\(\int (d+e x)^m (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [417]

Optimal result

Integrand size = 28, antiderivative size = 337 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=-\frac {(b d-a e)^5 (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (1+m) (a+b x)}+\frac {5 b (b d-a e)^4 (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (2+m) (a+b x)}-\frac {10 b^2 (b d-a e)^3 (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (3+m) (a+b x)}+\frac {10 b^3 (b d-a e)^2 (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (4+m) (a+b x)}-\frac {5 b^4 (b d-a e) (d+e x)^{5+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (5+m) (a+b x)}+\frac {b^5 (d+e x)^{6+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (6+m) (a+b x)} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.50 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {\left ((a+b x)^2\right )^{5/2} (d+e x)^{1+m} \left (-\frac {(b d-a e)^5}{1+m}+\frac {5 b (b d-a e)^4 (d+e x)}{2+m}-\frac {10 b^2 (b d-a e)^3 (d+e x)^2}{3+m}+\frac {10 b^3 (b d-a e)^2 (d+e x)^3}{4+m}-\frac {5 b^4 (b d-a e) (d+e x)^4}{5+m}+\frac {b^5 (d+e x)^5}{6+m}\right )}{e^6 (a+b x)^5} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 203, 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, 53, 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.

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

\(\Big \downarrow \) 1102

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 53

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(a e-b d)^5 (d+e x)^m}{e^5}+\frac {5 b (b d-a e)^4 (d+e x)^{m+1}}{e^5}-\frac {10 b^2 (b d-a e)^3 (d+e x)^{m+2}}{e^5}+\frac {10 b^3 (b d-a e)^2 (d+e x)^{m+3}}{e^5}-\frac {5 b^4 (b d-a e) (d+e x)^{m+4}}{e^5}+\frac {b^5 (d+e x)^{m+5}}{e^5}\right )dx}{a+b x}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {5 b^4 (b d-a e) (d+e x)^{m+5}}{e^6 (m+5)}+\frac {10 b^3 (b d-a e)^2 (d+e x)^{m+4}}{e^6 (m+4)}-\frac {10 b^2 (b d-a e)^3 (d+e x)^{m+3}}{e^6 (m+3)}-\frac {(b d-a e)^5 (d+e x)^{m+1}}{e^6 (m+1)}+\frac {5 b (b d-a e)^4 (d+e x)^{m+2}}{e^6 (m+2)}+\frac {b^5 (d+e x)^{m+6}}{e^6 (m+6)}\right )}{a+b x}\)

Input:

Int[(d + e*x)^m*(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)^5*(d + e*x)^(1 + m))/(e^6*( 
1 + m))) + (5*b*(b*d - a*e)^4*(d + e*x)^(2 + m))/(e^6*(2 + m)) - (10*b^2*( 
b*d - a*e)^3*(d + e*x)^(3 + m))/(e^6*(3 + m)) + (10*b^3*(b*d - a*e)^2*(d + 
 e*x)^(4 + m))/(e^6*(4 + m)) - (5*b^4*(b*d - a*e)*(d + e*x)^(5 + m))/(e^6* 
(5 + m)) + (b^5*(d + e*x)^(6 + m))/(e^6*(6 + m))))/(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 53

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, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[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. \(1360\) vs. \(2(271)=542\).

Time = 1.24 (sec) , antiderivative size = 1361, normalized size of antiderivative = 4.04

method	result	size
gosper	\(\text {Expression too large to display}\)	\(1361\)
orering	\(\text {Expression too large to display}\)	\(1373\)
risch	\(\text {Expression too large to display}\)	\(1745\)

Input:

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

Output:

1/e^6*(e*x+d)^(1+m)/(b*x+a)^5*((b*x+a)^2)^(5/2)/(m^6+21*m^5+175*m^4+735*m^ 
3+1624*m^2+1764*m+720)*(b^5*e^5*m^5*x^5+5*a*b^4*e^5*m^5*x^4+15*b^5*e^5*m^4 
*x^5+10*a^2*b^3*e^5*m^5*x^3+80*a*b^4*e^5*m^4*x^4-5*b^5*d*e^4*m^4*x^4+85*b^ 
5*e^5*m^3*x^5+10*a^3*b^2*e^5*m^5*x^2+170*a^2*b^3*e^5*m^4*x^3-20*a*b^4*d*e^ 
4*m^4*x^3+475*a*b^4*e^5*m^3*x^4-50*b^5*d*e^4*m^3*x^4+225*b^5*e^5*m^2*x^5+5 
*a^4*b*e^5*m^5*x+180*a^3*b^2*e^5*m^4*x^2-30*a^2*b^3*d*e^4*m^4*x^2+1070*a^2 
*b^3*e^5*m^3*x^3-240*a*b^4*d*e^4*m^3*x^3+1300*a*b^4*e^5*m^2*x^4+20*b^5*d^2 
*e^3*m^3*x^3-175*b^5*d*e^4*m^2*x^4+274*b^5*e^5*m*x^5+a^5*e^5*m^5+95*a^4*b* 
e^5*m^4*x-20*a^3*b^2*d*e^4*m^4*x+1210*a^3*b^2*e^5*m^3*x^2-420*a^2*b^3*d*e^ 
4*m^3*x^2+3070*a^2*b^3*e^5*m^2*x^3+60*a*b^4*d^2*e^3*m^3*x^2-940*a*b^4*d*e^ 
4*m^2*x^3+1620*a*b^4*e^5*m*x^4+120*b^5*d^2*e^3*m^2*x^3-250*b^5*d*e^4*m*x^4 
+120*b^5*e^5*x^5+20*a^5*e^5*m^4-5*a^4*b*d*e^4*m^4+685*a^4*b*e^5*m^3*x-320* 
a^3*b^2*d*e^4*m^3*x+3720*a^3*b^2*e^5*m^2*x^2+60*a^2*b^3*d^2*e^3*m^3*x-1950 
*a^2*b^3*d*e^4*m^2*x^2+3960*a^2*b^3*e^5*m*x^3+540*a*b^4*d^2*e^3*m^2*x^2-14 
40*a*b^4*d*e^4*m*x^3+720*a*b^4*e^5*x^4-60*b^5*d^3*e^2*m^2*x^2+220*b^5*d^2* 
e^3*m*x^3-120*b^5*d*e^4*x^4+155*a^5*e^5*m^3-90*a^4*b*d*e^4*m^3+2305*a^4*b* 
e^5*m^2*x+20*a^3*b^2*d^2*e^3*m^3-1780*a^3*b^2*d*e^4*m^2*x+5080*a^3*b^2*e^5 
*m*x^2+720*a^2*b^3*d^2*e^3*m^2*x-3360*a^2*b^3*d*e^4*m*x^2+1800*a^2*b^3*e^5 
*x^3-120*a*b^4*d^3*e^2*m^2*x+1200*a*b^4*d^2*e^3*m*x^2-720*a*b^4*d*e^4*x^3- 
180*b^5*d^3*e^2*m*x^2+120*b^5*d^2*e^3*x^3+580*a^5*e^5*m^2-595*a^4*b*d*e...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1460 vs. \(2 (271) = 542\).

Time = 0.09 (sec) , antiderivative size = 1460, normalized size of antiderivative = 4.33 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

(a^5*d*e^5*m^5 - 120*b^5*d^6 + 720*a*b^4*d^5*e - 1800*a^2*b^3*d^4*e^2 + 24 
00*a^3*b^2*d^3*e^3 - 1800*a^4*b*d^2*e^4 + 720*a^5*d*e^5 + (b^5*e^6*m^5 + 1 
5*b^5*e^6*m^4 + 85*b^5*e^6*m^3 + 225*b^5*e^6*m^2 + 274*b^5*e^6*m + 120*b^5 
*e^6)*x^6 + (720*a*b^4*e^6 + (b^5*d*e^5 + 5*a*b^4*e^6)*m^5 + 10*(b^5*d*e^5 
 + 8*a*b^4*e^6)*m^4 + 5*(7*b^5*d*e^5 + 95*a*b^4*e^6)*m^3 + 50*(b^5*d*e^5 + 
 26*a*b^4*e^6)*m^2 + 12*(2*b^5*d*e^5 + 135*a*b^4*e^6)*m)*x^5 - 5*(a^4*b*d^ 
2*e^4 - 4*a^5*d*e^5)*m^4 + 5*(360*a^2*b^3*e^6 + (a*b^4*d*e^5 + 2*a^2*b^3*e 
^6)*m^5 - (b^5*d^2*e^4 - 12*a*b^4*d*e^5 - 34*a^2*b^3*e^6)*m^4 - (6*b^5*d^2 
*e^4 - 47*a*b^4*d*e^5 - 214*a^2*b^3*e^6)*m^3 - (11*b^5*d^2*e^4 - 72*a*b^4* 
d*e^5 - 614*a^2*b^3*e^6)*m^2 - 6*(b^5*d^2*e^4 - 6*a*b^4*d*e^5 - 132*a^2*b^ 
3*e^6)*m)*x^4 + 5*(4*a^3*b^2*d^3*e^3 - 18*a^4*b*d^2*e^4 + 31*a^5*d*e^5)*m^ 
3 + 10*(240*a^3*b^2*e^6 + (a^2*b^3*d*e^5 + a^3*b^2*e^6)*m^5 - 2*(a*b^4*d^2 
*e^4 - 7*a^2*b^3*d*e^5 - 9*a^3*b^2*e^6)*m^4 + (2*b^5*d^3*e^3 - 18*a*b^4*d^ 
2*e^4 + 65*a^2*b^3*d*e^5 + 121*a^3*b^2*e^6)*m^3 + 2*(3*b^5*d^3*e^3 - 20*a* 
b^4*d^2*e^4 + 56*a^2*b^3*d*e^5 + 186*a^3*b^2*e^6)*m^2 + 4*(b^5*d^3*e^3 - 6 
*a*b^4*d^2*e^4 + 15*a^2*b^3*d*e^5 + 127*a^3*b^2*e^6)*m)*x^3 - 5*(12*a^2*b^ 
3*d^4*e^2 - 60*a^3*b^2*d^3*e^3 + 119*a^4*b*d^2*e^4 - 116*a^5*d*e^5)*m^2 + 
5*(360*a^4*b*e^6 + (2*a^3*b^2*d*e^5 + a^4*b*e^6)*m^5 - (6*a^2*b^3*d^2*e^4 
- 32*a^3*b^2*d*e^5 - 19*a^4*b*e^6)*m^4 + (12*a*b^4*d^3*e^3 - 72*a^2*b^3*d^ 
2*e^4 + 178*a^3*b^2*d*e^5 + 137*a^4*b*e^6)*m^3 - (12*b^5*d^4*e^2 - 84*a...
 

Sympy [F(-2)]

Exception generated. \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 792 vs. \(2 (271) = 542\).

Time = 0.07 (sec) , antiderivative size = 792, normalized size of antiderivative = 2.35 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*b^5*e^6*x^6 - 60*(m^2 + 1 
1*m + 30)*a^2*b^3*d^4*e^2 + 20*(m^3 + 15*m^2 + 74*m + 120)*a^3*b^2*d^3*e^3 
 - 5*(m^4 + 18*m^3 + 119*m^2 + 342*m + 360)*a^4*b*d^2*e^4 + (m^5 + 20*m^4 
+ 155*m^3 + 580*m^2 + 1044*m + 720)*a^5*d*e^5 + 120*a*b^4*d^5*e*(m + 6) - 
120*b^5*d^6 + ((m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*b^5*d*e^5 + 5*(m^5 
+ 16*m^4 + 95*m^3 + 260*m^2 + 324*m + 144)*a*b^4*e^6)*x^5 - 5*((m^4 + 6*m^ 
3 + 11*m^2 + 6*m)*b^5*d^2*e^4 - (m^5 + 12*m^4 + 47*m^3 + 72*m^2 + 36*m)*a* 
b^4*d*e^5 - 2*(m^5 + 17*m^4 + 107*m^3 + 307*m^2 + 396*m + 180)*a^2*b^3*e^6 
)*x^4 + 10*(2*(m^3 + 3*m^2 + 2*m)*b^5*d^3*e^3 - 2*(m^4 + 9*m^3 + 20*m^2 + 
12*m)*a*b^4*d^2*e^4 + (m^5 + 14*m^4 + 65*m^3 + 112*m^2 + 60*m)*a^2*b^3*d*e 
^5 + (m^5 + 18*m^4 + 121*m^3 + 372*m^2 + 508*m + 240)*a^3*b^2*e^6)*x^3 - 5 
*(12*(m^2 + m)*b^5*d^4*e^2 - 12*(m^3 + 7*m^2 + 6*m)*a*b^4*d^3*e^3 + 6*(m^4 
 + 12*m^3 + 41*m^2 + 30*m)*a^2*b^3*d^2*e^4 - 2*(m^5 + 16*m^4 + 89*m^3 + 19 
4*m^2 + 120*m)*a^3*b^2*d*e^5 - (m^5 + 19*m^4 + 137*m^3 + 461*m^2 + 702*m + 
 360)*a^4*b*e^6)*x^2 - (120*(m^2 + 6*m)*a*b^4*d^4*e^2 - 60*(m^3 + 11*m^2 + 
 30*m)*a^2*b^3*d^3*e^3 + 20*(m^4 + 15*m^3 + 74*m^2 + 120*m)*a^3*b^2*d^2*e^ 
4 - 5*(m^5 + 18*m^4 + 119*m^3 + 342*m^2 + 360*m)*a^4*b*d*e^5 - (m^5 + 20*m 
^4 + 155*m^3 + 580*m^2 + 1044*m + 720)*a^5*e^6 - 120*b^5*d^5*e*m)*x)*(e*x 
+ d)^m/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^6)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3197 vs. \(2 (271) = 542\).

Time = 0.20 (sec) , antiderivative size = 3197, normalized size of antiderivative = 9.49 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

((e*x + d)^m*b^5*e^6*m^5*x^6*sgn(b*x + a) + (e*x + d)^m*b^5*d*e^5*m^5*x^5* 
sgn(b*x + a) + 5*(e*x + d)^m*a*b^4*e^6*m^5*x^5*sgn(b*x + a) + 15*(e*x + d) 
^m*b^5*e^6*m^4*x^6*sgn(b*x + a) + 5*(e*x + d)^m*a*b^4*d*e^5*m^5*x^4*sgn(b* 
x + a) + 10*(e*x + d)^m*a^2*b^3*e^6*m^5*x^4*sgn(b*x + a) + 10*(e*x + d)^m* 
b^5*d*e^5*m^4*x^5*sgn(b*x + a) + 80*(e*x + d)^m*a*b^4*e^6*m^4*x^5*sgn(b*x 
+ a) + 85*(e*x + d)^m*b^5*e^6*m^3*x^6*sgn(b*x + a) + 10*(e*x + d)^m*a^2*b^ 
3*d*e^5*m^5*x^3*sgn(b*x + a) + 10*(e*x + d)^m*a^3*b^2*e^6*m^5*x^3*sgn(b*x 
+ a) - 5*(e*x + d)^m*b^5*d^2*e^4*m^4*x^4*sgn(b*x + a) + 60*(e*x + d)^m*a*b 
^4*d*e^5*m^4*x^4*sgn(b*x + a) + 170*(e*x + d)^m*a^2*b^3*e^6*m^4*x^4*sgn(b* 
x + a) + 35*(e*x + d)^m*b^5*d*e^5*m^3*x^5*sgn(b*x + a) + 475*(e*x + d)^m*a 
*b^4*e^6*m^3*x^5*sgn(b*x + a) + 225*(e*x + d)^m*b^5*e^6*m^2*x^6*sgn(b*x + 
a) + 10*(e*x + d)^m*a^3*b^2*d*e^5*m^5*x^2*sgn(b*x + a) + 5*(e*x + d)^m*a^4 
*b*e^6*m^5*x^2*sgn(b*x + a) - 20*(e*x + d)^m*a*b^4*d^2*e^4*m^4*x^3*sgn(b*x 
 + a) + 140*(e*x + d)^m*a^2*b^3*d*e^5*m^4*x^3*sgn(b*x + a) + 180*(e*x + d) 
^m*a^3*b^2*e^6*m^4*x^3*sgn(b*x + a) - 30*(e*x + d)^m*b^5*d^2*e^4*m^3*x^4*s 
gn(b*x + a) + 235*(e*x + d)^m*a*b^4*d*e^5*m^3*x^4*sgn(b*x + a) + 1070*(e*x 
 + d)^m*a^2*b^3*e^6*m^3*x^4*sgn(b*x + a) + 50*(e*x + d)^m*b^5*d*e^5*m^2*x^ 
5*sgn(b*x + a) + 1300*(e*x + d)^m*a*b^4*e^6*m^2*x^5*sgn(b*x + a) + 274*(e* 
x + d)^m*b^5*e^6*m*x^6*sgn(b*x + a) + 5*(e*x + d)^m*a^4*b*d*e^5*m^5*x*sgn( 
b*x + a) + (e*x + d)^m*a^5*e^6*m^5*x*sgn(b*x + a) - 30*(e*x + d)^m*a^2*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1728, normalized size of antiderivative = 5.13 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

((d + e*x)**m*(a**5*d*e**5*m**5 + 20*a**5*d*e**5*m**4 + 155*a**5*d*e**5*m* 
*3 + 580*a**5*d*e**5*m**2 + 1044*a**5*d*e**5*m + 720*a**5*d*e**5 + a**5*e* 
*6*m**5*x + 20*a**5*e**6*m**4*x + 155*a**5*e**6*m**3*x + 580*a**5*e**6*m** 
2*x + 1044*a**5*e**6*m*x + 720*a**5*e**6*x - 5*a**4*b*d**2*e**4*m**4 - 90* 
a**4*b*d**2*e**4*m**3 - 595*a**4*b*d**2*e**4*m**2 - 1710*a**4*b*d**2*e**4* 
m - 1800*a**4*b*d**2*e**4 + 5*a**4*b*d*e**5*m**5*x + 90*a**4*b*d*e**5*m**4 
*x + 595*a**4*b*d*e**5*m**3*x + 1710*a**4*b*d*e**5*m**2*x + 1800*a**4*b*d* 
e**5*m*x + 5*a**4*b*e**6*m**5*x**2 + 95*a**4*b*e**6*m**4*x**2 + 685*a**4*b 
*e**6*m**3*x**2 + 2305*a**4*b*e**6*m**2*x**2 + 3510*a**4*b*e**6*m*x**2 + 1 
800*a**4*b*e**6*x**2 + 20*a**3*b**2*d**3*e**3*m**3 + 300*a**3*b**2*d**3*e* 
*3*m**2 + 1480*a**3*b**2*d**3*e**3*m + 2400*a**3*b**2*d**3*e**3 - 20*a**3* 
b**2*d**2*e**4*m**4*x - 300*a**3*b**2*d**2*e**4*m**3*x - 1480*a**3*b**2*d* 
*2*e**4*m**2*x - 2400*a**3*b**2*d**2*e**4*m*x + 10*a**3*b**2*d*e**5*m**5*x 
**2 + 160*a**3*b**2*d*e**5*m**4*x**2 + 890*a**3*b**2*d*e**5*m**3*x**2 + 19 
40*a**3*b**2*d*e**5*m**2*x**2 + 1200*a**3*b**2*d*e**5*m*x**2 + 10*a**3*b** 
2*e**6*m**5*x**3 + 180*a**3*b**2*e**6*m**4*x**3 + 1210*a**3*b**2*e**6*m**3 
*x**3 + 3720*a**3*b**2*e**6*m**2*x**3 + 5080*a**3*b**2*e**6*m*x**3 + 2400* 
a**3*b**2*e**6*x**3 - 60*a**2*b**3*d**4*e**2*m**2 - 660*a**2*b**3*d**4*e** 
2*m - 1800*a**2*b**3*d**4*e**2 + 60*a**2*b**3*d**3*e**3*m**3*x + 660*a**2* 
b**3*d**3*e**3*m**2*x + 1800*a**2*b**3*d**3*e**3*m*x - 30*a**2*b**3*d**...