Integrand size = 33, antiderivative size = 41 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (b d-a e) (d+e x)^7} \] Output:
1/7*(b^2*x^2+2*a*b*x+a^2)^(7/2)/(-a*e+b*d)/(e*x+d)^7
Leaf count is larger than twice the leaf count of optimal. \(289\) vs. \(2(41)=82\).
Time = 1.13 (sec) , antiderivative size = 289, normalized size of antiderivative = 7.05 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=-\frac {\sqrt {(a+b x)^2} \left (a^6 e^6+a^5 b e^5 (d+7 e x)+a^4 b^2 e^4 \left (d^2+7 d e x+21 e^2 x^2\right )+a^3 b^3 e^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+a^2 b^4 e^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+a b^5 e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+b^6 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )\right )}{7 e^7 (a+b x) (d+e x)^7} \] Input:
Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^8,x]
Output:
-1/7*(Sqrt[(a + b*x)^2]*(a^6*e^6 + a^5*b*e^5*(d + 7*e*x) + a^4*b^2*e^4*(d^ 2 + 7*d*e*x + 21*e^2*x^2) + a^3*b^3*e^3*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + a^2*b^4*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + a*b^5*e*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^ 3 + 35*d*e^4*x^4 + 21*e^5*x^5) + b^6*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 3 5*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6)))/(e^7*(a + b*x )*(d + e*x)^7)
Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {1185}
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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx\) |
| \(\Big \downarrow \) 1185 |
| \(\displaystyle \frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (d+e x)^7 (b d-a e)}\) |
Input:
Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^8,x]
Output:
(a^2 + 2*a*b*x + b^2*x^2)^(7/2)/(7*(b*d - a*e)*(d + e*x)^7)
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))*((a_) + (b_.)*(x_) + (c_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-f)*g*(d + e*x)^(m + 1)*((a + b*x + c* x^2)^(p + 1)/(b*(p + 1)*(e*f - d*g))), x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && EqQ[m + 2*p + 3, 0] && EqQ[2*c*f - b*g, 0 ]
Leaf count of result is larger than twice the leaf count of optimal. \(329\) vs. \(2(37)=74\).
Time = 2.96 (sec) , antiderivative size = 330, normalized size of antiderivative = 8.05
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{6} x^{6}}{e}-\frac {3 b^{5} \left (a e +b d \right ) x^{5}}{e^{2}}-\frac {5 b^{4} \left (e^{2} a^{2}+a b d e +b^{2} d^{2}\right ) x^{4}}{e^{3}}-\frac {5 b^{3} \left (e^{3} a^{3}+a^{2} b d \,e^{2}+a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{e^{4}}-\frac {3 b^{2} \left (a^{4} e^{4}+a^{3} b d \,e^{3}+a^{2} b^{2} d^{2} e^{2}+a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{e^{5}}-\frac {b \left (e^{5} a^{5}+a^{4} b d \,e^{4}+a^{3} b^{2} d^{2} e^{3}+a^{2} b^{3} d^{3} e^{2}+a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{e^{6}}-\frac {a^{6} e^{6}+a^{5} b d \,e^{5}+a^{4} b^{2} d^{2} e^{4}+a^{3} b^{3} d^{3} e^{3}+a^{2} b^{4} d^{4} e^{2}+a \,b^{5} d^{5} e +b^{6} d^{6}}{7 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{7}}\) | \(330\) |
| gosper | \(-\frac {\left (7 b^{6} x^{6} e^{6}+21 x^{5} a \,b^{5} e^{6}+21 x^{5} b^{6} d \,e^{5}+35 x^{4} a^{2} b^{4} e^{6}+35 x^{4} a \,b^{5} d \,e^{5}+35 x^{4} b^{6} d^{2} e^{4}+35 x^{3} a^{3} b^{3} e^{6}+35 x^{3} a^{2} b^{4} d \,e^{5}+35 x^{3} a \,b^{5} d^{2} e^{4}+35 x^{3} b^{6} d^{3} e^{3}+21 x^{2} a^{4} b^{2} e^{6}+21 x^{2} a^{3} b^{3} d \,e^{5}+21 x^{2} a^{2} b^{4} d^{2} e^{4}+21 x^{2} a \,b^{5} d^{3} e^{3}+21 x^{2} b^{6} d^{4} e^{2}+7 x \,a^{5} b \,e^{6}+7 x \,a^{4} b^{2} d \,e^{5}+7 x \,a^{3} b^{3} d^{2} e^{4}+7 x \,a^{2} b^{4} d^{3} e^{3}+7 x a \,b^{5} d^{4} e^{2}+7 x \,b^{6} d^{5} e +a^{6} e^{6}+a^{5} b d \,e^{5}+a^{4} b^{2} d^{2} e^{4}+a^{3} b^{3} d^{3} e^{3}+a^{2} b^{4} d^{4} e^{2}+a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{7 \left (e x +d \right )^{7} e^{7} \left (b x +a \right )^{5}}\) | \(386\) |
| default | \(-\frac {\left (7 b^{6} x^{6} e^{6}+21 x^{5} a \,b^{5} e^{6}+21 x^{5} b^{6} d \,e^{5}+35 x^{4} a^{2} b^{4} e^{6}+35 x^{4} a \,b^{5} d \,e^{5}+35 x^{4} b^{6} d^{2} e^{4}+35 x^{3} a^{3} b^{3} e^{6}+35 x^{3} a^{2} b^{4} d \,e^{5}+35 x^{3} a \,b^{5} d^{2} e^{4}+35 x^{3} b^{6} d^{3} e^{3}+21 x^{2} a^{4} b^{2} e^{6}+21 x^{2} a^{3} b^{3} d \,e^{5}+21 x^{2} a^{2} b^{4} d^{2} e^{4}+21 x^{2} a \,b^{5} d^{3} e^{3}+21 x^{2} b^{6} d^{4} e^{2}+7 x \,a^{5} b \,e^{6}+7 x \,a^{4} b^{2} d \,e^{5}+7 x \,a^{3} b^{3} d^{2} e^{4}+7 x \,a^{2} b^{4} d^{3} e^{3}+7 x a \,b^{5} d^{4} e^{2}+7 x \,b^{6} d^{5} e +a^{6} e^{6}+a^{5} b d \,e^{5}+a^{4} b^{2} d^{2} e^{4}+a^{3} b^{3} d^{3} e^{3}+a^{2} b^{4} d^{4} e^{2}+a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{7 \left (e x +d \right )^{7} e^{7} \left (b x +a \right )^{5}}\) | \(386\) |
| orering | \(-\frac {\left (7 b^{6} x^{6} e^{6}+21 x^{5} a \,b^{5} e^{6}+21 x^{5} b^{6} d \,e^{5}+35 x^{4} a^{2} b^{4} e^{6}+35 x^{4} a \,b^{5} d \,e^{5}+35 x^{4} b^{6} d^{2} e^{4}+35 x^{3} a^{3} b^{3} e^{6}+35 x^{3} a^{2} b^{4} d \,e^{5}+35 x^{3} a \,b^{5} d^{2} e^{4}+35 x^{3} b^{6} d^{3} e^{3}+21 x^{2} a^{4} b^{2} e^{6}+21 x^{2} a^{3} b^{3} d \,e^{5}+21 x^{2} a^{2} b^{4} d^{2} e^{4}+21 x^{2} a \,b^{5} d^{3} e^{3}+21 x^{2} b^{6} d^{4} e^{2}+7 x \,a^{5} b \,e^{6}+7 x \,a^{4} b^{2} d \,e^{5}+7 x \,a^{3} b^{3} d^{2} e^{4}+7 x \,a^{2} b^{4} d^{3} e^{3}+7 x a \,b^{5} d^{4} e^{2}+7 x \,b^{6} d^{5} e +a^{6} e^{6}+a^{5} b d \,e^{5}+a^{4} b^{2} d^{2} e^{4}+a^{3} b^{3} d^{3} e^{3}+a^{2} b^{4} d^{4} e^{2}+a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{7 e^{7} \left (b x +a \right )^{5} \left (e x +d \right )^{7}}\) | \(395\) |
Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^8,x,method=_RETURNVERBOSE)
Output:
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/e*b^6*x^6-3*b^5*(a*e+b*d)/e^2*x^5-5*b^4*(a^2 *e^2+a*b*d*e+b^2*d^2)/e^3*x^4-5*b^3*(a^3*e^3+a^2*b*d*e^2+a*b^2*d^2*e+b^3*d ^3)/e^4*x^3-3*b^2*(a^4*e^4+a^3*b*d*e^3+a^2*b^2*d^2*e^2+a*b^3*d^3*e+b^4*d^4 )/e^5*x^2-b*(a^5*e^5+a^4*b*d*e^4+a^3*b^2*d^2*e^3+a^2*b^3*d^3*e^2+a*b^4*d^4 *e+b^5*d^5)/e^6*x-1/7*(a^6*e^6+a^5*b*d*e^5+a^4*b^2*d^2*e^4+a^3*b^3*d^3*e^3 +a^2*b^4*d^4*e^2+a*b^5*d^5*e+b^6*d^6)/e^7)/(e*x+d)^7
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (37) = 74\).
Time = 0.08 (sec) , antiderivative size = 398, normalized size of antiderivative = 9.71 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=-\frac {7 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} + a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} + a^{5} b d e^{5} + a^{6} e^{6} + 21 \, {\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{5} + 35 \, {\left (b^{6} d^{2} e^{4} + a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 35 \, {\left (b^{6} d^{3} e^{3} + a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 21 \, {\left (b^{6} d^{4} e^{2} + a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4} + a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 7 \, {\left (b^{6} d^{5} e + a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3} + a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x}{7 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^8,x, algorithm="fric as")
Output:
-1/7*(7*b^6*e^6*x^6 + b^6*d^6 + a*b^5*d^5*e + a^2*b^4*d^4*e^2 + a^3*b^3*d^ 3*e^3 + a^4*b^2*d^2*e^4 + a^5*b*d*e^5 + a^6*e^6 + 21*(b^6*d*e^5 + a*b^5*e^ 6)*x^5 + 35*(b^6*d^2*e^4 + a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 35*(b^6*d^3*e^ 3 + a*b^5*d^2*e^4 + a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 21*(b^6*d^4*e^2 + a *b^5*d^3*e^3 + a^2*b^4*d^2*e^4 + a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 7*(b^6 *d^5*e + a*b^5*d^4*e^2 + a^2*b^4*d^3*e^3 + a^3*b^3*d^2*e^4 + a^4*b^2*d*e^5 + a^5*b*e^6)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*e^11* x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**8,x)
Output:
Timed out
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^8,x, algorithm="maxi ma")
Output:
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. 564 vs. \(2 (37) = 74\).
Time = 0.16 (sec) , antiderivative size = 564, normalized size of antiderivative = 13.76 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\frac {b^{7} \mathrm {sgn}\left (b x + a\right )}{7 \, {\left (b d e^{7} - a e^{8}\right )}} - \frac {7 \, b^{6} e^{6} x^{6} \mathrm {sgn}\left (b x + a\right ) + 21 \, b^{6} d e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 21 \, a b^{5} e^{6} x^{5} \mathrm {sgn}\left (b x + a\right ) + 35 \, b^{6} d^{2} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, a b^{5} d e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{2} b^{4} e^{6} x^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, b^{6} d^{3} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a b^{5} d^{2} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{2} b^{4} d e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{3} b^{3} e^{6} x^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, b^{6} d^{4} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, a b^{5} d^{3} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{4} d^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{3} b^{3} d e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{4} b^{2} e^{6} x^{2} \mathrm {sgn}\left (b x + a\right ) + 7 \, b^{6} d^{5} e x \mathrm {sgn}\left (b x + a\right ) + 7 \, a b^{5} d^{4} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{2} b^{4} d^{3} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{3} b^{3} d^{2} e^{4} x \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{4} b^{2} d e^{5} x \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{5} b e^{6} x \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )}{7 \, {\left (e x + d\right )}^{7} e^{7}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^8,x, algorithm="giac ")
Output:
1/7*b^7*sgn(b*x + a)/(b*d*e^7 - a*e^8) - 1/7*(7*b^6*e^6*x^6*sgn(b*x + a) + 21*b^6*d*e^5*x^5*sgn(b*x + a) + 21*a*b^5*e^6*x^5*sgn(b*x + a) + 35*b^6*d^ 2*e^4*x^4*sgn(b*x + a) + 35*a*b^5*d*e^5*x^4*sgn(b*x + a) + 35*a^2*b^4*e^6* x^4*sgn(b*x + a) + 35*b^6*d^3*e^3*x^3*sgn(b*x + a) + 35*a*b^5*d^2*e^4*x^3* sgn(b*x + a) + 35*a^2*b^4*d*e^5*x^3*sgn(b*x + a) + 35*a^3*b^3*e^6*x^3*sgn( b*x + a) + 21*b^6*d^4*e^2*x^2*sgn(b*x + a) + 21*a*b^5*d^3*e^3*x^2*sgn(b*x + a) + 21*a^2*b^4*d^2*e^4*x^2*sgn(b*x + a) + 21*a^3*b^3*d*e^5*x^2*sgn(b*x + a) + 21*a^4*b^2*e^6*x^2*sgn(b*x + a) + 7*b^6*d^5*e*x*sgn(b*x + a) + 7*a* b^5*d^4*e^2*x*sgn(b*x + a) + 7*a^2*b^4*d^3*e^3*x*sgn(b*x + a) + 7*a^3*b^3* d^2*e^4*x*sgn(b*x + a) + 7*a^4*b^2*d*e^5*x*sgn(b*x + a) + 7*a^5*b*e^6*x*sg n(b*x + a) + b^6*d^6*sgn(b*x + a) + a*b^5*d^5*e*sgn(b*x + a) + a^2*b^4*d^4 *e^2*sgn(b*x + a) + a^3*b^3*d^3*e^3*sgn(b*x + a) + a^4*b^2*d^2*e^4*sgn(b*x + a) + a^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a))/((e*x + d)^7*e^7)
Time = 11.71 (sec) , antiderivative size = 1010, normalized size of antiderivative = 24.63 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx =\text {Too large to display} \] Input:
int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^8,x)
Output:
(((b^6*d^5 - 6*a^5*b*e^5 + 15*a^4*b^2*d*e^4 + 15*a^2*b^4*d^3*e^2 - 20*a^3* b^3*d^2*e^3 - 6*a*b^5*d^4*e)/(6*e^7) + (d*((b^6*d^4*e + 15*a^4*b^2*e^5 - 6 *a*b^5*d^3*e^2 - 20*a^3*b^3*d*e^4 + 15*a^2*b^4*d^2*e^3)/(6*e^7) - (d*((20* a^3*b^3*e^5 - b^6*d^3*e^2 + 6*a*b^5*d^2*e^3 - 15*a^2*b^4*d*e^4)/(6*e^7) - (d*((d*((b^6*d)/(6*e^3) - (b^5*(6*a*e - b*d))/(6*e^3)))/e + (b^4*(15*a^2*e ^2 + b^2*d^2 - 6*a*b*d*e))/(6*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^ (1/2))/((a + b*x)*(d + e*x)^6) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 24*a*b^5 *d*e)/(3*e^7) + (d*((b^6*d)/(3*e^6) - (2*b^5*(3*a*e - 2*b*d))/(3*e^6)))/e) *(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^3) - ((a^6/(7*e) - (d*((d*((d*((d*((d*((6*a*b^5)/(7*e) - (b^6*d)/(7*e^2)))/e - (15*a^2*b^4)/( 7*e)))/e + (20*a^3*b^3)/(7*e)))/e - (15*a^4*b^2)/(7*e)))/e + (6*a^5*b)/(7* e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^7) - (((5*b^ 6*d^4 + 15*a^4*b^2*e^4 - 40*a^3*b^3*d*e^3 + 45*a^2*b^4*d^2*e^2 - 24*a*b^5* d^3*e)/(5*e^7) + (d*((4*b^6*d^3*e - 20*a^3*b^3*e^4 - 18*a*b^5*d^2*e^2 + 30 *a^2*b^4*d*e^3)/(5*e^7) + (d*((d*((b^6*d)/(5*e^4) - (2*b^5*(3*a*e - b*d))/ (5*e^4)))/e + (3*b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(5*e^5)))/e))/e)*( a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^5) + (((5*b^6*d - 6*a *b^5*e)/(2*e^7) + (b^6*d)/(2*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^2) + (((10*b^6*d^3 - 20*a^3*b^3*e^3 + 45*a^2*b^4*d*e^2 - 36 *a*b^5*d^2*e)/(4*e^7) + (d*((d*((b^6*d)/(4*e^5) - (3*b^5*(2*a*e - b*d))...
Time = 0.23 (sec) , antiderivative size = 383, normalized size of antiderivative = 9.34 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\frac {b^{6} e^{6} x^{7}-21 a \,b^{5} d \,e^{5} x^{5}-35 a^{2} b^{4} d \,e^{5} x^{4}-35 a \,b^{5} d^{2} e^{4} x^{4}-35 a^{3} b^{3} d \,e^{5} x^{3}-35 a^{2} b^{4} d^{2} e^{4} x^{3}-35 a \,b^{5} d^{3} e^{3} x^{3}-21 a^{4} b^{2} d \,e^{5} x^{2}-21 a^{3} b^{3} d^{2} e^{4} x^{2}-21 a^{2} b^{4} d^{3} e^{3} x^{2}-21 a \,b^{5} d^{4} e^{2} x^{2}-7 a^{5} b d \,e^{5} x -7 a^{4} b^{2} d^{2} e^{4} x -7 a^{3} b^{3} d^{3} e^{3} x -7 a^{2} b^{4} d^{4} e^{2} x -7 a \,b^{5} d^{5} e x -a^{6} d \,e^{5}-a^{5} b \,d^{2} e^{4}-a^{4} b^{2} d^{3} e^{3}-a^{3} b^{3} d^{4} e^{2}-a^{2} b^{4} d^{5} e -a \,b^{5} d^{6}}{7 d \,e^{6} \left (e^{7} x^{7}+7 d \,e^{6} x^{6}+21 d^{2} e^{5} x^{5}+35 d^{3} e^{4} x^{4}+35 d^{4} e^{3} x^{3}+21 d^{5} e^{2} x^{2}+7 d^{6} e x +d^{7}\right )} \] Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^8,x)
Output:
( - a**6*d*e**5 - a**5*b*d**2*e**4 - 7*a**5*b*d*e**5*x - a**4*b**2*d**3*e* *3 - 7*a**4*b**2*d**2*e**4*x - 21*a**4*b**2*d*e**5*x**2 - a**3*b**3*d**4*e **2 - 7*a**3*b**3*d**3*e**3*x - 21*a**3*b**3*d**2*e**4*x**2 - 35*a**3*b**3 *d*e**5*x**3 - a**2*b**4*d**5*e - 7*a**2*b**4*d**4*e**2*x - 21*a**2*b**4*d **3*e**3*x**2 - 35*a**2*b**4*d**2*e**4*x**3 - 35*a**2*b**4*d*e**5*x**4 - a *b**5*d**6 - 7*a*b**5*d**5*e*x - 21*a*b**5*d**4*e**2*x**2 - 35*a*b**5*d**3 *e**3*x**3 - 35*a*b**5*d**2*e**4*x**4 - 21*a*b**5*d*e**5*x**5 + b**6*e**6* x**7)/(7*d*e**6*(d**7 + 7*d**6*e*x + 21*d**5*e**2*x**2 + 35*d**4*e**3*x**3 + 35*d**3*e**4*x**4 + 21*d**2*e**5*x**5 + 7*d*e**6*x**6 + e**7*x**7))