Integrand size = 33, antiderivative size = 344 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=-\frac {b^5 (5 b d-6 a e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {b^6 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)}-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}+\frac {2 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^3}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}+\frac {20 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}+\frac {15 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \]
-b^5*(-6*a*e+5*b*d)*x*((b*x+a)^2)^(1/2)/e^6/(b*x+a)+1/2*b^6*x^2*((b*x+a)^2 )^(1/2)/e^5/(b*x+a)-1/4*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d) ^4+2*b*(-a*e+b*d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^3-15/2*b^2*(-a*e +b*d)^4*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^2+20*b^3*(-a*e+b*d)^3*((b*x+ a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)+15*b^4*(-a*e+b*d)^2*ln(e*x+d)*((b*x+a)^2)^ (1/2)/e^7/(b*x+a)
Time = 1.10 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=-\frac {\sqrt {(a+b x)^2} \left (a^6 e^6+2 a^5 b e^5 (d+4 e x)+5 a^4 b^2 e^4 \left (d^2+4 d e x+6 e^2 x^2\right )+20 a^3 b^3 e^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-5 a^2 b^4 d e^2 \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+2 a b^5 e \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )+b^6 \left (-57 d^6-168 d^5 e x-132 d^4 e^2 x^2+32 d^3 e^3 x^3+68 d^2 e^4 x^4+12 d e^5 x^5-2 e^6 x^6\right )-60 b^4 (b d-a e)^2 (d+e x)^4 \log (d+e x)\right )}{4 e^7 (a+b x) (d+e x)^4} \]
-1/4*(Sqrt[(a + b*x)^2]*(a^6*e^6 + 2*a^5*b*e^5*(d + 4*e*x) + 5*a^4*b^2*e^4 *(d^2 + 4*d*e*x + 6*e^2*x^2) + 20*a^3*b^3*e^3*(d^3 + 4*d^2*e*x + 6*d*e^2*x ^2 + 4*e^3*x^3) - 5*a^2*b^4*d*e^2*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 4 8*e^3*x^3) + 2*a*b^5*e*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^ 3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5) + b^6*(-57*d^6 - 168*d^5*e*x - 132*d^4* e^2*x^2 + 32*d^3*e^3*x^3 + 68*d^2*e^4*x^4 + 12*d*e^5*x^5 - 2*e^6*x^6) - 60 *b^4*(b*d - a*e)^2*(d + e*x)^4*Log[d + e*x]))/(e^7*(a + b*x)*(d + e*x)^4)
Time = 0.41 (sec) , antiderivative size = 183, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^5}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^5}dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {x b^6}{e^5}-\frac {(5 b d-6 a e) b^5}{e^6}+\frac {15 (b d-a e)^2 b^4}{e^6 (d+e x)}-\frac {20 (b d-a e)^3 b^3}{e^6 (d+e x)^2}+\frac {15 (b d-a e)^4 b^2}{e^6 (d+e x)^3}-\frac {6 (b d-a e)^5 b}{e^6 (d+e x)^4}+\frac {(a e-b d)^6}{e^6 (d+e x)^5}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {b^5 x (5 b d-6 a e)}{e^6}+\frac {15 b^4 (b d-a e)^2 \log (d+e x)}{e^7}+\frac {20 b^3 (b d-a e)^3}{e^7 (d+e x)}-\frac {15 b^2 (b d-a e)^4}{2 e^7 (d+e x)^2}+\frac {2 b (b d-a e)^5}{e^7 (d+e x)^3}-\frac {(b d-a e)^6}{4 e^7 (d+e x)^4}+\frac {b^6 x^2}{2 e^5}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-((b^5*(5*b*d - 6*a*e)*x)/e^6) + (b^6*x^2) /(2*e^5) - (b*d - a*e)^6/(4*e^7*(d + e*x)^4) + (2*b*(b*d - a*e)^5)/(e^7*(d + e*x)^3) - (15*b^2*(b*d - a*e)^4)/(2*e^7*(d + e*x)^2) + (20*b^3*(b*d - a *e)^3)/(e^7*(d + e*x)) + (15*b^4*(b*d - a*e)^2*Log[d + e*x])/e^7))/(a + b* x)
3.21.2.3.1 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 = 0.72 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.12
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{5} \left (\frac {1}{2} b e \,x^{2}+6 a e x -5 b d x \right )}{\left (b x +a \right ) e^{6}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (-20 e^{5} a^{3} b^{3}+60 d \,e^{4} b^{4} a^{2}-60 d^{2} b^{5} a \,e^{3}+20 b^{6} d^{3} e^{2}\right ) x^{3}-\frac {15 b^{2} e \left (e^{4} a^{4}+4 b d \,e^{3} a^{3}-18 b^{2} d^{2} e^{2} a^{2}+20 b^{3} d^{3} e a -7 b^{4} d^{4}\right ) x^{2}}{2}-b \left (2 e^{5} a^{5}+5 b d \,e^{4} a^{4}+20 b^{2} d^{2} e^{3} a^{3}-110 b^{3} d^{3} e^{2} a^{2}+130 b^{4} d^{4} e a -47 b^{5} d^{5}\right ) x -\frac {e^{6} a^{6}+2 b d \,e^{5} a^{5}+5 b^{2} d^{2} e^{4} a^{4}+20 b^{3} d^{3} e^{3} a^{3}-125 b^{4} d^{4} e^{2} a^{2}+154 b^{5} d^{5} e a -57 b^{6} d^{6}}{4 e}\right )}{\left (b x +a \right ) e^{6} \left (e x +d \right )^{4}}+\frac {15 \sqrt {\left (b x +a \right )^{2}}\, b^{4} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{7}}\) | \(384\) |
| default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (-5 b^{2} d^{2} e^{4} a^{4}-20 b^{3} d^{3} e^{3} a^{3}+125 b^{4} d^{4} e^{2} a^{2}-154 b^{5} d^{5} e a -2 b d \,e^{5} a^{5}-8 a^{5} b \,e^{6} x +168 b^{6} d^{5} e x +24 a \,b^{5} e^{6} x^{5}-12 b^{6} d \,e^{5} x^{5}-68 b^{6} d^{2} e^{4} x^{4}-80 a^{3} b^{3} e^{6} x^{3}-32 b^{6} d^{3} e^{3} x^{3}-30 a^{4} b^{2} e^{6} x^{2}+132 b^{6} d^{4} e^{2} x^{2}+96 a \,b^{5} d \,e^{5} x^{4}+240 a^{2} b^{4} d \,e^{5} x^{3}-96 a \,b^{5} d^{2} e^{4} x^{3}-120 a^{3} b^{3} d \,e^{5} x^{2}+540 a^{2} b^{4} d^{2} e^{4} x^{2}-504 a \,b^{5} d^{3} e^{3} x^{2}-20 a^{4} b^{2} d \,e^{5} x -120 \ln \left (e x +d \right ) a \,b^{5} d \,e^{5} x^{4}+360 \ln \left (e x +d \right ) a^{2} b^{4} d^{2} e^{4} x^{2}-720 \ln \left (e x +d \right ) a \,b^{5} d^{3} e^{3} x^{2}+240 \ln \left (e x +d \right ) a^{2} b^{4} d \,e^{5} x^{3}-480 \ln \left (e x +d \right ) a \,b^{5} d^{2} e^{4} x^{3}+240 \ln \left (e x +d \right ) a^{2} b^{4} d^{3} e^{3} x -480 \ln \left (e x +d \right ) a \,b^{5} d^{4} e^{2} x +60 \ln \left (e x +d \right ) a^{2} b^{4} e^{6} x^{4}+60 \ln \left (e x +d \right ) b^{6} d^{2} e^{4} x^{4}+360 \ln \left (e x +d \right ) b^{6} d^{4} e^{2} x^{2}+240 \ln \left (e x +d \right ) b^{6} d^{3} e^{3} x^{3}-e^{6} a^{6}+57 b^{6} d^{6}+60 \ln \left (e x +d \right ) a^{2} b^{4} d^{4} e^{2}-120 \ln \left (e x +d \right ) a \,b^{5} d^{5} e -80 a^{3} b^{3} d^{2} e^{4} x +440 a^{2} b^{4} d^{3} e^{3} x -496 a \,b^{5} d^{4} e^{2} x +240 \ln \left (e x +d \right ) b^{6} d^{5} e x +2 b^{6} e^{6} x^{6}+60 \ln \left (e x +d \right ) b^{6} d^{6}\right )}{4 \left (b x +a \right )^{5} e^{7} \left (e x +d \right )^{4}}\) | \(670\) |
((b*x+a)^2)^(1/2)/(b*x+a)*b^5/e^6*(1/2*b*e*x^2+6*a*e*x-5*b*d*x)+((b*x+a)^2 )^(1/2)/(b*x+a)*((-20*a^3*b^3*e^5+60*a^2*b^4*d*e^4-60*a*b^5*d^2*e^3+20*b^6 *d^3*e^2)*x^3-15/2*b^2*e*(a^4*e^4+4*a^3*b*d*e^3-18*a^2*b^2*d^2*e^2+20*a*b^ 3*d^3*e-7*b^4*d^4)*x^2-b*(2*a^5*e^5+5*a^4*b*d*e^4+20*a^3*b^2*d^2*e^3-110*a ^2*b^3*d^3*e^2+130*a*b^4*d^4*e-47*b^5*d^5)*x-1/4*(a^6*e^6+2*a^5*b*d*e^5+5* a^4*b^2*d^2*e^4+20*a^3*b^3*d^3*e^3-125*a^2*b^4*d^4*e^2+154*a*b^5*d^5*e-57* b^6*d^6)/e)/e^6/(e*x+d)^4+15*((b*x+a)^2)^(1/2)/(b*x+a)*b^4/e^7*(a^2*e^2-2* a*b*d*e+b^2*d^2)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 571 vs. \(2 (261) = 522\).
Time = 0.31 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.66 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {2 \, b^{6} e^{6} x^{6} + 57 \, b^{6} d^{6} - 154 \, a b^{5} d^{5} e + 125 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} - a^{6} e^{6} - 12 \, {\left (b^{6} d e^{5} - 2 \, a b^{5} e^{6}\right )} x^{5} - 4 \, {\left (17 \, b^{6} d^{2} e^{4} - 24 \, a b^{5} d e^{5}\right )} x^{4} - 16 \, {\left (2 \, b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} - 15 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 6 \, {\left (22 \, b^{6} d^{4} e^{2} - 84 \, a b^{5} d^{3} e^{3} + 90 \, a^{2} b^{4} d^{2} e^{4} - 20 \, a^{3} b^{3} d e^{5} - 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 4 \, {\left (42 \, b^{6} d^{5} e - 124 \, a b^{5} d^{4} e^{2} + 110 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 5 \, a^{4} b^{2} d e^{5} - 2 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} d^{6} - 2 \, a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} + {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 4 \, {\left (b^{6} d^{3} e^{3} - 2 \, a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5}\right )} x^{3} + 6 \, {\left (b^{6} d^{4} e^{2} - 2 \, a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4}\right )} x^{2} + 4 \, {\left (b^{6} d^{5} e - 2 \, a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \, {\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} \]
1/4*(2*b^6*e^6*x^6 + 57*b^6*d^6 - 154*a*b^5*d^5*e + 125*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 5*a^4*b^2*d^2*e^4 - 2*a^5*b*d*e^5 - a^6*e^6 - 12*(b^6 *d*e^5 - 2*a*b^5*e^6)*x^5 - 4*(17*b^6*d^2*e^4 - 24*a*b^5*d*e^5)*x^4 - 16*( 2*b^6*d^3*e^3 + 6*a*b^5*d^2*e^4 - 15*a^2*b^4*d*e^5 + 5*a^3*b^3*e^6)*x^3 + 6*(22*b^6*d^4*e^2 - 84*a*b^5*d^3*e^3 + 90*a^2*b^4*d^2*e^4 - 20*a^3*b^3*d*e ^5 - 5*a^4*b^2*e^6)*x^2 + 4*(42*b^6*d^5*e - 124*a*b^5*d^4*e^2 + 110*a^2*b^ 4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 5*a^4*b^2*d*e^5 - 2*a^5*b*e^6)*x + 60*(b^ 6*d^6 - 2*a*b^5*d^5*e + a^2*b^4*d^4*e^2 + (b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a ^2*b^4*e^6)*x^4 + 4*(b^6*d^3*e^3 - 2*a*b^5*d^2*e^4 + a^2*b^4*d*e^5)*x^3 + 6*(b^6*d^4*e^2 - 2*a*b^5*d^3*e^3 + a^2*b^4*d^2*e^4)*x^2 + 4*(b^6*d^5*e - 2 *a*b^5*d^4*e^2 + a^2*b^4*d^3*e^3)*x)*log(e*x + d))/(e^11*x^4 + 4*d*e^10*x^ 3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7)
\[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\int \frac {\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{5}}\, dx \]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\text {Exception raised: ValueError} \]
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. 525 vs. \(2 (261) = 522\).
Time = 0.28 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.53 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {15 \, {\left (b^{6} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{5} d e \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{4} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} + \frac {b^{6} e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, b^{6} d e^{4} x \mathrm {sgn}\left (b x + a\right ) + 12 \, a b^{5} e^{5} x \mathrm {sgn}\left (b x + a\right )}{2 \, e^{10}} + \frac {57 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 154 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 125 \, 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 ) - 5 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 2 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 80 \, {\left (b^{6} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{5} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{3} b^{3} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} + 30 \, {\left (7 \, b^{6} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a b^{5} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 18 \, 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} + 4 \, {\left (47 \, b^{6} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 130 \, a b^{5} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 110 \, a^{2} b^{4} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 20 \, 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 ) - 2 \, a^{5} b e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x}{4 \, {\left (e x + d\right )}^{4} e^{7}} \]
15*(b^6*d^2*sgn(b*x + a) - 2*a*b^5*d*e*sgn(b*x + a) + a^2*b^4*e^2*sgn(b*x + a))*log(abs(e*x + d))/e^7 + 1/2*(b^6*e^5*x^2*sgn(b*x + a) - 10*b^6*d*e^4 *x*sgn(b*x + a) + 12*a*b^5*e^5*x*sgn(b*x + a))/e^10 + 1/4*(57*b^6*d^6*sgn( b*x + a) - 154*a*b^5*d^5*e*sgn(b*x + a) + 125*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) - 5*a^4*b^2*d^2*e^4*sgn(b*x + a) - 2*a^ 5*b*d*e^5*sgn(b*x + a) - a^6*e^6*sgn(b*x + a) + 80*(b^6*d^3*e^3*sgn(b*x + a) - 3*a*b^5*d^2*e^4*sgn(b*x + a) + 3*a^2*b^4*d*e^5*sgn(b*x + a) - a^3*b^3 *e^6*sgn(b*x + a))*x^3 + 30*(7*b^6*d^4*e^2*sgn(b*x + a) - 20*a*b^5*d^3*e^3 *sgn(b*x + a) + 18*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 + 4*(47*b^6*d^5*e*sgn(b*x + a) - 130* a*b^5*d^4*e^2*sgn(b*x + a) + 110*a^2*b^4*d^3*e^3*sgn(b*x + a) - 20*a^3*b^3 *d^2*e^4*sgn(b*x + a) - 5*a^4*b^2*d*e^5*sgn(b*x + a) - 2*a^5*b*e^6*sgn(b*x + a))*x)/((e*x + d)^4*e^7)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\int \frac {\left (a+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^5} \,d x \]