Integrand size = 26, antiderivative size = 156 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx=\frac {15 b^4 (b d-a e)^2 x}{e^6}-\frac {(b d-a e)^6}{3 e^7 (d+e x)^3}+\frac {3 b (b d-a e)^5}{e^7 (d+e x)^2}-\frac {15 b^2 (b d-a e)^4}{e^7 (d+e x)}-\frac {3 b^5 (b d-a e) (d+e x)^2}{e^7}+\frac {b^6 (d+e x)^3}{3 e^7}-\frac {20 b^3 (b d-a e)^3 \log (d+e x)}{e^7} \] Output:
15*b^4*(-a*e+b*d)^2*x/e^6-1/3*(-a*e+b*d)^6/e^7/(e*x+d)^3+3*b*(-a*e+b*d)^5/ e^7/(e*x+d)^2-15*b^2*(-a*e+b*d)^4/e^7/(e*x+d)-3*b^5*(-a*e+b*d)*(e*x+d)^2/e ^7+1/3*b^6*(e*x+d)^3/e^7-20*b^3*(-a*e+b*d)^3*ln(e*x+d)/e^7
Time = 0.08 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx=\frac {-a^6 e^6-3 a^5 b e^5 (d+3 e x)-15 a^4 b^2 e^4 \left (d^2+3 d e x+3 e^2 x^2\right )+10 a^3 b^3 d e^3 \left (11 d^2+27 d e x+18 e^2 x^2\right )+15 a^2 b^4 e^2 \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )+3 a b^5 e \left (47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right )+b^6 \left (-37 d^6-51 d^5 e x+39 d^4 e^2 x^2+73 d^3 e^3 x^3+15 d^2 e^4 x^4-3 d e^5 x^5+e^6 x^6\right )-60 b^3 (b d-a e)^3 (d+e x)^3 \log (d+e x)}{3 e^7 (d+e x)^3} \] Input:
Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^4,x]
Output:
(-(a^6*e^6) - 3*a^5*b*e^5*(d + 3*e*x) - 15*a^4*b^2*e^4*(d^2 + 3*d*e*x + 3* e^2*x^2) + 10*a^3*b^3*d*e^3*(11*d^2 + 27*d*e*x + 18*e^2*x^2) + 15*a^2*b^4* e^2*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) + 3*a *b^5*e*(47*d^5 + 81*d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2*e^3*x^3 - 15*d*e^4*x^ 4 + 3*e^5*x^5) + b^6*(-37*d^6 - 51*d^5*e*x + 39*d^4*e^2*x^2 + 73*d^3*e^3*x ^3 + 15*d^2*e^4*x^4 - 3*d*e^5*x^5 + e^6*x^6) - 60*b^3*(b*d - a*e)^3*(d + e *x)^3*Log[d + e*x])/(3*e^7*(d + e*x)^3)
Time = 0.62 (sec) , antiderivative size = 156, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^6}{(d+e x)^4}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^6}{(d+e x)^4}dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (-\frac {6 b^5 (d+e x) (b d-a e)}{e^6}+\frac {15 b^4 (b d-a e)^2}{e^6}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^2}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^3}+\frac {(a e-b d)^6}{e^6 (d+e x)^4}+\frac {b^6 (d+e x)^2}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 b^5 (d+e x)^2 (b d-a e)}{e^7}+\frac {15 b^4 x (b d-a e)^2}{e^6}-\frac {20 b^3 (b d-a e)^3 \log (d+e x)}{e^7}-\frac {15 b^2 (b d-a e)^4}{e^7 (d+e x)}+\frac {3 b (b d-a e)^5}{e^7 (d+e x)^2}-\frac {(b d-a e)^6}{3 e^7 (d+e x)^3}+\frac {b^6 (d+e x)^3}{3 e^7}\) |
Input:
Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^4,x]
Output:
(15*b^4*(b*d - a*e)^2*x)/e^6 - (b*d - a*e)^6/(3*e^7*(d + e*x)^3) + (3*b*(b *d - a*e)^5)/(e^7*(d + e*x)^2) - (15*b^2*(b*d - a*e)^4)/(e^7*(d + e*x)) - (3*b^5*(b*d - a*e)*(d + e*x)^2)/e^7 + (b^6*(d + e*x)^3)/(3*e^7) - (20*b^3* (b*d - a*e)^3*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. \(343\) vs. \(2(152)=304\).
Time = 1.16 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.21
| method | result | size |
| norman | \(\frac {\frac {b^{5} \left (3 a e -b d \right ) x^{5}}{e^{2}}-\frac {a^{6} e^{6}+3 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-110 a^{3} b^{3} d^{3} e^{3}+330 a^{2} b^{4} d^{4} e^{2}-330 a \,b^{5} d^{5} e +110 b^{6} d^{6}}{3 e^{7}}+\frac {b^{6} x^{6}}{3 e}-\frac {3 \left (5 e^{4} a^{4} b^{2}-20 d \,e^{3} a^{3} b^{3}+60 d^{2} e^{2} a^{2} b^{4}-60 d^{3} e a \,b^{5}+20 d^{4} b^{6}\right ) x^{2}}{e^{5}}-\frac {3 \left (a^{5} b \,e^{5}+5 d \,e^{4} a^{4} b^{2}-30 d^{2} e^{3} a^{3} b^{3}+90 d^{3} e^{2} a^{2} b^{4}-90 d^{4} e a \,b^{5}+30 d^{5} b^{6}\right ) x}{e^{6}}+\frac {5 b^{4} \left (3 e^{2} a^{2}-3 a b d e +b^{2} d^{2}\right ) x^{4}}{e^{3}}}{\left (e x +d \right )^{3}}+\frac {20 b^{3} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(344\) |
| default | \(\frac {b^{4} \left (\frac {1}{3} b^{2} e^{2} x^{3}+3 x^{2} a b \,e^{2}-2 b^{2} d e \,x^{2}+15 a^{2} e^{2} x -24 a b d e x +10 b^{2} d^{2} x \right )}{e^{6}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{3 e^{7} \left (e x +d \right )^{3}}+\frac {20 b^{3} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \ln \left (e x +d \right )}{e^{7}}-\frac {15 b^{2} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{e^{7} \left (e x +d \right )}-\frac {3 b \left (e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{e^{7} \left (e x +d \right )^{2}}\) | \(346\) |
| risch | \(\frac {b^{6} x^{3}}{3 e^{4}}+\frac {3 b^{5} x^{2} a}{e^{4}}-\frac {2 b^{6} d \,x^{2}}{e^{5}}+\frac {15 b^{4} a^{2} x}{e^{4}}-\frac {24 b^{5} a d x}{e^{5}}+\frac {10 b^{6} d^{2} x}{e^{6}}+\frac {\left (-15 e^{5} a^{4} b^{2}+60 d \,e^{4} a^{3} b^{3}-90 d^{2} e^{3} a^{2} b^{4}+60 d^{3} e^{2} a \,b^{5}-15 d^{4} e \,b^{6}\right ) x^{2}-3 b \left (e^{5} a^{5}+5 a^{4} b d \,e^{4}-30 a^{3} b^{2} d^{2} e^{3}+50 a^{2} b^{3} d^{3} e^{2}-35 a \,b^{4} d^{4} e +9 b^{5} d^{5}\right ) x -\frac {a^{6} e^{6}+3 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-110 a^{3} b^{3} d^{3} e^{3}+195 a^{2} b^{4} d^{4} e^{2}-141 a \,b^{5} d^{5} e +37 b^{6} d^{6}}{3 e}}{e^{6} \left (e x +d \right )^{3}}+\frac {20 b^{3} \ln \left (e x +d \right ) a^{3}}{e^{4}}-\frac {60 b^{4} \ln \left (e x +d \right ) a^{2} d}{e^{5}}+\frac {60 b^{5} \ln \left (e x +d \right ) a \,d^{2}}{e^{6}}-\frac {20 b^{6} \ln \left (e x +d \right ) d^{3}}{e^{7}}\) | \(368\) |
| parallelrisch | \(\frac {330 a \,b^{5} d^{5} e -3 a^{5} b d \,e^{5}+110 a^{3} b^{3} d^{3} e^{3}+180 \ln \left (e x +d \right ) x^{2} a^{3} b^{3} d \,e^{5}-15 a^{4} b^{2} d^{2} e^{4}-540 \ln \left (e x +d \right ) x \,a^{2} b^{4} d^{3} e^{3}-330 a^{2} b^{4} d^{4} e^{2}-180 \ln \left (e x +d \right ) x^{2} b^{6} d^{4} e^{2}-180 \ln \left (e x +d \right ) x \,b^{6} d^{5} e -180 \ln \left (e x +d \right ) x^{3} a^{2} b^{4} d \,e^{5}-110 b^{6} d^{6}+180 \ln \left (e x +d \right ) x^{3} a \,b^{5} d^{2} e^{4}+180 \ln \left (e x +d \right ) x \,a^{3} b^{3} d^{2} e^{4}-540 \ln \left (e x +d \right ) x^{2} a^{2} b^{4} d^{2} e^{4}+540 \ln \left (e x +d \right ) x^{2} a \,b^{5} d^{3} e^{3}+540 \ln \left (e x +d \right ) x a \,b^{5} d^{4} e^{2}+180 x^{2} a^{3} b^{3} d \,e^{5}-540 x^{2} a^{2} b^{4} d^{2} e^{4}+540 x^{2} a \,b^{5} d^{3} e^{3}-45 x \,a^{4} b^{2} d \,e^{5}+270 x \,a^{3} b^{3} d^{2} e^{4}-810 x \,a^{2} b^{4} d^{3} e^{3}+810 x a \,b^{5} d^{4} e^{2}+60 \ln \left (e x +d \right ) a^{3} b^{3} d^{3} e^{3}-180 \ln \left (e x +d \right ) a^{2} b^{4} d^{4} e^{2}+180 \ln \left (e x +d \right ) a \,b^{5} d^{5} e -45 x^{2} a^{4} b^{2} e^{6}-180 x^{2} b^{6} d^{4} e^{2}-9 x \,a^{5} b \,e^{6}-270 x \,b^{6} d^{5} e -60 \ln \left (e x +d \right ) x^{3} b^{6} d^{3} e^{3}+60 \ln \left (e x +d \right ) x^{3} a^{3} b^{3} e^{6}-45 x^{4} a \,b^{5} d \,e^{5}-a^{6} e^{6}+9 x^{5} a \,b^{5} e^{6}-3 x^{5} b^{6} d \,e^{5}+45 x^{4} a^{2} b^{4} e^{6}+15 x^{4} b^{6} d^{2} e^{4}-60 \ln \left (e x +d \right ) b^{6} d^{6}+x^{6} b^{6} e^{6}}{3 e^{7} \left (e x +d \right )^{3}}\) | \(632\) |
Input:
int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^4,x,method=_RETURNVERBOSE)
Output:
(b^5*(3*a*e-b*d)/e^2*x^5-1/3*(a^6*e^6+3*a^5*b*d*e^5+15*a^4*b^2*d^2*e^4-110 *a^3*b^3*d^3*e^3+330*a^2*b^4*d^4*e^2-330*a*b^5*d^5*e+110*b^6*d^6)/e^7+1/3/ e*b^6*x^6-3*(5*a^4*b^2*e^4-20*a^3*b^3*d*e^3+60*a^2*b^4*d^2*e^2-60*a*b^5*d^ 3*e+20*b^6*d^4)/e^5*x^2-3*(a^5*b*e^5+5*a^4*b^2*d*e^4-30*a^3*b^3*d^2*e^3+90 *a^2*b^4*d^3*e^2-90*a*b^5*d^4*e+30*b^6*d^5)/e^6*x+5*b^4*(3*a^2*e^2-3*a*b*d *e+b^2*d^2)/e^3*x^4)/(e*x+d)^3+20*b^3/e^7*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d ^2*e-b^3*d^3)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (152) = 304\).
Time = 0.08 (sec) , antiderivative size = 576, normalized size of antiderivative = 3.69 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx=\frac {b^{6} e^{6} x^{6} - 37 \, b^{6} d^{6} + 141 \, a b^{5} d^{5} e - 195 \, a^{2} b^{4} d^{4} e^{2} + 110 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - a^{6} e^{6} - 3 \, {\left (b^{6} d e^{5} - 3 \, a b^{5} e^{6}\right )} x^{5} + 15 \, {\left (b^{6} d^{2} e^{4} - 3 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + {\left (73 \, b^{6} d^{3} e^{3} - 189 \, a b^{5} d^{2} e^{4} + 135 \, a^{2} b^{4} d e^{5}\right )} x^{3} + 3 \, {\left (13 \, b^{6} d^{4} e^{2} - 9 \, a b^{5} d^{3} e^{3} - 45 \, a^{2} b^{4} d^{2} e^{4} + 60 \, a^{3} b^{3} d e^{5} - 15 \, a^{4} b^{2} e^{6}\right )} x^{2} - 3 \, {\left (17 \, b^{6} d^{5} e - 81 \, a b^{5} d^{4} e^{2} + 135 \, a^{2} b^{4} d^{3} e^{3} - 90 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \, {\left (b^{6} d^{6} - 3 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} - a^{3} b^{3} d^{3} e^{3} + {\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 3 \, {\left (b^{6} d^{4} e^{2} - 3 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} - a^{3} b^{3} d e^{5}\right )} x^{2} + 3 \, {\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} - a^{3} b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^4,x, algorithm="fricas")
Output:
1/3*(b^6*e^6*x^6 - 37*b^6*d^6 + 141*a*b^5*d^5*e - 195*a^2*b^4*d^4*e^2 + 11 0*a^3*b^3*d^3*e^3 - 15*a^4*b^2*d^2*e^4 - 3*a^5*b*d*e^5 - a^6*e^6 - 3*(b^6* d*e^5 - 3*a*b^5*e^6)*x^5 + 15*(b^6*d^2*e^4 - 3*a*b^5*d*e^5 + 3*a^2*b^4*e^6 )*x^4 + (73*b^6*d^3*e^3 - 189*a*b^5*d^2*e^4 + 135*a^2*b^4*d*e^5)*x^3 + 3*( 13*b^6*d^4*e^2 - 9*a*b^5*d^3*e^3 - 45*a^2*b^4*d^2*e^4 + 60*a^3*b^3*d*e^5 - 15*a^4*b^2*e^6)*x^2 - 3*(17*b^6*d^5*e - 81*a*b^5*d^4*e^2 + 135*a^2*b^4*d^ 3*e^3 - 90*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5 + 3*a^5*b*e^6)*x - 60*(b^6*d ^6 - 3*a*b^5*d^5*e + 3*a^2*b^4*d^4*e^2 - a^3*b^3*d^3*e^3 + (b^6*d^3*e^3 - 3*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 3*(b^6*d^4*e^2 - 3* a*b^5*d^3*e^3 + 3*a^2*b^4*d^2*e^4 - a^3*b^3*d*e^5)*x^2 + 3*(b^6*d^5*e - 3* a*b^5*d^4*e^2 + 3*a^2*b^4*d^3*e^3 - a^3*b^3*d^2*e^4)*x)*log(e*x + d))/(e^1 0*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)
Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (143) = 286\).
Time = 2.46 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.35 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx=\frac {b^{6} x^{3}}{3 e^{4}} + \frac {20 b^{3} \left (a e - b d\right )^{3} \log {\left (d + e x \right )}}{e^{7}} + x^{2} \cdot \left (\frac {3 a b^{5}}{e^{4}} - \frac {2 b^{6} d}{e^{5}}\right ) + x \left (\frac {15 a^{2} b^{4}}{e^{4}} - \frac {24 a b^{5} d}{e^{5}} + \frac {10 b^{6} d^{2}}{e^{6}}\right ) + \frac {- a^{6} e^{6} - 3 a^{5} b d e^{5} - 15 a^{4} b^{2} d^{2} e^{4} + 110 a^{3} b^{3} d^{3} e^{3} - 195 a^{2} b^{4} d^{4} e^{2} + 141 a b^{5} d^{5} e - 37 b^{6} d^{6} + x^{2} \left (- 45 a^{4} b^{2} e^{6} + 180 a^{3} b^{3} d e^{5} - 270 a^{2} b^{4} d^{2} e^{4} + 180 a b^{5} d^{3} e^{3} - 45 b^{6} d^{4} e^{2}\right ) + x \left (- 9 a^{5} b e^{6} - 45 a^{4} b^{2} d e^{5} + 270 a^{3} b^{3} d^{2} e^{4} - 450 a^{2} b^{4} d^{3} e^{3} + 315 a b^{5} d^{4} e^{2} - 81 b^{6} d^{5} e\right )}{3 d^{3} e^{7} + 9 d^{2} e^{8} x + 9 d e^{9} x^{2} + 3 e^{10} x^{3}} \] Input:
integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**4,x)
Output:
b**6*x**3/(3*e**4) + 20*b**3*(a*e - b*d)**3*log(d + e*x)/e**7 + x**2*(3*a* b**5/e**4 - 2*b**6*d/e**5) + x*(15*a**2*b**4/e**4 - 24*a*b**5*d/e**5 + 10* b**6*d**2/e**6) + (-a**6*e**6 - 3*a**5*b*d*e**5 - 15*a**4*b**2*d**2*e**4 + 110*a**3*b**3*d**3*e**3 - 195*a**2*b**4*d**4*e**2 + 141*a*b**5*d**5*e - 3 7*b**6*d**6 + x**2*(-45*a**4*b**2*e**6 + 180*a**3*b**3*d*e**5 - 270*a**2*b **4*d**2*e**4 + 180*a*b**5*d**3*e**3 - 45*b**6*d**4*e**2) + x*(-9*a**5*b*e **6 - 45*a**4*b**2*d*e**5 + 270*a**3*b**3*d**2*e**4 - 450*a**2*b**4*d**3*e **3 + 315*a*b**5*d**4*e**2 - 81*b**6*d**5*e))/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3)
Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (152) = 304\).
Time = 0.05 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.40 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx=-\frac {37 \, b^{6} d^{6} - 141 \, a b^{5} d^{5} e + 195 \, a^{2} b^{4} d^{4} e^{2} - 110 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} + a^{6} e^{6} + 45 \, {\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 9 \, {\left (9 \, b^{6} d^{5} e - 35 \, a b^{5} d^{4} e^{2} + 50 \, a^{2} b^{4} d^{3} e^{3} - 30 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x}{3 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} + \frac {b^{6} e^{2} x^{3} - 3 \, {\left (2 \, b^{6} d e - 3 \, a b^{5} e^{2}\right )} x^{2} + 3 \, {\left (10 \, b^{6} d^{2} - 24 \, a b^{5} d e + 15 \, a^{2} b^{4} e^{2}\right )} x}{3 \, e^{6}} - \frac {20 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} \log \left (e x + d\right )}{e^{7}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^4,x, algorithm="maxima")
Output:
-1/3*(37*b^6*d^6 - 141*a*b^5*d^5*e + 195*a^2*b^4*d^4*e^2 - 110*a^3*b^3*d^3 *e^3 + 15*a^4*b^2*d^2*e^4 + 3*a^5*b*d*e^5 + a^6*e^6 + 45*(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 + 9 *(9*b^6*d^5*e - 35*a*b^5*d^4*e^2 + 50*a^2*b^4*d^3*e^3 - 30*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 + a^5*b*e^6)*x)/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7) + 1/3*(b^6*e^2*x^3 - 3*(2*b^6*d*e - 3*a*b^5*e^2)*x^2 + 3*(10*b^6 *d^2 - 24*a*b^5*d*e + 15*a^2*b^4*e^2)*x)/e^6 - 20*(b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*log(e*x + d)/e^7
Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (152) = 304\).
Time = 0.19 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.28 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx=-\frac {20 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} - \frac {37 \, b^{6} d^{6} - 141 \, a b^{5} d^{5} e + 195 \, a^{2} b^{4} d^{4} e^{2} - 110 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} + a^{6} e^{6} + 45 \, {\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 9 \, {\left (9 \, b^{6} d^{5} e - 35 \, a b^{5} d^{4} e^{2} + 50 \, a^{2} b^{4} d^{3} e^{3} - 30 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x}{3 \, {\left (e x + d\right )}^{3} e^{7}} + \frac {b^{6} e^{8} x^{3} - 6 \, b^{6} d e^{7} x^{2} + 9 \, a b^{5} e^{8} x^{2} + 30 \, b^{6} d^{2} e^{6} x - 72 \, a b^{5} d e^{7} x + 45 \, a^{2} b^{4} e^{8} x}{3 \, e^{12}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^4,x, algorithm="giac")
Output:
-20*(b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*log(abs(e*x + d))/e^7 - 1/3*(37*b^6*d^6 - 141*a*b^5*d^5*e + 195*a^2*b^4*d^4*e^2 - 110* a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 + 3*a^5*b*d*e^5 + a^6*e^6 + 45*(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 + 9*(9*b^6*d^5*e - 35*a*b^5*d^4*e^2 + 50*a^2*b^4*d^3*e^3 - 30*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)^3*e^7) + 1/3*(b^6 *e^8*x^3 - 6*b^6*d*e^7*x^2 + 9*a*b^5*e^8*x^2 + 30*b^6*d^2*e^6*x - 72*a*b^5 *d*e^7*x + 45*a^2*b^4*e^8*x)/e^12
Time = 0.12 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.52 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx=x^2\,\left (\frac {3\,a\,b^5}{e^4}-\frac {2\,b^6\,d}{e^5}\right )-\frac {x^2\,\left (15\,a^4\,b^2\,e^5-60\,a^3\,b^3\,d\,e^4+90\,a^2\,b^4\,d^2\,e^3-60\,a\,b^5\,d^3\,e^2+15\,b^6\,d^4\,e\right )+\frac {a^6\,e^6+3\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-110\,a^3\,b^3\,d^3\,e^3+195\,a^2\,b^4\,d^4\,e^2-141\,a\,b^5\,d^5\,e+37\,b^6\,d^6}{3\,e}+x\,\left (3\,a^5\,b\,e^5+15\,a^4\,b^2\,d\,e^4-90\,a^3\,b^3\,d^2\,e^3+150\,a^2\,b^4\,d^3\,e^2-105\,a\,b^5\,d^4\,e+27\,b^6\,d^5\right )}{d^3\,e^6+3\,d^2\,e^7\,x+3\,d\,e^8\,x^2+e^9\,x^3}-x\,\left (\frac {4\,d\,\left (\frac {6\,a\,b^5}{e^4}-\frac {4\,b^6\,d}{e^5}\right )}{e}-\frac {15\,a^2\,b^4}{e^4}+\frac {6\,b^6\,d^2}{e^6}\right )-\frac {\ln \left (d+e\,x\right )\,\left (-20\,a^3\,b^3\,e^3+60\,a^2\,b^4\,d\,e^2-60\,a\,b^5\,d^2\,e+20\,b^6\,d^3\right )}{e^7}+\frac {b^6\,x^3}{3\,e^4} \] Input:
int((a^2 + b^2*x^2 + 2*a*b*x)^3/(d + e*x)^4,x)
Output:
x^2*((3*a*b^5)/e^4 - (2*b^6*d)/e^5) - (x^2*(15*b^6*d^4*e + 15*a^4*b^2*e^5 - 60*a*b^5*d^3*e^2 - 60*a^3*b^3*d*e^4 + 90*a^2*b^4*d^2*e^3) + (a^6*e^6 + 3 7*b^6*d^6 + 195*a^2*b^4*d^4*e^2 - 110*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 141*a*b^5*d^5*e + 3*a^5*b*d*e^5)/(3*e) + x*(27*b^6*d^5 + 3*a^5*b*e^5 + 15*a^4*b^2*d*e^4 + 150*a^2*b^4*d^3*e^2 - 90*a^3*b^3*d^2*e^3 - 105*a*b^5*d^ 4*e))/(d^3*e^6 + e^9*x^3 + 3*d^2*e^7*x + 3*d*e^8*x^2) - x*((4*d*((6*a*b^5) /e^4 - (4*b^6*d)/e^5))/e - (15*a^2*b^4)/e^4 + (6*b^6*d^2)/e^6) - (log(d + e*x)*(20*b^6*d^3 - 20*a^3*b^3*e^3 + 60*a^2*b^4*d*e^2 - 60*a*b^5*d^2*e))/e^ 7 + (b^6*x^3)/(3*e^4)
Time = 0.23 (sec) , antiderivative size = 645, normalized size of antiderivative = 4.13 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx=\frac {-a^{6} d \,e^{6}+60 \,\mathrm {log}\left (e x +d \right ) a^{3} b^{3} d^{4} e^{3}-180 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{4} d^{5} e^{2}+180 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{6} e -180 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{6} e x -9 a^{5} b d \,e^{6} x +90 a^{3} b^{3} d^{3} e^{4} x -60 a^{3} b^{3} d \,e^{6} x^{3}-270 a^{2} b^{4} d^{4} e^{3} x +180 a^{2} b^{4} d^{2} e^{5} x^{3}+45 a^{2} b^{4} d \,e^{6} x^{4}+270 a \,b^{5} d^{5} e^{2} x -180 a \,b^{5} d^{3} e^{4} x^{3}-45 a \,b^{5} d^{2} e^{5} x^{4}+9 a \,b^{5} d \,e^{6} x^{5}-90 b^{6} d^{6} e x +60 b^{6} d^{4} e^{3} x^{3}+15 b^{6} d^{3} e^{4} x^{4}-3 b^{6} d^{2} e^{5} x^{5}+b^{6} d \,e^{6} x^{6}-60 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{4} e^{3} x^{3}-60 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{7}+50 a^{3} b^{3} d^{4} e^{3}-150 a^{2} b^{4} d^{5} e^{2}+150 a \,b^{5} d^{6} e +60 \,\mathrm {log}\left (e x +d \right ) a^{3} b^{3} d \,e^{6} x^{3}-180 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{4} d^{2} e^{5} x^{3}+180 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{3} e^{4} x^{3}-3 a^{5} b \,d^{2} e^{5}+15 a^{4} b^{2} e^{7} x^{3}-50 b^{6} d^{7}-180 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{5} e^{2} x^{2}+180 \,\mathrm {log}\left (e x +d \right ) a^{3} b^{3} d^{2} e^{5} x^{2}-540 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{4} d^{3} e^{4} x^{2}+540 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{4} e^{3} x^{2}+180 \,\mathrm {log}\left (e x +d \right ) a^{3} b^{3} d^{3} e^{4} x -540 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{4} d^{4} e^{3} x +540 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{5} e^{2} x}{3 d \,e^{7} \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )} \] Input:
int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^4,x)
Output:
(60*log(d + e*x)*a**3*b**3*d**4*e**3 + 180*log(d + e*x)*a**3*b**3*d**3*e** 4*x + 180*log(d + e*x)*a**3*b**3*d**2*e**5*x**2 + 60*log(d + e*x)*a**3*b** 3*d*e**6*x**3 - 180*log(d + e*x)*a**2*b**4*d**5*e**2 - 540*log(d + e*x)*a* *2*b**4*d**4*e**3*x - 540*log(d + e*x)*a**2*b**4*d**3*e**4*x**2 - 180*log( d + e*x)*a**2*b**4*d**2*e**5*x**3 + 180*log(d + e*x)*a*b**5*d**6*e + 540*l og(d + e*x)*a*b**5*d**5*e**2*x + 540*log(d + e*x)*a*b**5*d**4*e**3*x**2 + 180*log(d + e*x)*a*b**5*d**3*e**4*x**3 - 60*log(d + e*x)*b**6*d**7 - 180*l og(d + e*x)*b**6*d**6*e*x - 180*log(d + e*x)*b**6*d**5*e**2*x**2 - 60*log( d + e*x)*b**6*d**4*e**3*x**3 - a**6*d*e**6 - 3*a**5*b*d**2*e**5 - 9*a**5*b *d*e**6*x + 15*a**4*b**2*e**7*x**3 + 50*a**3*b**3*d**4*e**3 + 90*a**3*b**3 *d**3*e**4*x - 60*a**3*b**3*d*e**6*x**3 - 150*a**2*b**4*d**5*e**2 - 270*a* *2*b**4*d**4*e**3*x + 180*a**2*b**4*d**2*e**5*x**3 + 45*a**2*b**4*d*e**6*x **4 + 150*a*b**5*d**6*e + 270*a*b**5*d**5*e**2*x - 180*a*b**5*d**3*e**4*x* *3 - 45*a*b**5*d**2*e**5*x**4 + 9*a*b**5*d*e**6*x**5 - 50*b**6*d**7 - 90*b **6*d**6*e*x + 60*b**6*d**4*e**3*x**3 + 15*b**6*d**3*e**4*x**4 - 3*b**6*d* *2*e**5*x**5 + b**6*d*e**6*x**6)/(3*d*e**7*(d**3 + 3*d**2*e*x + 3*d*e**2*x **2 + e**3*x**3))