Integrand size = 26, antiderivative size = 173 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{15}} \, dx=-\frac {(b d-a e)^6}{14 e^7 (d+e x)^{14}}+\frac {6 b (b d-a e)^5}{13 e^7 (d+e x)^{13}}-\frac {5 b^2 (b d-a e)^4}{4 e^7 (d+e x)^{12}}+\frac {20 b^3 (b d-a e)^3}{11 e^7 (d+e x)^{11}}-\frac {3 b^4 (b d-a e)^2}{2 e^7 (d+e x)^{10}}+\frac {2 b^5 (b d-a e)}{3 e^7 (d+e x)^9}-\frac {b^6}{8 e^7 (d+e x)^8} \] Output:
-1/14*(-a*e+b*d)^6/e^7/(e*x+d)^14+6/13*b*(-a*e+b*d)^5/e^7/(e*x+d)^13-5/4*b ^2*(-a*e+b*d)^4/e^7/(e*x+d)^12+20/11*b^3*(-a*e+b*d)^3/e^7/(e*x+d)^11-3/2*b ^4*(-a*e+b*d)^2/e^7/(e*x+d)^10+2/3*b^5*(-a*e+b*d)/e^7/(e*x+d)^9-1/8*b^6/e^ 7/(e*x+d)^8
Time = 0.08 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{15}} \, dx=-\frac {1716 a^6 e^6+792 a^5 b e^5 (d+14 e x)+330 a^4 b^2 e^4 \left (d^2+14 d e x+91 e^2 x^2\right )+120 a^3 b^3 e^3 \left (d^3+14 d^2 e x+91 d e^2 x^2+364 e^3 x^3\right )+36 a^2 b^4 e^2 \left (d^4+14 d^3 e x+91 d^2 e^2 x^2+364 d e^3 x^3+1001 e^4 x^4\right )+8 a b^5 e \left (d^5+14 d^4 e x+91 d^3 e^2 x^2+364 d^2 e^3 x^3+1001 d e^4 x^4+2002 e^5 x^5\right )+b^6 \left (d^6+14 d^5 e x+91 d^4 e^2 x^2+364 d^3 e^3 x^3+1001 d^2 e^4 x^4+2002 d e^5 x^5+3003 e^6 x^6\right )}{24024 e^7 (d+e x)^{14}} \] Input:
Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^15,x]
Output:
-1/24024*(1716*a^6*e^6 + 792*a^5*b*e^5*(d + 14*e*x) + 330*a^4*b^2*e^4*(d^2 + 14*d*e*x + 91*e^2*x^2) + 120*a^3*b^3*e^3*(d^3 + 14*d^2*e*x + 91*d*e^2*x ^2 + 364*e^3*x^3) + 36*a^2*b^4*e^2*(d^4 + 14*d^3*e*x + 91*d^2*e^2*x^2 + 36 4*d*e^3*x^3 + 1001*e^4*x^4) + 8*a*b^5*e*(d^5 + 14*d^4*e*x + 91*d^3*e^2*x^2 + 364*d^2*e^3*x^3 + 1001*d*e^4*x^4 + 2002*e^5*x^5) + b^6*(d^6 + 14*d^5*e* x + 91*d^4*e^2*x^2 + 364*d^3*e^3*x^3 + 1001*d^2*e^4*x^4 + 2002*d*e^5*x^5 + 3003*e^6*x^6))/(e^7*(d + e*x)^14)
Time = 0.62 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1098, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{15}} \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^6}{(d+e x)^{15}}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^6}{(d+e x)^{15}}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^{10}}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^{11}}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^{12}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{13}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{14}}+\frac {(a e-b d)^6}{e^6 (d+e x)^{15}}+\frac {b^6}{e^6 (d+e x)^9}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b^5 (b d-a e)}{3 e^7 (d+e x)^9}-\frac {3 b^4 (b d-a e)^2}{2 e^7 (d+e x)^{10}}+\frac {20 b^3 (b d-a e)^3}{11 e^7 (d+e x)^{11}}-\frac {5 b^2 (b d-a e)^4}{4 e^7 (d+e x)^{12}}+\frac {6 b (b d-a e)^5}{13 e^7 (d+e x)^{13}}-\frac {(b d-a e)^6}{14 e^7 (d+e x)^{14}}-\frac {b^6}{8 e^7 (d+e x)^8}\) |
Input:
Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^15,x]
Output:
-1/14*(b*d - a*e)^6/(e^7*(d + e*x)^14) + (6*b*(b*d - a*e)^5)/(13*e^7*(d + e*x)^13) - (5*b^2*(b*d - a*e)^4)/(4*e^7*(d + e*x)^12) + (20*b^3*(b*d - a*e )^3)/(11*e^7*(d + e*x)^11) - (3*b^4*(b*d - a*e)^2)/(2*e^7*(d + e*x)^10) + (2*b^5*(b*d - a*e))/(3*e^7*(d + e*x)^9) - b^6/(8*e^7*(d + e*x)^8)
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_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(334\) vs. \(2(159)=318\).
Time = 1.04 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.94
| method | result | size |
| risch | \(\frac {-\frac {b^{6} x^{6}}{8 e}-\frac {b^{5} \left (8 a e +b d \right ) x^{5}}{12 e^{2}}-\frac {b^{4} \left (36 e^{2} a^{2}+8 a b d e +b^{2} d^{2}\right ) x^{4}}{24 e^{3}}-\frac {b^{3} \left (120 e^{3} a^{3}+36 a^{2} b d \,e^{2}+8 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{66 e^{4}}-\frac {b^{2} \left (330 a^{4} e^{4}+120 a^{3} b d \,e^{3}+36 a^{2} b^{2} d^{2} e^{2}+8 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{264 e^{5}}-\frac {b \left (792 e^{5} a^{5}+330 a^{4} b d \,e^{4}+120 a^{3} b^{2} d^{2} e^{3}+36 a^{2} b^{3} d^{3} e^{2}+8 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{1716 e^{6}}-\frac {1716 a^{6} e^{6}+792 a^{5} b d \,e^{5}+330 a^{4} b^{2} d^{2} e^{4}+120 a^{3} b^{3} d^{3} e^{3}+36 a^{2} b^{4} d^{4} e^{2}+8 a \,b^{5} d^{5} e +b^{6} d^{6}}{24024 e^{7}}}{\left (e x +d \right )^{14}}\) | \(335\) |
| default | \(-\frac {20 b^{3} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{11 e^{7} \left (e x +d \right )^{11}}-\frac {5 b^{2} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{4 e^{7} \left (e x +d \right )^{12}}-\frac {6 b \left (e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{13 e^{7} \left (e x +d \right )^{13}}-\frac {2 b^{5} \left (a e -b d \right )}{3 e^{7} \left (e x +d \right )^{9}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{14 e^{7} \left (e x +d \right )^{14}}-\frac {b^{6}}{8 e^{7} \left (e x +d \right )^{8}}-\frac {3 b^{4} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )}{2 e^{7} \left (e x +d \right )^{10}}\) | \(357\) |
| norman | \(\frac {-\frac {1716 a^{6} e^{13}+792 a^{5} b d \,e^{12}+330 a^{4} b^{2} d^{2} e^{11}+120 a^{3} b^{3} d^{3} e^{10}+36 a^{2} b^{4} d^{4} e^{9}+8 a \,b^{5} d^{5} e^{8}+b^{6} d^{6} e^{7}}{24024 e^{14}}-\frac {\left (792 a^{5} b \,e^{12}+330 a^{4} b^{2} d \,e^{11}+120 a^{3} b^{3} d^{2} e^{10}+36 a^{2} b^{4} d^{3} e^{9}+8 a \,b^{5} d^{4} e^{8}+b^{6} d^{5} e^{7}\right ) x}{1716 e^{13}}-\frac {\left (330 a^{4} b^{2} e^{11}+120 a^{3} b^{3} d \,e^{10}+36 a^{2} b^{4} d^{2} e^{9}+8 a \,b^{5} d^{3} e^{8}+b^{6} d^{4} e^{7}\right ) x^{2}}{264 e^{12}}-\frac {\left (120 a^{3} b^{3} e^{10}+36 a^{2} b^{4} d \,e^{9}+8 a \,b^{5} d^{2} e^{8}+b^{6} d^{3} e^{7}\right ) x^{3}}{66 e^{11}}-\frac {\left (36 a^{2} b^{4} e^{9}+8 a \,b^{5} d \,e^{8}+b^{6} d^{2} e^{7}\right ) x^{4}}{24 e^{10}}-\frac {\left (8 a \,b^{5} e^{8}+b^{6} d \,e^{7}\right ) x^{5}}{12 e^{9}}-\frac {b^{6} x^{6}}{8 e}}{\left (e x +d \right )^{14}}\) | \(375\) |
| gosper | \(-\frac {3003 x^{6} b^{6} e^{6}+16016 x^{5} a \,b^{5} e^{6}+2002 x^{5} b^{6} d \,e^{5}+36036 x^{4} a^{2} b^{4} e^{6}+8008 x^{4} a \,b^{5} d \,e^{5}+1001 x^{4} b^{6} d^{2} e^{4}+43680 x^{3} a^{3} b^{3} e^{6}+13104 x^{3} a^{2} b^{4} d \,e^{5}+2912 x^{3} a \,b^{5} d^{2} e^{4}+364 x^{3} b^{6} d^{3} e^{3}+30030 x^{2} a^{4} b^{2} e^{6}+10920 x^{2} a^{3} b^{3} d \,e^{5}+3276 x^{2} a^{2} b^{4} d^{2} e^{4}+728 x^{2} a \,b^{5} d^{3} e^{3}+91 x^{2} b^{6} d^{4} e^{2}+11088 x \,a^{5} b \,e^{6}+4620 x \,a^{4} b^{2} d \,e^{5}+1680 x \,a^{3} b^{3} d^{2} e^{4}+504 x \,a^{2} b^{4} d^{3} e^{3}+112 x a \,b^{5} d^{4} e^{2}+14 x \,b^{6} d^{5} e +1716 a^{6} e^{6}+792 a^{5} b d \,e^{5}+330 a^{4} b^{2} d^{2} e^{4}+120 a^{3} b^{3} d^{3} e^{3}+36 a^{2} b^{4} d^{4} e^{2}+8 a \,b^{5} d^{5} e +b^{6} d^{6}}{24024 e^{7} \left (e x +d \right )^{14}}\) | \(376\) |
| parallelrisch | \(\frac {-3003 b^{6} x^{6} e^{13}-16016 a \,b^{5} e^{13} x^{5}-2002 b^{6} d \,e^{12} x^{5}-36036 a^{2} b^{4} e^{13} x^{4}-8008 a \,b^{5} d \,e^{12} x^{4}-1001 b^{6} d^{2} e^{11} x^{4}-43680 a^{3} b^{3} e^{13} x^{3}-13104 a^{2} b^{4} d \,e^{12} x^{3}-2912 a \,b^{5} d^{2} e^{11} x^{3}-364 b^{6} d^{3} e^{10} x^{3}-30030 a^{4} b^{2} e^{13} x^{2}-10920 a^{3} b^{3} d \,e^{12} x^{2}-3276 a^{2} b^{4} d^{2} e^{11} x^{2}-728 a \,b^{5} d^{3} e^{10} x^{2}-91 b^{6} d^{4} e^{9} x^{2}-11088 a^{5} b \,e^{13} x -4620 a^{4} b^{2} d \,e^{12} x -1680 a^{3} b^{3} d^{2} e^{11} x -504 a^{2} b^{4} d^{3} e^{10} x -112 a \,b^{5} d^{4} e^{9} x -14 b^{6} d^{5} e^{8} x -1716 a^{6} e^{13}-792 a^{5} b d \,e^{12}-330 a^{4} b^{2} d^{2} e^{11}-120 a^{3} b^{3} d^{3} e^{10}-36 a^{2} b^{4} d^{4} e^{9}-8 a \,b^{5} d^{5} e^{8}-b^{6} d^{6} e^{7}}{24024 e^{14} \left (e x +d \right )^{14}}\) | \(384\) |
| orering | \(-\frac {\left (3003 x^{6} b^{6} e^{6}+16016 x^{5} a \,b^{5} e^{6}+2002 x^{5} b^{6} d \,e^{5}+36036 x^{4} a^{2} b^{4} e^{6}+8008 x^{4} a \,b^{5} d \,e^{5}+1001 x^{4} b^{6} d^{2} e^{4}+43680 x^{3} a^{3} b^{3} e^{6}+13104 x^{3} a^{2} b^{4} d \,e^{5}+2912 x^{3} a \,b^{5} d^{2} e^{4}+364 x^{3} b^{6} d^{3} e^{3}+30030 x^{2} a^{4} b^{2} e^{6}+10920 x^{2} a^{3} b^{3} d \,e^{5}+3276 x^{2} a^{2} b^{4} d^{2} e^{4}+728 x^{2} a \,b^{5} d^{3} e^{3}+91 x^{2} b^{6} d^{4} e^{2}+11088 x \,a^{5} b \,e^{6}+4620 x \,a^{4} b^{2} d \,e^{5}+1680 x \,a^{3} b^{3} d^{2} e^{4}+504 x \,a^{2} b^{4} d^{3} e^{3}+112 x a \,b^{5} d^{4} e^{2}+14 x \,b^{6} d^{5} e +1716 a^{6} e^{6}+792 a^{5} b d \,e^{5}+330 a^{4} b^{2} d^{2} e^{4}+120 a^{3} b^{3} d^{3} e^{3}+36 a^{2} b^{4} d^{4} e^{2}+8 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{3}}{24024 e^{7} \left (b x +a \right )^{6} \left (e x +d \right )^{14}}\) | \(401\) |
Input:
int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^15,x,method=_RETURNVERBOSE)
Output:
(-1/8/e*b^6*x^6-1/12*b^5/e^2*(8*a*e+b*d)*x^5-1/24*b^4/e^3*(36*a^2*e^2+8*a* b*d*e+b^2*d^2)*x^4-1/66*b^3/e^4*(120*a^3*e^3+36*a^2*b*d*e^2+8*a*b^2*d^2*e+ b^3*d^3)*x^3-1/264*b^2/e^5*(330*a^4*e^4+120*a^3*b*d*e^3+36*a^2*b^2*d^2*e^2 +8*a*b^3*d^3*e+b^4*d^4)*x^2-1/1716*b/e^6*(792*a^5*e^5+330*a^4*b*d*e^4+120* a^3*b^2*d^2*e^3+36*a^2*b^3*d^3*e^2+8*a*b^4*d^4*e+b^5*d^5)*x-1/24024/e^7*(1 716*a^6*e^6+792*a^5*b*d*e^5+330*a^4*b^2*d^2*e^4+120*a^3*b^3*d^3*e^3+36*a^2 *b^4*d^4*e^2+8*a*b^5*d^5*e+b^6*d^6))/(e*x+d)^14
Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (159) = 318\).
Time = 0.09 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.87 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{15}} \, dx=-\frac {3003 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 8 \, a b^{5} d^{5} e + 36 \, a^{2} b^{4} d^{4} e^{2} + 120 \, a^{3} b^{3} d^{3} e^{3} + 330 \, a^{4} b^{2} d^{2} e^{4} + 792 \, a^{5} b d e^{5} + 1716 \, a^{6} e^{6} + 2002 \, {\left (b^{6} d e^{5} + 8 \, a b^{5} e^{6}\right )} x^{5} + 1001 \, {\left (b^{6} d^{2} e^{4} + 8 \, a b^{5} d e^{5} + 36 \, a^{2} b^{4} e^{6}\right )} x^{4} + 364 \, {\left (b^{6} d^{3} e^{3} + 8 \, a b^{5} d^{2} e^{4} + 36 \, a^{2} b^{4} d e^{5} + 120 \, a^{3} b^{3} e^{6}\right )} x^{3} + 91 \, {\left (b^{6} d^{4} e^{2} + 8 \, a b^{5} d^{3} e^{3} + 36 \, a^{2} b^{4} d^{2} e^{4} + 120 \, a^{3} b^{3} d e^{5} + 330 \, a^{4} b^{2} e^{6}\right )} x^{2} + 14 \, {\left (b^{6} d^{5} e + 8 \, a b^{5} d^{4} e^{2} + 36 \, a^{2} b^{4} d^{3} e^{3} + 120 \, a^{3} b^{3} d^{2} e^{4} + 330 \, a^{4} b^{2} d e^{5} + 792 \, a^{5} b e^{6}\right )} x}{24024 \, {\left (e^{21} x^{14} + 14 \, d e^{20} x^{13} + 91 \, d^{2} e^{19} x^{12} + 364 \, d^{3} e^{18} x^{11} + 1001 \, d^{4} e^{17} x^{10} + 2002 \, d^{5} e^{16} x^{9} + 3003 \, d^{6} e^{15} x^{8} + 3432 \, d^{7} e^{14} x^{7} + 3003 \, d^{8} e^{13} x^{6} + 2002 \, d^{9} e^{12} x^{5} + 1001 \, d^{10} e^{11} x^{4} + 364 \, d^{11} e^{10} x^{3} + 91 \, d^{12} e^{9} x^{2} + 14 \, d^{13} e^{8} x + d^{14} e^{7}\right )}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^15,x, algorithm="fricas")
Output:
-1/24024*(3003*b^6*e^6*x^6 + b^6*d^6 + 8*a*b^5*d^5*e + 36*a^2*b^4*d^4*e^2 + 120*a^3*b^3*d^3*e^3 + 330*a^4*b^2*d^2*e^4 + 792*a^5*b*d*e^5 + 1716*a^6*e ^6 + 2002*(b^6*d*e^5 + 8*a*b^5*e^6)*x^5 + 1001*(b^6*d^2*e^4 + 8*a*b^5*d*e^ 5 + 36*a^2*b^4*e^6)*x^4 + 364*(b^6*d^3*e^3 + 8*a*b^5*d^2*e^4 + 36*a^2*b^4* d*e^5 + 120*a^3*b^3*e^6)*x^3 + 91*(b^6*d^4*e^2 + 8*a*b^5*d^3*e^3 + 36*a^2* b^4*d^2*e^4 + 120*a^3*b^3*d*e^5 + 330*a^4*b^2*e^6)*x^2 + 14*(b^6*d^5*e + 8 *a*b^5*d^4*e^2 + 36*a^2*b^4*d^3*e^3 + 120*a^3*b^3*d^2*e^4 + 330*a^4*b^2*d* e^5 + 792*a^5*b*e^6)*x)/(e^21*x^14 + 14*d*e^20*x^13 + 91*d^2*e^19*x^12 + 3 64*d^3*e^18*x^11 + 1001*d^4*e^17*x^10 + 2002*d^5*e^16*x^9 + 3003*d^6*e^15* x^8 + 3432*d^7*e^14*x^7 + 3003*d^8*e^13*x^6 + 2002*d^9*e^12*x^5 + 1001*d^1 0*e^11*x^4 + 364*d^11*e^10*x^3 + 91*d^12*e^9*x^2 + 14*d^13*e^8*x + d^14*e^ 7)
Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{15}} \, dx=\text {Timed out} \] Input:
integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**15,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (159) = 318\).
Time = 0.07 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.87 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{15}} \, dx=-\frac {3003 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 8 \, a b^{5} d^{5} e + 36 \, a^{2} b^{4} d^{4} e^{2} + 120 \, a^{3} b^{3} d^{3} e^{3} + 330 \, a^{4} b^{2} d^{2} e^{4} + 792 \, a^{5} b d e^{5} + 1716 \, a^{6} e^{6} + 2002 \, {\left (b^{6} d e^{5} + 8 \, a b^{5} e^{6}\right )} x^{5} + 1001 \, {\left (b^{6} d^{2} e^{4} + 8 \, a b^{5} d e^{5} + 36 \, a^{2} b^{4} e^{6}\right )} x^{4} + 364 \, {\left (b^{6} d^{3} e^{3} + 8 \, a b^{5} d^{2} e^{4} + 36 \, a^{2} b^{4} d e^{5} + 120 \, a^{3} b^{3} e^{6}\right )} x^{3} + 91 \, {\left (b^{6} d^{4} e^{2} + 8 \, a b^{5} d^{3} e^{3} + 36 \, a^{2} b^{4} d^{2} e^{4} + 120 \, a^{3} b^{3} d e^{5} + 330 \, a^{4} b^{2} e^{6}\right )} x^{2} + 14 \, {\left (b^{6} d^{5} e + 8 \, a b^{5} d^{4} e^{2} + 36 \, a^{2} b^{4} d^{3} e^{3} + 120 \, a^{3} b^{3} d^{2} e^{4} + 330 \, a^{4} b^{2} d e^{5} + 792 \, a^{5} b e^{6}\right )} x}{24024 \, {\left (e^{21} x^{14} + 14 \, d e^{20} x^{13} + 91 \, d^{2} e^{19} x^{12} + 364 \, d^{3} e^{18} x^{11} + 1001 \, d^{4} e^{17} x^{10} + 2002 \, d^{5} e^{16} x^{9} + 3003 \, d^{6} e^{15} x^{8} + 3432 \, d^{7} e^{14} x^{7} + 3003 \, d^{8} e^{13} x^{6} + 2002 \, d^{9} e^{12} x^{5} + 1001 \, d^{10} e^{11} x^{4} + 364 \, d^{11} e^{10} x^{3} + 91 \, d^{12} e^{9} x^{2} + 14 \, d^{13} e^{8} x + d^{14} e^{7}\right )}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^15,x, algorithm="maxima")
Output:
-1/24024*(3003*b^6*e^6*x^6 + b^6*d^6 + 8*a*b^5*d^5*e + 36*a^2*b^4*d^4*e^2 + 120*a^3*b^3*d^3*e^3 + 330*a^4*b^2*d^2*e^4 + 792*a^5*b*d*e^5 + 1716*a^6*e ^6 + 2002*(b^6*d*e^5 + 8*a*b^5*e^6)*x^5 + 1001*(b^6*d^2*e^4 + 8*a*b^5*d*e^ 5 + 36*a^2*b^4*e^6)*x^4 + 364*(b^6*d^3*e^3 + 8*a*b^5*d^2*e^4 + 36*a^2*b^4* d*e^5 + 120*a^3*b^3*e^6)*x^3 + 91*(b^6*d^4*e^2 + 8*a*b^5*d^3*e^3 + 36*a^2* b^4*d^2*e^4 + 120*a^3*b^3*d*e^5 + 330*a^4*b^2*e^6)*x^2 + 14*(b^6*d^5*e + 8 *a*b^5*d^4*e^2 + 36*a^2*b^4*d^3*e^3 + 120*a^3*b^3*d^2*e^4 + 330*a^4*b^2*d* e^5 + 792*a^5*b*e^6)*x)/(e^21*x^14 + 14*d*e^20*x^13 + 91*d^2*e^19*x^12 + 3 64*d^3*e^18*x^11 + 1001*d^4*e^17*x^10 + 2002*d^5*e^16*x^9 + 3003*d^6*e^15* x^8 + 3432*d^7*e^14*x^7 + 3003*d^8*e^13*x^6 + 2002*d^9*e^12*x^5 + 1001*d^1 0*e^11*x^4 + 364*d^11*e^10*x^3 + 91*d^12*e^9*x^2 + 14*d^13*e^8*x + d^14*e^ 7)
Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (159) = 318\).
Time = 0.22 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.17 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{15}} \, dx=-\frac {3003 \, b^{6} e^{6} x^{6} + 2002 \, b^{6} d e^{5} x^{5} + 16016 \, a b^{5} e^{6} x^{5} + 1001 \, b^{6} d^{2} e^{4} x^{4} + 8008 \, a b^{5} d e^{5} x^{4} + 36036 \, a^{2} b^{4} e^{6} x^{4} + 364 \, b^{6} d^{3} e^{3} x^{3} + 2912 \, a b^{5} d^{2} e^{4} x^{3} + 13104 \, a^{2} b^{4} d e^{5} x^{3} + 43680 \, a^{3} b^{3} e^{6} x^{3} + 91 \, b^{6} d^{4} e^{2} x^{2} + 728 \, a b^{5} d^{3} e^{3} x^{2} + 3276 \, a^{2} b^{4} d^{2} e^{4} x^{2} + 10920 \, a^{3} b^{3} d e^{5} x^{2} + 30030 \, a^{4} b^{2} e^{6} x^{2} + 14 \, b^{6} d^{5} e x + 112 \, a b^{5} d^{4} e^{2} x + 504 \, a^{2} b^{4} d^{3} e^{3} x + 1680 \, a^{3} b^{3} d^{2} e^{4} x + 4620 \, a^{4} b^{2} d e^{5} x + 11088 \, a^{5} b e^{6} x + b^{6} d^{6} + 8 \, a b^{5} d^{5} e + 36 \, a^{2} b^{4} d^{4} e^{2} + 120 \, a^{3} b^{3} d^{3} e^{3} + 330 \, a^{4} b^{2} d^{2} e^{4} + 792 \, a^{5} b d e^{5} + 1716 \, a^{6} e^{6}}{24024 \, {\left (e x + d\right )}^{14} e^{7}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^15,x, algorithm="giac")
Output:
-1/24024*(3003*b^6*e^6*x^6 + 2002*b^6*d*e^5*x^5 + 16016*a*b^5*e^6*x^5 + 10 01*b^6*d^2*e^4*x^4 + 8008*a*b^5*d*e^5*x^4 + 36036*a^2*b^4*e^6*x^4 + 364*b^ 6*d^3*e^3*x^3 + 2912*a*b^5*d^2*e^4*x^3 + 13104*a^2*b^4*d*e^5*x^3 + 43680*a ^3*b^3*e^6*x^3 + 91*b^6*d^4*e^2*x^2 + 728*a*b^5*d^3*e^3*x^2 + 3276*a^2*b^4 *d^2*e^4*x^2 + 10920*a^3*b^3*d*e^5*x^2 + 30030*a^4*b^2*e^6*x^2 + 14*b^6*d^ 5*e*x + 112*a*b^5*d^4*e^2*x + 504*a^2*b^4*d^3*e^3*x + 1680*a^3*b^3*d^2*e^4 *x + 4620*a^4*b^2*d*e^5*x + 11088*a^5*b*e^6*x + b^6*d^6 + 8*a*b^5*d^5*e + 36*a^2*b^4*d^4*e^2 + 120*a^3*b^3*d^3*e^3 + 330*a^4*b^2*d^2*e^4 + 792*a^5*b *d*e^5 + 1716*a^6*e^6)/((e*x + d)^14*e^7)
Time = 9.85 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.76 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{15}} \, dx=-\frac {\frac {1716\,a^6\,e^6+792\,a^5\,b\,d\,e^5+330\,a^4\,b^2\,d^2\,e^4+120\,a^3\,b^3\,d^3\,e^3+36\,a^2\,b^4\,d^4\,e^2+8\,a\,b^5\,d^5\,e+b^6\,d^6}{24024\,e^7}+\frac {b^6\,x^6}{8\,e}+\frac {b^3\,x^3\,\left (120\,a^3\,e^3+36\,a^2\,b\,d\,e^2+8\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{66\,e^4}+\frac {b\,x\,\left (792\,a^5\,e^5+330\,a^4\,b\,d\,e^4+120\,a^3\,b^2\,d^2\,e^3+36\,a^2\,b^3\,d^3\,e^2+8\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{1716\,e^6}+\frac {b^5\,x^5\,\left (8\,a\,e+b\,d\right )}{12\,e^2}+\frac {b^2\,x^2\,\left (330\,a^4\,e^4+120\,a^3\,b\,d\,e^3+36\,a^2\,b^2\,d^2\,e^2+8\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{264\,e^5}+\frac {b^4\,x^4\,\left (36\,a^2\,e^2+8\,a\,b\,d\,e+b^2\,d^2\right )}{24\,e^3}}{d^{14}+14\,d^{13}\,e\,x+91\,d^{12}\,e^2\,x^2+364\,d^{11}\,e^3\,x^3+1001\,d^{10}\,e^4\,x^4+2002\,d^9\,e^5\,x^5+3003\,d^8\,e^6\,x^6+3432\,d^7\,e^7\,x^7+3003\,d^6\,e^8\,x^8+2002\,d^5\,e^9\,x^9+1001\,d^4\,e^{10}\,x^{10}+364\,d^3\,e^{11}\,x^{11}+91\,d^2\,e^{12}\,x^{12}+14\,d\,e^{13}\,x^{13}+e^{14}\,x^{14}} \] Input:
int((a^2 + b^2*x^2 + 2*a*b*x)^3/(d + e*x)^15,x)
Output:
-((1716*a^6*e^6 + b^6*d^6 + 36*a^2*b^4*d^4*e^2 + 120*a^3*b^3*d^3*e^3 + 330 *a^4*b^2*d^2*e^4 + 8*a*b^5*d^5*e + 792*a^5*b*d*e^5)/(24024*e^7) + (b^6*x^6 )/(8*e) + (b^3*x^3*(120*a^3*e^3 + b^3*d^3 + 8*a*b^2*d^2*e + 36*a^2*b*d*e^2 ))/(66*e^4) + (b*x*(792*a^5*e^5 + b^5*d^5 + 36*a^2*b^3*d^3*e^2 + 120*a^3*b ^2*d^2*e^3 + 8*a*b^4*d^4*e + 330*a^4*b*d*e^4))/(1716*e^6) + (b^5*x^5*(8*a* e + b*d))/(12*e^2) + (b^2*x^2*(330*a^4*e^4 + b^4*d^4 + 36*a^2*b^2*d^2*e^2 + 8*a*b^3*d^3*e + 120*a^3*b*d*e^3))/(264*e^5) + (b^4*x^4*(36*a^2*e^2 + b^2 *d^2 + 8*a*b*d*e))/(24*e^3))/(d^14 + e^14*x^14 + 14*d*e^13*x^13 + 91*d^12* e^2*x^2 + 364*d^11*e^3*x^3 + 1001*d^10*e^4*x^4 + 2002*d^9*e^5*x^5 + 3003*d ^8*e^6*x^6 + 3432*d^7*e^7*x^7 + 3003*d^6*e^8*x^8 + 2002*d^5*e^9*x^9 + 1001 *d^4*e^10*x^10 + 364*d^3*e^11*x^11 + 91*d^2*e^12*x^12 + 14*d^13*e*x)
Time = 0.25 (sec) , antiderivative size = 519, normalized size of antiderivative = 3.00 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{15}} \, dx=\frac {-3003 b^{6} e^{6} x^{6}-16016 a \,b^{5} e^{6} x^{5}-2002 b^{6} d \,e^{5} x^{5}-36036 a^{2} b^{4} e^{6} x^{4}-8008 a \,b^{5} d \,e^{5} x^{4}-1001 b^{6} d^{2} e^{4} x^{4}-43680 a^{3} b^{3} e^{6} x^{3}-13104 a^{2} b^{4} d \,e^{5} x^{3}-2912 a \,b^{5} d^{2} e^{4} x^{3}-364 b^{6} d^{3} e^{3} x^{3}-30030 a^{4} b^{2} e^{6} x^{2}-10920 a^{3} b^{3} d \,e^{5} x^{2}-3276 a^{2} b^{4} d^{2} e^{4} x^{2}-728 a \,b^{5} d^{3} e^{3} x^{2}-91 b^{6} d^{4} e^{2} x^{2}-11088 a^{5} b \,e^{6} x -4620 a^{4} b^{2} d \,e^{5} x -1680 a^{3} b^{3} d^{2} e^{4} x -504 a^{2} b^{4} d^{3} e^{3} x -112 a \,b^{5} d^{4} e^{2} x -14 b^{6} d^{5} e x -1716 a^{6} e^{6}-792 a^{5} b d \,e^{5}-330 a^{4} b^{2} d^{2} e^{4}-120 a^{3} b^{3} d^{3} e^{3}-36 a^{2} b^{4} d^{4} e^{2}-8 a \,b^{5} d^{5} e -b^{6} d^{6}}{24024 e^{7} \left (e^{14} x^{14}+14 d \,e^{13} x^{13}+91 d^{2} e^{12} x^{12}+364 d^{3} e^{11} x^{11}+1001 d^{4} e^{10} x^{10}+2002 d^{5} e^{9} x^{9}+3003 d^{6} e^{8} x^{8}+3432 d^{7} e^{7} x^{7}+3003 d^{8} e^{6} x^{6}+2002 d^{9} e^{5} x^{5}+1001 d^{10} e^{4} x^{4}+364 d^{11} e^{3} x^{3}+91 d^{12} e^{2} x^{2}+14 d^{13} e x +d^{14}\right )} \] Input:
int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^15,x)
Output:
( - 1716*a**6*e**6 - 792*a**5*b*d*e**5 - 11088*a**5*b*e**6*x - 330*a**4*b* *2*d**2*e**4 - 4620*a**4*b**2*d*e**5*x - 30030*a**4*b**2*e**6*x**2 - 120*a **3*b**3*d**3*e**3 - 1680*a**3*b**3*d**2*e**4*x - 10920*a**3*b**3*d*e**5*x **2 - 43680*a**3*b**3*e**6*x**3 - 36*a**2*b**4*d**4*e**2 - 504*a**2*b**4*d **3*e**3*x - 3276*a**2*b**4*d**2*e**4*x**2 - 13104*a**2*b**4*d*e**5*x**3 - 36036*a**2*b**4*e**6*x**4 - 8*a*b**5*d**5*e - 112*a*b**5*d**4*e**2*x - 72 8*a*b**5*d**3*e**3*x**2 - 2912*a*b**5*d**2*e**4*x**3 - 8008*a*b**5*d*e**5* x**4 - 16016*a*b**5*e**6*x**5 - b**6*d**6 - 14*b**6*d**5*e*x - 91*b**6*d** 4*e**2*x**2 - 364*b**6*d**3*e**3*x**3 - 1001*b**6*d**2*e**4*x**4 - 2002*b* *6*d*e**5*x**5 - 3003*b**6*e**6*x**6)/(24024*e**7*(d**14 + 14*d**13*e*x + 91*d**12*e**2*x**2 + 364*d**11*e**3*x**3 + 1001*d**10*e**4*x**4 + 2002*d** 9*e**5*x**5 + 3003*d**8*e**6*x**6 + 3432*d**7*e**7*x**7 + 3003*d**6*e**8*x **8 + 2002*d**5*e**9*x**9 + 1001*d**4*e**10*x**10 + 364*d**3*e**11*x**11 + 91*d**2*e**12*x**12 + 14*d*e**13*x**13 + e**14*x**14))