Integrand size = 33, antiderivative size = 345 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {15 b^2 (b d-a e)^4 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac {10 b^3 (b d-a e)^3 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {5 b^4 (b d-a e)^2 (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {3 b^5 (b d-a e) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x)}+\frac {b^6 (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}-\frac {6 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \]
15*b^2*(-a*e+b*d)^4*x*((b*x+a)^2)^(1/2)/e^6/(b*x+a)-(-a*e+b*d)^6*((b*x+a)^ 2)^(1/2)/e^7/(b*x+a)/(e*x+d)-10*b^3*(-a*e+b*d)^3*(e*x+d)^2*((b*x+a)^2)^(1/ 2)/e^7/(b*x+a)+5*b^4*(-a*e+b*d)^2*(e*x+d)^3*((b*x+a)^2)^(1/2)/e^7/(b*x+a)- 3/2*b^5*(-a*e+b*d)*(e*x+d)^4*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+1/5*b^6*(e*x+d) ^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-6*b*(-a*e+b*d)^5*ln(e*x+d)*((b*x+a)^2)^(1 /2)/e^7/(b*x+a)
Time = 1.14 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {\sqrt {(a+b x)^2} \left (60 a^5 b d e^5-10 a^6 e^6+150 a^4 b^2 e^4 \left (-d^2+d e x+e^2 x^2\right )+100 a^3 b^3 e^3 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+50 a^2 b^4 e^2 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+5 a b^5 e \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )+b^6 \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )-60 b (b d-a e)^5 (d+e x) \log (d+e x)\right )}{10 e^7 (a+b x) (d+e x)} \]
(Sqrt[(a + b*x)^2]*(60*a^5*b*d*e^5 - 10*a^6*e^6 + 150*a^4*b^2*e^4*(-d^2 + d*e*x + e^2*x^2) + 100*a^3*b^3*e^3*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3* x^3) + 50*a^2*b^4*e^2*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + 5*a*b^5*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^ 3 - 5*d*e^4*x^4 + 3*e^5*x^5) + b^6*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) - 60*b*(b*d - a*e)^5*(d + e*x)*Log[d + e*x]))/(10*e^7*(a + b*x)*(d + e*x))
Time = 0.45 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.53, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^2}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^2}dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(d+e x)^4 b^6}{e^6}-\frac {6 (b d-a e) (d+e x)^3 b^5}{e^6}+\frac {15 (b d-a e)^2 (d+e x)^2 b^4}{e^6}-\frac {20 (b d-a e)^3 (d+e x) b^3}{e^6}+\frac {15 (b d-a e)^4 b^2}{e^6}-\frac {6 (b d-a e)^5 b}{e^6 (d+e x)}+\frac {(a e-b d)^6}{e^6 (d+e x)^2}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {3 b^5 (d+e x)^4 (b d-a e)}{2 e^7}+\frac {5 b^4 (d+e x)^3 (b d-a e)^2}{e^7}-\frac {10 b^3 (d+e x)^2 (b d-a e)^3}{e^7}+\frac {15 b^2 x (b d-a e)^4}{e^6}-\frac {(b d-a e)^6}{e^7 (d+e x)}-\frac {6 b (b d-a e)^5 \log (d+e x)}{e^7}+\frac {b^6 (d+e x)^5}{5 e^7}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((15*b^2*(b*d - a*e)^4*x)/e^6 - (b*d - a*e) ^6/(e^7*(d + e*x)) - (10*b^3*(b*d - a*e)^3*(d + e*x)^2)/e^7 + (5*b^4*(b*d - a*e)^2*(d + e*x)^3)/e^7 - (3*b^5*(b*d - a*e)*(d + e*x)^4)/(2*e^7) + (b^6 *(d + e*x)^5)/(5*e^7) - (6*b*(b*d - a*e)^5*Log[d + e*x])/e^7))/(a + b*x)
3.20.99.3.1 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_))^(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]
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]
Time = 0.36 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} \left (\frac {1}{5} b^{4} x^{5} e^{4}+\frac {3}{2} a \,b^{3} e^{4} x^{4}-\frac {1}{2} b^{4} d \,e^{3} x^{4}+5 a^{2} b^{2} e^{4} x^{3}-4 a \,b^{3} d \,e^{3} x^{3}+b^{4} d^{2} e^{2} x^{3}+10 a^{3} b \,e^{4} x^{2}-15 a^{2} b^{2} d \,e^{3} x^{2}+9 a \,b^{3} d^{2} e^{2} x^{2}-2 b^{4} d^{3} e \,x^{2}+15 e^{4} a^{4} x -40 b d \,e^{3} a^{3} x +45 b^{2} d^{2} e^{2} a^{2} x -24 b^{3} d^{3} e a x +5 b^{4} d^{4} x \right )}{\left (b x +a \right ) e^{6}}-\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e^{6} a^{6}-6 b d \,e^{5} a^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}-6 b^{5} d^{5} e a +b^{6} d^{6}\right )}{\left (b x +a \right ) e^{7} \left (e x +d \right )}+\frac {6 \sqrt {\left (b x +a \right )^{2}}\, b \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{7}}\) | \(407\) |
| default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (-150 b^{2} d^{2} e^{4} a^{4}+200 b^{3} d^{3} e^{3} a^{3}-150 b^{4} d^{4} e^{2} a^{2}+60 b^{5} d^{5} e a +60 b d \,e^{5} a^{5}+50 b^{6} d^{5} e x +15 a \,b^{5} e^{6} x^{5}-3 b^{6} d \,e^{5} x^{5}+50 a^{2} b^{4} e^{6} x^{4}+5 b^{6} d^{2} e^{4} x^{4}+100 a^{3} b^{3} e^{6} x^{3}-10 b^{6} d^{3} e^{3} x^{3}+150 a^{4} b^{2} e^{6} x^{2}+30 b^{6} d^{4} e^{2} x^{2}-25 a \,b^{5} d \,e^{5} x^{4}-100 a^{2} b^{4} d \,e^{5} x^{3}+50 a \,b^{5} d^{2} e^{4} x^{3}-300 a^{3} b^{3} d \,e^{5} x^{2}+300 a^{2} b^{4} d^{2} e^{4} x^{2}-150 a \,b^{5} d^{3} e^{3} x^{2}+150 a^{4} b^{2} d \,e^{5} x -300 \ln \left (e x +d \right ) a^{4} b^{2} d \,e^{5} x +600 \ln \left (e x +d \right ) a^{3} b^{3} d^{2} e^{4} x -600 \ln \left (e x +d \right ) a^{2} b^{4} d^{3} e^{3} x +300 \ln \left (e x +d \right ) a \,b^{5} d^{4} e^{2} x -10 e^{6} a^{6}-10 b^{6} d^{6}+600 \ln \left (e x +d \right ) a^{3} b^{3} d^{3} e^{3}-600 \ln \left (e x +d \right ) a^{2} b^{4} d^{4} e^{2}+300 \ln \left (e x +d \right ) a \,b^{5} d^{5} e +60 \ln \left (e x +d \right ) a^{5} b d \,e^{5}-300 \ln \left (e x +d \right ) a^{4} b^{2} d^{2} e^{4}-400 a^{3} b^{3} d^{2} e^{4} x +450 a^{2} b^{4} d^{3} e^{3} x -240 a \,b^{5} d^{4} e^{2} x +60 \ln \left (e x +d \right ) a^{5} b \,e^{6} x -60 \ln \left (e x +d \right ) b^{6} d^{5} e x +2 b^{6} e^{6} x^{6}-60 \ln \left (e x +d \right ) b^{6} d^{6}\right )}{10 \left (b x +a \right )^{5} e^{7} \left (e x +d \right )}\) | \(601\) |
((b*x+a)^2)^(1/2)/(b*x+a)*b^2/e^6*(1/5*b^4*x^5*e^4+3/2*a*b^3*e^4*x^4-1/2*b ^4*d*e^3*x^4+5*a^2*b^2*e^4*x^3-4*a*b^3*d*e^3*x^3+b^4*d^2*e^2*x^3+10*a^3*b* e^4*x^2-15*a^2*b^2*d*e^3*x^2+9*a*b^3*d^2*e^2*x^2-2*b^4*d^3*e*x^2+15*e^4*a^ 4*x-40*b*d*e^3*a^3*x+45*b^2*d^2*e^2*a^2*x-24*b^3*d^3*e*a*x+5*b^4*d^4*x)-(( b*x+a)^2)^(1/2)/(b*x+a)/e^7*(a^6*e^6-6*a^5*b*d*e^5+15*a^4*b^2*d^2*e^4-20*a ^3*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2-6*a*b^5*d^5*e+b^6*d^6)/(e*x+d)+6*((b*x+a )^2)^(1/2)/(b*x+a)*b/e^7*(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2* b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)*ln(e*x+d)
Time = 0.32 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.44 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {2 \, b^{6} e^{6} x^{6} - 10 \, b^{6} d^{6} + 60 \, a b^{5} d^{5} e - 150 \, a^{2} b^{4} d^{4} e^{2} + 200 \, a^{3} b^{3} d^{3} e^{3} - 150 \, a^{4} b^{2} d^{2} e^{4} + 60 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} - 3 \, {\left (b^{6} d e^{5} - 5 \, a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (b^{6} d^{2} e^{4} - 5 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} - 10 \, {\left (b^{6} d^{3} e^{3} - 5 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} - 10 \, a^{3} b^{3} e^{6}\right )} x^{3} + 30 \, {\left (b^{6} d^{4} e^{2} - 5 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} - 10 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \, {\left (5 \, b^{6} d^{5} e - 24 \, a b^{5} d^{4} e^{2} + 45 \, a^{2} b^{4} d^{3} e^{3} - 40 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5}\right )} x - 60 \, {\left (b^{6} d^{6} - 5 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} - 10 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} - a^{5} b d e^{5} + {\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} - 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x\right )} \log \left (e x + d\right )}{10 \, {\left (e^{8} x + d e^{7}\right )}} \]
1/10*(2*b^6*e^6*x^6 - 10*b^6*d^6 + 60*a*b^5*d^5*e - 150*a^2*b^4*d^4*e^2 + 200*a^3*b^3*d^3*e^3 - 150*a^4*b^2*d^2*e^4 + 60*a^5*b*d*e^5 - 10*a^6*e^6 - 3*(b^6*d*e^5 - 5*a*b^5*e^6)*x^5 + 5*(b^6*d^2*e^4 - 5*a*b^5*d*e^5 + 10*a^2* b^4*e^6)*x^4 - 10*(b^6*d^3*e^3 - 5*a*b^5*d^2*e^4 + 10*a^2*b^4*d*e^5 - 10*a ^3*b^3*e^6)*x^3 + 30*(b^6*d^4*e^2 - 5*a*b^5*d^3*e^3 + 10*a^2*b^4*d^2*e^4 - 10*a^3*b^3*d*e^5 + 5*a^4*b^2*e^6)*x^2 + 10*(5*b^6*d^5*e - 24*a*b^5*d^4*e^ 2 + 45*a^2*b^4*d^3*e^3 - 40*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5)*x - 60*(b^ 6*d^6 - 5*a*b^5*d^5*e + 10*a^2*b^4*d^4*e^2 - 10*a^3*b^3*d^3*e^3 + 5*a^4*b^ 2*d^2*e^4 - a^5*b*d*e^5 + (b^6*d^5*e - 5*a*b^5*d^4*e^2 + 10*a^2*b^4*d^3*e^ 3 - 10*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 - a^5*b*e^6)*x)*log(e*x + d))/(e^ 8*x + d*e^7)
\[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\int \frac {\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (264) = 528\).
Time = 0.27 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.57 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {6 \, {\left (b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} - \frac {b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )}{{\left (e x + d\right )} e^{7}} + \frac {2 \, b^{6} e^{8} x^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, b^{6} d e^{7} x^{4} \mathrm {sgn}\left (b x + a\right ) + 15 \, a b^{5} e^{8} x^{4} \mathrm {sgn}\left (b x + a\right ) + 10 \, b^{6} d^{2} e^{6} x^{3} \mathrm {sgn}\left (b x + a\right ) - 40 \, a b^{5} d e^{7} x^{3} \mathrm {sgn}\left (b x + a\right ) + 50 \, a^{2} b^{4} e^{8} x^{3} \mathrm {sgn}\left (b x + a\right ) - 20 \, b^{6} d^{3} e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 90 \, a b^{5} d^{2} e^{6} x^{2} \mathrm {sgn}\left (b x + a\right ) - 150 \, a^{2} b^{4} d e^{7} x^{2} \mathrm {sgn}\left (b x + a\right ) + 100 \, a^{3} b^{3} e^{8} x^{2} \mathrm {sgn}\left (b x + a\right ) + 50 \, b^{6} d^{4} e^{4} x \mathrm {sgn}\left (b x + a\right ) - 240 \, a b^{5} d^{3} e^{5} x \mathrm {sgn}\left (b x + a\right ) + 450 \, a^{2} b^{4} d^{2} e^{6} x \mathrm {sgn}\left (b x + a\right ) - 400 \, a^{3} b^{3} d e^{7} x \mathrm {sgn}\left (b x + a\right ) + 150 \, a^{4} b^{2} e^{8} x \mathrm {sgn}\left (b x + a\right )}{10 \, e^{10}} \]
-6*(b^6*d^5*sgn(b*x + a) - 5*a*b^5*d^4*e*sgn(b*x + a) + 10*a^2*b^4*d^3*e^2 *sgn(b*x + a) - 10*a^3*b^3*d^2*e^3*sgn(b*x + a) + 5*a^4*b^2*d*e^4*sgn(b*x + a) - a^5*b*e^5*sgn(b*x + a))*log(abs(e*x + d))/e^7 - (b^6*d^6*sgn(b*x + a) - 6*a*b^5*d^5*e*sgn(b*x + a) + 15*a^2*b^4*d^4*e^2*sgn(b*x + a) - 20*a^3 *b^3*d^3*e^3*sgn(b*x + a) + 15*a^4*b^2*d^2*e^4*sgn(b*x + a) - 6*a^5*b*d*e^ 5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a))/((e*x + d)*e^7) + 1/10*(2*b^6*e^8*x ^5*sgn(b*x + a) - 5*b^6*d*e^7*x^4*sgn(b*x + a) + 15*a*b^5*e^8*x^4*sgn(b*x + a) + 10*b^6*d^2*e^6*x^3*sgn(b*x + a) - 40*a*b^5*d*e^7*x^3*sgn(b*x + a) + 50*a^2*b^4*e^8*x^3*sgn(b*x + a) - 20*b^6*d^3*e^5*x^2*sgn(b*x + a) + 90*a* b^5*d^2*e^6*x^2*sgn(b*x + a) - 150*a^2*b^4*d*e^7*x^2*sgn(b*x + a) + 100*a^ 3*b^3*e^8*x^2*sgn(b*x + a) + 50*b^6*d^4*e^4*x*sgn(b*x + a) - 240*a*b^5*d^3 *e^5*x*sgn(b*x + a) + 450*a^2*b^4*d^2*e^6*x*sgn(b*x + a) - 400*a^3*b^3*d*e ^7*x*sgn(b*x + a) + 150*a^4*b^2*e^8*x*sgn(b*x + a))/e^10
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\int \frac {\left (a+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^2} \,d x \]