\(\int (d+e x)^2 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [294]

Optimal result

Integrand size = 28, antiderivative size = 125 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(b d-a e)^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^3}+\frac {2 e (b d-a e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^3}+\frac {e^2 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^3} \] Output:

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

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.50 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (56 a^5 \left (3 d^2+3 d e x+e^2 x^2\right )+70 a^4 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+56 a^3 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+28 a^2 b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+8 a b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )+b^5 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )}{168 (a+b x)} \] Input:

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

Output:

(x*Sqrt[(a + b*x)^2]*(56*a^5*(3*d^2 + 3*d*e*x + e^2*x^2) + 70*a^4*b*x*(6*d 
^2 + 8*d*e*x + 3*e^2*x^2) + 56*a^3*b^2*x^2*(10*d^2 + 15*d*e*x + 6*e^2*x^2) 
 + 28*a^2*b^3*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 8*a*b^4*x^4*(21*d^2 + 
 35*d*e*x + 15*e^2*x^2) + b^5*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2)))/(168* 
(a + b*x))
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.74, 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.

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

\(\Big \downarrow \) 1102

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

Input:

Int[(d + e*x)^2*(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)^2*(a + b*x)^6)/(6*b^3) + (2*e 
*(b*d - a*e)*(a + b*x)^7)/(7*b^3) + (e^2*(a + b*x)^8)/(8*b^3)))/(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. \(229\) vs. \(2(86)=172\).

Time = 1.12 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.84

method	result	size
gosper	\(\frac {x \left (21 b^{5} e^{2} x^{7}+120 x^{6} a \,b^{4} e^{2}+48 x^{6} b^{5} d e +280 x^{5} a^{2} b^{3} e^{2}+280 x^{5} a \,b^{4} d e +28 x^{5} b^{5} d^{2}+336 a^{3} b^{2} e^{2} x^{4}+672 a^{2} b^{3} d e \,x^{4}+168 a \,b^{4} d^{2} x^{4}+210 x^{3} a^{4} b \,e^{2}+840 x^{3} a^{3} b^{2} d e +420 x^{3} a^{2} b^{3} d^{2}+56 x^{2} a^{5} e^{2}+560 x^{2} a^{4} b d e +560 x^{2} a^{3} b^{2} d^{2}+168 x \,a^{5} d e +420 x \,a^{4} d^{2} b +168 d^{2} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{168 \left (b x +a \right )^{5}}\)	\(230\)
default	\(\frac {x \left (21 b^{5} e^{2} x^{7}+120 x^{6} a \,b^{4} e^{2}+48 x^{6} b^{5} d e +280 x^{5} a^{2} b^{3} e^{2}+280 x^{5} a \,b^{4} d e +28 x^{5} b^{5} d^{2}+336 a^{3} b^{2} e^{2} x^{4}+672 a^{2} b^{3} d e \,x^{4}+168 a \,b^{4} d^{2} x^{4}+210 x^{3} a^{4} b \,e^{2}+840 x^{3} a^{3} b^{2} d e +420 x^{3} a^{2} b^{3} d^{2}+56 x^{2} a^{5} e^{2}+560 x^{2} a^{4} b d e +560 x^{2} a^{3} b^{2} d^{2}+168 x \,a^{5} d e +420 x \,a^{4} d^{2} b +168 d^{2} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{168 \left (b x +a \right )^{5}}\)	\(230\)
orering	\(\frac {x \left (21 b^{5} e^{2} x^{7}+120 x^{6} a \,b^{4} e^{2}+48 x^{6} b^{5} d e +280 x^{5} a^{2} b^{3} e^{2}+280 x^{5} a \,b^{4} d e +28 x^{5} b^{5} d^{2}+336 a^{3} b^{2} e^{2} x^{4}+672 a^{2} b^{3} d e \,x^{4}+168 a \,b^{4} d^{2} x^{4}+210 x^{3} a^{4} b \,e^{2}+840 x^{3} a^{3} b^{2} d e +420 x^{3} a^{2} b^{3} d^{2}+56 x^{2} a^{5} e^{2}+560 x^{2} a^{4} b d e +560 x^{2} a^{3} b^{2} d^{2}+168 x \,a^{5} d e +420 x \,a^{4} d^{2} b +168 d^{2} a^{5}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{168 \left (b x +a \right )^{5}}\)	\(239\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{5} e^{2} x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a \,b^{4} e^{2}+2 b^{5} d e \right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{2} b^{3} e^{2}+10 a \,b^{4} d e +b^{5} d^{2}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{3} b^{2} e^{2}+20 a^{2} b^{3} d e +5 a \,b^{4} d^{2}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a^{4} b \,e^{2}+20 a^{3} b^{2} d e +10 a^{2} b^{3} d^{2}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{5} e^{2}+10 a^{4} b d e +10 a^{3} b^{2} d^{2}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (2 a^{5} d e +5 a^{4} d^{2} b \right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, d^{2} a^{5} x}{b x +a}\)	\(329\)

Input:

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

Output:

1/168*x*(21*b^5*e^2*x^7+120*a*b^4*e^2*x^6+48*b^5*d*e*x^6+280*a^2*b^3*e^2*x 
^5+280*a*b^4*d*e*x^5+28*b^5*d^2*x^5+336*a^3*b^2*e^2*x^4+672*a^2*b^3*d*e*x^ 
4+168*a*b^4*d^2*x^4+210*a^4*b*e^2*x^3+840*a^3*b^2*d*e*x^3+420*a^2*b^3*d^2* 
x^3+56*a^5*e^2*x^2+560*a^4*b*d*e*x^2+560*a^3*b^2*d^2*x^2+168*a^5*d*e*x+420 
*a^4*b*d^2*x+168*a^5*d^2)*((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. 197 vs. \(2 (86) = 172\).

Time = 0.08 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.58 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{8} \, b^{5} e^{2} x^{8} + a^{5} d^{2} x + \frac {1}{7} \, {\left (2 \, b^{5} d e + 5 \, a b^{4} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{2} + 10 \, a b^{4} d e + 10 \, a^{2} b^{3} e^{2}\right )} x^{6} + {\left (a b^{4} d^{2} + 4 \, a^{2} b^{3} d e + 2 \, a^{3} b^{2} e^{2}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} d^{2} + 4 \, a^{3} b^{2} d e + a^{4} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} d^{2} + 10 \, a^{4} b d e + a^{5} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d^{2} + 2 \, a^{5} d e\right )} x^{2} \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4855 vs. \(2 (87) = 174\).

Time = 0.90 (sec) , antiderivative size = 4855, normalized size of antiderivative = 38.84 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)**2*(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**2*x**7/8 + x**6*(33*a 
*b**5*e**2/8 + 2*b**6*d*e)/(7*b**2) + x**5*(113*a**2*b**4*e**2/8 + 12*a*b* 
*5*d*e - 13*a*(33*a*b**5*e**2/8 + 2*b**6*d*e)/(7*b) + b**6*d**2)/(6*b**2) 
+ x**4*(20*a**3*b**3*e**2 + 30*a**2*b**4*d*e - 6*a**2*(33*a*b**5*e**2/8 + 
2*b**6*d*e)/(7*b**2) + 6*a*b**5*d**2 - 11*a*(113*a**2*b**4*e**2/8 + 12*a*b 
**5*d*e - 13*a*(33*a*b**5*e**2/8 + 2*b**6*d*e)/(7*b) + b**6*d**2)/(6*b))/( 
5*b**2) + x**3*(15*a**4*b**2*e**2 + 40*a**3*b**3*d*e + 15*a**2*b**4*d**2 - 
 5*a**2*(113*a**2*b**4*e**2/8 + 12*a*b**5*d*e - 13*a*(33*a*b**5*e**2/8 + 2 
*b**6*d*e)/(7*b) + b**6*d**2)/(6*b**2) - 9*a*(20*a**3*b**3*e**2 + 30*a**2* 
b**4*d*e - 6*a**2*(33*a*b**5*e**2/8 + 2*b**6*d*e)/(7*b**2) + 6*a*b**5*d**2 
 - 11*a*(113*a**2*b**4*e**2/8 + 12*a*b**5*d*e - 13*a*(33*a*b**5*e**2/8 + 2 
*b**6*d*e)/(7*b) + b**6*d**2)/(6*b))/(5*b))/(4*b**2) + x**2*(6*a**5*b*e**2 
 + 30*a**4*b**2*d*e + 20*a**3*b**3*d**2 - 4*a**2*(20*a**3*b**3*e**2 + 30*a 
**2*b**4*d*e - 6*a**2*(33*a*b**5*e**2/8 + 2*b**6*d*e)/(7*b**2) + 6*a*b**5* 
d**2 - 11*a*(113*a**2*b**4*e**2/8 + 12*a*b**5*d*e - 13*a*(33*a*b**5*e**2/8 
 + 2*b**6*d*e)/(7*b) + b**6*d**2)/(6*b))/(5*b**2) - 7*a*(15*a**4*b**2*e**2 
 + 40*a**3*b**3*d*e + 15*a**2*b**4*d**2 - 5*a**2*(113*a**2*b**4*e**2/8 + 1 
2*a*b**5*d*e - 13*a*(33*a*b**5*e**2/8 + 2*b**6*d*e)/(7*b) + b**6*d**2)/(6* 
b**2) - 9*a*(20*a**3*b**3*e**2 + 30*a**2*b**4*d*e - 6*a**2*(33*a*b**5*e**2 
/8 + 2*b**6*d*e)/(7*b**2) + 6*a*b**5*d**2 - 11*a*(113*a**2*b**4*e**2/8 ...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (86) = 172\).

Time = 0.04 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.96 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{2} x - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d e x}{3 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} e^{2} x}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{2}}{6 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d e}{3 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} e^{2}}{6 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} e^{2} x}{8 \, b^{2}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d e}{7 \, b^{2}} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a e^{2}}{56 \, b^{3}} \] Input:

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

Output:

1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*d^2*x - 1/3*(b^2*x^2 + 2*a*b*x + a^2)^ 
(5/2)*a*d*e*x/b + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*e^2*x/b^2 + 1/6* 
(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^2/b - 1/3*(b^2*x^2 + 2*a*b*x + a^2)^(5 
/2)*a^2*d*e/b^2 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*e^2/b^3 + 1/8*(b 
^2*x^2 + 2*a*b*x + a^2)^(7/2)*e^2*x/b^2 + 2/7*(b^2*x^2 + 2*a*b*x + a^2)^(7 
/2)*d*e/b^2 - 9/56*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*e^2/b^3
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (86) = 172\).

Time = 0.28 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.86 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{8} \, b^{5} e^{2} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{7} \, b^{5} d e x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, a b^{4} e^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, b^{5} d^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, a b^{4} d e x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, a^{2} b^{3} e^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + a b^{4} d^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{2} b^{3} d e x^{5} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b^{2} e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{2} b^{3} d^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{3} b^{2} d e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, a^{4} b e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{3} b^{2} d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{4} b d e x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, a^{5} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} b d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{5} d e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{5} d^{2} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (28 \, a^{6} b^{2} d^{2} - 8 \, a^{7} b d e + a^{8} e^{2}\right )} \mathrm {sgn}\left (b x + a\right )}{168 \, b^{3}} \] Input:

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

Output:

1/8*b^5*e^2*x^8*sgn(b*x + a) + 2/7*b^5*d*e*x^7*sgn(b*x + a) + 5/7*a*b^4*e^ 
2*x^7*sgn(b*x + a) + 1/6*b^5*d^2*x^6*sgn(b*x + a) + 5/3*a*b^4*d*e*x^6*sgn( 
b*x + a) + 5/3*a^2*b^3*e^2*x^6*sgn(b*x + a) + a*b^4*d^2*x^5*sgn(b*x + a) + 
 4*a^2*b^3*d*e*x^5*sgn(b*x + a) + 2*a^3*b^2*e^2*x^5*sgn(b*x + a) + 5/2*a^2 
*b^3*d^2*x^4*sgn(b*x + a) + 5*a^3*b^2*d*e*x^4*sgn(b*x + a) + 5/4*a^4*b*e^2 
*x^4*sgn(b*x + a) + 10/3*a^3*b^2*d^2*x^3*sgn(b*x + a) + 10/3*a^4*b*d*e*x^3 
*sgn(b*x + a) + 1/3*a^5*e^2*x^3*sgn(b*x + a) + 5/2*a^4*b*d^2*x^2*sgn(b*x + 
 a) + a^5*d*e*x^2*sgn(b*x + a) + a^5*d^2*x*sgn(b*x + a) + 1/168*(28*a^6*b^ 
2*d^2 - 8*a^7*b*d*e + a^8*e^2)*sgn(b*x + a)/b^3
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.70 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \left (21 b^{5} e^{2} x^{7}+120 a \,b^{4} e^{2} x^{6}+48 b^{5} d e \,x^{6}+280 a^{2} b^{3} e^{2} x^{5}+280 a \,b^{4} d e \,x^{5}+28 b^{5} d^{2} x^{5}+336 a^{3} b^{2} e^{2} x^{4}+672 a^{2} b^{3} d e \,x^{4}+168 a \,b^{4} d^{2} x^{4}+210 a^{4} b \,e^{2} x^{3}+840 a^{3} b^{2} d e \,x^{3}+420 a^{2} b^{3} d^{2} x^{3}+56 a^{5} e^{2} x^{2}+560 a^{4} b d e \,x^{2}+560 a^{3} b^{2} d^{2} x^{2}+168 a^{5} d e x +420 a^{4} b \,d^{2} x +168 a^{5} d^{2}\right )}{168} \] Input:

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

Output:

(x*(168*a**5*d**2 + 168*a**5*d*e*x + 56*a**5*e**2*x**2 + 420*a**4*b*d**2*x 
 + 560*a**4*b*d*e*x**2 + 210*a**4*b*e**2*x**3 + 560*a**3*b**2*d**2*x**2 + 
840*a**3*b**2*d*e*x**3 + 336*a**3*b**2*e**2*x**4 + 420*a**2*b**3*d**2*x**3 
 + 672*a**2*b**3*d*e*x**4 + 280*a**2*b**3*e**2*x**5 + 168*a*b**4*d**2*x**4 
 + 280*a*b**4*d*e*x**5 + 120*a*b**4*e**2*x**6 + 28*b**5*d**2*x**5 + 48*b** 
5*d*e*x**6 + 21*b**5*e**2*x**7))/168