Integrand size = 33, antiderivative size = 216 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{10}} \, dx=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{9 (b d-a e) (d+e x)^9}+\frac {b \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{18 (b d-a e)^2 (d+e x)^8}+\frac {b^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{42 (b d-a e)^3 (d+e x)^7}+\frac {b^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{126 (b d-a e)^4 (d+e x)^6}+\frac {b^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{630 (b d-a e)^5 (d+e x)^5} \] Output:
1/9*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(-a*e+b*d)/(e*x+d)^9+1/18*b*(b^2*x^2+2*a*b *x+a^2)^(5/2)/(-a*e+b*d)^2/(e*x+d)^8+1/42*b^2*(b^2*x^2+2*a*b*x+a^2)^(5/2)/ (-a*e+b*d)^3/(e*x+d)^7+1/126*b^3*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(-a*e+b*d)^4/ (e*x+d)^6+1/630*b^4*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(-a*e+b*d)^5/(e*x+d)^5
Time = 1.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.75 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{10}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (70 a^4 e^4+35 a^3 b e^3 (d+9 e x)+15 a^2 b^2 e^2 \left (d^2+9 d e x+36 e^2 x^2\right )+5 a b^3 e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+b^4 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )}{630 e^5 (a+b x) (d+e x)^9} \] Input:
Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^10,x]
Output:
-1/630*(Sqrt[(a + b*x)^2]*(70*a^4*e^4 + 35*a^3*b*e^3*(d + 9*e*x) + 15*a^2* b^2*e^2*(d^2 + 9*d*e*x + 36*e^2*x^2) + 5*a*b^3*e*(d^3 + 9*d^2*e*x + 36*d*e ^2*x^2 + 84*e^3*x^3) + b^4*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^ 3 + 126*e^4*x^4)))/(e^5*(a + b*x)*(d + e*x)^9)
Time = 0.54 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.68, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 53, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{10}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^3 (a+b x)^4}{(d+e x)^{10}}dx}{b^3 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^4}{(d+e x)^{10}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^4}{e^4 (d+e x)^6}-\frac {4 (b d-a e) b^3}{e^4 (d+e x)^7}+\frac {6 (b d-a e)^2 b^2}{e^4 (d+e x)^8}-\frac {4 (b d-a e)^3 b}{e^4 (d+e x)^9}+\frac {(a e-b d)^4}{e^4 (d+e x)^{10}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {2 b^3 (b d-a e)}{3 e^5 (d+e x)^6}-\frac {6 b^2 (b d-a e)^2}{7 e^5 (d+e x)^7}+\frac {b (b d-a e)^3}{2 e^5 (d+e x)^8}-\frac {(b d-a e)^4}{9 e^5 (d+e x)^9}-\frac {b^4}{5 e^5 (d+e x)^5}\right )}{a+b x}\) |
Input:
Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^10,x]
Output:
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/9*(b*d - a*e)^4/(e^5*(d + e*x)^9) + (b* (b*d - a*e)^3)/(2*e^5*(d + e*x)^8) - (6*b^2*(b*d - a*e)^2)/(7*e^5*(d + e*x )^7) + (2*b^3*(b*d - a*e))/(3*e^5*(d + e*x)^6) - b^4/(5*e^5*(d + e*x)^5))) /(a + b*x)
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, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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 = 2.94 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.87
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{4} x^{4}}{5 e}-\frac {2 b^{3} \left (5 a e +b d \right ) x^{3}}{15 e^{2}}-\frac {2 b^{2} \left (15 e^{2} a^{2}+5 a b d e +b^{2} d^{2}\right ) x^{2}}{35 e^{3}}-\frac {b \left (35 e^{3} a^{3}+15 a^{2} b d \,e^{2}+5 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x}{70 e^{4}}-\frac {70 a^{4} e^{4}+35 a^{3} b d \,e^{3}+15 a^{2} b^{2} d^{2} e^{2}+5 a \,b^{3} d^{3} e +b^{4} d^{4}}{630 e^{5}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{9}}\) | \(187\) |
| gosper | \(-\frac {\left (126 b^{4} x^{4} e^{4}+420 x^{3} a \,b^{3} e^{4}+84 x^{3} b^{4} d \,e^{3}+540 x^{2} a^{2} b^{2} e^{4}+180 x^{2} a \,b^{3} d \,e^{3}+36 x^{2} b^{4} d^{2} e^{2}+315 x \,a^{3} b \,e^{4}+135 x \,a^{2} b^{2} d \,e^{3}+45 x a \,b^{3} d^{2} e^{2}+9 x \,b^{4} d^{3} e +70 a^{4} e^{4}+35 a^{3} b d \,e^{3}+15 a^{2} b^{2} d^{2} e^{2}+5 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{630 e^{5} \left (e x +d \right )^{9} \left (b x +a \right )^{3}}\) | \(201\) |
| default | \(-\frac {\left (126 b^{4} x^{4} e^{4}+420 x^{3} a \,b^{3} e^{4}+84 x^{3} b^{4} d \,e^{3}+540 x^{2} a^{2} b^{2} e^{4}+180 x^{2} a \,b^{3} d \,e^{3}+36 x^{2} b^{4} d^{2} e^{2}+315 x \,a^{3} b \,e^{4}+135 x \,a^{2} b^{2} d \,e^{3}+45 x a \,b^{3} d^{2} e^{2}+9 x \,b^{4} d^{3} e +70 a^{4} e^{4}+35 a^{3} b d \,e^{3}+15 a^{2} b^{2} d^{2} e^{2}+5 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{630 e^{5} \left (e x +d \right )^{9} \left (b x +a \right )^{3}}\) | \(201\) |
| orering | \(-\frac {\left (126 b^{4} x^{4} e^{4}+420 x^{3} a \,b^{3} e^{4}+84 x^{3} b^{4} d \,e^{3}+540 x^{2} a^{2} b^{2} e^{4}+180 x^{2} a \,b^{3} d \,e^{3}+36 x^{2} b^{4} d^{2} e^{2}+315 x \,a^{3} b \,e^{4}+135 x \,a^{2} b^{2} d \,e^{3}+45 x a \,b^{3} d^{2} e^{2}+9 x \,b^{4} d^{3} e +70 a^{4} e^{4}+35 a^{3} b d \,e^{3}+15 a^{2} b^{2} d^{2} e^{2}+5 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}{630 e^{5} \left (b x +a \right )^{3} \left (e x +d \right )^{9}}\) | \(210\) |
Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^10,x,method=_RETURNVERBOSE )
Output:
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/5/e*b^4*x^4-2/15/e^2*b^3*(5*a*e+b*d)*x^3-2/3 5*b^2/e^3*(15*a^2*e^2+5*a*b*d*e+b^2*d^2)*x^2-1/70*b/e^4*(35*a^3*e^3+15*a^2 *b*d*e^2+5*a*b^2*d^2*e+b^3*d^3)*x-1/630/e^5*(70*a^4*e^4+35*a^3*b*d*e^3+15* a^2*b^2*d^2*e^2+5*a*b^3*d^3*e+b^4*d^4))/(e*x+d)^9
Time = 0.07 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.25 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{10}} \, dx=-\frac {126 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 5 \, a b^{3} d^{3} e + 15 \, a^{2} b^{2} d^{2} e^{2} + 35 \, a^{3} b d e^{3} + 70 \, a^{4} e^{4} + 84 \, {\left (b^{4} d e^{3} + 5 \, a b^{3} e^{4}\right )} x^{3} + 36 \, {\left (b^{4} d^{2} e^{2} + 5 \, a b^{3} d e^{3} + 15 \, a^{2} b^{2} e^{4}\right )} x^{2} + 9 \, {\left (b^{4} d^{3} e + 5 \, a b^{3} d^{2} e^{2} + 15 \, a^{2} b^{2} d e^{3} + 35 \, a^{3} b e^{4}\right )} x}{630 \, {\left (e^{14} x^{9} + 9 \, d e^{13} x^{8} + 36 \, d^{2} e^{12} x^{7} + 84 \, d^{3} e^{11} x^{6} + 126 \, d^{4} e^{10} x^{5} + 126 \, d^{5} e^{9} x^{4} + 84 \, d^{6} e^{8} x^{3} + 36 \, d^{7} e^{7} x^{2} + 9 \, d^{8} e^{6} x + d^{9} e^{5}\right )}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^10,x, algorithm="fri cas")
Output:
-1/630*(126*b^4*e^4*x^4 + b^4*d^4 + 5*a*b^3*d^3*e + 15*a^2*b^2*d^2*e^2 + 3 5*a^3*b*d*e^3 + 70*a^4*e^4 + 84*(b^4*d*e^3 + 5*a*b^3*e^4)*x^3 + 36*(b^4*d^ 2*e^2 + 5*a*b^3*d*e^3 + 15*a^2*b^2*e^4)*x^2 + 9*(b^4*d^3*e + 5*a*b^3*d^2*e ^2 + 15*a^2*b^2*d*e^3 + 35*a^3*b*e^4)*x)/(e^14*x^9 + 9*d*e^13*x^8 + 36*d^2 *e^12*x^7 + 84*d^3*e^11*x^6 + 126*d^4*e^10*x^5 + 126*d^5*e^9*x^4 + 84*d^6* e^8*x^3 + 36*d^7*e^7*x^2 + 9*d^8*e^6*x + d^9*e^5)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{10}} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**10,x)
Output:
Timed out
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{10}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^10,x, algorithm="max ima")
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
Time = 0.18 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.65 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{10}} \, dx=\frac {b^{9} \mathrm {sgn}\left (b x + a\right )}{630 \, {\left (b^{5} d^{5} e^{5} - 5 \, a b^{4} d^{4} e^{6} + 10 \, a^{2} b^{3} d^{3} e^{7} - 10 \, a^{3} b^{2} d^{2} e^{8} + 5 \, a^{4} b d e^{9} - a^{5} e^{10}\right )}} - \frac {126 \, b^{4} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 84 \, b^{4} d e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 420 \, a b^{3} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 36 \, b^{4} d^{2} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 180 \, a b^{3} d e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 540 \, a^{2} b^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 9 \, b^{4} d^{3} e x \mathrm {sgn}\left (b x + a\right ) + 45 \, a b^{3} d^{2} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 135 \, a^{2} b^{2} d e^{3} x \mathrm {sgn}\left (b x + a\right ) + 315 \, a^{3} b e^{4} x \mathrm {sgn}\left (b x + a\right ) + b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 70 \, a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )}{630 \, {\left (e x + d\right )}^{9} e^{5}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^10,x, algorithm="gia c")
Output:
1/630*b^9*sgn(b*x + a)/(b^5*d^5*e^5 - 5*a*b^4*d^4*e^6 + 10*a^2*b^3*d^3*e^7 - 10*a^3*b^2*d^2*e^8 + 5*a^4*b*d*e^9 - a^5*e^10) - 1/630*(126*b^4*e^4*x^4 *sgn(b*x + a) + 84*b^4*d*e^3*x^3*sgn(b*x + a) + 420*a*b^3*e^4*x^3*sgn(b*x + a) + 36*b^4*d^2*e^2*x^2*sgn(b*x + a) + 180*a*b^3*d*e^3*x^2*sgn(b*x + a) + 540*a^2*b^2*e^4*x^2*sgn(b*x + a) + 9*b^4*d^3*e*x*sgn(b*x + a) + 45*a*b^3 *d^2*e^2*x*sgn(b*x + a) + 135*a^2*b^2*d*e^3*x*sgn(b*x + a) + 315*a^3*b*e^4 *x*sgn(b*x + a) + b^4*d^4*sgn(b*x + a) + 5*a*b^3*d^3*e*sgn(b*x + a) + 15*a ^2*b^2*d^2*e^2*sgn(b*x + a) + 35*a^3*b*d*e^3*sgn(b*x + a) + 70*a^4*e^4*sgn (b*x + a))/((e*x + d)^9*e^5)
Time = 11.59 (sec) , antiderivative size = 449, normalized size of antiderivative = 2.08 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{10}} \, dx=\frac {\left (\frac {-4\,a^3\,b\,e^3+6\,a^2\,b^2\,d\,e^2-4\,a\,b^3\,d^2\,e+b^4\,d^3}{8\,e^5}+\frac {d\,\left (\frac {d\,\left (\frac {b^4\,d}{8\,e^3}-\frac {b^3\,\left (4\,a\,e-b\,d\right )}{8\,e^3}\right )}{e}+\frac {b^2\,\left (6\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{8\,e^4}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}-\frac {\left (\frac {a^4}{9\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {4\,a\,b^3}{9\,e}-\frac {b^4\,d}{9\,e^2}\right )}{e}-\frac {2\,a^2\,b^2}{3\,e}\right )}{e}+\frac {4\,a^3\,b}{9\,e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9}-\frac {\left (\frac {6\,a^2\,b^2\,e^2-8\,a\,b^3\,d\,e+3\,b^4\,d^2}{7\,e^5}+\frac {d\,\left (\frac {b^4\,d}{7\,e^4}-\frac {2\,b^3\,\left (2\,a\,e-b\,d\right )}{7\,e^4}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}+\frac {\left (\frac {3\,b^4\,d-4\,a\,b^3\,e}{6\,e^5}+\frac {b^4\,d}{6\,e^5}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {b^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,e^5\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5} \] Input:
int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(d + e*x)^10,x)
Output:
(((b^4*d^3 - 4*a^3*b*e^3 + 6*a^2*b^2*d*e^2 - 4*a*b^3*d^2*e)/(8*e^5) + (d*( (d*((b^4*d)/(8*e^3) - (b^3*(4*a*e - b*d))/(8*e^3)))/e + (b^2*(6*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(8*e^4)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^8) - ((a^4/(9*e) - (d*((d*((d*((4*a*b^3)/(9*e) - (b^4*d)/(9 *e^2)))/e - (2*a^2*b^2)/(3*e)))/e + (4*a^3*b)/(9*e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^9) - (((3*b^4*d^2 + 6*a^2*b^2*e^2 - 8 *a*b^3*d*e)/(7*e^5) + (d*((b^4*d)/(7*e^4) - (2*b^3*(2*a*e - b*d))/(7*e^4)) )/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^7) + (((3*b^4*d - 4*a*b^3*e)/(6*e^5) + (b^4*d)/(6*e^5))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/ ((a + b*x)*(d + e*x)^6) - (b^4*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(5*e^5*(a + b*x)*(d + e*x)^5)
Time = 0.21 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.26 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{10}} \, dx=\frac {-126 b^{4} e^{4} x^{4}-420 a \,b^{3} e^{4} x^{3}-84 b^{4} d \,e^{3} x^{3}-540 a^{2} b^{2} e^{4} x^{2}-180 a \,b^{3} d \,e^{3} x^{2}-36 b^{4} d^{2} e^{2} x^{2}-315 a^{3} b \,e^{4} x -135 a^{2} b^{2} d \,e^{3} x -45 a \,b^{3} d^{2} e^{2} x -9 b^{4} d^{3} e x -70 a^{4} e^{4}-35 a^{3} b d \,e^{3}-15 a^{2} b^{2} d^{2} e^{2}-5 a \,b^{3} d^{3} e -b^{4} d^{4}}{630 e^{5} \left (e^{9} x^{9}+9 d \,e^{8} x^{8}+36 d^{2} e^{7} x^{7}+84 d^{3} e^{6} x^{6}+126 d^{4} e^{5} x^{5}+126 d^{5} e^{4} x^{4}+84 d^{6} e^{3} x^{3}+36 d^{7} e^{2} x^{2}+9 d^{8} e x +d^{9}\right )} \] Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^10,x)
Output:
( - 70*a**4*e**4 - 35*a**3*b*d*e**3 - 315*a**3*b*e**4*x - 15*a**2*b**2*d** 2*e**2 - 135*a**2*b**2*d*e**3*x - 540*a**2*b**2*e**4*x**2 - 5*a*b**3*d**3* e - 45*a*b**3*d**2*e**2*x - 180*a*b**3*d*e**3*x**2 - 420*a*b**3*e**4*x**3 - b**4*d**4 - 9*b**4*d**3*e*x - 36*b**4*d**2*e**2*x**2 - 84*b**4*d*e**3*x* *3 - 126*b**4*e**4*x**4)/(630*e**5*(d**9 + 9*d**8*e*x + 36*d**7*e**2*x**2 + 84*d**6*e**3*x**3 + 126*d**5*e**4*x**4 + 126*d**4*e**5*x**5 + 84*d**3*e* *6*x**6 + 36*d**2*e**7*x**7 + 9*d*e**8*x**8 + e**9*x**9))