\(\int (d+e x)^4 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [292]

Optimal result

Integrand size = 28, antiderivative size = 219 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(b d-a e)^4 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^5}+\frac {4 e (b d-a e)^3 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac {3 e^2 (b d-a e)^2 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^5}+\frac {4 e^3 (b d-a e) (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^5}+\frac {e^4 (a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{10 b^5} \] Output:

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

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.46 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (252 a^5 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+210 a^4 b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+120 a^3 b^2 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+45 a^2 b^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )+10 a b^4 x^4 \left (126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right )+b^5 x^5 \left (210 d^4+720 d^3 e x+945 d^2 e^2 x^2+560 d e^3 x^3+126 e^4 x^4\right )\right )}{1260 (a+b x)} \] Input:

Integrate[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
 

Output:

(x*Sqrt[(a + b*x)^2]*(252*a^5*(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) + 210*a^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) + 120*a^3*b^2*x^2*(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) + 45*a^2*b^3*x^3*(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) + 10*a*b^4*x^4*(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) + b^5*x^5*(2 
10*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)))/(1 
260*(a + b*x))
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.67, 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.

\(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^4 \, dx\)

\(\Big \downarrow \) 1102

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

Input:

Int[(d + e*x)^4*(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)^4*(a + b*x)^6)/(6*b^5) + (4*e 
*(b*d - a*e)^3*(a + b*x)^7)/(7*b^5) + (3*e^2*(b*d - a*e)^2*(a + b*x)^8)/(4 
*b^5) + (4*e^3*(b*d - a*e)*(a + b*x)^9)/(9*b^5) + (e^4*(a + b*x)^10)/(10*b 
^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 1102

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]
 

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. \(2(154)=308\).

Time = 1.34 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.89

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

Input:

int((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/1260*x*(126*b^5*e^4*x^9+700*a*b^4*e^4*x^8+560*b^5*d*e^3*x^8+1575*a^2*b^3 
*e^4*x^7+3150*a*b^4*d*e^3*x^7+945*b^5*d^2*e^2*x^7+1800*a^3*b^2*e^4*x^6+720 
0*a^2*b^3*d*e^3*x^6+5400*a*b^4*d^2*e^2*x^6+720*b^5*d^3*e*x^6+1050*a^4*b*e^ 
4*x^5+8400*a^3*b^2*d*e^3*x^5+12600*a^2*b^3*d^2*e^2*x^5+4200*a*b^4*d^3*e*x^ 
5+210*b^5*d^4*x^5+252*a^5*e^4*x^4+5040*a^4*b*d*e^3*x^4+15120*a^3*b^2*d^2*e 
^2*x^4+10080*a^2*b^3*d^3*e*x^4+1260*a*b^4*d^4*x^4+1260*a^5*d*e^3*x^3+9450* 
a^4*b*d^2*e^2*x^3+12600*a^3*b^2*d^3*e*x^3+3150*a^2*b^3*d^4*x^3+2520*a^5*d^ 
2*e^2*x^2+8400*a^4*b*d^3*e*x^2+4200*a^3*b^2*d^4*x^2+2520*a^5*d^3*e*x+3150* 
a^4*b*d^4*x+1260*a^5*d^4)*((b*x+a)^2)^(5/2)/(b*x+a)^5
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (154) = 308\).

Time = 0.08 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.64 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{10} \, b^{5} e^{4} x^{10} + a^{5} d^{4} x + \frac {1}{9} \, {\left (4 \, b^{5} d e^{3} + 5 \, a b^{4} e^{4}\right )} x^{9} + \frac {1}{4} \, {\left (3 \, b^{5} d^{2} e^{2} + 10 \, a b^{4} d e^{3} + 5 \, a^{2} b^{3} e^{4}\right )} x^{8} + \frac {2}{7} \, {\left (2 \, b^{5} d^{3} e + 15 \, a b^{4} d^{2} e^{2} + 20 \, a^{2} b^{3} d e^{3} + 5 \, a^{3} b^{2} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{4} + 20 \, a b^{4} d^{3} e + 60 \, a^{2} b^{3} d^{2} e^{2} + 40 \, a^{3} b^{2} d e^{3} + 5 \, a^{4} b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, a b^{4} d^{4} + 40 \, a^{2} b^{3} d^{3} e + 60 \, a^{3} b^{2} d^{2} e^{2} + 20 \, a^{4} b d e^{3} + a^{5} e^{4}\right )} x^{5} + \frac {1}{2} \, {\left (5 \, a^{2} b^{3} d^{4} + 20 \, a^{3} b^{2} d^{3} e + 15 \, a^{4} b d^{2} e^{2} + 2 \, a^{5} d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (5 \, a^{3} b^{2} d^{4} + 10 \, a^{4} b d^{3} e + 3 \, a^{5} d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d^{4} + 4 \, a^{5} d^{3} e\right )} x^{2} \] Input:

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 16842 vs. \(2 (158) = 316\).

Time = 1.22 (sec) , antiderivative size = 16842, normalized size of antiderivative = 76.90 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)**4*(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**4*e**4*x**9/10 + x**8*(41* 
a*b**5*e**4/10 + 4*b**6*d*e**3)/(9*b**2) + x**7*(141*a**2*b**4*e**4/10 + 2 
4*a*b**5*d*e**3 - 17*a*(41*a*b**5*e**4/10 + 4*b**6*d*e**3)/(9*b) + 6*b**6* 
d**2*e**2)/(8*b**2) + x**6*(20*a**3*b**3*e**4 + 60*a**2*b**4*d*e**3 - 8*a* 
*2*(41*a*b**5*e**4/10 + 4*b**6*d*e**3)/(9*b**2) + 36*a*b**5*d**2*e**2 - 15 
*a*(141*a**2*b**4*e**4/10 + 24*a*b**5*d*e**3 - 17*a*(41*a*b**5*e**4/10 + 4 
*b**6*d*e**3)/(9*b) + 6*b**6*d**2*e**2)/(8*b) + 4*b**6*d**3*e)/(7*b**2) + 
x**5*(15*a**4*b**2*e**4 + 80*a**3*b**3*d*e**3 + 90*a**2*b**4*d**2*e**2 - 7 
*a**2*(141*a**2*b**4*e**4/10 + 24*a*b**5*d*e**3 - 17*a*(41*a*b**5*e**4/10 
+ 4*b**6*d*e**3)/(9*b) + 6*b**6*d**2*e**2)/(8*b**2) + 24*a*b**5*d**3*e - 1 
3*a*(20*a**3*b**3*e**4 + 60*a**2*b**4*d*e**3 - 8*a**2*(41*a*b**5*e**4/10 + 
 4*b**6*d*e**3)/(9*b**2) + 36*a*b**5*d**2*e**2 - 15*a*(141*a**2*b**4*e**4/ 
10 + 24*a*b**5*d*e**3 - 17*a*(41*a*b**5*e**4/10 + 4*b**6*d*e**3)/(9*b) + 6 
*b**6*d**2*e**2)/(8*b) + 4*b**6*d**3*e)/(7*b) + b**6*d**4)/(6*b**2) + x**4 
*(6*a**5*b*e**4 + 60*a**4*b**2*d*e**3 + 120*a**3*b**3*d**2*e**2 + 60*a**2* 
b**4*d**3*e - 6*a**2*(20*a**3*b**3*e**4 + 60*a**2*b**4*d*e**3 - 8*a**2*(41 
*a*b**5*e**4/10 + 4*b**6*d*e**3)/(9*b**2) + 36*a*b**5*d**2*e**2 - 15*a*(14 
1*a**2*b**4*e**4/10 + 24*a*b**5*d*e**3 - 17*a*(41*a*b**5*e**4/10 + 4*b**6* 
d*e**3)/(9*b) + 6*b**6*d**2*e**2)/(8*b) + 4*b**6*d**3*e)/(7*b**2) + 6*a*b* 
*5*d**4 - 11*a*(15*a**4*b**2*e**4 + 80*a**3*b**3*d*e**3 + 90*a**2*b**4*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (154) = 308\).

Time = 0.04 (sec) , antiderivative size = 588, normalized size of antiderivative = 2.68 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} e^{4} x^{3}}{10 \, b^{2}} + \frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{4} x - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{3} e x}{3 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{2} e^{2} x}{b^{2}} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} d e^{3} x}{3 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} e^{4} x}{6 \, b^{4}} + \frac {4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d e^{3} x^{2}}{9 \, b^{2}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a e^{4} x^{2}}{90 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{4}}{6 \, b} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{3} e}{3 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} d^{2} e^{2}}{b^{3}} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} d e^{3}}{3 \, b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{5} e^{4}}{6 \, b^{5}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d^{2} e^{2} x}{4 \, b^{2}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a d e^{3} x}{18 \, b^{3}} + \frac {29 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} e^{4} x}{180 \, b^{4}} + \frac {4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d^{3} e}{7 \, b^{2}} - \frac {27 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a d^{2} e^{2}}{28 \, b^{3}} + \frac {83 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} d e^{3}}{126 \, b^{4}} - \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{3} e^{4}}{1260 \, b^{5}} \] Input:

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")
 

Output:

1/10*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*e^4*x^3/b^2 + 1/6*(b^2*x^2 + 2*a*b*x 
+ a^2)^(5/2)*d^4*x - 2/3*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^3*e*x/b + (b^ 
2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d^2*e^2*x/b^2 - 2/3*(b^2*x^2 + 2*a*b*x + 
a^2)^(5/2)*a^3*d*e^3*x/b^3 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*e^4*x 
/b^4 + 4/9*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*d*e^3*x^2/b^2 - 13/90*(b^2*x^2 
+ 2*a*b*x + a^2)^(7/2)*a*e^4*x^2/b^3 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2) 
*a*d^4/b - 2/3*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d^3*e/b^2 + (b^2*x^2 + 
2*a*b*x + a^2)^(5/2)*a^3*d^2*e^2/b^3 - 2/3*(b^2*x^2 + 2*a*b*x + a^2)^(5/2) 
*a^4*d*e^3/b^4 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5*e^4/b^5 + 3/4*(b^ 
2*x^2 + 2*a*b*x + a^2)^(7/2)*d^2*e^2*x/b^2 - 11/18*(b^2*x^2 + 2*a*b*x + a^ 
2)^(7/2)*a*d*e^3*x/b^3 + 29/180*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*e^4*x/ 
b^4 + 4/7*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*d^3*e/b^2 - 27/28*(b^2*x^2 + 2*a 
*b*x + a^2)^(7/2)*a*d^2*e^2/b^3 + 83/126*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a 
^2*d*e^3/b^4 - 209/1260*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*e^4/b^5
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 642 vs. \(2 (154) = 308\).

Time = 0.23 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.93 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int {\left (d+e\,x\right )}^4\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \] Input:

int((d + e*x)^4*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)
 

Output:

int((d + e*x)^4*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.81 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \left (126 b^{5} e^{4} x^{9}+700 a \,b^{4} e^{4} x^{8}+560 b^{5} d \,e^{3} x^{8}+1575 a^{2} b^{3} e^{4} x^{7}+3150 a \,b^{4} d \,e^{3} x^{7}+945 b^{5} d^{2} e^{2} x^{7}+1800 a^{3} b^{2} e^{4} x^{6}+7200 a^{2} b^{3} d \,e^{3} x^{6}+5400 a \,b^{4} d^{2} e^{2} x^{6}+720 b^{5} d^{3} e \,x^{6}+1050 a^{4} b \,e^{4} x^{5}+8400 a^{3} b^{2} d \,e^{3} x^{5}+12600 a^{2} b^{3} d^{2} e^{2} x^{5}+4200 a \,b^{4} d^{3} e \,x^{5}+210 b^{5} d^{4} x^{5}+252 a^{5} e^{4} x^{4}+5040 a^{4} b d \,e^{3} x^{4}+15120 a^{3} b^{2} d^{2} e^{2} x^{4}+10080 a^{2} b^{3} d^{3} e \,x^{4}+1260 a \,b^{4} d^{4} x^{4}+1260 a^{5} d \,e^{3} x^{3}+9450 a^{4} b \,d^{2} e^{2} x^{3}+12600 a^{3} b^{2} d^{3} e \,x^{3}+3150 a^{2} b^{3} d^{4} x^{3}+2520 a^{5} d^{2} e^{2} x^{2}+8400 a^{4} b \,d^{3} e \,x^{2}+4200 a^{3} b^{2} d^{4} x^{2}+2520 a^{5} d^{3} e x +3150 a^{4} b \,d^{4} x +1260 a^{5} d^{4}\right )}{1260} \] Input:

int((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
 

Output:

(x*(1260*a**5*d**4 + 2520*a**5*d**3*e*x + 2520*a**5*d**2*e**2*x**2 + 1260* 
a**5*d*e**3*x**3 + 252*a**5*e**4*x**4 + 3150*a**4*b*d**4*x + 8400*a**4*b*d 
**3*e*x**2 + 9450*a**4*b*d**2*e**2*x**3 + 5040*a**4*b*d*e**3*x**4 + 1050*a 
**4*b*e**4*x**5 + 4200*a**3*b**2*d**4*x**2 + 12600*a**3*b**2*d**3*e*x**3 + 
 15120*a**3*b**2*d**2*e**2*x**4 + 8400*a**3*b**2*d*e**3*x**5 + 1800*a**3*b 
**2*e**4*x**6 + 3150*a**2*b**3*d**4*x**3 + 10080*a**2*b**3*d**3*e*x**4 + 1 
2600*a**2*b**3*d**2*e**2*x**5 + 7200*a**2*b**3*d*e**3*x**6 + 1575*a**2*b** 
3*e**4*x**7 + 1260*a*b**4*d**4*x**4 + 4200*a*b**4*d**3*e*x**5 + 5400*a*b** 
4*d**2*e**2*x**6 + 3150*a*b**4*d*e**3*x**7 + 700*a*b**4*e**4*x**8 + 210*b* 
*5*d**4*x**5 + 720*b**5*d**3*e*x**6 + 945*b**5*d**2*e**2*x**7 + 560*b**5*d 
*e**3*x**8 + 126*b**5*e**4*x**9))/1260