Integrand size = 33, antiderivative size = 471 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(b d-a e)^5 (B d-A e) (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (1+m) (a+b x)}-\frac {(b d-a e)^4 (6 b B d-5 A b e-a B e) (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (2+m) (a+b x)}+\frac {5 b (b d-a e)^3 (3 b B d-2 A b e-a B e) (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (3+m) (a+b x)}-\frac {10 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (4+m) (a+b x)}+\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^{5+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (5+m) (a+b x)}-\frac {b^4 (6 b B d-A b e-5 a B e) (d+e x)^{6+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (6+m) (a+b x)}+\frac {b^5 B (d+e x)^{7+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (7+m) (a+b x)} \]
(-a*e+b*d)^5*(-A*e+B*d)*(e*x+d)^(1+m)*((b*x+a)^2)^(1/2)/e^7/(1+m)/(b*x+a)- (-a*e+b*d)^4*(-5*A*b*e-B*a*e+6*B*b*d)*(e*x+d)^(2+m)*((b*x+a)^2)^(1/2)/e^7/ (2+m)/(b*x+a)+5*b*(-a*e+b*d)^3*(-2*A*b*e-B*a*e+3*B*b*d)*(e*x+d)^(3+m)*((b* x+a)^2)^(1/2)/e^7/(3+m)/(b*x+a)-10*b^2*(-a*e+b*d)^2*(-A*b*e-B*a*e+2*B*b*d) *(e*x+d)^(4+m)*((b*x+a)^2)^(1/2)/e^7/(4+m)/(b*x+a)+5*b^3*(-a*e+b*d)*(-A*b* e-2*B*a*e+3*B*b*d)*(e*x+d)^(5+m)*((b*x+a)^2)^(1/2)/e^7/(5+m)/(b*x+a)-b^4*( -A*b*e-5*B*a*e+6*B*b*d)*(e*x+d)^(6+m)*((b*x+a)^2)^(1/2)/e^7/(6+m)/(b*x+a)+ b^5*B*(e*x+d)^(7+m)*((b*x+a)^2)^(1/2)/e^7/(7+m)/(b*x+a)
Time = 0.36 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.57 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {\sqrt {(a+b x)^2} (d+e x)^{1+m} \left (\frac {(b d-a e)^5 (B d-A e)}{1+m}-\frac {(b d-a e)^4 (6 b B d-5 A b e-a B e) (d+e x)}{2+m}+\frac {5 b (b d-a e)^3 (3 b B d-2 A b e-a B e) (d+e x)^2}{3+m}-\frac {10 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) (d+e x)^3}{4+m}+\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^4}{5+m}-\frac {b^4 (6 b B d-A b e-5 a B e) (d+e x)^5}{6+m}+\frac {b^5 B (d+e x)^6}{7+m}\right )}{e^7 (a+b x)} \]
(Sqrt[(a + b*x)^2]*(d + e*x)^(1 + m)*(((b*d - a*e)^5*(B*d - A*e))/(1 + m) - ((b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x))/(2 + m) + (5*b*(b* d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2)/(3 + m) - (10*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^3)/(4 + m) + (5*b^3*(b*d - a *e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^4)/(5 + m) - (b^4*(6*b*B*d - A*b *e - 5*a*B*e)*(d + e*x)^5)/(6 + m) + (b^5*B*(d + e*x)^6)/(7 + m)))/(e^7*(a + b*x))
Time = 0.51 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.66, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 86, 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.
\(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x) (d+e x)^m \, dx\) |
\(\Big \downarrow \) 1187 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 (a+b x)^5 (A+B x) (d+e x)^mdx}{b^5 (a+b x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^5 (A+B x) (d+e x)^mdx}{a+b x}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(a e-b d)^5 (A e-B d) (d+e x)^m}{e^6}+\frac {(a e-b d)^4 (-6 b B d+5 A b e+a B e) (d+e x)^{m+1}}{e^6}-\frac {5 b (b d-a e)^3 (-3 b B d+2 A b e+a B e) (d+e x)^{m+2}}{e^6}+\frac {10 b^2 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^{m+3}}{e^6}-\frac {5 b^3 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{m+4}}{e^6}+\frac {b^4 (-6 b B d+A b e+5 a B e) (d+e x)^{m+5}}{e^6}+\frac {b^5 B (d+e x)^{m+6}}{e^6}\right )dx}{a+b x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {b^4 (d+e x)^{m+6} (-5 a B e-A b e+6 b B d)}{e^7 (m+6)}+\frac {5 b^3 (b d-a e) (d+e x)^{m+5} (-2 a B e-A b e+3 b B d)}{e^7 (m+5)}-\frac {10 b^2 (b d-a e)^2 (d+e x)^{m+4} (-a B e-A b e+2 b B d)}{e^7 (m+4)}+\frac {(b d-a e)^5 (B d-A e) (d+e x)^{m+1}}{e^7 (m+1)}-\frac {(b d-a e)^4 (d+e x)^{m+2} (-a B e-5 A b e+6 b B d)}{e^7 (m+2)}+\frac {5 b (b d-a e)^3 (d+e x)^{m+3} (-a B e-2 A b e+3 b B d)}{e^7 (m+3)}+\frac {b^5 B (d+e x)^{m+7}}{e^7 (m+7)}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((b*d - a*e)^5*(B*d - A*e)*(d + e*x)^(1 + m))/(e^7*(1 + m)) - ((b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^( 2 + m))/(e^7*(2 + m)) + (5*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(3 + m))/(e^7*(3 + m)) - (10*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a *B*e)*(d + e*x)^(4 + m))/(e^7*(4 + m)) + (5*b^3*(b*d - a*e)*(3*b*B*d - A*b *e - 2*a*B*e)*(d + e*x)^(5 + m))/(e^7*(5 + m)) - (b^4*(6*b*B*d - A*b*e - 5 *a*B*e)*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (b^5*B*(d + e*x)^(7 + m))/(e^7* (7 + m))))/(a + b*x)
3.19.88.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(3930\) vs. \(2(394)=788\).
Time = 0.36 (sec) , antiderivative size = 3931, normalized size of antiderivative = 8.35
method | result | size |
gosper | \(\text {Expression too large to display}\) | \(3931\) |
risch | \(\text {Expression too large to display}\) | \(4897\) |
1/e^7*(e*x+d)^(1+m)/(b*x+a)^5*((b*x+a)^2)^(5/2)/(m^7+28*m^6+322*m^5+1960*m ^4+6769*m^3+13132*m^2+13068*m+5040)*(B*b^5*e^6*m^6*x^6+A*b^5*e^6*m^6*x^5+5 *B*a*b^4*e^6*m^6*x^5+21*B*b^5*e^6*m^5*x^6+5*A*a*b^4*e^6*m^6*x^4+22*A*b^5*e ^6*m^5*x^5+10*B*a^2*b^3*e^6*m^6*x^4+110*B*a*b^4*e^6*m^5*x^5-6*B*b^5*d*e^5* m^5*x^5+175*B*b^5*e^6*m^4*x^6+10*A*a^2*b^3*e^6*m^6*x^3+115*A*a*b^4*e^6*m^5 *x^4-5*A*b^5*d*e^5*m^5*x^4+190*A*b^5*e^6*m^4*x^5+10*B*a^3*b^2*e^6*m^6*x^3+ 230*B*a^2*b^3*e^6*m^5*x^4-25*B*a*b^4*d*e^5*m^5*x^4+950*B*a*b^4*e^6*m^4*x^5 -90*B*b^5*d*e^5*m^4*x^5+735*B*b^5*e^6*m^3*x^6+10*A*a^3*b^2*e^6*m^6*x^2+240 *A*a^2*b^3*e^6*m^5*x^3-20*A*a*b^4*d*e^5*m^5*x^3+1035*A*a*b^4*e^6*m^4*x^4-8 5*A*b^5*d*e^5*m^4*x^4+820*A*b^5*e^6*m^3*x^5+5*B*a^4*b*e^6*m^6*x^2+240*B*a^ 3*b^2*e^6*m^5*x^3-40*B*a^2*b^3*d*e^5*m^5*x^3+2070*B*a^2*b^3*e^6*m^4*x^4-42 5*B*a*b^4*d*e^5*m^4*x^4+4100*B*a*b^4*e^6*m^3*x^5+30*B*b^5*d^2*e^4*m^4*x^4- 510*B*b^5*d*e^5*m^3*x^5+1624*B*b^5*e^6*m^2*x^6+5*A*a^4*b*e^6*m^6*x+250*A*a ^3*b^2*e^6*m^5*x^2-30*A*a^2*b^3*d*e^5*m^5*x^2+2260*A*a^2*b^3*e^6*m^4*x^3-3 80*A*a*b^4*d*e^5*m^4*x^3+4625*A*a*b^4*e^6*m^3*x^4+20*A*b^5*d^2*e^4*m^4*x^3 -525*A*b^5*d*e^5*m^3*x^4+1849*A*b^5*e^6*m^2*x^5+B*a^5*e^6*m^6*x+125*B*a^4* b*e^6*m^5*x^2-30*B*a^3*b^2*d*e^5*m^5*x^2+2260*B*a^3*b^2*e^6*m^4*x^3-760*B* a^2*b^3*d*e^5*m^4*x^3+9250*B*a^2*b^3*e^6*m^3*x^4+100*B*a*b^4*d^2*e^4*m^4*x ^3-2625*B*a*b^4*d*e^5*m^3*x^4+9245*B*a*b^4*e^6*m^2*x^5+300*B*b^5*d^2*e^4*m ^3*x^4-1350*B*b^5*d*e^5*m^2*x^5+1764*B*b^5*e^6*m*x^6+A*a^5*e^6*m^6+130*...
Leaf count of result is larger than twice the leaf count of optimal. 3485 vs. \(2 (394) = 788\).
Time = 0.48 (sec) , antiderivative size = 3485, normalized size of antiderivative = 7.40 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
(A*a^5*d*e^6*m^6 + 720*B*b^5*d^7 + 5040*A*a^5*d*e^6 - 840*(5*B*a*b^4 + A*b ^5)*d^6*e + 5040*(2*B*a^2*b^3 + A*a*b^4)*d^5*e^2 - 12600*(B*a^3*b^2 + A*a^ 2*b^3)*d^4*e^3 + 8400*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^4 - 2520*(B*a^5 + 5*A* a^4*b)*d^2*e^5 + (B*b^5*e^7*m^6 + 21*B*b^5*e^7*m^5 + 175*B*b^5*e^7*m^4 + 7 35*B*b^5*e^7*m^3 + 1624*B*b^5*e^7*m^2 + 1764*B*b^5*e^7*m + 720*B*b^5*e^7)* x^7 + (840*(5*B*a*b^4 + A*b^5)*e^7 + (B*b^5*d*e^6 + (5*B*a*b^4 + A*b^5)*e^ 7)*m^6 + (15*B*b^5*d*e^6 + 22*(5*B*a*b^4 + A*b^5)*e^7)*m^5 + 5*(17*B*b^5*d *e^6 + 38*(5*B*a*b^4 + A*b^5)*e^7)*m^4 + 5*(45*B*b^5*d*e^6 + 164*(5*B*a*b^ 4 + A*b^5)*e^7)*m^3 + (274*B*b^5*d*e^6 + 1849*(5*B*a*b^4 + A*b^5)*e^7)*m^2 + 2*(60*B*b^5*d*e^6 + 1019*(5*B*a*b^4 + A*b^5)*e^7)*m)*x^6 + (27*A*a^5*d* e^6 - (B*a^5 + 5*A*a^4*b)*d^2*e^5)*m^5 + (5040*(2*B*a^2*b^3 + A*a*b^4)*e^7 + ((5*B*a*b^4 + A*b^5)*d*e^6 + 5*(2*B*a^2*b^3 + A*a*b^4)*e^7)*m^6 - (6*B* b^5*d^2*e^5 - 17*(5*B*a*b^4 + A*b^5)*d*e^6 - 115*(2*B*a^2*b^3 + A*a*b^4)*e ^7)*m^5 - 15*(4*B*b^5*d^2*e^5 - 7*(5*B*a*b^4 + A*b^5)*d*e^6 - 69*(2*B*a^2* b^3 + A*a*b^4)*e^7)*m^4 - 5*(42*B*b^5*d^2*e^5 - 59*(5*B*a*b^4 + A*b^5)*d*e ^6 - 925*(2*B*a^2*b^3 + A*a*b^4)*e^7)*m^3 - 2*(150*B*b^5*d^2*e^5 - 187*(5* B*a*b^4 + A*b^5)*d*e^6 - 5360*(2*B*a^2*b^3 + A*a*b^4)*e^7)*m^2 - 12*(12*B* b^5*d^2*e^5 - 14*(5*B*a*b^4 + A*b^5)*d*e^6 - 1005*(2*B*a^2*b^3 + A*a*b^4)* e^7)*m)*x^5 + 5*(59*A*a^5*d*e^6 + 2*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^4 - 5*(B *a^5 + 5*A*a^4*b)*d^2*e^5)*m^4 + 5*(2520*(B*a^3*b^2 + A*a^2*b^3)*e^7 + ...
Exception generated. \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
Leaf count of result is larger than twice the leaf count of optimal. 1864 vs. \(2 (394) = 788\).
Time = 0.22 (sec) , antiderivative size = 1864, normalized size of antiderivative = 3.96 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
((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*A/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^6 ) + ((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*b^5*e...
Leaf count of result is larger than twice the leaf count of optimal. 8705 vs. \(2 (394) = 788\).
Time = 0.45 (sec) , antiderivative size = 8705, normalized size of antiderivative = 18.48 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
((e*x + d)^m*B*b^5*e^7*m^6*x^7*sgn(b*x + a) + (e*x + d)^m*B*b^5*d*e^6*m^6* x^6*sgn(b*x + a) + 5*(e*x + d)^m*B*a*b^4*e^7*m^6*x^6*sgn(b*x + a) + (e*x + d)^m*A*b^5*e^7*m^6*x^6*sgn(b*x + a) + 21*(e*x + d)^m*B*b^5*e^7*m^5*x^7*sg n(b*x + a) + 5*(e*x + d)^m*B*a*b^4*d*e^6*m^6*x^5*sgn(b*x + a) + (e*x + d)^ m*A*b^5*d*e^6*m^6*x^5*sgn(b*x + a) + 10*(e*x + d)^m*B*a^2*b^3*e^7*m^6*x^5* sgn(b*x + a) + 5*(e*x + d)^m*A*a*b^4*e^7*m^6*x^5*sgn(b*x + a) + 15*(e*x + d)^m*B*b^5*d*e^6*m^5*x^6*sgn(b*x + a) + 110*(e*x + d)^m*B*a*b^4*e^7*m^5*x^ 6*sgn(b*x + a) + 22*(e*x + d)^m*A*b^5*e^7*m^5*x^6*sgn(b*x + a) + 175*(e*x + d)^m*B*b^5*e^7*m^4*x^7*sgn(b*x + a) + 10*(e*x + d)^m*B*a^2*b^3*d*e^6*m^6 *x^4*sgn(b*x + a) + 5*(e*x + d)^m*A*a*b^4*d*e^6*m^6*x^4*sgn(b*x + a) + 10* (e*x + d)^m*B*a^3*b^2*e^7*m^6*x^4*sgn(b*x + a) + 10*(e*x + d)^m*A*a^2*b^3* e^7*m^6*x^4*sgn(b*x + a) - 6*(e*x + d)^m*B*b^5*d^2*e^5*m^5*x^5*sgn(b*x + a ) + 85*(e*x + d)^m*B*a*b^4*d*e^6*m^5*x^5*sgn(b*x + a) + 17*(e*x + d)^m*A*b ^5*d*e^6*m^5*x^5*sgn(b*x + a) + 230*(e*x + d)^m*B*a^2*b^3*e^7*m^5*x^5*sgn( b*x + a) + 115*(e*x + d)^m*A*a*b^4*e^7*m^5*x^5*sgn(b*x + a) + 85*(e*x + d) ^m*B*b^5*d*e^6*m^4*x^6*sgn(b*x + a) + 950*(e*x + d)^m*B*a*b^4*e^7*m^4*x^6* sgn(b*x + a) + 190*(e*x + d)^m*A*b^5*e^7*m^4*x^6*sgn(b*x + a) + 735*(e*x + d)^m*B*b^5*e^7*m^3*x^7*sgn(b*x + a) + 10*(e*x + d)^m*B*a^3*b^2*d*e^6*m^6* x^3*sgn(b*x + a) + 10*(e*x + d)^m*A*a^2*b^3*d*e^6*m^6*x^3*sgn(b*x + a) + 5 *(e*x + d)^m*B*a^4*b*e^7*m^6*x^3*sgn(b*x + a) + 10*(e*x + d)^m*A*a^3*b^...
Timed out. \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^m\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]