Integrand size = 26, antiderivative size = 158 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^3} \, dx=-\frac {20 b^3 (b d-a e)^3 x}{e^6}-\frac {(b d-a e)^6}{2 e^7 (d+e x)^2}+\frac {6 b (b d-a e)^5}{e^7 (d+e x)}+\frac {15 b^4 (b d-a e)^2 (d+e x)^2}{2 e^7}-\frac {2 b^5 (b d-a e) (d+e x)^3}{e^7}+\frac {b^6 (d+e x)^4}{4 e^7}+\frac {15 b^2 (b d-a e)^4 \log (d+e x)}{e^7} \] Output:
-20*b^3*(-a*e+b*d)^3*x/e^6-1/2*(-a*e+b*d)^6/e^7/(e*x+d)^2+6*b*(-a*e+b*d)^5 /e^7/(e*x+d)+15/2*b^4*(-a*e+b*d)^2*(e*x+d)^2/e^7-2*b^5*(-a*e+b*d)*(e*x+d)^ 3/e^7+1/4*b^6*(e*x+d)^4/e^7+15*b^2*(-a*e+b*d)^4*ln(e*x+d)/e^7
Time = 0.09 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.92 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^3} \, dx=\frac {-2 a^6 e^6-12 a^5 b e^5 (d+2 e x)+30 a^4 b^2 d e^4 (3 d+4 e x)+40 a^3 b^3 e^3 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+30 a^2 b^4 e^2 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+4 a b^5 e \left (-27 d^5+6 d^4 e x+63 d^3 e^2 x^2+20 d^2 e^3 x^3-5 d e^4 x^4+2 e^5 x^5\right )+b^6 \left (22 d^6-16 d^5 e x-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-2 d e^5 x^5+e^6 x^6\right )+60 b^2 (b d-a e)^4 (d+e x)^2 \log (d+e x)}{4 e^7 (d+e x)^2} \] Input:
Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^3,x]
Output:
(-2*a^6*e^6 - 12*a^5*b*e^5*(d + 2*e*x) + 30*a^4*b^2*d*e^4*(3*d + 4*e*x) + 40*a^3*b^3*e^3*(-5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3) + 30*a^2*b^4 *e^2*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + 4*a*b^ 5*e*(-27*d^5 + 6*d^4*e*x + 63*d^3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5) + b^6*(22*d^6 - 16*d^5*e*x - 68*d^4*e^2*x^2 - 20*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 2*d*e^5*x^5 + e^6*x^6) + 60*b^2*(b*d - a*e)^4*(d + e*x)^2 *Log[d + e*x])/(4*e^7*(d + e*x)^2)
Time = 0.64 (sec) , antiderivative size = 158, 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, 49, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^6}{(d+e x)^3}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^6}{(d+e x)^3}dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (-\frac {6 b^5 (d+e x)^2 (b d-a e)}{e^6}+\frac {15 b^4 (d+e x) (b d-a e)^2}{e^6}-\frac {20 b^3 (b d-a e)^3}{e^6}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^2}+\frac {(a e-b d)^6}{e^6 (d+e x)^3}+\frac {b^6 (d+e x)^3}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^5 (d+e x)^3 (b d-a e)}{e^7}+\frac {15 b^4 (d+e x)^2 (b d-a e)^2}{2 e^7}-\frac {20 b^3 x (b d-a e)^3}{e^6}+\frac {15 b^2 (b d-a e)^4 \log (d+e x)}{e^7}+\frac {6 b (b d-a e)^5}{e^7 (d+e x)}-\frac {(b d-a e)^6}{2 e^7 (d+e x)^2}+\frac {b^6 (d+e x)^4}{4 e^7}\) |
Input:
Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^3,x]
Output:
(-20*b^3*(b*d - a*e)^3*x)/e^6 - (b*d - a*e)^6/(2*e^7*(d + e*x)^2) + (6*b*( b*d - a*e)^5)/(e^7*(d + e*x)) + (15*b^4*(b*d - a*e)^2*(d + e*x)^2)/(2*e^7) - (2*b^5*(b*d - a*e)*(d + e*x)^3)/e^7 + (b^6*(d + e*x)^4)/(4*e^7) + (15*b ^2*(b*d - a*e)^4*Log[d + e*x])/e^7
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}, x] && IGtQ[m, 0] && IGtQ[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. \(342\) vs. \(2(152)=304\).
Time = 1.16 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.17
| method | result | size |
| norman | \(\frac {-\frac {a^{6} e^{6}+6 a^{5} b d \,e^{5}-45 a^{4} b^{2} d^{2} e^{4}+180 a^{3} b^{3} d^{3} e^{3}-270 a^{2} b^{4} d^{4} e^{2}+180 a \,b^{5} d^{5} e -45 b^{6} d^{6}}{2 e^{7}}+\frac {b^{6} x^{6}}{4 e}-\frac {2 \left (3 a^{5} b \,e^{5}-15 d \,e^{4} a^{4} b^{2}+60 d^{2} e^{3} a^{3} b^{3}-90 d^{3} e^{2} a^{2} b^{4}+60 d^{4} e a \,b^{5}-15 d^{5} b^{6}\right ) x}{e^{6}}+\frac {5 b^{3} \left (4 e^{3} a^{3}-6 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) x^{3}}{e^{4}}+\frac {5 b^{4} \left (6 e^{2} a^{2}-4 a b d e +b^{2} d^{2}\right ) x^{4}}{4 e^{3}}+\frac {b^{5} \left (4 a e -b d \right ) x^{5}}{2 e^{2}}}{\left (e x +d \right )^{2}}+\frac {15 b^{2} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(343\) |
| default | \(\frac {b^{3} \left (\frac {1}{4} b^{3} x^{4} e^{3}+2 a \,b^{2} e^{3} x^{3}-b^{3} d \,e^{2} x^{3}+\frac {15}{2} a^{2} b \,e^{3} x^{2}-9 a \,b^{2} d \,e^{2} x^{2}+3 b^{3} d^{2} e \,x^{2}+20 e^{3} a^{3} x -45 a^{2} b d \,e^{2} x +36 a \,b^{2} d^{2} e x -10 b^{3} d^{3} x \right )}{e^{6}}+\frac {15 b^{2} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \ln \left (e x +d \right )}{e^{7}}-\frac {6 b \left (e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{e^{7} \left (e x +d \right )}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{2 e^{7} \left (e x +d \right )^{2}}\) | \(351\) |
| risch | \(\frac {b^{6} x^{4}}{4 e^{3}}+\frac {2 b^{5} a \,x^{3}}{e^{3}}-\frac {b^{6} d \,x^{3}}{e^{4}}+\frac {15 b^{4} a^{2} x^{2}}{2 e^{3}}-\frac {9 b^{5} a d \,x^{2}}{e^{4}}+\frac {3 b^{6} d^{2} x^{2}}{e^{5}}+\frac {20 b^{3} a^{3} x}{e^{3}}-\frac {45 b^{4} a^{2} d x}{e^{4}}+\frac {36 b^{5} a \,d^{2} x}{e^{5}}-\frac {10 b^{6} d^{3} x}{e^{6}}+\frac {\left (-6 a^{5} b \,e^{5}+30 d \,e^{4} a^{4} b^{2}-60 d^{2} e^{3} a^{3} b^{3}+60 d^{3} e^{2} a^{2} b^{4}-30 d^{4} e a \,b^{5}+6 d^{5} b^{6}\right ) x -\frac {a^{6} e^{6}+6 a^{5} b d \,e^{5}-45 a^{4} b^{2} d^{2} e^{4}+100 a^{3} b^{3} d^{3} e^{3}-105 a^{2} b^{4} d^{4} e^{2}+54 a \,b^{5} d^{5} e -11 b^{6} d^{6}}{2 e}}{e^{6} \left (e x +d \right )^{2}}+\frac {15 b^{2} \ln \left (e x +d \right ) a^{4}}{e^{3}}-\frac {60 b^{3} \ln \left (e x +d \right ) a^{3} d}{e^{4}}+\frac {90 b^{4} \ln \left (e x +d \right ) a^{2} d^{2}}{e^{5}}-\frac {60 b^{5} \ln \left (e x +d \right ) a \,d^{3}}{e^{6}}+\frac {15 b^{6} \ln \left (e x +d \right ) d^{4}}{e^{7}}\) | \(383\) |
| parallelrisch | \(\frac {-360 a \,b^{5} d^{5} e -12 a^{5} b d \,e^{5}-360 a^{3} b^{3} d^{3} e^{3}-240 \ln \left (e x +d \right ) x^{2} a^{3} b^{3} d \,e^{5}+90 a^{4} b^{2} d^{2} e^{4}+720 \ln \left (e x +d \right ) x \,a^{2} b^{4} d^{3} e^{3}+540 a^{2} b^{4} d^{4} e^{2}+60 \ln \left (e x +d \right ) x^{2} a^{4} b^{2} e^{6}+60 \ln \left (e x +d \right ) x^{2} b^{6} d^{4} e^{2}+120 \ln \left (e x +d \right ) x \,b^{6} d^{5} e +90 b^{6} d^{6}-480 \ln \left (e x +d \right ) x \,a^{3} b^{3} d^{2} e^{4}+120 \ln \left (e x +d \right ) x \,a^{4} b^{2} d \,e^{5}+360 \ln \left (e x +d \right ) x^{2} a^{2} b^{4} d^{2} e^{4}-240 \ln \left (e x +d \right ) x^{2} a \,b^{5} d^{3} e^{3}-480 \ln \left (e x +d \right ) x a \,b^{5} d^{4} e^{2}+80 x^{3} a \,b^{5} d^{2} e^{4}+120 x \,a^{4} b^{2} d \,e^{5}-480 x \,a^{3} b^{3} d^{2} e^{4}+720 x \,a^{2} b^{4} d^{3} e^{3}-480 x a \,b^{5} d^{4} e^{2}+60 \ln \left (e x +d \right ) a^{4} b^{2} d^{2} e^{4}-240 \ln \left (e x +d \right ) a^{3} b^{3} d^{3} e^{3}+360 \ln \left (e x +d \right ) a^{2} b^{4} d^{4} e^{2}-240 \ln \left (e x +d \right ) a \,b^{5} d^{5} e +80 x^{3} a^{3} b^{3} e^{6}-20 x^{3} b^{6} d^{3} e^{3}-24 x \,a^{5} b \,e^{6}+120 x \,b^{6} d^{5} e -20 x^{4} a \,b^{5} d \,e^{5}-120 x^{3} a^{2} b^{4} d \,e^{5}-2 a^{6} e^{6}+8 x^{5} a \,b^{5} e^{6}-2 x^{5} b^{6} d \,e^{5}+30 x^{4} a^{2} b^{4} e^{6}+5 x^{4} b^{6} d^{2} e^{4}+60 \ln \left (e x +d \right ) b^{6} d^{6}+x^{6} b^{6} e^{6}}{4 e^{7} \left (e x +d \right )^{2}}\) | \(592\) |
Input:
int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^3,x,method=_RETURNVERBOSE)
Output:
(-1/2*(a^6*e^6+6*a^5*b*d*e^5-45*a^4*b^2*d^2*e^4+180*a^3*b^3*d^3*e^3-270*a^ 2*b^4*d^4*e^2+180*a*b^5*d^5*e-45*b^6*d^6)/e^7+1/4/e*b^6*x^6-2*(3*a^5*b*e^5 -15*a^4*b^2*d*e^4+60*a^3*b^3*d^2*e^3-90*a^2*b^4*d^3*e^2+60*a*b^5*d^4*e-15* b^6*d^5)/e^6*x+5*b^3*(4*a^3*e^3-6*a^2*b*d*e^2+4*a*b^2*d^2*e-b^3*d^3)/e^4*x ^3+5/4*b^4*(6*a^2*e^2-4*a*b*d*e+b^2*d^2)/e^3*x^4+1/2*b^5*(4*a*e-b*d)/e^2*x ^5)/(e*x+d)^2+15*b^2/e^7*(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)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (152) = 304\).
Time = 0.08 (sec) , antiderivative size = 548, normalized size of antiderivative = 3.47 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^3} \, dx=\frac {b^{6} e^{6} x^{6} + 22 \, b^{6} d^{6} - 108 \, a b^{5} d^{5} e + 210 \, a^{2} b^{4} d^{4} e^{2} - 200 \, a^{3} b^{3} d^{3} e^{3} + 90 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 2 \, {\left (b^{6} d e^{5} - 4 \, a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (b^{6} d^{2} e^{4} - 4 \, a b^{5} d e^{5} + 6 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{3} - 4 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} - 4 \, a^{3} b^{3} e^{6}\right )} x^{3} - 2 \, {\left (34 \, b^{6} d^{4} e^{2} - 126 \, a b^{5} d^{3} e^{3} + 165 \, a^{2} b^{4} d^{2} e^{4} - 80 \, a^{3} b^{3} d e^{5}\right )} x^{2} - 4 \, {\left (4 \, b^{6} d^{5} e - 6 \, a b^{5} d^{4} e^{2} - 15 \, a^{2} b^{4} d^{3} e^{3} + 40 \, a^{3} b^{3} d^{2} e^{4} - 30 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} d^{6} - 4 \, a b^{5} d^{5} e + 6 \, a^{2} b^{4} d^{4} e^{2} - 4 \, a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} + {\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 2 \, {\left (b^{6} d^{5} e - 4 \, a b^{5} d^{4} e^{2} + 6 \, a^{2} b^{4} d^{3} e^{3} - 4 \, a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{4 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^3,x, algorithm="fricas")
Output:
1/4*(b^6*e^6*x^6 + 22*b^6*d^6 - 108*a*b^5*d^5*e + 210*a^2*b^4*d^4*e^2 - 20 0*a^3*b^3*d^3*e^3 + 90*a^4*b^2*d^2*e^4 - 12*a^5*b*d*e^5 - 2*a^6*e^6 - 2*(b ^6*d*e^5 - 4*a*b^5*e^6)*x^5 + 5*(b^6*d^2*e^4 - 4*a*b^5*d*e^5 + 6*a^2*b^4*e ^6)*x^4 - 20*(b^6*d^3*e^3 - 4*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 - 4*a^3*b^3* e^6)*x^3 - 2*(34*b^6*d^4*e^2 - 126*a*b^5*d^3*e^3 + 165*a^2*b^4*d^2*e^4 - 8 0*a^3*b^3*d*e^5)*x^2 - 4*(4*b^6*d^5*e - 6*a*b^5*d^4*e^2 - 15*a^2*b^4*d^3*e ^3 + 40*a^3*b^3*d^2*e^4 - 30*a^4*b^2*d*e^5 + 6*a^5*b*e^6)*x + 60*(b^6*d^6 - 4*a*b^5*d^5*e + 6*a^2*b^4*d^4*e^2 - 4*a^3*b^3*d^3*e^3 + a^4*b^2*d^2*e^4 + (b^6*d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 + a ^4*b^2*e^6)*x^2 + 2*(b^6*d^5*e - 4*a*b^5*d^4*e^2 + 6*a^2*b^4*d^3*e^3 - 4*a ^3*b^3*d^2*e^4 + a^4*b^2*d*e^5)*x)*log(e*x + d))/(e^9*x^2 + 2*d*e^8*x + d^ 2*e^7)
Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (144) = 288\).
Time = 1.10 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.15 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^3} \, dx=\frac {b^{6} x^{4}}{4 e^{3}} + \frac {15 b^{2} \left (a e - b d\right )^{4} \log {\left (d + e x \right )}}{e^{7}} + x^{3} \cdot \left (\frac {2 a b^{5}}{e^{3}} - \frac {b^{6} d}{e^{4}}\right ) + x^{2} \cdot \left (\frac {15 a^{2} b^{4}}{2 e^{3}} - \frac {9 a b^{5} d}{e^{4}} + \frac {3 b^{6} d^{2}}{e^{5}}\right ) + x \left (\frac {20 a^{3} b^{3}}{e^{3}} - \frac {45 a^{2} b^{4} d}{e^{4}} + \frac {36 a b^{5} d^{2}}{e^{5}} - \frac {10 b^{6} d^{3}}{e^{6}}\right ) + \frac {- a^{6} e^{6} - 6 a^{5} b d e^{5} + 45 a^{4} b^{2} d^{2} e^{4} - 100 a^{3} b^{3} d^{3} e^{3} + 105 a^{2} b^{4} d^{4} e^{2} - 54 a b^{5} d^{5} e + 11 b^{6} d^{6} + x \left (- 12 a^{5} b e^{6} + 60 a^{4} b^{2} d e^{5} - 120 a^{3} b^{3} d^{2} e^{4} + 120 a^{2} b^{4} d^{3} e^{3} - 60 a b^{5} d^{4} e^{2} + 12 b^{6} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} \] Input:
integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**3,x)
Output:
b**6*x**4/(4*e**3) + 15*b**2*(a*e - b*d)**4*log(d + e*x)/e**7 + x**3*(2*a* b**5/e**3 - b**6*d/e**4) + x**2*(15*a**2*b**4/(2*e**3) - 9*a*b**5*d/e**4 + 3*b**6*d**2/e**5) + x*(20*a**3*b**3/e**3 - 45*a**2*b**4*d/e**4 + 36*a*b** 5*d**2/e**5 - 10*b**6*d**3/e**6) + (-a**6*e**6 - 6*a**5*b*d*e**5 + 45*a**4 *b**2*d**2*e**4 - 100*a**3*b**3*d**3*e**3 + 105*a**2*b**4*d**4*e**2 - 54*a *b**5*d**5*e + 11*b**6*d**6 + x*(-12*a**5*b*e**6 + 60*a**4*b**2*d*e**5 - 1 20*a**3*b**3*d**2*e**4 + 120*a**2*b**4*d**3*e**3 - 60*a*b**5*d**4*e**2 + 1 2*b**6*d**5*e))/(2*d**2*e**7 + 4*d*e**8*x + 2*e**9*x**2)
Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (152) = 304\).
Time = 0.04 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.30 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^3} \, dx=\frac {11 \, b^{6} d^{6} - 54 \, a b^{5} d^{5} e + 105 \, a^{2} b^{4} d^{4} e^{2} - 100 \, a^{3} b^{3} d^{3} e^{3} + 45 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} - a^{6} e^{6} + 12 \, {\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} - 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x}{2 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac {b^{6} e^{3} x^{4} - 4 \, {\left (b^{6} d e^{2} - 2 \, a b^{5} e^{3}\right )} x^{3} + 6 \, {\left (2 \, b^{6} d^{2} e - 6 \, a b^{5} d e^{2} + 5 \, a^{2} b^{4} e^{3}\right )} x^{2} - 4 \, {\left (10 \, b^{6} d^{3} - 36 \, a b^{5} d^{2} e + 45 \, a^{2} b^{4} d e^{2} - 20 \, a^{3} b^{3} e^{3}\right )} x}{4 \, e^{6}} + \frac {15 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^3,x, algorithm="maxima")
Output:
1/2*(11*b^6*d^6 - 54*a*b^5*d^5*e + 105*a^2*b^4*d^4*e^2 - 100*a^3*b^3*d^3*e ^3 + 45*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 - a^6*e^6 + 12*(b^6*d^5*e - 5*a*b^ 5*d^4*e^2 + 10*a^2*b^4*d^3*e^3 - 10*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 - a^ 5*b*e^6)*x)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7) + 1/4*(b^6*e^3*x^4 - 4*(b^6*d* e^2 - 2*a*b^5*e^3)*x^3 + 6*(2*b^6*d^2*e - 6*a*b^5*d*e^2 + 5*a^2*b^4*e^3)*x ^2 - 4*(10*b^6*d^3 - 36*a*b^5*d^2*e + 45*a^2*b^4*d*e^2 - 20*a^3*b^3*e^3)*x )/e^6 + 15*(b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*log(e*x + d)/e^7
Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (152) = 304\).
Time = 0.18 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.29 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^3} \, dx=\frac {15 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} + \frac {11 \, b^{6} d^{6} - 54 \, a b^{5} d^{5} e + 105 \, a^{2} b^{4} d^{4} e^{2} - 100 \, a^{3} b^{3} d^{3} e^{3} + 45 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} - a^{6} e^{6} + 12 \, {\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} - 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{7}} + \frac {b^{6} e^{9} x^{4} - 4 \, b^{6} d e^{8} x^{3} + 8 \, a b^{5} e^{9} x^{3} + 12 \, b^{6} d^{2} e^{7} x^{2} - 36 \, a b^{5} d e^{8} x^{2} + 30 \, a^{2} b^{4} e^{9} x^{2} - 40 \, b^{6} d^{3} e^{6} x + 144 \, a b^{5} d^{2} e^{7} x - 180 \, a^{2} b^{4} d e^{8} x + 80 \, a^{3} b^{3} e^{9} x}{4 \, e^{12}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^3,x, algorithm="giac")
Output:
15*(b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^ 2*e^4)*log(abs(e*x + d))/e^7 + 1/2*(11*b^6*d^6 - 54*a*b^5*d^5*e + 105*a^2* b^4*d^4*e^2 - 100*a^3*b^3*d^3*e^3 + 45*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 - a ^6*e^6 + 12*(b^6*d^5*e - 5*a*b^5*d^4*e^2 + 10*a^2*b^4*d^3*e^3 - 10*a^3*b^3 *d^2*e^4 + 5*a^4*b^2*d*e^5 - a^5*b*e^6)*x)/((e*x + d)^2*e^7) + 1/4*(b^6*e^ 9*x^4 - 4*b^6*d*e^8*x^3 + 8*a*b^5*e^9*x^3 + 12*b^6*d^2*e^7*x^2 - 36*a*b^5* d*e^8*x^2 + 30*a^2*b^4*e^9*x^2 - 40*b^6*d^3*e^6*x + 144*a*b^5*d^2*e^7*x - 180*a^2*b^4*d*e^8*x + 80*a^3*b^3*e^9*x)/e^12
Time = 8.71 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.79 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^3} \, dx=x\,\left (\frac {20\,a^3\,b^3}{e^3}-\frac {b^6\,d^3}{e^6}+\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {6\,a\,b^5}{e^3}-\frac {3\,b^6\,d}{e^4}\right )}{e}-\frac {15\,a^2\,b^4}{e^3}+\frac {3\,b^6\,d^2}{e^5}\right )}{e}-\frac {3\,d^2\,\left (\frac {6\,a\,b^5}{e^3}-\frac {3\,b^6\,d}{e^4}\right )}{e^2}\right )-\frac {\frac {a^6\,e^6+6\,a^5\,b\,d\,e^5-45\,a^4\,b^2\,d^2\,e^4+100\,a^3\,b^3\,d^3\,e^3-105\,a^2\,b^4\,d^4\,e^2+54\,a\,b^5\,d^5\,e-11\,b^6\,d^6}{2\,e}-x\,\left (-6\,a^5\,b\,e^5+30\,a^4\,b^2\,d\,e^4-60\,a^3\,b^3\,d^2\,e^3+60\,a^2\,b^4\,d^3\,e^2-30\,a\,b^5\,d^4\,e+6\,b^6\,d^5\right )}{d^2\,e^6+2\,d\,e^7\,x+e^8\,x^2}+x^3\,\left (\frac {2\,a\,b^5}{e^3}-\frac {b^6\,d}{e^4}\right )-x^2\,\left (\frac {3\,d\,\left (\frac {6\,a\,b^5}{e^3}-\frac {3\,b^6\,d}{e^4}\right )}{2\,e}-\frac {15\,a^2\,b^4}{2\,e^3}+\frac {3\,b^6\,d^2}{2\,e^5}\right )+\frac {\ln \left (d+e\,x\right )\,\left (15\,a^4\,b^2\,e^4-60\,a^3\,b^3\,d\,e^3+90\,a^2\,b^4\,d^2\,e^2-60\,a\,b^5\,d^3\,e+15\,b^6\,d^4\right )}{e^7}+\frac {b^6\,x^4}{4\,e^3} \] Input:
int((a^2 + b^2*x^2 + 2*a*b*x)^3/(d + e*x)^3,x)
Output:
x*((20*a^3*b^3)/e^3 - (b^6*d^3)/e^6 + (3*d*((3*d*((6*a*b^5)/e^3 - (3*b^6*d )/e^4))/e - (15*a^2*b^4)/e^3 + (3*b^6*d^2)/e^5))/e - (3*d^2*((6*a*b^5)/e^3 - (3*b^6*d)/e^4))/e^2) - ((a^6*e^6 - 11*b^6*d^6 - 105*a^2*b^4*d^4*e^2 + 1 00*a^3*b^3*d^3*e^3 - 45*a^4*b^2*d^2*e^4 + 54*a*b^5*d^5*e + 6*a^5*b*d*e^5)/ (2*e) - x*(6*b^6*d^5 - 6*a^5*b*e^5 + 30*a^4*b^2*d*e^4 + 60*a^2*b^4*d^3*e^2 - 60*a^3*b^3*d^2*e^3 - 30*a*b^5*d^4*e))/(d^2*e^6 + e^8*x^2 + 2*d*e^7*x) + x^3*((2*a*b^5)/e^3 - (b^6*d)/e^4) - x^2*((3*d*((6*a*b^5)/e^3 - (3*b^6*d)/ e^4))/(2*e) - (15*a^2*b^4)/(2*e^3) + (3*b^6*d^2)/(2*e^5)) + (log(d + e*x)* (15*b^6*d^4 + 15*a^4*b^2*e^4 - 60*a^3*b^3*d*e^3 + 90*a^2*b^4*d^2*e^2 - 60* a*b^5*d^3*e))/e^7 + (b^6*x^4)/(4*e^3)
Time = 0.22 (sec) , antiderivative size = 625, normalized size of antiderivative = 3.96 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^3} \, dx=\frac {-2 a^{6} d \,e^{6}+60 \,\mathrm {log}\left (e x +d \right ) a^{4} b^{2} d^{3} e^{4}-240 \,\mathrm {log}\left (e x +d \right ) a^{3} b^{3} d^{4} e^{3}+360 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{4} d^{5} e^{2}-240 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{6} e +120 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{6} e x -60 a^{4} b^{2} d \,e^{6} x^{2}+240 a^{3} b^{3} d^{2} e^{5} x^{2}+80 a^{3} b^{3} d \,e^{6} x^{3}-360 a^{2} b^{4} d^{3} e^{4} x^{2}-120 a^{2} b^{4} d^{2} e^{5} x^{3}+30 a^{2} b^{4} d \,e^{6} x^{4}+240 a \,b^{5} d^{4} e^{3} x^{2}+80 a \,b^{5} d^{3} e^{4} x^{3}-20 a \,b^{5} d^{2} e^{5} x^{4}+8 a \,b^{5} d \,e^{6} x^{5}-60 b^{6} d^{5} e^{2} x^{2}-20 b^{6} d^{4} e^{3} x^{3}+5 b^{6} d^{3} e^{4} x^{4}-2 b^{6} d^{2} e^{5} x^{5}+b^{6} d \,e^{6} x^{6}+60 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{7}+12 a^{5} b \,e^{7} x^{2}+30 a^{4} b^{2} d^{3} e^{4}-120 a^{3} b^{3} d^{4} e^{3}+180 a^{2} b^{4} d^{5} e^{2}-120 a \,b^{5} d^{6} e +30 b^{6} d^{7}+60 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{5} e^{2} x^{2}+60 \,\mathrm {log}\left (e x +d \right ) a^{4} b^{2} d \,e^{6} x^{2}-240 \,\mathrm {log}\left (e x +d \right ) a^{3} b^{3} d^{2} e^{5} x^{2}+360 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{4} d^{3} e^{4} x^{2}-240 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{4} e^{3} x^{2}+120 \,\mathrm {log}\left (e x +d \right ) a^{4} b^{2} d^{2} e^{5} x -480 \,\mathrm {log}\left (e x +d \right ) a^{3} b^{3} d^{3} e^{4} x +720 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{4} d^{4} e^{3} x -480 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{5} e^{2} x}{4 d \,e^{7} \left (e^{2} x^{2}+2 d e x +d^{2}\right )} \] Input:
int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^3,x)
Output:
(60*log(d + e*x)*a**4*b**2*d**3*e**4 + 120*log(d + e*x)*a**4*b**2*d**2*e** 5*x + 60*log(d + e*x)*a**4*b**2*d*e**6*x**2 - 240*log(d + e*x)*a**3*b**3*d **4*e**3 - 480*log(d + e*x)*a**3*b**3*d**3*e**4*x - 240*log(d + e*x)*a**3* b**3*d**2*e**5*x**2 + 360*log(d + e*x)*a**2*b**4*d**5*e**2 + 720*log(d + e *x)*a**2*b**4*d**4*e**3*x + 360*log(d + e*x)*a**2*b**4*d**3*e**4*x**2 - 24 0*log(d + e*x)*a*b**5*d**6*e - 480*log(d + e*x)*a*b**5*d**5*e**2*x - 240*l og(d + e*x)*a*b**5*d**4*e**3*x**2 + 60*log(d + e*x)*b**6*d**7 + 120*log(d + e*x)*b**6*d**6*e*x + 60*log(d + e*x)*b**6*d**5*e**2*x**2 - 2*a**6*d*e**6 + 12*a**5*b*e**7*x**2 + 30*a**4*b**2*d**3*e**4 - 60*a**4*b**2*d*e**6*x**2 - 120*a**3*b**3*d**4*e**3 + 240*a**3*b**3*d**2*e**5*x**2 + 80*a**3*b**3*d *e**6*x**3 + 180*a**2*b**4*d**5*e**2 - 360*a**2*b**4*d**3*e**4*x**2 - 120* a**2*b**4*d**2*e**5*x**3 + 30*a**2*b**4*d*e**6*x**4 - 120*a*b**5*d**6*e + 240*a*b**5*d**4*e**3*x**2 + 80*a*b**5*d**3*e**4*x**3 - 20*a*b**5*d**2*e**5 *x**4 + 8*a*b**5*d*e**6*x**5 + 30*b**6*d**7 - 60*b**6*d**5*e**2*x**2 - 20* b**6*d**4*e**3*x**3 + 5*b**6*d**3*e**4*x**4 - 2*b**6*d**2*e**5*x**5 + b**6 *d*e**6*x**6)/(4*d*e**7*(d**2 + 2*d*e*x + e**2*x**2))