Integrand size = 33, antiderivative size = 254 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11}} \, dx=-\frac {(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^5 (a+b x) (d+e x)^{10}}+\frac {4 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x) (d+e x)^9}-\frac {3 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^8}+\frac {4 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^5 (a+b x) (d+e x)^6} \] Output:
-1/10*(-a*e+b*d)^4*((b*x+a)^2)^(1/2)/e^5/(b*x+a)/(e*x+d)^10+4/9*b*(-a*e+b* d)^3*((b*x+a)^2)^(1/2)/e^5/(b*x+a)/(e*x+d)^9-3/4*b^2*(-a*e+b*d)^2*((b*x+a) ^2)^(1/2)/e^5/(b*x+a)/(e*x+d)^8+4/7*b^3*(-a*e+b*d)*((b*x+a)^2)^(1/2)/e^5/( b*x+a)/(e*x+d)^7-1/6*b^4*((b*x+a)^2)^(1/2)/e^5/(b*x+a)/(e*x+d)^6
Time = 1.07 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.64 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (126 a^4 e^4+56 a^3 b e^3 (d+10 e x)+21 a^2 b^2 e^2 \left (d^2+10 d e x+45 e^2 x^2\right )+6 a b^3 e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+b^4 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )\right )}{1260 e^5 (a+b x) (d+e x)^{10}} \] Input:
Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^11,x]
Output:
-1/1260*(Sqrt[(a + b*x)^2]*(126*a^4*e^4 + 56*a^3*b*e^3*(d + 10*e*x) + 21*a ^2*b^2*e^2*(d^2 + 10*d*e*x + 45*e^2*x^2) + 6*a*b^3*e*(d^3 + 10*d^2*e*x + 4 5*d*e^2*x^2 + 120*e^3*x^3) + b^4*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120* d*e^3*x^3 + 210*e^4*x^4)))/(e^5*(a + b*x)*(d + e*x)^10)
Time = 0.53 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.58, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^3 (a+b x)^4}{(d+e x)^{11}}dx}{b^3 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^4}{(d+e x)^{11}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^4}{e^4 (d+e x)^7}-\frac {4 (b d-a e) b^3}{e^4 (d+e x)^8}+\frac {6 (b d-a e)^2 b^2}{e^4 (d+e x)^9}-\frac {4 (b d-a e)^3 b}{e^4 (d+e x)^{10}}+\frac {(a e-b d)^4}{e^4 (d+e x)^{11}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {4 b^3 (b d-a e)}{7 e^5 (d+e x)^7}-\frac {3 b^2 (b d-a e)^2}{4 e^5 (d+e x)^8}+\frac {4 b (b d-a e)^3}{9 e^5 (d+e x)^9}-\frac {(b d-a e)^4}{10 e^5 (d+e x)^{10}}-\frac {b^4}{6 e^5 (d+e x)^6}\right )}{a+b x}\) |
Input:
Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^11,x]
Output:
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/10*(b*d - a*e)^4/(e^5*(d + e*x)^10) + ( 4*b*(b*d - a*e)^3)/(9*e^5*(d + e*x)^9) - (3*b^2*(b*d - a*e)^2)/(4*e^5*(d + e*x)^8) + (4*b^3*(b*d - a*e))/(7*e^5*(d + e*x)^7) - b^4/(6*e^5*(d + e*x)^ 6)))/(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 = 3.67 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.74
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{4} x^{4}}{6 e}-\frac {2 b^{3} \left (6 a e +b d \right ) x^{3}}{21 e^{2}}-\frac {b^{2} \left (21 e^{2} a^{2}+6 a b d e +b^{2} d^{2}\right ) x^{2}}{28 e^{3}}-\frac {b \left (56 e^{3} a^{3}+21 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x}{126 e^{4}}-\frac {126 a^{4} e^{4}+56 a^{3} b d \,e^{3}+21 a^{2} b^{2} d^{2} e^{2}+6 a \,b^{3} d^{3} e +b^{4} d^{4}}{1260 e^{5}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{10}}\) | \(187\) |
| gosper | \(-\frac {\left (210 b^{4} x^{4} e^{4}+720 x^{3} a \,b^{3} e^{4}+120 x^{3} b^{4} d \,e^{3}+945 x^{2} a^{2} b^{2} e^{4}+270 x^{2} a \,b^{3} d \,e^{3}+45 x^{2} b^{4} d^{2} e^{2}+560 x \,a^{3} b \,e^{4}+210 x \,a^{2} b^{2} d \,e^{3}+60 x a \,b^{3} d^{2} e^{2}+10 x \,b^{4} d^{3} e +126 a^{4} e^{4}+56 a^{3} b d \,e^{3}+21 a^{2} b^{2} d^{2} e^{2}+6 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{1260 e^{5} \left (e x +d \right )^{10} \left (b x +a \right )^{3}}\) | \(201\) |
| default | \(-\frac {\left (210 b^{4} x^{4} e^{4}+720 x^{3} a \,b^{3} e^{4}+120 x^{3} b^{4} d \,e^{3}+945 x^{2} a^{2} b^{2} e^{4}+270 x^{2} a \,b^{3} d \,e^{3}+45 x^{2} b^{4} d^{2} e^{2}+560 x \,a^{3} b \,e^{4}+210 x \,a^{2} b^{2} d \,e^{3}+60 x a \,b^{3} d^{2} e^{2}+10 x \,b^{4} d^{3} e +126 a^{4} e^{4}+56 a^{3} b d \,e^{3}+21 a^{2} b^{2} d^{2} e^{2}+6 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{1260 e^{5} \left (e x +d \right )^{10} \left (b x +a \right )^{3}}\) | \(201\) |
| orering | \(-\frac {\left (210 b^{4} x^{4} e^{4}+720 x^{3} a \,b^{3} e^{4}+120 x^{3} b^{4} d \,e^{3}+945 x^{2} a^{2} b^{2} e^{4}+270 x^{2} a \,b^{3} d \,e^{3}+45 x^{2} b^{4} d^{2} e^{2}+560 x \,a^{3} b \,e^{4}+210 x \,a^{2} b^{2} d \,e^{3}+60 x a \,b^{3} d^{2} e^{2}+10 x \,b^{4} d^{3} e +126 a^{4} e^{4}+56 a^{3} b d \,e^{3}+21 a^{2} b^{2} d^{2} e^{2}+6 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}{1260 e^{5} \left (b x +a \right )^{3} \left (e x +d \right )^{10}}\) | \(210\) |
Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^11,x,method=_RETURNVERBOSE )
Output:
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/6/e*b^4*x^4-2/21*b^3/e^2*(6*a*e+b*d)*x^3-1/2 8*b^2/e^3*(21*a^2*e^2+6*a*b*d*e+b^2*d^2)*x^2-1/126*b/e^4*(56*a^3*e^3+21*a^ 2*b*d*e^2+6*a*b^2*d^2*e+b^3*d^3)*x-1/1260/e^5*(126*a^4*e^4+56*a^3*b*d*e^3+ 21*a^2*b^2*d^2*e^2+6*a*b^3*d^3*e+b^4*d^4))/(e*x+d)^10
Time = 0.08 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11}} \, dx=-\frac {210 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 6 \, a b^{3} d^{3} e + 21 \, a^{2} b^{2} d^{2} e^{2} + 56 \, a^{3} b d e^{3} + 126 \, a^{4} e^{4} + 120 \, {\left (b^{4} d e^{3} + 6 \, a b^{3} e^{4}\right )} x^{3} + 45 \, {\left (b^{4} d^{2} e^{2} + 6 \, a b^{3} d e^{3} + 21 \, a^{2} b^{2} e^{4}\right )} x^{2} + 10 \, {\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} + 21 \, a^{2} b^{2} d e^{3} + 56 \, a^{3} b e^{4}\right )} x}{1260 \, {\left (e^{15} x^{10} + 10 \, d e^{14} x^{9} + 45 \, d^{2} e^{13} x^{8} + 120 \, d^{3} e^{12} x^{7} + 210 \, d^{4} e^{11} x^{6} + 252 \, d^{5} e^{10} x^{5} + 210 \, d^{6} e^{9} x^{4} + 120 \, d^{7} e^{8} x^{3} + 45 \, d^{8} e^{7} x^{2} + 10 \, d^{9} e^{6} x + d^{10} e^{5}\right )}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^11,x, algorithm="fri cas")
Output:
-1/1260*(210*b^4*e^4*x^4 + b^4*d^4 + 6*a*b^3*d^3*e + 21*a^2*b^2*d^2*e^2 + 56*a^3*b*d*e^3 + 126*a^4*e^4 + 120*(b^4*d*e^3 + 6*a*b^3*e^4)*x^3 + 45*(b^4 *d^2*e^2 + 6*a*b^3*d*e^3 + 21*a^2*b^2*e^4)*x^2 + 10*(b^4*d^3*e + 6*a*b^3*d ^2*e^2 + 21*a^2*b^2*d*e^3 + 56*a^3*b*e^4)*x)/(e^15*x^10 + 10*d*e^14*x^9 + 45*d^2*e^13*x^8 + 120*d^3*e^12*x^7 + 210*d^4*e^11*x^6 + 252*d^5*e^10*x^5 + 210*d^6*e^9*x^4 + 120*d^7*e^8*x^3 + 45*d^8*e^7*x^2 + 10*d^9*e^6*x + d^10* e^5)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11}} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**11,x)
Output:
Timed out
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^11,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 = 0.15 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.46 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11}} \, dx=\frac {b^{10} \mathrm {sgn}\left (b x + a\right )}{1260 \, {\left (b^{6} d^{6} e^{5} - 6 \, a b^{5} d^{5} e^{6} + 15 \, a^{2} b^{4} d^{4} e^{7} - 20 \, a^{3} b^{3} d^{3} e^{8} + 15 \, a^{4} b^{2} d^{2} e^{9} - 6 \, a^{5} b d e^{10} + a^{6} e^{11}\right )}} - \frac {210 \, b^{4} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 120 \, b^{4} d e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 720 \, a b^{3} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 45 \, b^{4} d^{2} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 270 \, a b^{3} d e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 945 \, a^{2} b^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, b^{4} d^{3} e x \mathrm {sgn}\left (b x + a\right ) + 60 \, a b^{3} d^{2} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 210 \, a^{2} b^{2} d e^{3} x \mathrm {sgn}\left (b x + a\right ) + 560 \, a^{3} b e^{4} x \mathrm {sgn}\left (b x + a\right ) + b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) + 6 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 56 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 126 \, a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )}{1260 \, {\left (e x + d\right )}^{10} e^{5}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^11,x, algorithm="gia c")
Output:
1/1260*b^10*sgn(b*x + a)/(b^6*d^6*e^5 - 6*a*b^5*d^5*e^6 + 15*a^2*b^4*d^4*e ^7 - 20*a^3*b^3*d^3*e^8 + 15*a^4*b^2*d^2*e^9 - 6*a^5*b*d*e^10 + a^6*e^11) - 1/1260*(210*b^4*e^4*x^4*sgn(b*x + a) + 120*b^4*d*e^3*x^3*sgn(b*x + a) + 720*a*b^3*e^4*x^3*sgn(b*x + a) + 45*b^4*d^2*e^2*x^2*sgn(b*x + a) + 270*a*b ^3*d*e^3*x^2*sgn(b*x + a) + 945*a^2*b^2*e^4*x^2*sgn(b*x + a) + 10*b^4*d^3* e*x*sgn(b*x + a) + 60*a*b^3*d^2*e^2*x*sgn(b*x + a) + 210*a^2*b^2*d*e^3*x*s gn(b*x + a) + 560*a^3*b*e^4*x*sgn(b*x + a) + b^4*d^4*sgn(b*x + a) + 6*a*b^ 3*d^3*e*sgn(b*x + a) + 21*a^2*b^2*d^2*e^2*sgn(b*x + a) + 56*a^3*b*d*e^3*sg n(b*x + a) + 126*a^4*e^4*sgn(b*x + a))/((e*x + d)^10*e^5)
Time = 11.54 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.77 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11}} \, dx=\frac {\left (\frac {-4\,a^3\,b\,e^3+6\,a^2\,b^2\,d\,e^2-4\,a\,b^3\,d^2\,e+b^4\,d^3}{9\,e^5}+\frac {d\,\left (\frac {d\,\left (\frac {b^4\,d}{9\,e^3}-\frac {b^3\,\left (4\,a\,e-b\,d\right )}{9\,e^3}\right )}{e}+\frac {b^2\,\left (6\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{9\,e^4}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9}-\frac {\left (\frac {a^4}{10\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {2\,a\,b^3}{5\,e}-\frac {b^4\,d}{10\,e^2}\right )}{e}-\frac {3\,a^2\,b^2}{5\,e}\right )}{e}+\frac {2\,a^3\,b}{5\,e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{10}}-\frac {\left (\frac {6\,a^2\,b^2\,e^2-8\,a\,b^3\,d\,e+3\,b^4\,d^2}{8\,e^5}+\frac {d\,\left (\frac {b^4\,d}{8\,e^4}-\frac {b^3\,\left (2\,a\,e-b\,d\right )}{4\,e^4}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}+\frac {\left (\frac {3\,b^4\,d-4\,a\,b^3\,e}{7\,e^5}+\frac {b^4\,d}{7\,e^5}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {b^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,e^5\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6} \] Input:
int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(d + e*x)^11,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)/(9*e^5) + (d*( (d*((b^4*d)/(9*e^3) - (b^3*(4*a*e - b*d))/(9*e^3)))/e + (b^2*(6*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(9*e^4)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^9) - ((a^4/(10*e) - (d*((d*((d*((2*a*b^3)/(5*e) - (b^4*d)/( 10*e^2)))/e - (3*a^2*b^2)/(5*e)))/e + (2*a^3*b)/(5*e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^10) - (((3*b^4*d^2 + 6*a^2*b^2*e^2 - 8*a*b^3*d*e)/(8*e^5) + (d*((b^4*d)/(8*e^4) - (b^3*(2*a*e - b*d))/(4*e^4) ))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^8) + (((3*b^4* d - 4*a*b^3*e)/(7*e^5) + (b^4*d)/(7*e^5))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)) /((a + b*x)*(d + e*x)^7) - (b^4*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(6*e^5*(a + b*x)*(d + e*x)^6)
Time = 0.20 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11}} \, dx=\frac {-210 b^{4} e^{4} x^{4}-720 a \,b^{3} e^{4} x^{3}-120 b^{4} d \,e^{3} x^{3}-945 a^{2} b^{2} e^{4} x^{2}-270 a \,b^{3} d \,e^{3} x^{2}-45 b^{4} d^{2} e^{2} x^{2}-560 a^{3} b \,e^{4} x -210 a^{2} b^{2} d \,e^{3} x -60 a \,b^{3} d^{2} e^{2} x -10 b^{4} d^{3} e x -126 a^{4} e^{4}-56 a^{3} b d \,e^{3}-21 a^{2} b^{2} d^{2} e^{2}-6 a \,b^{3} d^{3} e -b^{4} d^{4}}{1260 e^{5} \left (e^{10} x^{10}+10 d \,e^{9} x^{9}+45 d^{2} e^{8} x^{8}+120 d^{3} e^{7} x^{7}+210 d^{4} e^{6} x^{6}+252 d^{5} e^{5} x^{5}+210 d^{6} e^{4} x^{4}+120 d^{7} e^{3} x^{3}+45 d^{8} e^{2} x^{2}+10 d^{9} e x +d^{10}\right )} \] Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^11,x)
Output:
( - 126*a**4*e**4 - 56*a**3*b*d*e**3 - 560*a**3*b*e**4*x - 21*a**2*b**2*d* *2*e**2 - 210*a**2*b**2*d*e**3*x - 945*a**2*b**2*e**4*x**2 - 6*a*b**3*d**3 *e - 60*a*b**3*d**2*e**2*x - 270*a*b**3*d*e**3*x**2 - 720*a*b**3*e**4*x**3 - b**4*d**4 - 10*b**4*d**3*e*x - 45*b**4*d**2*e**2*x**2 - 120*b**4*d*e**3 *x**3 - 210*b**4*e**4*x**4)/(1260*e**5*(d**10 + 10*d**9*e*x + 45*d**8*e**2 *x**2 + 120*d**7*e**3*x**3 + 210*d**6*e**4*x**4 + 252*d**5*e**5*x**5 + 210 *d**4*e**6*x**6 + 120*d**3*e**7*x**7 + 45*d**2*e**8*x**8 + 10*d*e**9*x**9 + e**10*x**10))