Integrand size = 33, antiderivative size = 345 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {15 b^4 (b d-a e)^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^3}+\frac {3 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^2}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac {3 b^5 (b d-a e) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {b^6 (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {20 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \]
15*b^4*(-a*e+b*d)^2*x*((b*x+a)^2)^(1/2)/e^6/(b*x+a)-1/3*(-a*e+b*d)^6*((b*x +a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^3+3*b*(-a*e+b*d)^5*((b*x+a)^2)^(1/2)/e^7/ (b*x+a)/(e*x+d)^2-15*b^2*(-a*e+b*d)^4*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d )-3*b^5*(-a*e+b*d)*(e*x+d)^2*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+1/3*b^6*(e*x+d) ^3*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-20*b^3*(-a*e+b*d)^3*ln(e*x+d)*((b*x+a)^2) ^(1/2)/e^7/(b*x+a)
Time = 1.10 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=-\frac {\sqrt {(a+b x)^2} \left (a^6 e^6+3 a^5 b e^5 (d+3 e x)+15 a^4 b^2 e^4 \left (d^2+3 d e x+3 e^2 x^2\right )-10 a^3 b^3 d e^3 \left (11 d^2+27 d e x+18 e^2 x^2\right )+15 a^2 b^4 e^2 \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )-3 a b^5 e \left (47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right )+b^6 \left (37 d^6+51 d^5 e x-39 d^4 e^2 x^2-73 d^3 e^3 x^3-15 d^2 e^4 x^4+3 d e^5 x^5-e^6 x^6\right )+60 b^3 (b d-a e)^3 (d+e x)^3 \log (d+e x)\right )}{3 e^7 (a+b x) (d+e x)^3} \]
-1/3*(Sqrt[(a + b*x)^2]*(a^6*e^6 + 3*a^5*b*e^5*(d + 3*e*x) + 15*a^4*b^2*e^ 4*(d^2 + 3*d*e*x + 3*e^2*x^2) - 10*a^3*b^3*d*e^3*(11*d^2 + 27*d*e*x + 18*e ^2*x^2) + 15*a^2*b^4*e^2*(13*d^4 + 27*d^3*e*x + 9*d^2*e^2*x^2 - 9*d*e^3*x^ 3 - 3*e^4*x^4) - 3*a*b^5*e*(47*d^5 + 81*d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2*e ^3*x^3 - 15*d*e^4*x^4 + 3*e^5*x^5) + b^6*(37*d^6 + 51*d^5*e*x - 39*d^4*e^2 *x^2 - 73*d^3*e^3*x^3 - 15*d^2*e^4*x^4 + 3*d*e^5*x^5 - e^6*x^6) + 60*b^3*( b*d - a*e)^3*(d + e*x)^3*Log[d + e*x]))/(e^7*(a + b*x)*(d + e*x)^3)
Time = 0.41 (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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^4}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^4}dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(d+e x)^2 b^6}{e^6}-\frac {6 (b d-a e) (d+e x) b^5}{e^6}+\frac {15 (b d-a e)^2 b^4}{e^6}-\frac {20 (b d-a e)^3 b^3}{e^6 (d+e x)}+\frac {15 (b d-a e)^4 b^2}{e^6 (d+e x)^2}-\frac {6 (b d-a e)^5 b}{e^6 (d+e x)^3}+\frac {(a e-b d)^6}{e^6 (d+e x)^4}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {3 b^5 (d+e x)^2 (b d-a e)}{e^7}+\frac {15 b^4 x (b d-a e)^2}{e^6}-\frac {20 b^3 (b d-a e)^3 \log (d+e x)}{e^7}-\frac {15 b^2 (b d-a e)^4}{e^7 (d+e x)}+\frac {3 b (b d-a e)^5}{e^7 (d+e x)^2}-\frac {(b d-a e)^6}{3 e^7 (d+e x)^3}+\frac {b^6 (d+e x)^3}{3 e^7}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((15*b^4*(b*d - a*e)^2*x)/e^6 - (b*d - a*e) ^6/(3*e^7*(d + e*x)^3) + (3*b*(b*d - a*e)^5)/(e^7*(d + e*x)^2) - (15*b^2*( b*d - a*e)^4)/(e^7*(d + e*x)) - (3*b^5*(b*d - a*e)*(d + e*x)^2)/e^7 + (b^6 *(d + e*x)^3)/(3*e^7) - (20*b^3*(b*d - a*e)^3*Log[d + e*x])/e^7))/(a + b*x )
3.21.1.3.1 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 = 0.61 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.13
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{4} \left (\frac {1}{3} b^{2} e^{2} x^{3}+3 a b \,e^{2} x^{2}-2 b^{2} d e \,x^{2}+15 e^{2} a^{2} x -24 a b d e x +10 b^{2} d^{2} x \right )}{\left (b x +a \right ) e^{6}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (-15 e^{5} a^{4} b^{2}+60 d \,e^{4} a^{3} b^{3}-90 d^{2} b^{4} a^{2} e^{3}+60 d^{3} e^{2} b^{5} a -15 d^{4} e \,b^{6}\right ) x^{2}-3 b \left (e^{5} a^{5}+5 b d \,e^{4} a^{4}-30 b^{2} d^{2} e^{3} a^{3}+50 b^{3} d^{3} e^{2} a^{2}-35 b^{4} d^{4} e a +9 b^{5} d^{5}\right ) x -\frac {e^{6} a^{6}+3 b d \,e^{5} a^{5}+15 b^{2} d^{2} e^{4} a^{4}-110 b^{3} d^{3} e^{3} a^{3}+195 b^{4} d^{4} e^{2} a^{2}-141 b^{5} d^{5} e a +37 b^{6} d^{6}}{3 e}\right )}{\left (b x +a \right ) e^{6} \left (e x +d \right )^{3}}+\frac {20 \sqrt {\left (b x +a \right )^{2}}\, b^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{7}}\) | \(390\) |
| default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (-15 b^{2} d^{2} e^{4} a^{4}+110 b^{3} d^{3} e^{3} a^{3}-195 b^{4} d^{4} e^{2} a^{2}+141 b^{5} d^{5} e a -3 b d \,e^{5} a^{5}-9 a^{5} b \,e^{6} x -51 b^{6} d^{5} e x +9 a \,b^{5} e^{6} x^{5}-3 b^{6} d \,e^{5} x^{5}+45 a^{2} b^{4} e^{6} x^{4}+15 b^{6} d^{2} e^{4} x^{4}+73 b^{6} d^{3} e^{3} x^{3}-45 a^{4} b^{2} e^{6} x^{2}+39 b^{6} d^{4} e^{2} x^{2}-45 a \,b^{5} d \,e^{5} x^{4}+135 a^{2} b^{4} d \,e^{5} x^{3}-189 a \,b^{5} d^{2} e^{4} x^{3}+180 a^{3} b^{3} d \,e^{5} x^{2}-135 a^{2} b^{4} d^{2} e^{4} x^{2}-27 a \,b^{5} d^{3} e^{3} x^{2}-45 a^{4} b^{2} d \,e^{5} x +180 \ln \left (e x +d \right ) a^{3} b^{3} d \,e^{5} x^{2}-540 \ln \left (e x +d \right ) a^{2} b^{4} d^{2} e^{4} x^{2}+540 \ln \left (e x +d \right ) a \,b^{5} d^{3} e^{3} x^{2}-180 \ln \left (e x +d \right ) a^{2} b^{4} d \,e^{5} x^{3}+180 \ln \left (e x +d \right ) a \,b^{5} d^{2} e^{4} x^{3}+180 \ln \left (e x +d \right ) a^{3} b^{3} d^{2} e^{4} x -540 \ln \left (e x +d \right ) a^{2} b^{4} d^{3} e^{3} x +540 \ln \left (e x +d \right ) a \,b^{5} d^{4} e^{2} x -180 \ln \left (e x +d \right ) b^{6} d^{4} e^{2} x^{2}+60 \ln \left (e x +d \right ) a^{3} b^{3} e^{6} x^{3}-60 \ln \left (e x +d \right ) b^{6} d^{3} e^{3} x^{3}-e^{6} a^{6}-37 b^{6} d^{6}+60 \ln \left (e x +d \right ) a^{3} b^{3} d^{3} e^{3}-180 \ln \left (e x +d \right ) a^{2} b^{4} d^{4} e^{2}+180 \ln \left (e x +d \right ) a \,b^{5} d^{5} e +270 a^{3} b^{3} d^{2} e^{4} x -405 a^{2} b^{4} d^{3} e^{3} x +243 a \,b^{5} d^{4} e^{2} x -180 \ln \left (e x +d \right ) b^{6} d^{5} e x +b^{6} e^{6} x^{6}-60 \ln \left (e x +d \right ) b^{6} d^{6}\right )}{3 \left (b x +a \right )^{5} e^{7} \left (e x +d \right )^{3}}\) | \(692\) |
((b*x+a)^2)^(1/2)/(b*x+a)*b^4/e^6*(1/3*b^2*e^2*x^3+3*a*b*e^2*x^2-2*b^2*d*e *x^2+15*e^2*a^2*x-24*a*b*d*e*x+10*b^2*d^2*x)+((b*x+a)^2)^(1/2)/(b*x+a)*((- 15*a^4*b^2*e^5+60*a^3*b^3*d*e^4-90*a^2*b^4*d^2*e^3+60*a*b^5*d^3*e^2-15*b^6 *d^4*e)*x^2-3*b*(a^5*e^5+5*a^4*b*d*e^4-30*a^3*b^2*d^2*e^3+50*a^2*b^3*d^3*e ^2-35*a*b^4*d^4*e+9*b^5*d^5)*x-1/3*(a^6*e^6+3*a^5*b*d*e^5+15*a^4*b^2*d^2*e ^4-110*a^3*b^3*d^3*e^3+195*a^2*b^4*d^4*e^2-141*a*b^5*d^5*e+37*b^6*d^6)/e)/ e^6/(e*x+d)^3+20*((b*x+a)^2)^(1/2)/(b*x+a)*b^3/e^7*(a^3*e^3-3*a^2*b*d*e^2+ 3*a*b^2*d^2*e-b^3*d^3)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (264) = 528\).
Time = 0.31 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.67 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {b^{6} e^{6} x^{6} - 37 \, b^{6} d^{6} + 141 \, a b^{5} d^{5} e - 195 \, a^{2} b^{4} d^{4} e^{2} + 110 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - a^{6} e^{6} - 3 \, {\left (b^{6} d e^{5} - 3 \, a b^{5} e^{6}\right )} x^{5} + 15 \, {\left (b^{6} d^{2} e^{4} - 3 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + {\left (73 \, b^{6} d^{3} e^{3} - 189 \, a b^{5} d^{2} e^{4} + 135 \, a^{2} b^{4} d e^{5}\right )} x^{3} + 3 \, {\left (13 \, b^{6} d^{4} e^{2} - 9 \, a b^{5} d^{3} e^{3} - 45 \, a^{2} b^{4} d^{2} e^{4} + 60 \, a^{3} b^{3} d e^{5} - 15 \, a^{4} b^{2} e^{6}\right )} x^{2} - 3 \, {\left (17 \, b^{6} d^{5} e - 81 \, a b^{5} d^{4} e^{2} + 135 \, a^{2} b^{4} d^{3} e^{3} - 90 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \, {\left (b^{6} d^{6} - 3 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} - a^{3} b^{3} d^{3} e^{3} + {\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 3 \, {\left (b^{6} d^{4} e^{2} - 3 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} - a^{3} b^{3} d e^{5}\right )} x^{2} + 3 \, {\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} - a^{3} b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \]
1/3*(b^6*e^6*x^6 - 37*b^6*d^6 + 141*a*b^5*d^5*e - 195*a^2*b^4*d^4*e^2 + 11 0*a^3*b^3*d^3*e^3 - 15*a^4*b^2*d^2*e^4 - 3*a^5*b*d*e^5 - a^6*e^6 - 3*(b^6* d*e^5 - 3*a*b^5*e^6)*x^5 + 15*(b^6*d^2*e^4 - 3*a*b^5*d*e^5 + 3*a^2*b^4*e^6 )*x^4 + (73*b^6*d^3*e^3 - 189*a*b^5*d^2*e^4 + 135*a^2*b^4*d*e^5)*x^3 + 3*( 13*b^6*d^4*e^2 - 9*a*b^5*d^3*e^3 - 45*a^2*b^4*d^2*e^4 + 60*a^3*b^3*d*e^5 - 15*a^4*b^2*e^6)*x^2 - 3*(17*b^6*d^5*e - 81*a*b^5*d^4*e^2 + 135*a^2*b^4*d^ 3*e^3 - 90*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5 + 3*a^5*b*e^6)*x - 60*(b^6*d ^6 - 3*a*b^5*d^5*e + 3*a^2*b^4*d^4*e^2 - a^3*b^3*d^3*e^3 + (b^6*d^3*e^3 - 3*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 3*(b^6*d^4*e^2 - 3* a*b^5*d^3*e^3 + 3*a^2*b^4*d^2*e^4 - a^3*b^3*d*e^5)*x^2 + 3*(b^6*d^5*e - 3* a*b^5*d^4*e^2 + 3*a^2*b^4*d^3*e^3 - a^3*b^3*d^2*e^4)*x)*log(e*x + d))/(e^1 0*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)
\[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.28 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=-\frac {20 \, {\left (b^{6} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{5} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} b^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} - \frac {37 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 141 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 195 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 110 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 45 \, {\left (b^{6} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{5} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{4} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b^{3} d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{4} b^{2} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 9 \, {\left (9 \, b^{6} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 35 \, a b^{5} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 50 \, a^{2} b^{4} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 30 \, a^{3} b^{3} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{5} b e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x}{3 \, {\left (e x + d\right )}^{3} e^{7}} + \frac {b^{6} e^{8} x^{3} \mathrm {sgn}\left (b x + a\right ) - 6 \, b^{6} d e^{7} x^{2} \mathrm {sgn}\left (b x + a\right ) + 9 \, a b^{5} e^{8} x^{2} \mathrm {sgn}\left (b x + a\right ) + 30 \, b^{6} d^{2} e^{6} x \mathrm {sgn}\left (b x + a\right ) - 72 \, a b^{5} d e^{7} x \mathrm {sgn}\left (b x + a\right ) + 45 \, a^{2} b^{4} e^{8} x \mathrm {sgn}\left (b x + a\right )}{3 \, e^{12}} \]
-20*(b^6*d^3*sgn(b*x + a) - 3*a*b^5*d^2*e*sgn(b*x + a) + 3*a^2*b^4*d*e^2*s gn(b*x + a) - a^3*b^3*e^3*sgn(b*x + a))*log(abs(e*x + d))/e^7 - 1/3*(37*b^ 6*d^6*sgn(b*x + a) - 141*a*b^5*d^5*e*sgn(b*x + a) + 195*a^2*b^4*d^4*e^2*sg n(b*x + a) - 110*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) + 3*a^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a) + 45*(b^6*d^4*e^ 2*sgn(b*x + a) - 4*a*b^5*d^3*e^3*sgn(b*x + a) + 6*a^2*b^4*d^2*e^4*sgn(b*x + a) - 4*a^3*b^3*d*e^5*sgn(b*x + a) + a^4*b^2*e^6*sgn(b*x + a))*x^2 + 9*(9 *b^6*d^5*e*sgn(b*x + a) - 35*a*b^5*d^4*e^2*sgn(b*x + a) + 50*a^2*b^4*d^3*e ^3*sgn(b*x + a) - 30*a^3*b^3*d^2*e^4*sgn(b*x + a) + 5*a^4*b^2*d*e^5*sgn(b* x + a) + a^5*b*e^6*sgn(b*x + a))*x)/((e*x + d)^3*e^7) + 1/3*(b^6*e^8*x^3*s gn(b*x + a) - 6*b^6*d*e^7*x^2*sgn(b*x + a) + 9*a*b^5*e^8*x^2*sgn(b*x + a) + 30*b^6*d^2*e^6*x*sgn(b*x + a) - 72*a*b^5*d*e^7*x*sgn(b*x + a) + 45*a^2*b ^4*e^8*x*sgn(b*x + a))/e^12
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {\left (a+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \]