Integrand size = 33, antiderivative size = 216 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{11 (b d-a e) (d+e x)^{11}}+\frac {2 b \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{55 (b d-a e)^2 (d+e x)^{10}}+\frac {2 b^2 \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{165 (b d-a e)^3 (d+e x)^9}+\frac {b^3 \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{330 (b d-a e)^4 (d+e x)^8}+\frac {b^4 \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{2310 (b d-a e)^5 (d+e x)^7} \] Output:
1/11*(b^2*x^2+2*a*b*x+a^2)^(7/2)/(-a*e+b*d)/(e*x+d)^11+2/55*b*(b^2*x^2+2*a *b*x+a^2)^(7/2)/(-a*e+b*d)^2/(e*x+d)^10+2/165*b^2*(b^2*x^2+2*a*b*x+a^2)^(7 /2)/(-a*e+b*d)^3/(e*x+d)^9+1/330*b^3*(b^2*x^2+2*a*b*x+a^2)^(7/2)/(-a*e+b*d )^4/(e*x+d)^8+1/2310*b^4*(b^2*x^2+2*a*b*x+a^2)^(7/2)/(-a*e+b*d)^5/(e*x+d)^ 7
Time = 1.13 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.37 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (210 a^6 e^6+126 a^5 b e^5 (d+11 e x)+70 a^4 b^2 e^4 \left (d^2+11 d e x+55 e^2 x^2\right )+35 a^3 b^3 e^3 \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+15 a^2 b^4 e^2 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 a b^5 e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+b^6 \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )\right )}{2310 e^7 (a+b x) (d+e x)^{11}} \] Input:
Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^12,x]
Output:
-1/2310*(Sqrt[(a + b*x)^2]*(210*a^6*e^6 + 126*a^5*b*e^5*(d + 11*e*x) + 70* a^4*b^2*e^4*(d^2 + 11*d*e*x + 55*e^2*x^2) + 35*a^3*b^3*e^3*(d^3 + 11*d^2*e *x + 55*d*e^2*x^2 + 165*e^3*x^3) + 15*a^2*b^4*e^2*(d^4 + 11*d^3*e*x + 55*d ^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4) + 5*a*b^5*e*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5) + b^6*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d *e^5*x^5 + 462*e^6*x^6)))/(e^7*(a + b*x)*(d + e*x)^11)
Time = 0.68 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.92, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^{12}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^{12}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^6}{e^6 (d+e x)^6}-\frac {6 (b d-a e) b^5}{e^6 (d+e x)^7}+\frac {15 (b d-a e)^2 b^4}{e^6 (d+e x)^8}-\frac {20 (b d-a e)^3 b^3}{e^6 (d+e x)^9}+\frac {15 (b d-a e)^4 b^2}{e^6 (d+e x)^{10}}-\frac {6 (b d-a e)^5 b}{e^6 (d+e x)^{11}}+\frac {(a e-b d)^6}{e^6 (d+e x)^{12}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {b^5 (b d-a e)}{e^7 (d+e x)^6}-\frac {15 b^4 (b d-a e)^2}{7 e^7 (d+e x)^7}+\frac {5 b^3 (b d-a e)^3}{2 e^7 (d+e x)^8}-\frac {5 b^2 (b d-a e)^4}{3 e^7 (d+e x)^9}+\frac {3 b (b d-a e)^5}{5 e^7 (d+e x)^{10}}-\frac {(b d-a e)^6}{11 e^7 (d+e x)^{11}}-\frac {b^6}{5 e^7 (d+e x)^5}\right )}{a+b x}\) |
Input:
Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^12,x]
Output:
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/11*(b*d - a*e)^6/(e^7*(d + e*x)^11) + ( 3*b*(b*d - a*e)^5)/(5*e^7*(d + e*x)^10) - (5*b^2*(b*d - a*e)^4)/(3*e^7*(d + e*x)^9) + (5*b^3*(b*d - a*e)^3)/(2*e^7*(d + e*x)^8) - (15*b^4*(b*d - a*e )^2)/(7*e^7*(d + e*x)^7) + (b^5*(b*d - a*e))/(e^7*(d + e*x)^6) - b^6/(5*e^ 7*(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 = 9.83 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.62
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{6} x^{6}}{5 e}-\frac {b^{5} \left (5 a e +b d \right ) x^{5}}{5 e^{2}}-\frac {b^{4} \left (15 e^{2} a^{2}+5 a b d e +b^{2} d^{2}\right ) x^{4}}{7 e^{3}}-\frac {b^{3} \left (35 e^{3} a^{3}+15 a^{2} b d \,e^{2}+5 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{14 e^{4}}-\frac {b^{2} \left (70 a^{4} e^{4}+35 a^{3} b d \,e^{3}+15 a^{2} b^{2} d^{2} e^{2}+5 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{42 e^{5}}-\frac {b \left (126 e^{5} a^{5}+70 a^{4} b d \,e^{4}+35 a^{3} b^{2} d^{2} e^{3}+15 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{210 e^{6}}-\frac {210 a^{6} e^{6}+126 a^{5} b d \,e^{5}+70 a^{4} b^{2} d^{2} e^{4}+35 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}+5 a \,b^{5} d^{5} e +b^{6} d^{6}}{2310 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{11}}\) | \(351\) |
| gosper | \(-\frac {\left (462 b^{6} x^{6} e^{6}+2310 x^{5} a \,b^{5} e^{6}+462 x^{5} b^{6} d \,e^{5}+4950 x^{4} a^{2} b^{4} e^{6}+1650 x^{4} a \,b^{5} d \,e^{5}+330 x^{4} b^{6} d^{2} e^{4}+5775 x^{3} a^{3} b^{3} e^{6}+2475 x^{3} a^{2} b^{4} d \,e^{5}+825 x^{3} a \,b^{5} d^{2} e^{4}+165 x^{3} b^{6} d^{3} e^{3}+3850 x^{2} a^{4} b^{2} e^{6}+1925 x^{2} a^{3} b^{3} d \,e^{5}+825 x^{2} a^{2} b^{4} d^{2} e^{4}+275 x^{2} a \,b^{5} d^{3} e^{3}+55 x^{2} b^{6} d^{4} e^{2}+1386 x \,a^{5} b \,e^{6}+770 x \,a^{4} b^{2} d \,e^{5}+385 x \,a^{3} b^{3} d^{2} e^{4}+165 x \,a^{2} b^{4} d^{3} e^{3}+55 x a \,b^{5} d^{4} e^{2}+11 x \,b^{6} d^{5} e +210 a^{6} e^{6}+126 a^{5} b d \,e^{5}+70 a^{4} b^{2} d^{2} e^{4}+35 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}+5 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2310 e^{7} \left (e x +d \right )^{11} \left (b x +a \right )^{5}}\) | \(392\) |
| default | \(-\frac {\left (462 b^{6} x^{6} e^{6}+2310 x^{5} a \,b^{5} e^{6}+462 x^{5} b^{6} d \,e^{5}+4950 x^{4} a^{2} b^{4} e^{6}+1650 x^{4} a \,b^{5} d \,e^{5}+330 x^{4} b^{6} d^{2} e^{4}+5775 x^{3} a^{3} b^{3} e^{6}+2475 x^{3} a^{2} b^{4} d \,e^{5}+825 x^{3} a \,b^{5} d^{2} e^{4}+165 x^{3} b^{6} d^{3} e^{3}+3850 x^{2} a^{4} b^{2} e^{6}+1925 x^{2} a^{3} b^{3} d \,e^{5}+825 x^{2} a^{2} b^{4} d^{2} e^{4}+275 x^{2} a \,b^{5} d^{3} e^{3}+55 x^{2} b^{6} d^{4} e^{2}+1386 x \,a^{5} b \,e^{6}+770 x \,a^{4} b^{2} d \,e^{5}+385 x \,a^{3} b^{3} d^{2} e^{4}+165 x \,a^{2} b^{4} d^{3} e^{3}+55 x a \,b^{5} d^{4} e^{2}+11 x \,b^{6} d^{5} e +210 a^{6} e^{6}+126 a^{5} b d \,e^{5}+70 a^{4} b^{2} d^{2} e^{4}+35 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}+5 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2310 e^{7} \left (e x +d \right )^{11} \left (b x +a \right )^{5}}\) | \(392\) |
| orering | \(-\frac {\left (462 b^{6} x^{6} e^{6}+2310 x^{5} a \,b^{5} e^{6}+462 x^{5} b^{6} d \,e^{5}+4950 x^{4} a^{2} b^{4} e^{6}+1650 x^{4} a \,b^{5} d \,e^{5}+330 x^{4} b^{6} d^{2} e^{4}+5775 x^{3} a^{3} b^{3} e^{6}+2475 x^{3} a^{2} b^{4} d \,e^{5}+825 x^{3} a \,b^{5} d^{2} e^{4}+165 x^{3} b^{6} d^{3} e^{3}+3850 x^{2} a^{4} b^{2} e^{6}+1925 x^{2} a^{3} b^{3} d \,e^{5}+825 x^{2} a^{2} b^{4} d^{2} e^{4}+275 x^{2} a \,b^{5} d^{3} e^{3}+55 x^{2} b^{6} d^{4} e^{2}+1386 x \,a^{5} b \,e^{6}+770 x \,a^{4} b^{2} d \,e^{5}+385 x \,a^{3} b^{3} d^{2} e^{4}+165 x \,a^{2} b^{4} d^{3} e^{3}+55 x a \,b^{5} d^{4} e^{2}+11 x \,b^{6} d^{5} e +210 a^{6} e^{6}+126 a^{5} b d \,e^{5}+70 a^{4} b^{2} d^{2} e^{4}+35 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}+5 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{2310 e^{7} \left (b x +a \right )^{5} \left (e x +d \right )^{11}}\) | \(401\) |
Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,x,method=_RETURNVERBOSE )
Output:
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/5/e*b^6*x^6-1/5*b^5/e^2*(5*a*e+b*d)*x^5-1/7* b^4/e^3*(15*a^2*e^2+5*a*b*d*e+b^2*d^2)*x^4-1/14/e^4*b^3*(35*a^3*e^3+15*a^2 *b*d*e^2+5*a*b^2*d^2*e+b^3*d^3)*x^3-1/42*b^2/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)*x^2-1/210*b/e^6*(126*a^5*e^5+7 0*a^4*b*d*e^4+35*a^3*b^2*d^2*e^3+15*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e+b^5*d^5) *x-1/2310/e^7*(210*a^6*e^6+126*a^5*b*d*e^5+70*a^4*b^2*d^2*e^4+35*a^3*b^3*d ^3*e^3+15*a^2*b^4*d^4*e^2+5*a*b^5*d^5*e+b^6*d^6))/(e*x+d)^11
Leaf count of result is larger than twice the leaf count of optimal. 463 vs. \(2 (196) = 392\).
Time = 0.08 (sec) , antiderivative size = 463, normalized size of antiderivative = 2.14 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=-\frac {462 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 5 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} + 35 \, a^{3} b^{3} d^{3} e^{3} + 70 \, a^{4} b^{2} d^{2} e^{4} + 126 \, a^{5} b d e^{5} + 210 \, a^{6} e^{6} + 462 \, {\left (b^{6} d e^{5} + 5 \, a b^{5} e^{6}\right )} x^{5} + 330 \, {\left (b^{6} d^{2} e^{4} + 5 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 165 \, {\left (b^{6} d^{3} e^{3} + 5 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} + 35 \, a^{3} b^{3} e^{6}\right )} x^{3} + 55 \, {\left (b^{6} d^{4} e^{2} + 5 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 35 \, a^{3} b^{3} d e^{5} + 70 \, a^{4} b^{2} e^{6}\right )} x^{2} + 11 \, {\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} + 35 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 126 \, a^{5} b e^{6}\right )} x}{2310 \, {\left (e^{18} x^{11} + 11 \, d e^{17} x^{10} + 55 \, d^{2} e^{16} x^{9} + 165 \, d^{3} e^{15} x^{8} + 330 \, d^{4} e^{14} x^{7} + 462 \, d^{5} e^{13} x^{6} + 462 \, d^{6} e^{12} x^{5} + 330 \, d^{7} e^{11} x^{4} + 165 \, d^{8} e^{10} x^{3} + 55 \, d^{9} e^{9} x^{2} + 11 \, d^{10} e^{8} x + d^{11} e^{7}\right )}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,x, algorithm="fri cas")
Output:
-1/2310*(462*b^6*e^6*x^6 + b^6*d^6 + 5*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 + 35*a^3*b^3*d^3*e^3 + 70*a^4*b^2*d^2*e^4 + 126*a^5*b*d*e^5 + 210*a^6*e^6 + 462*(b^6*d*e^5 + 5*a*b^5*e^6)*x^5 + 330*(b^6*d^2*e^4 + 5*a*b^5*d*e^5 + 15* a^2*b^4*e^6)*x^4 + 165*(b^6*d^3*e^3 + 5*a*b^5*d^2*e^4 + 15*a^2*b^4*d*e^5 + 35*a^3*b^3*e^6)*x^3 + 55*(b^6*d^4*e^2 + 5*a*b^5*d^3*e^3 + 15*a^2*b^4*d^2* e^4 + 35*a^3*b^3*d*e^5 + 70*a^4*b^2*e^6)*x^2 + 11*(b^6*d^5*e + 5*a*b^5*d^4 *e^2 + 15*a^2*b^4*d^3*e^3 + 35*a^3*b^3*d^2*e^4 + 70*a^4*b^2*d*e^5 + 126*a^ 5*b*e^6)*x)/(e^18*x^11 + 11*d*e^17*x^10 + 55*d^2*e^16*x^9 + 165*d^3*e^15*x ^8 + 330*d^4*e^14*x^7 + 462*d^5*e^13*x^6 + 462*d^6*e^12*x^5 + 330*d^7*e^11 *x^4 + 165*d^8*e^10*x^3 + 55*d^9*e^9*x^2 + 11*d^10*e^8*x + d^11*e^7)
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**12,x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,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
Leaf count of result is larger than twice the leaf count of optimal. 626 vs. \(2 (196) = 392\).
Time = 0.19 (sec) , antiderivative size = 626, normalized size of antiderivative = 2.90 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,x, algorithm="gia c")
Output:
1/2310*b^11*sgn(b*x + a)/(b^5*d^5*e^7 - 5*a*b^4*d^4*e^8 + 10*a^2*b^3*d^3*e ^9 - 10*a^3*b^2*d^2*e^10 + 5*a^4*b*d*e^11 - a^5*e^12) - 1/2310*(462*b^6*e^ 6*x^6*sgn(b*x + a) + 462*b^6*d*e^5*x^5*sgn(b*x + a) + 2310*a*b^5*e^6*x^5*s gn(b*x + a) + 330*b^6*d^2*e^4*x^4*sgn(b*x + a) + 1650*a*b^5*d*e^5*x^4*sgn( b*x + a) + 4950*a^2*b^4*e^6*x^4*sgn(b*x + a) + 165*b^6*d^3*e^3*x^3*sgn(b*x + a) + 825*a*b^5*d^2*e^4*x^3*sgn(b*x + a) + 2475*a^2*b^4*d*e^5*x^3*sgn(b* x + a) + 5775*a^3*b^3*e^6*x^3*sgn(b*x + a) + 55*b^6*d^4*e^2*x^2*sgn(b*x + a) + 275*a*b^5*d^3*e^3*x^2*sgn(b*x + a) + 825*a^2*b^4*d^2*e^4*x^2*sgn(b*x + a) + 1925*a^3*b^3*d*e^5*x^2*sgn(b*x + a) + 3850*a^4*b^2*e^6*x^2*sgn(b*x + a) + 11*b^6*d^5*e*x*sgn(b*x + a) + 55*a*b^5*d^4*e^2*x*sgn(b*x + a) + 165 *a^2*b^4*d^3*e^3*x*sgn(b*x + a) + 385*a^3*b^3*d^2*e^4*x*sgn(b*x + a) + 770 *a^4*b^2*d*e^5*x*sgn(b*x + a) + 1386*a^5*b*e^6*x*sgn(b*x + a) + b^6*d^6*sg n(b*x + a) + 5*a*b^5*d^5*e*sgn(b*x + a) + 15*a^2*b^4*d^4*e^2*sgn(b*x + a) + 35*a^3*b^3*d^3*e^3*sgn(b*x + a) + 70*a^4*b^2*d^2*e^4*sgn(b*x + a) + 126* a^5*b*d*e^5*sgn(b*x + a) + 210*a^6*e^6*sgn(b*x + a))/((e*x + d)^11*e^7)
Time = 11.54 (sec) , antiderivative size = 1010, normalized size of antiderivative = 4.68 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx =\text {Too large to display} \] Input:
int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^12,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)/(10*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)/(10*e^7) - (d*((2 0*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)/(10*e^7) - (d*((d*((b^6*d)/(10*e^3) - (b^5*(6*a*e - b*d))/(10*e^3)))/e + (b^4*(15* a^2*e^2 + b^2*d^2 - 6*a*b*d*e))/(10*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 + 2*a *b*x)^(1/2))/((a + b*x)*(d + e*x)^10) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 2 4*a*b^5*d*e)/(7*e^7) + (d*((b^6*d)/(7*e^6) - (2*b^5*(3*a*e - 2*b*d))/(7*e^ 6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^7) - ((a^6/( 11*e) - (d*((d*((d*((d*((d*((6*a*b^5)/(11*e) - (b^6*d)/(11*e^2)))/e - (15* a^2*b^4)/(11*e)))/e + (20*a^3*b^3)/(11*e)))/e - (15*a^4*b^2)/(11*e)))/e + (6*a^5*b)/(11*e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x )^11) - (((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)/(9*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)/(9*e^7) + (d*((d*((b^6*d)/(9*e^4) - (2*b^5* (3*a*e - b*d))/(9*e^4)))/e + (b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(3*e^ 5)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^9) + ((( 5*b^6*d - 6*a*b^5*e)/(6*e^7) + (b^6*d)/(6*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^ (1/2))/((a + b*x)*(d + e*x)^6) + (((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)/(8*e^7) + (d*((d*((b^6*d)/(8*e^5) - (3*b^5*(...
Time = 0.29 (sec) , antiderivative size = 486, normalized size of antiderivative = 2.25 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\frac {-462 b^{6} e^{6} x^{6}-2310 a \,b^{5} e^{6} x^{5}-462 b^{6} d \,e^{5} x^{5}-4950 a^{2} b^{4} e^{6} x^{4}-1650 a \,b^{5} d \,e^{5} x^{4}-330 b^{6} d^{2} e^{4} x^{4}-5775 a^{3} b^{3} e^{6} x^{3}-2475 a^{2} b^{4} d \,e^{5} x^{3}-825 a \,b^{5} d^{2} e^{4} x^{3}-165 b^{6} d^{3} e^{3} x^{3}-3850 a^{4} b^{2} e^{6} x^{2}-1925 a^{3} b^{3} d \,e^{5} x^{2}-825 a^{2} b^{4} d^{2} e^{4} x^{2}-275 a \,b^{5} d^{3} e^{3} x^{2}-55 b^{6} d^{4} e^{2} x^{2}-1386 a^{5} b \,e^{6} x -770 a^{4} b^{2} d \,e^{5} x -385 a^{3} b^{3} d^{2} e^{4} x -165 a^{2} b^{4} d^{3} e^{3} x -55 a \,b^{5} d^{4} e^{2} x -11 b^{6} d^{5} e x -210 a^{6} e^{6}-126 a^{5} b d \,e^{5}-70 a^{4} b^{2} d^{2} e^{4}-35 a^{3} b^{3} d^{3} e^{3}-15 a^{2} b^{4} d^{4} e^{2}-5 a \,b^{5} d^{5} e -b^{6} d^{6}}{2310 e^{7} \left (e^{11} x^{11}+11 d \,e^{10} x^{10}+55 d^{2} e^{9} x^{9}+165 d^{3} e^{8} x^{8}+330 d^{4} e^{7} x^{7}+462 d^{5} e^{6} x^{6}+462 d^{6} e^{5} x^{5}+330 d^{7} e^{4} x^{4}+165 d^{8} e^{3} x^{3}+55 d^{9} e^{2} x^{2}+11 d^{10} e x +d^{11}\right )} \] Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,x)
Output:
( - 210*a**6*e**6 - 126*a**5*b*d*e**5 - 1386*a**5*b*e**6*x - 70*a**4*b**2* d**2*e**4 - 770*a**4*b**2*d*e**5*x - 3850*a**4*b**2*e**6*x**2 - 35*a**3*b* *3*d**3*e**3 - 385*a**3*b**3*d**2*e**4*x - 1925*a**3*b**3*d*e**5*x**2 - 57 75*a**3*b**3*e**6*x**3 - 15*a**2*b**4*d**4*e**2 - 165*a**2*b**4*d**3*e**3* x - 825*a**2*b**4*d**2*e**4*x**2 - 2475*a**2*b**4*d*e**5*x**3 - 4950*a**2* b**4*e**6*x**4 - 5*a*b**5*d**5*e - 55*a*b**5*d**4*e**2*x - 275*a*b**5*d**3 *e**3*x**2 - 825*a*b**5*d**2*e**4*x**3 - 1650*a*b**5*d*e**5*x**4 - 2310*a* b**5*e**6*x**5 - b**6*d**6 - 11*b**6*d**5*e*x - 55*b**6*d**4*e**2*x**2 - 1 65*b**6*d**3*e**3*x**3 - 330*b**6*d**2*e**4*x**4 - 462*b**6*d*e**5*x**5 - 462*b**6*e**6*x**6)/(2310*e**7*(d**11 + 11*d**10*e*x + 55*d**9*e**2*x**2 + 165*d**8*e**3*x**3 + 330*d**7*e**4*x**4 + 462*d**6*e**5*x**5 + 462*d**5*e **6*x**6 + 330*d**4*e**7*x**7 + 165*d**3*e**8*x**8 + 55*d**2*e**9*x**9 + 1 1*d*e**10*x**10 + e**11*x**11))