a+b x) (d+e x)^8 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [148]

Optimal result

Integrand size = 33, antiderivative size = 362 \[ \int (a+b x) (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(b d-a e)^6 (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}-\frac {3 b (b d-a e)^5 (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac {5 b^3 (b d-a e)^3 (d+e x)^{12} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{13} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}-\frac {3 b^5 (b d-a e) (d+e x)^{14} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}+\frac {b^6 (d+e x)^{15} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x)} \] Output:

1/9*(-a*e+b*d)^6*(e*x+d)^9*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-3/5*b*(-a*e+b*d)^ 
5*(e*x+d)^10*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+15/11*b^2*(-a*e+b*d)^4*(e*x+d)^ 
11*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-5/3*b^3*(-a*e+b*d)^3*(e*x+d)^12*((b*x+a)^ 
2)^(1/2)/e^7/(b*x+a)+15/13*b^4*(-a*e+b*d)^2*(e*x+d)^13*((b*x+a)^2)^(1/2)/e 
^7/(b*x+a)-3/7*b^5*(-a*e+b*d)*(e*x+d)^14*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+1/1 
5*b^6*(e*x+d)^15*((b*x+a)^2)^(1/2)/e^7/(b*x+a)
 

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 679, normalized size of antiderivative = 1.88 \[ \int (a+b x) (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (5005 a^6 \left (9 d^8+36 d^7 e x+84 d^6 e^2 x^2+126 d^5 e^3 x^3+126 d^4 e^4 x^4+84 d^3 e^5 x^5+36 d^2 e^6 x^6+9 d e^7 x^7+e^8 x^8\right )+3003 a^5 b x \left (45 d^8+240 d^7 e x+630 d^6 e^2 x^2+1008 d^5 e^3 x^3+1050 d^4 e^4 x^4+720 d^3 e^5 x^5+315 d^2 e^6 x^6+80 d e^7 x^7+9 e^8 x^8\right )+1365 a^4 b^2 x^2 \left (165 d^8+990 d^7 e x+2772 d^6 e^2 x^2+4620 d^5 e^3 x^3+4950 d^4 e^4 x^4+3465 d^3 e^5 x^5+1540 d^2 e^6 x^6+396 d e^7 x^7+45 e^8 x^8\right )+455 a^3 b^3 x^3 \left (495 d^8+3168 d^7 e x+9240 d^6 e^2 x^2+15840 d^5 e^3 x^3+17325 d^4 e^4 x^4+12320 d^3 e^5 x^5+5544 d^2 e^6 x^6+1440 d e^7 x^7+165 e^8 x^8\right )+105 a^2 b^4 x^4 \left (1287 d^8+8580 d^7 e x+25740 d^6 e^2 x^2+45045 d^5 e^3 x^3+50050 d^4 e^4 x^4+36036 d^3 e^5 x^5+16380 d^2 e^6 x^6+4290 d e^7 x^7+495 e^8 x^8\right )+15 a b^5 x^5 \left (3003 d^8+20592 d^7 e x+63063 d^6 e^2 x^2+112112 d^5 e^3 x^3+126126 d^4 e^4 x^4+91728 d^3 e^5 x^5+42042 d^2 e^6 x^6+11088 d e^7 x^7+1287 e^8 x^8\right )+b^6 x^6 \left (6435 d^8+45045 d^7 e x+140140 d^6 e^2 x^2+252252 d^5 e^3 x^3+286650 d^4 e^4 x^4+210210 d^3 e^5 x^5+97020 d^2 e^6 x^6+25740 d e^7 x^7+3003 e^8 x^8\right )\right )}{45045 (a+b x)} \] Input:

Integrate[(a + b*x)*(d + e*x)^8*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
 

Output:

(x*Sqrt[(a + b*x)^2]*(5005*a^6*(9*d^8 + 36*d^7*e*x + 84*d^6*e^2*x^2 + 126* 
d^5*e^3*x^3 + 126*d^4*e^4*x^4 + 84*d^3*e^5*x^5 + 36*d^2*e^6*x^6 + 9*d*e^7* 
x^7 + e^8*x^8) + 3003*a^5*b*x*(45*d^8 + 240*d^7*e*x + 630*d^6*e^2*x^2 + 10 
08*d^5*e^3*x^3 + 1050*d^4*e^4*x^4 + 720*d^3*e^5*x^5 + 315*d^2*e^6*x^6 + 80 
*d*e^7*x^7 + 9*e^8*x^8) + 1365*a^4*b^2*x^2*(165*d^8 + 990*d^7*e*x + 2772*d 
^6*e^2*x^2 + 4620*d^5*e^3*x^3 + 4950*d^4*e^4*x^4 + 3465*d^3*e^5*x^5 + 1540 
*d^2*e^6*x^6 + 396*d*e^7*x^7 + 45*e^8*x^8) + 455*a^3*b^3*x^3*(495*d^8 + 31 
68*d^7*e*x + 9240*d^6*e^2*x^2 + 15840*d^5*e^3*x^3 + 17325*d^4*e^4*x^4 + 12 
320*d^3*e^5*x^5 + 5544*d^2*e^6*x^6 + 1440*d*e^7*x^7 + 165*e^8*x^8) + 105*a 
^2*b^4*x^4*(1287*d^8 + 8580*d^7*e*x + 25740*d^6*e^2*x^2 + 45045*d^5*e^3*x^ 
3 + 50050*d^4*e^4*x^4 + 36036*d^3*e^5*x^5 + 16380*d^2*e^6*x^6 + 4290*d*e^7 
*x^7 + 495*e^8*x^8) + 15*a*b^5*x^5*(3003*d^8 + 20592*d^7*e*x + 63063*d^6*e 
^2*x^2 + 112112*d^5*e^3*x^3 + 126126*d^4*e^4*x^4 + 91728*d^3*e^5*x^5 + 420 
42*d^2*e^6*x^6 + 11088*d*e^7*x^7 + 1287*e^8*x^8) + b^6*x^6*(6435*d^8 + 450 
45*d^7*e*x + 140140*d^6*e^2*x^2 + 252252*d^5*e^3*x^3 + 286650*d^4*e^4*x^4 
+ 210210*d^3*e^5*x^5 + 97020*d^2*e^6*x^6 + 25740*d*e^7*x^7 + 3003*e^8*x^8) 
))/(45045*(a + b*x))
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.56, 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.

\(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^8 \, dx\)

\(\Big \downarrow \) 1187

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 (a+b x)^6 (d+e x)^8dx}{b^5 (a+b x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^6 (d+e x)^8dx}{a+b x}\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^6 (d+e x)^{14}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{13}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{12}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{11}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{10}}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^9}{e^6}+\frac {(a e-b d)^6 (d+e x)^8}{e^6}\right )dx}{a+b x}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {3 b^5 (d+e x)^{14} (b d-a e)}{7 e^7}+\frac {15 b^4 (d+e x)^{13} (b d-a e)^2}{13 e^7}-\frac {5 b^3 (d+e x)^{12} (b d-a e)^3}{3 e^7}+\frac {15 b^2 (d+e x)^{11} (b d-a e)^4}{11 e^7}-\frac {3 b (d+e x)^{10} (b d-a e)^5}{5 e^7}+\frac {(d+e x)^9 (b d-a e)^6}{9 e^7}+\frac {b^6 (d+e x)^{15}}{15 e^7}\right )}{a+b x}\)

Input:

Int[(a + b*x)*(d + e*x)^8*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
 

Output:

(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((b*d - a*e)^6*(d + e*x)^9)/(9*e^7) - (3*b 
*(b*d - a*e)^5*(d + e*x)^10)/(5*e^7) + (15*b^2*(b*d - a*e)^4*(d + e*x)^11) 
/(11*e^7) - (5*b^3*(b*d - a*e)^3*(d + e*x)^12)/(3*e^7) + (15*b^4*(b*d - a* 
e)^2*(d + e*x)^13)/(13*e^7) - (3*b^5*(b*d - a*e)*(d + e*x)^14)/(7*e^7) + ( 
b^6*(d + e*x)^15)/(15*e^7)))/(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. \(924\) vs. \(2(271)=542\).

Time = 2.75 (sec) , antiderivative size = 925, normalized size of antiderivative = 2.56

method	result	size
gosper	\(\frac {x \left (3003 b^{6} e^{8} x^{14}+19305 x^{13} a \,b^{5} e^{8}+25740 x^{13} b^{6} d \,e^{7}+51975 x^{12} a^{2} b^{4} e^{8}+166320 x^{12} a \,b^{5} d \,e^{7}+97020 x^{12} b^{6} d^{2} e^{6}+75075 x^{11} a^{3} b^{3} e^{8}+450450 x^{11} a^{2} b^{4} d \,e^{7}+630630 x^{11} a \,b^{5} d^{2} e^{6}+210210 x^{11} b^{6} d^{3} e^{5}+61425 x^{10} a^{4} b^{2} e^{8}+655200 x^{10} a^{3} b^{3} d \,e^{7}+1719900 x^{10} a^{2} b^{4} d^{2} e^{6}+1375920 x^{10} a \,b^{5} d^{3} e^{5}+286650 x^{10} b^{6} d^{4} e^{4}+27027 x^{9} a^{5} b \,e^{8}+540540 x^{9} a^{4} b^{2} d \,e^{7}+2522520 x^{9} a^{3} b^{3} d^{2} e^{6}+3783780 x^{9} a^{2} b^{4} d^{3} e^{5}+1891890 x^{9} a \,b^{5} d^{4} e^{4}+252252 x^{9} b^{6} d^{5} e^{3}+5005 x^{8} a^{6} e^{8}+240240 x^{8} a^{5} b d \,e^{7}+2102100 x^{8} a^{4} b^{2} d^{2} e^{6}+5605600 x^{8} a^{3} b^{3} d^{3} e^{5}+5255250 x^{8} a^{2} b^{4} d^{4} e^{4}+1681680 x^{8} a \,b^{5} d^{5} e^{3}+140140 x^{8} b^{6} e^{2} d^{6}+45045 a^{6} d \,e^{7} x^{7}+945945 a^{5} b \,d^{2} e^{6} x^{7}+4729725 a^{4} b^{2} d^{3} e^{5} x^{7}+7882875 a^{3} b^{3} d^{4} e^{4} x^{7}+4729725 a^{2} b^{4} d^{5} e^{3} x^{7}+945945 a \,b^{5} d^{6} e^{2} x^{7}+45045 b^{6} d^{7} e \,x^{7}+180180 x^{6} a^{6} d^{2} e^{6}+2162160 x^{6} a^{5} b \,d^{3} e^{5}+6756750 x^{6} a^{4} b^{2} d^{4} e^{4}+7207200 x^{6} a^{3} b^{3} d^{5} e^{3}+2702700 x^{6} a^{2} b^{4} e^{2} d^{6}+308880 x^{6} a \,b^{5} d^{7} e +6435 x^{6} b^{6} d^{8}+420420 x^{5} a^{6} d^{3} e^{5}+3153150 x^{5} a^{5} b \,d^{4} e^{4}+6306300 x^{5} a^{4} b^{2} d^{5} e^{3}+4204200 x^{5} a^{3} b^{3} e^{2} d^{6}+900900 x^{5} a^{2} b^{4} d^{7} e +45045 x^{5} a \,b^{5} d^{8}+630630 x^{4} a^{6} d^{4} e^{4}+3027024 x^{4} a^{5} b \,d^{5} e^{3}+3783780 x^{4} a^{4} b^{2} e^{2} d^{6}+1441440 x^{4} a^{3} b^{3} d^{7} e +135135 x^{4} a^{2} b^{4} d^{8}+630630 a^{6} d^{5} e^{3} x^{3}+1891890 a^{5} b \,d^{6} e^{2} x^{3}+1351350 a^{4} b^{2} d^{7} e \,x^{3}+225225 a^{3} b^{3} d^{8} x^{3}+420420 x^{2} a^{6} e^{2} d^{6}+720720 x^{2} a^{5} b \,d^{7} e +225225 x^{2} a^{4} b^{2} d^{8}+180180 a^{6} d^{7} e x +135135 a^{5} b \,d^{8} x +45045 a^{6} d^{8}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 \left (b x +a \right )^{5}}\)	\(925\)
default	\(\frac {x \left (3003 b^{6} e^{8} x^{14}+19305 x^{13} a \,b^{5} e^{8}+25740 x^{13} b^{6} d \,e^{7}+51975 x^{12} a^{2} b^{4} e^{8}+166320 x^{12} a \,b^{5} d \,e^{7}+97020 x^{12} b^{6} d^{2} e^{6}+75075 x^{11} a^{3} b^{3} e^{8}+450450 x^{11} a^{2} b^{4} d \,e^{7}+630630 x^{11} a \,b^{5} d^{2} e^{6}+210210 x^{11} b^{6} d^{3} e^{5}+61425 x^{10} a^{4} b^{2} e^{8}+655200 x^{10} a^{3} b^{3} d \,e^{7}+1719900 x^{10} a^{2} b^{4} d^{2} e^{6}+1375920 x^{10} a \,b^{5} d^{3} e^{5}+286650 x^{10} b^{6} d^{4} e^{4}+27027 x^{9} a^{5} b \,e^{8}+540540 x^{9} a^{4} b^{2} d \,e^{7}+2522520 x^{9} a^{3} b^{3} d^{2} e^{6}+3783780 x^{9} a^{2} b^{4} d^{3} e^{5}+1891890 x^{9} a \,b^{5} d^{4} e^{4}+252252 x^{9} b^{6} d^{5} e^{3}+5005 x^{8} a^{6} e^{8}+240240 x^{8} a^{5} b d \,e^{7}+2102100 x^{8} a^{4} b^{2} d^{2} e^{6}+5605600 x^{8} a^{3} b^{3} d^{3} e^{5}+5255250 x^{8} a^{2} b^{4} d^{4} e^{4}+1681680 x^{8} a \,b^{5} d^{5} e^{3}+140140 x^{8} b^{6} e^{2} d^{6}+45045 a^{6} d \,e^{7} x^{7}+945945 a^{5} b \,d^{2} e^{6} x^{7}+4729725 a^{4} b^{2} d^{3} e^{5} x^{7}+7882875 a^{3} b^{3} d^{4} e^{4} x^{7}+4729725 a^{2} b^{4} d^{5} e^{3} x^{7}+945945 a \,b^{5} d^{6} e^{2} x^{7}+45045 b^{6} d^{7} e \,x^{7}+180180 x^{6} a^{6} d^{2} e^{6}+2162160 x^{6} a^{5} b \,d^{3} e^{5}+6756750 x^{6} a^{4} b^{2} d^{4} e^{4}+7207200 x^{6} a^{3} b^{3} d^{5} e^{3}+2702700 x^{6} a^{2} b^{4} e^{2} d^{6}+308880 x^{6} a \,b^{5} d^{7} e +6435 x^{6} b^{6} d^{8}+420420 x^{5} a^{6} d^{3} e^{5}+3153150 x^{5} a^{5} b \,d^{4} e^{4}+6306300 x^{5} a^{4} b^{2} d^{5} e^{3}+4204200 x^{5} a^{3} b^{3} e^{2} d^{6}+900900 x^{5} a^{2} b^{4} d^{7} e +45045 x^{5} a \,b^{5} d^{8}+630630 x^{4} a^{6} d^{4} e^{4}+3027024 x^{4} a^{5} b \,d^{5} e^{3}+3783780 x^{4} a^{4} b^{2} e^{2} d^{6}+1441440 x^{4} a^{3} b^{3} d^{7} e +135135 x^{4} a^{2} b^{4} d^{8}+630630 a^{6} d^{5} e^{3} x^{3}+1891890 a^{5} b \,d^{6} e^{2} x^{3}+1351350 a^{4} b^{2} d^{7} e \,x^{3}+225225 a^{3} b^{3} d^{8} x^{3}+420420 x^{2} a^{6} e^{2} d^{6}+720720 x^{2} a^{5} b \,d^{7} e +225225 x^{2} a^{4} b^{2} d^{8}+180180 a^{6} d^{7} e x +135135 a^{5} b \,d^{8} x +45045 a^{6} d^{8}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 \left (b x +a \right )^{5}}\)	\(925\)
orering	\(\frac {x \left (3003 b^{6} e^{8} x^{14}+19305 x^{13} a \,b^{5} e^{8}+25740 x^{13} b^{6} d \,e^{7}+51975 x^{12} a^{2} b^{4} e^{8}+166320 x^{12} a \,b^{5} d \,e^{7}+97020 x^{12} b^{6} d^{2} e^{6}+75075 x^{11} a^{3} b^{3} e^{8}+450450 x^{11} a^{2} b^{4} d \,e^{7}+630630 x^{11} a \,b^{5} d^{2} e^{6}+210210 x^{11} b^{6} d^{3} e^{5}+61425 x^{10} a^{4} b^{2} e^{8}+655200 x^{10} a^{3} b^{3} d \,e^{7}+1719900 x^{10} a^{2} b^{4} d^{2} e^{6}+1375920 x^{10} a \,b^{5} d^{3} e^{5}+286650 x^{10} b^{6} d^{4} e^{4}+27027 x^{9} a^{5} b \,e^{8}+540540 x^{9} a^{4} b^{2} d \,e^{7}+2522520 x^{9} a^{3} b^{3} d^{2} e^{6}+3783780 x^{9} a^{2} b^{4} d^{3} e^{5}+1891890 x^{9} a \,b^{5} d^{4} e^{4}+252252 x^{9} b^{6} d^{5} e^{3}+5005 x^{8} a^{6} e^{8}+240240 x^{8} a^{5} b d \,e^{7}+2102100 x^{8} a^{4} b^{2} d^{2} e^{6}+5605600 x^{8} a^{3} b^{3} d^{3} e^{5}+5255250 x^{8} a^{2} b^{4} d^{4} e^{4}+1681680 x^{8} a \,b^{5} d^{5} e^{3}+140140 x^{8} b^{6} e^{2} d^{6}+45045 a^{6} d \,e^{7} x^{7}+945945 a^{5} b \,d^{2} e^{6} x^{7}+4729725 a^{4} b^{2} d^{3} e^{5} x^{7}+7882875 a^{3} b^{3} d^{4} e^{4} x^{7}+4729725 a^{2} b^{4} d^{5} e^{3} x^{7}+945945 a \,b^{5} d^{6} e^{2} x^{7}+45045 b^{6} d^{7} e \,x^{7}+180180 x^{6} a^{6} d^{2} e^{6}+2162160 x^{6} a^{5} b \,d^{3} e^{5}+6756750 x^{6} a^{4} b^{2} d^{4} e^{4}+7207200 x^{6} a^{3} b^{3} d^{5} e^{3}+2702700 x^{6} a^{2} b^{4} e^{2} d^{6}+308880 x^{6} a \,b^{5} d^{7} e +6435 x^{6} b^{6} d^{8}+420420 x^{5} a^{6} d^{3} e^{5}+3153150 x^{5} a^{5} b \,d^{4} e^{4}+6306300 x^{5} a^{4} b^{2} d^{5} e^{3}+4204200 x^{5} a^{3} b^{3} e^{2} d^{6}+900900 x^{5} a^{2} b^{4} d^{7} e +45045 x^{5} a \,b^{5} d^{8}+630630 x^{4} a^{6} d^{4} e^{4}+3027024 x^{4} a^{5} b \,d^{5} e^{3}+3783780 x^{4} a^{4} b^{2} e^{2} d^{6}+1441440 x^{4} a^{3} b^{3} d^{7} e +135135 x^{4} a^{2} b^{4} d^{8}+630630 a^{6} d^{5} e^{3} x^{3}+1891890 a^{5} b \,d^{6} e^{2} x^{3}+1351350 a^{4} b^{2} d^{7} e \,x^{3}+225225 a^{3} b^{3} d^{8} x^{3}+420420 x^{2} a^{6} e^{2} d^{6}+720720 x^{2} a^{5} b \,d^{7} e +225225 x^{2} a^{4} b^{2} d^{8}+180180 a^{6} d^{7} e x +135135 a^{5} b \,d^{8} x +45045 a^{6} d^{8}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{45045 \left (b x +a \right )^{5}}\)	\(934\)
risch	\(\text {Expression too large to display}\)	\(1043\)

Input:

int((b*x+a)*(e*x+d)^8*(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/45045*x*(3003*b^6*e^8*x^14+19305*a*b^5*e^8*x^13+25740*b^6*d*e^7*x^13+519 
75*a^2*b^4*e^8*x^12+166320*a*b^5*d*e^7*x^12+97020*b^6*d^2*e^6*x^12+75075*a 
^3*b^3*e^8*x^11+450450*a^2*b^4*d*e^7*x^11+630630*a*b^5*d^2*e^6*x^11+210210 
*b^6*d^3*e^5*x^11+61425*a^4*b^2*e^8*x^10+655200*a^3*b^3*d*e^7*x^10+1719900 
*a^2*b^4*d^2*e^6*x^10+1375920*a*b^5*d^3*e^5*x^10+286650*b^6*d^4*e^4*x^10+2 
7027*a^5*b*e^8*x^9+540540*a^4*b^2*d*e^7*x^9+2522520*a^3*b^3*d^2*e^6*x^9+37 
83780*a^2*b^4*d^3*e^5*x^9+1891890*a*b^5*d^4*e^4*x^9+252252*b^6*d^5*e^3*x^9 
+5005*a^6*e^8*x^8+240240*a^5*b*d*e^7*x^8+2102100*a^4*b^2*d^2*e^6*x^8+56056 
00*a^3*b^3*d^3*e^5*x^8+5255250*a^2*b^4*d^4*e^4*x^8+1681680*a*b^5*d^5*e^3*x 
^8+140140*b^6*d^6*e^2*x^8+45045*a^6*d*e^7*x^7+945945*a^5*b*d^2*e^6*x^7+472 
9725*a^4*b^2*d^3*e^5*x^7+7882875*a^3*b^3*d^4*e^4*x^7+4729725*a^2*b^4*d^5*e 
^3*x^7+945945*a*b^5*d^6*e^2*x^7+45045*b^6*d^7*e*x^7+180180*a^6*d^2*e^6*x^6 
+2162160*a^5*b*d^3*e^5*x^6+6756750*a^4*b^2*d^4*e^4*x^6+7207200*a^3*b^3*d^5 
*e^3*x^6+2702700*a^2*b^4*d^6*e^2*x^6+308880*a*b^5*d^7*e*x^6+6435*b^6*d^8*x 
^6+420420*a^6*d^3*e^5*x^5+3153150*a^5*b*d^4*e^4*x^5+6306300*a^4*b^2*d^5*e^ 
3*x^5+4204200*a^3*b^3*d^6*e^2*x^5+900900*a^2*b^4*d^7*e*x^5+45045*a*b^5*d^8 
*x^5+630630*a^6*d^4*e^4*x^4+3027024*a^5*b*d^5*e^3*x^4+3783780*a^4*b^2*d^6* 
e^2*x^4+1441440*a^3*b^3*d^7*e*x^4+135135*a^2*b^4*d^8*x^4+630630*a^6*d^5*e^ 
3*x^3+1891890*a^5*b*d^6*e^2*x^3+1351350*a^4*b^2*d^7*e*x^3+225225*a^3*b^3*d 
^8*x^3+420420*a^6*d^6*e^2*x^2+720720*a^5*b*d^7*e*x^2+225225*a^4*b^2*d^8...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 797 vs. \(2 (271) = 542\).

Time = 0.08 (sec) , antiderivative size = 797, normalized size of antiderivative = 2.20 \[ \int (a+b x) (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)*(e*x+d)^8*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fric 
as")
 

Output:

1/15*b^6*e^8*x^15 + a^6*d^8*x + 1/7*(4*b^6*d*e^7 + 3*a*b^5*e^8)*x^14 + 1/1 
3*(28*b^6*d^2*e^6 + 48*a*b^5*d*e^7 + 15*a^2*b^4*e^8)*x^13 + 1/3*(14*b^6*d^ 
3*e^5 + 42*a*b^5*d^2*e^6 + 30*a^2*b^4*d*e^7 + 5*a^3*b^3*e^8)*x^12 + 1/11*( 
70*b^6*d^4*e^4 + 336*a*b^5*d^3*e^5 + 420*a^2*b^4*d^2*e^6 + 160*a^3*b^3*d*e 
^7 + 15*a^4*b^2*e^8)*x^11 + 1/5*(28*b^6*d^5*e^3 + 210*a*b^5*d^4*e^4 + 420* 
a^2*b^4*d^3*e^5 + 280*a^3*b^3*d^2*e^6 + 60*a^4*b^2*d*e^7 + 3*a^5*b*e^8)*x^ 
10 + 1/9*(28*b^6*d^6*e^2 + 336*a*b^5*d^5*e^3 + 1050*a^2*b^4*d^4*e^4 + 1120 
*a^3*b^3*d^3*e^5 + 420*a^4*b^2*d^2*e^6 + 48*a^5*b*d*e^7 + a^6*e^8)*x^9 + ( 
b^6*d^7*e + 21*a*b^5*d^6*e^2 + 105*a^2*b^4*d^5*e^3 + 175*a^3*b^3*d^4*e^4 + 
 105*a^4*b^2*d^3*e^5 + 21*a^5*b*d^2*e^6 + a^6*d*e^7)*x^8 + 1/7*(b^6*d^8 + 
48*a*b^5*d^7*e + 420*a^2*b^4*d^6*e^2 + 1120*a^3*b^3*d^5*e^3 + 1050*a^4*b^2 
*d^4*e^4 + 336*a^5*b*d^3*e^5 + 28*a^6*d^2*e^6)*x^7 + 1/3*(3*a*b^5*d^8 + 60 
*a^2*b^4*d^7*e + 280*a^3*b^3*d^6*e^2 + 420*a^4*b^2*d^5*e^3 + 210*a^5*b*d^4 
*e^4 + 28*a^6*d^3*e^5)*x^6 + 1/5*(15*a^2*b^4*d^8 + 160*a^3*b^3*d^7*e + 420 
*a^4*b^2*d^6*e^2 + 336*a^5*b*d^5*e^3 + 70*a^6*d^4*e^4)*x^5 + (5*a^3*b^3*d^ 
8 + 30*a^4*b^2*d^7*e + 42*a^5*b*d^6*e^2 + 14*a^6*d^5*e^3)*x^4 + 1/3*(15*a^ 
4*b^2*d^8 + 48*a^5*b*d^7*e + 28*a^6*d^6*e^2)*x^3 + (3*a^5*b*d^8 + 4*a^6*d^ 
7*e)*x^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 216311 vs. \(2 (265) = 530\).

Time = 2.67 (sec) , antiderivative size = 216311, normalized size of antiderivative = 597.54 \[ \int (a+b x) (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)*(e*x+d)**8*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
 

Output:

Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**5*e**8*x**14/15 + x**13*(7 
6*a*b**6*e**8/15 + 8*b**7*d*e**7)/(14*b**2) + x**12*(301*a**2*b**5*e**8/15 
 + 56*a*b**6*d*e**7 - 27*a*(76*a*b**6*e**8/15 + 8*b**7*d*e**7)/(14*b) + 28 
*b**7*d**2*e**6)/(13*b**2) + x**11*(35*a**3*b**4*e**8 + 168*a**2*b**5*d*e* 
*7 - 13*a**2*(76*a*b**6*e**8/15 + 8*b**7*d*e**7)/(14*b**2) + 196*a*b**6*d* 
*2*e**6 - 25*a*(301*a**2*b**5*e**8/15 + 56*a*b**6*d*e**7 - 27*a*(76*a*b**6 
*e**8/15 + 8*b**7*d*e**7)/(14*b) + 28*b**7*d**2*e**6)/(13*b) + 56*b**7*d** 
3*e**5)/(12*b**2) + x**10*(35*a**4*b**3*e**8 + 280*a**3*b**4*d*e**7 + 588* 
a**2*b**5*d**2*e**6 - 12*a**2*(301*a**2*b**5*e**8/15 + 56*a*b**6*d*e**7 - 
27*a*(76*a*b**6*e**8/15 + 8*b**7*d*e**7)/(14*b) + 28*b**7*d**2*e**6)/(13*b 
**2) + 392*a*b**6*d**3*e**5 - 23*a*(35*a**3*b**4*e**8 + 168*a**2*b**5*d*e* 
*7 - 13*a**2*(76*a*b**6*e**8/15 + 8*b**7*d*e**7)/(14*b**2) + 196*a*b**6*d* 
*2*e**6 - 25*a*(301*a**2*b**5*e**8/15 + 56*a*b**6*d*e**7 - 27*a*(76*a*b**6 
*e**8/15 + 8*b**7*d*e**7)/(14*b) + 28*b**7*d**2*e**6)/(13*b) + 56*b**7*d** 
3*e**5)/(12*b) + 70*b**7*d**4*e**4)/(11*b**2) + x**9*(21*a**5*b**2*e**8 + 
280*a**4*b**3*d*e**7 + 980*a**3*b**4*d**2*e**6 + 1176*a**2*b**5*d**3*e**5 
- 11*a**2*(35*a**3*b**4*e**8 + 168*a**2*b**5*d*e**7 - 13*a**2*(76*a*b**6*e 
**8/15 + 8*b**7*d*e**7)/(14*b**2) + 196*a*b**6*d**2*e**6 - 25*a*(301*a**2* 
b**5*e**8/15 + 56*a*b**6*d*e**7 - 27*a*(76*a*b**6*e**8/15 + 8*b**7*d*e**7) 
/(14*b) + 28*b**7*d**2*e**6)/(13*b) + 56*b**7*d**3*e**5)/(12*b**2) + 49...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2653 vs. \(2 (271) = 542\).

Time = 0.06 (sec) , antiderivative size = 2653, normalized size of antiderivative = 7.33 \[ \int (a+b x) (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)*(e*x+d)^8*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxi 
ma")
 

Output:

1/15*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*e^8*x^8/b - 23/210*(b^2*x^2 + 2*a*b*x 
 + a^2)^(7/2)*a*e^8*x^7/b^2 + 53/390*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*e 
^8*x^6/b^3 - 59/390*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*e^8*x^5/b^4 + 137/ 
858*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4*e^8*x^4/b^5 - 703/4290*(b^2*x^2 + 
2*a*b*x + a^2)^(7/2)*a^5*e^8*x^3/b^6 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2) 
*a*d^8*x - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^9*e^8*x/b^8 + 237/1430*(b 
^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^6*e^8*x^2/b^7 + 1/6*(b^2*x^2 + 2*a*b*x + a 
^2)^(5/2)*a^2*d^8/b - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^10*e^8/b^9 - 1 
19/715*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^7*e^8*x/b^8 + 834/5005*(b^2*x^2 + 
 2*a*b*x + a^2)^(7/2)*a^8*e^8/b^9 + 1/14*(8*b*d*e^7 + a*e^8)*(b^2*x^2 + 2* 
a*b*x + a^2)^(7/2)*x^7/b^2 - 3/26*(8*b*d*e^7 + a*e^8)*(b^2*x^2 + 2*a*b*x + 
 a^2)^(7/2)*a*x^6/b^3 + 4/13*(7*b*d^2*e^6 + 2*a*d*e^7)*(b^2*x^2 + 2*a*b*x 
+ a^2)^(7/2)*x^6/b^2 + 11/78*(8*b*d*e^7 + a*e^8)*(b^2*x^2 + 2*a*b*x + a^2) 
^(7/2)*a^2*x^5/b^4 - 19/39*(7*b*d^2*e^6 + 2*a*d*e^7)*(b^2*x^2 + 2*a*b*x + 
a^2)^(7/2)*a*x^5/b^3 + 7/3*(2*b*d^3*e^5 + a*d^2*e^6)*(b^2*x^2 + 2*a*b*x + 
a^2)^(7/2)*x^5/b^2 - 133/858*(8*b*d*e^7 + a*e^8)*(b^2*x^2 + 2*a*b*x + a^2) 
^(7/2)*a^3*x^4/b^5 + 251/429*(7*b*d^2*e^6 + 2*a*d*e^7)*(b^2*x^2 + 2*a*b*x 
+ a^2)^(7/2)*a^2*x^4/b^4 - 119/33*(2*b*d^3*e^5 + a*d^2*e^6)*(b^2*x^2 + 2*a 
*b*x + a^2)^(7/2)*a*x^4/b^3 + 14/11*(5*b*d^4*e^4 + 4*a*d^3*e^5)*(b^2*x^2 + 
 2*a*b*x + a^2)^(7/2)*x^4/b^2 + 139/858*(8*b*d*e^7 + a*e^8)*(b^2*x^2 + ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1406 vs. \(2 (271) = 542\).

Time = 0.21 (sec) , antiderivative size = 1406, normalized size of antiderivative = 3.88 \[ \int (a+b x) (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)*(e*x+d)^8*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac 
")
 

Output:

1/15*b^6*e^8*x^15*sgn(b*x + a) + 4/7*b^6*d*e^7*x^14*sgn(b*x + a) + 3/7*a*b 
^5*e^8*x^14*sgn(b*x + a) + 28/13*b^6*d^2*e^6*x^13*sgn(b*x + a) + 48/13*a*b 
^5*d*e^7*x^13*sgn(b*x + a) + 15/13*a^2*b^4*e^8*x^13*sgn(b*x + a) + 14/3*b^ 
6*d^3*e^5*x^12*sgn(b*x + a) + 14*a*b^5*d^2*e^6*x^12*sgn(b*x + a) + 10*a^2* 
b^4*d*e^7*x^12*sgn(b*x + a) + 5/3*a^3*b^3*e^8*x^12*sgn(b*x + a) + 70/11*b^ 
6*d^4*e^4*x^11*sgn(b*x + a) + 336/11*a*b^5*d^3*e^5*x^11*sgn(b*x + a) + 420 
/11*a^2*b^4*d^2*e^6*x^11*sgn(b*x + a) + 160/11*a^3*b^3*d*e^7*x^11*sgn(b*x 
+ a) + 15/11*a^4*b^2*e^8*x^11*sgn(b*x + a) + 28/5*b^6*d^5*e^3*x^10*sgn(b*x 
 + a) + 42*a*b^5*d^4*e^4*x^10*sgn(b*x + a) + 84*a^2*b^4*d^3*e^5*x^10*sgn(b 
*x + a) + 56*a^3*b^3*d^2*e^6*x^10*sgn(b*x + a) + 12*a^4*b^2*d*e^7*x^10*sgn 
(b*x + a) + 3/5*a^5*b*e^8*x^10*sgn(b*x + a) + 28/9*b^6*d^6*e^2*x^9*sgn(b*x 
 + a) + 112/3*a*b^5*d^5*e^3*x^9*sgn(b*x + a) + 350/3*a^2*b^4*d^4*e^4*x^9*s 
gn(b*x + a) + 1120/9*a^3*b^3*d^3*e^5*x^9*sgn(b*x + a) + 140/3*a^4*b^2*d^2* 
e^6*x^9*sgn(b*x + a) + 16/3*a^5*b*d*e^7*x^9*sgn(b*x + a) + 1/9*a^6*e^8*x^9 
*sgn(b*x + a) + b^6*d^7*e*x^8*sgn(b*x + a) + 21*a*b^5*d^6*e^2*x^8*sgn(b*x 
+ a) + 105*a^2*b^4*d^5*e^3*x^8*sgn(b*x + a) + 175*a^3*b^3*d^4*e^4*x^8*sgn( 
b*x + a) + 105*a^4*b^2*d^3*e^5*x^8*sgn(b*x + a) + 21*a^5*b*d^2*e^6*x^8*sgn 
(b*x + a) + a^6*d*e^7*x^8*sgn(b*x + a) + 1/7*b^6*d^8*x^7*sgn(b*x + a) + 48 
/7*a*b^5*d^7*e*x^7*sgn(b*x + a) + 60*a^2*b^4*d^6*e^2*x^7*sgn(b*x + a) + 16 
0*a^3*b^3*d^5*e^3*x^7*sgn(b*x + a) + 150*a^4*b^2*d^4*e^4*x^7*sgn(b*x + ...
 

Mupad [F(-1)]

Timed out. \[ \int (a+b x) (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^8\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \] Input:

int((a + b*x)*(d + e*x)^8*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)
 

Output:

int((a + b*x)*(d + e*x)^8*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 908, normalized size of antiderivative = 2.51 \[ \int (a+b x) (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

int((b*x+a)*(e*x+d)^8*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
 

Output:

(x*(45045*a**6*d**8 + 180180*a**6*d**7*e*x + 420420*a**6*d**6*e**2*x**2 + 
630630*a**6*d**5*e**3*x**3 + 630630*a**6*d**4*e**4*x**4 + 420420*a**6*d**3 
*e**5*x**5 + 180180*a**6*d**2*e**6*x**6 + 45045*a**6*d*e**7*x**7 + 5005*a* 
*6*e**8*x**8 + 135135*a**5*b*d**8*x + 720720*a**5*b*d**7*e*x**2 + 1891890* 
a**5*b*d**6*e**2*x**3 + 3027024*a**5*b*d**5*e**3*x**4 + 3153150*a**5*b*d** 
4*e**4*x**5 + 2162160*a**5*b*d**3*e**5*x**6 + 945945*a**5*b*d**2*e**6*x**7 
 + 240240*a**5*b*d*e**7*x**8 + 27027*a**5*b*e**8*x**9 + 225225*a**4*b**2*d 
**8*x**2 + 1351350*a**4*b**2*d**7*e*x**3 + 3783780*a**4*b**2*d**6*e**2*x** 
4 + 6306300*a**4*b**2*d**5*e**3*x**5 + 6756750*a**4*b**2*d**4*e**4*x**6 + 
4729725*a**4*b**2*d**3*e**5*x**7 + 2102100*a**4*b**2*d**2*e**6*x**8 + 5405 
40*a**4*b**2*d*e**7*x**9 + 61425*a**4*b**2*e**8*x**10 + 225225*a**3*b**3*d 
**8*x**3 + 1441440*a**3*b**3*d**7*e*x**4 + 4204200*a**3*b**3*d**6*e**2*x** 
5 + 7207200*a**3*b**3*d**5*e**3*x**6 + 7882875*a**3*b**3*d**4*e**4*x**7 + 
5605600*a**3*b**3*d**3*e**5*x**8 + 2522520*a**3*b**3*d**2*e**6*x**9 + 6552 
00*a**3*b**3*d*e**7*x**10 + 75075*a**3*b**3*e**8*x**11 + 135135*a**2*b**4* 
d**8*x**4 + 900900*a**2*b**4*d**7*e*x**5 + 2702700*a**2*b**4*d**6*e**2*x** 
6 + 4729725*a**2*b**4*d**5*e**3*x**7 + 5255250*a**2*b**4*d**4*e**4*x**8 + 
3783780*a**2*b**4*d**3*e**5*x**9 + 1719900*a**2*b**4*d**2*e**6*x**10 + 450 
450*a**2*b**4*d*e**7*x**11 + 51975*a**2*b**4*e**8*x**12 + 45045*a*b**5*d** 
8*x**5 + 308880*a*b**5*d**7*e*x**6 + 945945*a*b**5*d**6*e**2*x**7 + 168...