Integrand size = 33, antiderivative size = 344 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {b^6 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}+\frac {3 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^4}-\frac {5 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^3}+\frac {10 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^2}-\frac {15 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac {6 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \]
b^6*x*((b*x+a)^2)^(1/2)/e^6/(b*x+a)-1/5*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7 /(b*x+a)/(e*x+d)^5+3/2*b*(-a*e+b*d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d )^4-5*b^2*(-a*e+b*d)^4*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^3+10*b^3*(-a* e+b*d)^3*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^2-15*b^4*(-a*e+b*d)^2*((b*x +a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)-6*b^5*(-a*e+b*d)*ln(e*x+d)*((b*x+a)^2)^(1 /2)/e^7/(b*x+a)
Time = 1.11 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=-\frac {\sqrt {(a+b x)^2} \left (2 a^6 e^6+3 a^5 b e^5 (d+5 e x)+5 a^4 b^2 e^4 \left (d^2+5 d e x+10 e^2 x^2\right )+10 a^3 b^3 e^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+30 a^2 b^4 e^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )-a b^5 d e \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )+b^6 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )+60 b^5 (b d-a e) (d+e x)^5 \log (d+e x)\right )}{10 e^7 (a+b x) (d+e x)^5} \]
-1/10*(Sqrt[(a + b*x)^2]*(2*a^6*e^6 + 3*a^5*b*e^5*(d + 5*e*x) + 5*a^4*b^2* e^4*(d^2 + 5*d*e*x + 10*e^2*x^2) + 10*a^3*b^3*e^3*(d^3 + 5*d^2*e*x + 10*d* e^2*x^2 + 10*e^3*x^3) + 30*a^2*b^4*e^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) - a*b^5*d*e*(137*d^4 + 625*d^3*e*x + 1100*d^2*e ^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4) + b^6*(87*d^6 + 375*d^5*e*x + 600*d^ 4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6) + 60*b^5*(b*d - a*e)*(d + e*x)^5*Log[d + e*x]))/(e^7*(a + b*x)*(d + e*x)^5 )
Time = 0.40 (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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^6}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^6}dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^6}{e^6}-\frac {6 (b d-a e) b^5}{e^6 (d+e x)}+\frac {15 (b d-a e)^2 b^4}{e^6 (d+e x)^2}-\frac {20 (b d-a e)^3 b^3}{e^6 (d+e x)^3}+\frac {15 (b d-a e)^4 b^2}{e^6 (d+e x)^4}-\frac {6 (b d-a e)^5 b}{e^6 (d+e x)^5}+\frac {(a e-b d)^6}{e^6 (d+e x)^6}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {6 b^5 (b d-a e) \log (d+e x)}{e^7}-\frac {15 b^4 (b d-a e)^2}{e^7 (d+e x)}+\frac {10 b^3 (b d-a e)^3}{e^7 (d+e x)^2}-\frac {5 b^2 (b d-a e)^4}{e^7 (d+e x)^3}+\frac {3 b (b d-a e)^5}{2 e^7 (d+e x)^4}-\frac {(b d-a e)^6}{5 e^7 (d+e x)^5}+\frac {b^6 x}{e^6}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((b^6*x)/e^6 - (b*d - a*e)^6/(5*e^7*(d + e* x)^5) + (3*b*(b*d - a*e)^5)/(2*e^7*(d + e*x)^4) - (5*b^2*(b*d - a*e)^4)/(e ^7*(d + e*x)^3) + (10*b^3*(b*d - a*e)^3)/(e^7*(d + e*x)^2) - (15*b^4*(b*d - a*e)^2)/(e^7*(d + e*x)) - (6*b^5*(b*d - a*e)*Log[d + e*x])/e^7))/(a + b* x)
3.21.3.3.1 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 = 0.97 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.13
| method | result | size |
| risch | \(\frac {b^{6} x \sqrt {\left (b x +a \right )^{2}}}{e^{6} \left (b x +a \right )}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (-15 e^{5} b^{4} a^{2}+30 a \,b^{5} d \,e^{4}-15 d^{2} e^{3} b^{6}\right ) x^{4}-10 b^{3} e^{2} \left (a^{3} e^{3}+3 a^{2} b d \,e^{2}-9 a \,b^{2} d^{2} e +5 b^{3} d^{3}\right ) x^{3}-5 b^{2} e \left (e^{4} a^{4}+2 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-22 b^{3} d^{3} e a +13 b^{4} d^{4}\right ) x^{2}-\frac {b \left (3 e^{5} a^{5}+5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}+30 b^{3} d^{3} e^{2} a^{2}-125 b^{4} d^{4} e a +77 b^{5} d^{5}\right ) x}{2}-\frac {2 e^{6} a^{6}+3 b d \,e^{5} a^{5}+5 b^{2} d^{2} e^{4} a^{4}+10 b^{3} d^{3} e^{3} a^{3}+30 b^{4} d^{4} e^{2} a^{2}-137 b^{5} d^{5} e a +87 b^{6} d^{6}}{10 e}\right )}{\left (b x +a \right ) e^{6} \left (e x +d \right )^{5}}+\frac {6 \sqrt {\left (b x +a \right )^{2}}\, b^{5} \left (a e -b d \right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{7}}\) | \(388\) |
| default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (-5 b^{2} d^{2} e^{4} a^{4}-10 b^{3} d^{3} e^{3} a^{3}-30 b^{4} d^{4} e^{2} a^{2}+137 b^{5} d^{5} e a -3 b d \,e^{5} a^{5}-15 a^{5} b \,e^{6} x -375 b^{6} d^{5} e x +50 b^{6} d \,e^{5} x^{5}-150 a^{2} b^{4} e^{6} x^{4}-50 b^{6} d^{2} e^{4} x^{4}-100 a^{3} b^{3} e^{6} x^{3}-400 b^{6} d^{3} e^{3} x^{3}-50 a^{4} b^{2} e^{6} x^{2}-600 b^{6} d^{4} e^{2} x^{2}+300 a \,b^{5} d \,e^{5} x^{4}-300 a^{2} b^{4} d \,e^{5} x^{3}+900 a \,b^{5} d^{2} e^{4} x^{3}-100 a^{3} b^{3} d \,e^{5} x^{2}-300 a^{2} b^{4} d^{2} e^{4} x^{2}+1100 a \,b^{5} d^{3} e^{3} x^{2}-25 a^{4} b^{2} d \,e^{5} x +300 \ln \left (e x +d \right ) a \,b^{5} d \,e^{5} x^{4}+600 \ln \left (e x +d \right ) a \,b^{5} d^{3} e^{3} x^{2}+600 \ln \left (e x +d \right ) a \,b^{5} d^{2} e^{4} x^{3}+300 \ln \left (e x +d \right ) a \,b^{5} d^{4} e^{2} x -300 \ln \left (e x +d \right ) b^{6} d^{2} e^{4} x^{4}-600 \ln \left (e x +d \right ) b^{6} d^{4} e^{2} x^{2}-600 \ln \left (e x +d \right ) b^{6} d^{3} e^{3} x^{3}-2 e^{6} a^{6}-87 b^{6} d^{6}+60 \ln \left (e x +d \right ) a \,b^{5} d^{5} e +60 \ln \left (e x +d \right ) a \,b^{5} e^{6} x^{5}-60 \ln \left (e x +d \right ) b^{6} d \,e^{5} x^{5}-50 a^{3} b^{3} d^{2} e^{4} x -150 a^{2} b^{4} d^{3} e^{3} x +625 a \,b^{5} d^{4} e^{2} x -300 \ln \left (e x +d \right ) b^{6} d^{5} e x +10 b^{6} e^{6} x^{6}-60 \ln \left (e x +d \right ) b^{6} d^{6}\right )}{10 \left (b x +a \right )^{5} e^{7} \left (e x +d \right )^{5}}\) | \(603\) |
b^6*x*((b*x+a)^2)^(1/2)/e^6/(b*x+a)+((b*x+a)^2)^(1/2)/(b*x+a)*((-15*a^2*b^ 4*e^5+30*a*b^5*d*e^4-15*b^6*d^2*e^3)*x^4-10*b^3*e^2*(a^3*e^3+3*a^2*b*d*e^2 -9*a*b^2*d^2*e+5*b^3*d^3)*x^3-5*b^2*e*(a^4*e^4+2*a^3*b*d*e^3+6*a^2*b^2*d^2 *e^2-22*a*b^3*d^3*e+13*b^4*d^4)*x^2-1/2*b*(3*a^5*e^5+5*a^4*b*d*e^4+10*a^3* b^2*d^2*e^3+30*a^2*b^3*d^3*e^2-125*a*b^4*d^4*e+77*b^5*d^5)*x-1/10/e*(2*a^6 *e^6+3*a^5*b*d*e^5+5*a^4*b^2*d^2*e^4+10*a^3*b^3*d^3*e^3+30*a^2*b^4*d^4*e^2 -137*a*b^5*d^5*e+87*b^6*d^6))/e^6/(e*x+d)^5+6*((b*x+a)^2)^(1/2)/(b*x+a)*b^ 5/e^7*(a*e-b*d)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (263) = 526\).
Time = 0.29 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {10 \, b^{6} e^{6} x^{6} + 50 \, b^{6} d e^{5} x^{5} - 87 \, b^{6} d^{6} + 137 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 10 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 50 \, {\left (b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} - 100 \, {\left (4 \, b^{6} d^{3} e^{3} - 9 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} - 50 \, {\left (12 \, b^{6} d^{4} e^{2} - 22 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} - 5 \, {\left (75 \, b^{6} d^{5} e - 125 \, a b^{5} d^{4} e^{2} + 30 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \, {\left (b^{6} d^{6} - a b^{5} d^{5} e + {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (b^{6} d^{2} e^{4} - a b^{5} d e^{5}\right )} x^{4} + 10 \, {\left (b^{6} d^{3} e^{3} - a b^{5} d^{2} e^{4}\right )} x^{3} + 10 \, {\left (b^{6} d^{4} e^{2} - a b^{5} d^{3} e^{3}\right )} x^{2} + 5 \, {\left (b^{6} d^{5} e - a b^{5} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{10 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \]
1/10*(10*b^6*e^6*x^6 + 50*b^6*d*e^5*x^5 - 87*b^6*d^6 + 137*a*b^5*d^5*e - 3 0*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 - 3*a^5*b*d*e^5 - 2*a^6*e^6 - 50*(b^6*d^2*e^4 - 6*a*b^5*d*e^5 + 3*a^2*b^4*e^6)*x^4 - 100* (4*b^6*d^3*e^3 - 9*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 - 50 *(12*b^6*d^4*e^2 - 22*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 + 2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 - 5*(75*b^6*d^5*e - 125*a*b^5*d^4*e^2 + 30*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 + 3*a^5*b*e^6)*x - 60*(b^6*d^6 - a*b^5*d^5*e + (b^6*d*e^5 - a*b^5*e^6)*x^5 + 5*(b^6*d^2*e^4 - a*b^5*d*e^5 )*x^4 + 10*(b^6*d^3*e^3 - a*b^5*d^2*e^4)*x^3 + 10*(b^6*d^4*e^2 - a*b^5*d^3 *e^3)*x^2 + 5*(b^6*d^5*e - a*b^5*d^4*e^2)*x)*log(e*x + d))/(e^12*x^5 + 5*d *e^11*x^4 + 10*d^2*e^10*x^3 + 10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*e^7)
\[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\int \frac {\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{6}}\, dx \]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \]
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 = 519, normalized size of antiderivative = 1.51 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {b^{6} x \mathrm {sgn}\left (b x + a\right )}{e^{6}} - \frac {6 \, {\left (b^{6} d \mathrm {sgn}\left (b x + a\right ) - a b^{5} e \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} - \frac {87 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 137 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 30 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, 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 ) + 3 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 150 \, {\left (b^{6} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{5} d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{4} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x^{4} + 100 \, {\left (5 \, b^{6} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 9 \, 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} + 50 \, {\left (13 \, b^{6} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 22 \, 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 ) + 2 \, 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} + 5 \, {\left (77 \, b^{6} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 125 \, a b^{5} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 30 \, a^{2} b^{4} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, 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 ) + 3 \, a^{5} b e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x}{10 \, {\left (e x + d\right )}^{5} e^{7}} \]
b^6*x*sgn(b*x + a)/e^6 - 6*(b^6*d*sgn(b*x + a) - a*b^5*e*sgn(b*x + a))*log (abs(e*x + d))/e^7 - 1/10*(87*b^6*d^6*sgn(b*x + a) - 137*a*b^5*d^5*e*sgn(b *x + a) + 30*a^2*b^4*d^4*e^2*sgn(b*x + a) + 10*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) + 3*a^5*b*d*e^5*sgn(b*x + a) + 2*a^6*e^ 6*sgn(b*x + a) + 150*(b^6*d^2*e^4*sgn(b*x + a) - 2*a*b^5*d*e^5*sgn(b*x + a ) + a^2*b^4*e^6*sgn(b*x + a))*x^4 + 100*(5*b^6*d^3*e^3*sgn(b*x + a) - 9*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 + 50*(13*b^6*d^4*e^2*sgn(b*x + a) - 22*a*b^5*d^3*e^3*sgn(b*x + a) + 6*a^2*b^4*d^2*e^4*sgn(b*x + a) + 2*a^3*b^3*d*e^5*sgn(b*x + a) + a^ 4*b^2*e^6*sgn(b*x + a))*x^2 + 5*(77*b^6*d^5*e*sgn(b*x + a) - 125*a*b^5*d^4 *e^2*sgn(b*x + a) + 30*a^2*b^4*d^3*e^3*sgn(b*x + a) + 10*a^3*b^3*d^2*e^4*s gn(b*x + a) + 5*a^4*b^2*d*e^5*sgn(b*x + a) + 3*a^5*b*e^6*sgn(b*x + a))*x)/ ((e*x + d)^5*e^7)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\int \frac {\left (a+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^6} \,d x \]