Integrand size = 26, antiderivative size = 212 \[ \int \frac {(d+e x)^8}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^6 \left (28 b^2 d^2-48 a b d e+21 a^2 e^2\right ) x}{b^8}+\frac {e^7 (4 b d-3 a e) x^2}{b^7}+\frac {e^8 x^3}{3 b^6}-\frac {(b d-a e)^8}{5 b^9 (a+b x)^5}-\frac {2 e (b d-a e)^7}{b^9 (a+b x)^4}-\frac {28 e^2 (b d-a e)^6}{3 b^9 (a+b x)^3}-\frac {28 e^3 (b d-a e)^5}{b^9 (a+b x)^2}-\frac {70 e^4 (b d-a e)^4}{b^9 (a+b x)}+\frac {56 e^5 (b d-a e)^3 \log (a+b x)}{b^9} \] Output:
e^6*(21*a^2*e^2-48*a*b*d*e+28*b^2*d^2)*x/b^8+e^7*(-3*a*e+4*b*d)*x^2/b^7+1/ 3*e^8*x^3/b^6-1/5*(-a*e+b*d)^8/b^9/(b*x+a)^5-2*e*(-a*e+b*d)^7/b^9/(b*x+a)^ 4-28/3*e^2*(-a*e+b*d)^6/b^9/(b*x+a)^3-28*e^3*(-a*e+b*d)^5/b^9/(b*x+a)^2-70 *e^4*(-a*e+b*d)^4/b^9/(b*x+a)+56*e^5*(-a*e+b*d)^3*ln(b*x+a)/b^9
Time = 0.18 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^8}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {15 b e^6 \left (28 b^2 d^2-48 a b d e+21 a^2 e^2\right ) x+15 b^2 e^7 (4 b d-3 a e) x^2+5 b^3 e^8 x^3-\frac {3 (b d-a e)^8}{(a+b x)^5}+\frac {30 e (-b d+a e)^7}{(a+b x)^4}-\frac {140 e^2 (b d-a e)^6}{(a+b x)^3}+\frac {420 e^3 (-b d+a e)^5}{(a+b x)^2}-\frac {1050 e^4 (b d-a e)^4}{a+b x}+840 e^5 (b d-a e)^3 \log (a+b x)}{15 b^9} \] Input:
Integrate[(d + e*x)^8/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
Output:
(15*b*e^6*(28*b^2*d^2 - 48*a*b*d*e + 21*a^2*e^2)*x + 15*b^2*e^7*(4*b*d - 3 *a*e)*x^2 + 5*b^3*e^8*x^3 - (3*(b*d - a*e)^8)/(a + b*x)^5 + (30*e*(-(b*d) + a*e)^7)/(a + b*x)^4 - (140*e^2*(b*d - a*e)^6)/(a + b*x)^3 + (420*e^3*(-( b*d) + a*e)^5)/(a + b*x)^2 - (1050*e^4*(b*d - a*e)^4)/(a + b*x) + 840*e^5* (b*d - a*e)^3*Log[a + b*x])/(15*b^9)
Time = 0.91 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.98, 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.
| \(\displaystyle \int \frac {(d+e x)^8}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle b^6 \int \frac {(d+e x)^8}{b^6 (a+b x)^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(d+e x)^8}{(a+b x)^6}dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {8 e^7 (a+b x) (b d-a e)}{b^8}+\frac {28 e^6 (b d-a e)^2}{b^8}+\frac {56 e^5 (b d-a e)^3}{b^8 (a+b x)}+\frac {70 e^4 (b d-a e)^4}{b^8 (a+b x)^2}+\frac {56 e^3 (b d-a e)^5}{b^8 (a+b x)^3}+\frac {28 e^2 (b d-a e)^6}{b^8 (a+b x)^4}+\frac {8 e (b d-a e)^7}{b^8 (a+b x)^5}+\frac {(b d-a e)^8}{b^8 (a+b x)^6}+\frac {e^8 (a+b x)^2}{b^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 e^7 (a+b x)^2 (b d-a e)}{b^9}+\frac {56 e^5 (b d-a e)^3 \log (a+b x)}{b^9}-\frac {70 e^4 (b d-a e)^4}{b^9 (a+b x)}-\frac {28 e^3 (b d-a e)^5}{b^9 (a+b x)^2}-\frac {28 e^2 (b d-a e)^6}{3 b^9 (a+b x)^3}-\frac {2 e (b d-a e)^7}{b^9 (a+b x)^4}-\frac {(b d-a e)^8}{5 b^9 (a+b x)^5}+\frac {e^8 (a+b x)^3}{3 b^9}+\frac {28 e^6 x (b d-a e)^2}{b^8}\) |
Input:
Int[(d + e*x)^8/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
Output:
(28*e^6*(b*d - a*e)^2*x)/b^8 - (b*d - a*e)^8/(5*b^9*(a + b*x)^5) - (2*e*(b *d - a*e)^7)/(b^9*(a + b*x)^4) - (28*e^2*(b*d - a*e)^6)/(3*b^9*(a + b*x)^3 ) - (28*e^3*(b*d - a*e)^5)/(b^9*(a + b*x)^2) - (70*e^4*(b*d - a*e)^4)/(b^9 *(a + b*x)) + (4*e^7*(b*d - a*e)*(a + b*x)^2)/b^9 + (e^8*(a + b*x)^3)/(3*b ^9) + (56*e^5*(b*d - a*e)^3*Log[a + b*x])/b^9
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. \(571\) vs. \(2(206)=412\).
Time = 1.10 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.70
| method | result | size |
| norman | \(\frac {-\frac {1918 a^{8} e^{8}-5754 a^{7} b d \,e^{7}+5754 a^{6} b^{2} d^{2} e^{6}-1918 b^{3} d^{3} e^{5} a^{5}+210 b^{4} d^{4} e^{4} a^{4}+42 b^{5} d^{5} e^{3} a^{3}+14 b^{6} d^{6} e^{2} a^{2}+6 b^{7} d^{7} e a +3 b^{8} d^{8}}{15 b^{9}}+\frac {e^{8} x^{8}}{3 b}-\frac {5 \left (56 a^{4} e^{8}-168 a^{3} b d \,e^{7}+168 a^{2} b^{2} d^{2} e^{6}-56 a \,b^{3} d^{3} e^{5}+14 d^{4} e^{4} b^{4}\right ) x^{4}}{b^{5}}-\frac {2 \left (420 e^{8} a^{5}-1260 a^{4} b d \,e^{7}+1260 a^{3} b^{2} d^{2} e^{6}-420 a^{2} b^{3} d^{3} e^{5}+70 a \,b^{4} d^{4} e^{4}+14 d^{5} e^{3} b^{5}\right ) x^{3}}{b^{6}}-\frac {2 \left (1540 a^{6} e^{8}-4620 a^{5} b d \,e^{7}+4620 a^{4} b^{2} d^{2} e^{6}-1540 a^{3} b^{3} d^{3} e^{5}+210 a^{2} b^{4} d^{4} e^{4}+42 a \,b^{5} d^{5} e^{3}+14 b^{6} e^{2} d^{6}\right ) x^{2}}{3 b^{7}}-\frac {\left (1750 e^{8} a^{7}-5250 a^{6} b d \,e^{7}+5250 a^{5} b^{2} d^{2} e^{6}-1750 a^{4} b^{3} d^{3} e^{5}+210 a^{3} b^{4} d^{4} e^{4}+42 a^{2} b^{5} d^{5} e^{3}+14 a \,b^{6} d^{6} e^{2}+6 d^{7} e \,b^{7}\right ) x}{3 b^{8}}+\frac {28 e^{6} \left (e^{2} a^{2}-3 a b d e +3 b^{2} d^{2}\right ) x^{6}}{3 b^{3}}-\frac {4 e^{7} \left (a e -3 b d \right ) x^{7}}{3 b^{2}}}{\left (b x +a \right )^{5}}-\frac {56 e^{5} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \ln \left (b x +a \right )}{b^{9}}\) | \(572\) |
| default | \(\frac {e^{6} \left (\frac {1}{3} b^{2} e^{2} x^{3}-3 x^{2} a b \,e^{2}+4 b^{2} d e \,x^{2}+21 a^{2} e^{2} x -48 a b d e x +28 b^{2} d^{2} x \right )}{b^{8}}-\frac {a^{8} e^{8}-8 a^{7} b d \,e^{7}+28 a^{6} b^{2} d^{2} e^{6}-56 b^{3} d^{3} e^{5} a^{5}+70 b^{4} d^{4} e^{4} a^{4}-56 b^{5} d^{5} e^{3} a^{3}+28 b^{6} d^{6} e^{2} a^{2}-8 b^{7} d^{7} e a +b^{8} d^{8}}{5 b^{9} \left (b x +a \right )^{5}}-\frac {28 e^{2} \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}{3 b^{9} \left (b x +a \right )^{3}}-\frac {70 e^{4} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{b^{9} \left (b x +a \right )}-\frac {56 e^{5} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \ln \left (b x +a \right )}{b^{9}}+\frac {28 e^{3} \left (e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{b^{9} \left (b x +a \right )^{2}}+\frac {2 e \left (a^{7} e^{7}-7 a^{6} b d \,e^{6}+21 a^{5} b^{2} d^{2} e^{5}-35 b^{3} d^{3} e^{4} a^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a -b^{7} d^{7}\right )}{b^{9} \left (b x +a \right )^{4}}\) | \(573\) |
| risch | \(\frac {e^{8} x^{3}}{3 b^{6}}-\frac {3 e^{8} x^{2} a}{b^{7}}+\frac {4 e^{7} d \,x^{2}}{b^{6}}+\frac {21 e^{8} a^{2} x}{b^{8}}-\frac {48 e^{7} a d x}{b^{7}}+\frac {28 e^{6} d^{2} x}{b^{6}}+\frac {\left (-70 a^{4} b^{3} e^{8}+280 a^{3} d \,e^{7} b^{4}-420 a^{2} d^{2} e^{6} b^{5}+280 a \,b^{6} d^{3} e^{5}-70 d^{4} e^{4} b^{7}\right ) x^{4}-28 b^{2} e^{3} \left (9 e^{5} a^{5}-35 a^{4} b d \,e^{4}+50 a^{3} b^{2} d^{2} e^{3}-30 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x^{3}-\frac {28 b \,e^{2} \left (37 a^{6} e^{6}-141 a^{5} b d \,e^{5}+195 a^{4} b^{2} d^{2} e^{4}-110 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}+3 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) x^{2}}{3}-\frac {2 e \left (319 a^{7} e^{7}-1197 a^{6} b d \,e^{6}+1617 a^{5} b^{2} d^{2} e^{5}-875 b^{3} d^{3} e^{4} a^{4}+105 b^{4} d^{4} e^{3} a^{3}+21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a +3 b^{7} d^{7}\right ) x}{3}-\frac {743 a^{8} e^{8}-2754 a^{7} b d \,e^{7}+3654 a^{6} b^{2} d^{2} e^{6}-1918 b^{3} d^{3} e^{5} a^{5}+210 b^{4} d^{4} e^{4} a^{4}+42 b^{5} d^{5} e^{3} a^{3}+14 b^{6} d^{6} e^{2} a^{2}+6 b^{7} d^{7} e a +3 b^{8} d^{8}}{15 b}}{b^{8} \left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}-\frac {56 e^{8} \ln \left (b x +a \right ) a^{3}}{b^{9}}+\frac {168 e^{7} \ln \left (b x +a \right ) a^{2} d}{b^{8}}-\frac {168 e^{6} \ln \left (b x +a \right ) a \,d^{2}}{b^{7}}+\frac {56 e^{5} \ln \left (b x +a \right ) d^{3}}{b^{6}}\) | \(608\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1084\) |
Input:
int((e*x+d)^8/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
Output:
(-1/15*(1918*a^8*e^8-5754*a^7*b*d*e^7+5754*a^6*b^2*d^2*e^6-1918*a^5*b^3*d^ 3*e^5+210*a^4*b^4*d^4*e^4+42*a^3*b^5*d^5*e^3+14*a^2*b^6*d^6*e^2+6*a*b^7*d^ 7*e+3*b^8*d^8)/b^9+1/3*e^8/b*x^8-5*(56*a^4*e^8-168*a^3*b*d*e^7+168*a^2*b^2 *d^2*e^6-56*a*b^3*d^3*e^5+14*b^4*d^4*e^4)/b^5*x^4-2*(420*a^5*e^8-1260*a^4* b*d*e^7+1260*a^3*b^2*d^2*e^6-420*a^2*b^3*d^3*e^5+70*a*b^4*d^4*e^4+14*b^5*d ^5*e^3)/b^6*x^3-2/3*(1540*a^6*e^8-4620*a^5*b*d*e^7+4620*a^4*b^2*d^2*e^6-15 40*a^3*b^3*d^3*e^5+210*a^2*b^4*d^4*e^4+42*a*b^5*d^5*e^3+14*b^6*d^6*e^2)/b^ 7*x^2-1/3*(1750*a^7*e^8-5250*a^6*b*d*e^7+5250*a^5*b^2*d^2*e^6-1750*a^4*b^3 *d^3*e^5+210*a^3*b^4*d^4*e^4+42*a^2*b^5*d^5*e^3+14*a*b^6*d^6*e^2+6*b^7*d^7 *e)/b^8*x+28/3*e^6*(a^2*e^2-3*a*b*d*e+3*b^2*d^2)/b^3*x^6-4/3*e^7*(a*e-3*b* d)/b^2*x^7)/(b*x+a)^5-56/b^9*e^5*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3* d^3)*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 943 vs. \(2 (206) = 412\).
Time = 0.09 (sec) , antiderivative size = 943, normalized size of antiderivative = 4.45 \[ \int \frac {(d+e x)^8}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^8/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
Output:
1/15*(5*b^8*e^8*x^8 - 3*b^8*d^8 - 6*a*b^7*d^7*e - 14*a^2*b^6*d^6*e^2 - 42* a^3*b^5*d^5*e^3 - 210*a^4*b^4*d^4*e^4 + 1918*a^5*b^3*d^3*e^5 - 3654*a^6*b^ 2*d^2*e^6 + 2754*a^7*b*d*e^7 - 743*a^8*e^8 + 20*(3*b^8*d*e^7 - a*b^7*e^8)* x^7 + 140*(3*b^8*d^2*e^6 - 3*a*b^7*d*e^7 + a^2*b^6*e^8)*x^6 + 25*(84*a*b^7 *d^2*e^6 - 120*a^2*b^6*d*e^7 + 47*a^3*b^5*e^8)*x^5 - 25*(42*b^8*d^4*e^4 - 168*a*b^7*d^3*e^5 + 84*a^2*b^6*d^2*e^6 + 96*a^3*b^5*d*e^7 - 67*a^4*b^4*e^8 )*x^4 - 10*(42*b^8*d^5*e^3 + 210*a*b^7*d^4*e^4 - 1260*a^2*b^6*d^3*e^5 + 16 80*a^3*b^5*d^2*e^6 - 780*a^4*b^4*d*e^7 + 85*a^5*b^3*e^8)*x^3 - 10*(14*b^8* d^6*e^2 + 42*a*b^7*d^5*e^3 + 210*a^2*b^6*d^4*e^4 - 1540*a^3*b^5*d^3*e^5 + 2520*a^4*b^4*d^2*e^6 - 1620*a^5*b^3*d*e^7 + 365*a^6*b^2*e^8)*x^2 - 5*(6*b^ 8*d^7*e + 14*a*b^7*d^6*e^2 + 42*a^2*b^6*d^5*e^3 + 210*a^3*b^5*d^4*e^4 - 17 50*a^4*b^4*d^3*e^5 + 3150*a^5*b^3*d^2*e^6 - 2250*a^6*b^2*d*e^7 + 575*a^7*b *e^8)*x + 840*(a^5*b^3*d^3*e^5 - 3*a^6*b^2*d^2*e^6 + 3*a^7*b*d*e^7 - a^8*e ^8 + (b^8*d^3*e^5 - 3*a*b^7*d^2*e^6 + 3*a^2*b^6*d*e^7 - a^3*b^5*e^8)*x^5 + 5*(a*b^7*d^3*e^5 - 3*a^2*b^6*d^2*e^6 + 3*a^3*b^5*d*e^7 - a^4*b^4*e^8)*x^4 + 10*(a^2*b^6*d^3*e^5 - 3*a^3*b^5*d^2*e^6 + 3*a^4*b^4*d*e^7 - a^5*b^3*e^8 )*x^3 + 10*(a^3*b^5*d^3*e^5 - 3*a^4*b^4*d^2*e^6 + 3*a^5*b^3*d*e^7 - a^6*b^ 2*e^8)*x^2 + 5*(a^4*b^4*d^3*e^5 - 3*a^5*b^3*d^2*e^6 + 3*a^6*b^2*d*e^7 - a^ 7*b*e^8)*x)*log(b*x + a))/(b^14*x^5 + 5*a*b^13*x^4 + 10*a^2*b^12*x^3 + 10* a^3*b^11*x^2 + 5*a^4*b^10*x + a^5*b^9)
Timed out. \[ \int \frac {(d+e x)^8}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**8/(b**2*x**2+2*a*b*x+a**2)**3,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 626 vs. \(2 (206) = 412\).
Time = 0.08 (sec) , antiderivative size = 626, normalized size of antiderivative = 2.95 \[ \int \frac {(d+e x)^8}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {3 \, b^{8} d^{8} + 6 \, a b^{7} d^{7} e + 14 \, a^{2} b^{6} d^{6} e^{2} + 42 \, a^{3} b^{5} d^{5} e^{3} + 210 \, a^{4} b^{4} d^{4} e^{4} - 1918 \, a^{5} b^{3} d^{3} e^{5} + 3654 \, a^{6} b^{2} d^{2} e^{6} - 2754 \, a^{7} b d e^{7} + 743 \, a^{8} e^{8} + 1050 \, {\left (b^{8} d^{4} e^{4} - 4 \, a b^{7} d^{3} e^{5} + 6 \, a^{2} b^{6} d^{2} e^{6} - 4 \, a^{3} b^{5} d e^{7} + a^{4} b^{4} e^{8}\right )} x^{4} + 420 \, {\left (b^{8} d^{5} e^{3} + 5 \, a b^{7} d^{4} e^{4} - 30 \, a^{2} b^{6} d^{3} e^{5} + 50 \, a^{3} b^{5} d^{2} e^{6} - 35 \, a^{4} b^{4} d e^{7} + 9 \, a^{5} b^{3} e^{8}\right )} x^{3} + 140 \, {\left (b^{8} d^{6} e^{2} + 3 \, a b^{7} d^{5} e^{3} + 15 \, a^{2} b^{6} d^{4} e^{4} - 110 \, a^{3} b^{5} d^{3} e^{5} + 195 \, a^{4} b^{4} d^{2} e^{6} - 141 \, a^{5} b^{3} d e^{7} + 37 \, a^{6} b^{2} e^{8}\right )} x^{2} + 10 \, {\left (3 \, b^{8} d^{7} e + 7 \, a b^{7} d^{6} e^{2} + 21 \, a^{2} b^{6} d^{5} e^{3} + 105 \, a^{3} b^{5} d^{4} e^{4} - 875 \, a^{4} b^{4} d^{3} e^{5} + 1617 \, a^{5} b^{3} d^{2} e^{6} - 1197 \, a^{6} b^{2} d e^{7} + 319 \, a^{7} b e^{8}\right )} x}{15 \, {\left (b^{14} x^{5} + 5 \, a b^{13} x^{4} + 10 \, a^{2} b^{12} x^{3} + 10 \, a^{3} b^{11} x^{2} + 5 \, a^{4} b^{10} x + a^{5} b^{9}\right )}} + \frac {b^{2} e^{8} x^{3} + 3 \, {\left (4 \, b^{2} d e^{7} - 3 \, a b e^{8}\right )} x^{2} + 3 \, {\left (28 \, b^{2} d^{2} e^{6} - 48 \, a b d e^{7} + 21 \, a^{2} e^{8}\right )} x}{3 \, b^{8}} + \frac {56 \, {\left (b^{3} d^{3} e^{5} - 3 \, a b^{2} d^{2} e^{6} + 3 \, a^{2} b d e^{7} - a^{3} e^{8}\right )} \log \left (b x + a\right )}{b^{9}} \] Input:
integrate((e*x+d)^8/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
Output:
-1/15*(3*b^8*d^8 + 6*a*b^7*d^7*e + 14*a^2*b^6*d^6*e^2 + 42*a^3*b^5*d^5*e^3 + 210*a^4*b^4*d^4*e^4 - 1918*a^5*b^3*d^3*e^5 + 3654*a^6*b^2*d^2*e^6 - 275 4*a^7*b*d*e^7 + 743*a^8*e^8 + 1050*(b^8*d^4*e^4 - 4*a*b^7*d^3*e^5 + 6*a^2* b^6*d^2*e^6 - 4*a^3*b^5*d*e^7 + a^4*b^4*e^8)*x^4 + 420*(b^8*d^5*e^3 + 5*a* b^7*d^4*e^4 - 30*a^2*b^6*d^3*e^5 + 50*a^3*b^5*d^2*e^6 - 35*a^4*b^4*d*e^7 + 9*a^5*b^3*e^8)*x^3 + 140*(b^8*d^6*e^2 + 3*a*b^7*d^5*e^3 + 15*a^2*b^6*d^4* e^4 - 110*a^3*b^5*d^3*e^5 + 195*a^4*b^4*d^2*e^6 - 141*a^5*b^3*d*e^7 + 37*a ^6*b^2*e^8)*x^2 + 10*(3*b^8*d^7*e + 7*a*b^7*d^6*e^2 + 21*a^2*b^6*d^5*e^3 + 105*a^3*b^5*d^4*e^4 - 875*a^4*b^4*d^3*e^5 + 1617*a^5*b^3*d^2*e^6 - 1197*a ^6*b^2*d*e^7 + 319*a^7*b*e^8)*x)/(b^14*x^5 + 5*a*b^13*x^4 + 10*a^2*b^12*x^ 3 + 10*a^3*b^11*x^2 + 5*a^4*b^10*x + a^5*b^9) + 1/3*(b^2*e^8*x^3 + 3*(4*b^ 2*d*e^7 - 3*a*b*e^8)*x^2 + 3*(28*b^2*d^2*e^6 - 48*a*b*d*e^7 + 21*a^2*e^8)* x)/b^8 + 56*(b^3*d^3*e^5 - 3*a*b^2*d^2*e^6 + 3*a^2*b*d*e^7 - a^3*e^8)*log( b*x + a)/b^9
Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (206) = 412\).
Time = 0.16 (sec) , antiderivative size = 586, normalized size of antiderivative = 2.76 \[ \int \frac {(d+e x)^8}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {56 \, {\left (b^{3} d^{3} e^{5} - 3 \, a b^{2} d^{2} e^{6} + 3 \, a^{2} b d e^{7} - a^{3} e^{8}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{9}} - \frac {3 \, b^{8} d^{8} + 6 \, a b^{7} d^{7} e + 14 \, a^{2} b^{6} d^{6} e^{2} + 42 \, a^{3} b^{5} d^{5} e^{3} + 210 \, a^{4} b^{4} d^{4} e^{4} - 1918 \, a^{5} b^{3} d^{3} e^{5} + 3654 \, a^{6} b^{2} d^{2} e^{6} - 2754 \, a^{7} b d e^{7} + 743 \, a^{8} e^{8} + 1050 \, {\left (b^{8} d^{4} e^{4} - 4 \, a b^{7} d^{3} e^{5} + 6 \, a^{2} b^{6} d^{2} e^{6} - 4 \, a^{3} b^{5} d e^{7} + a^{4} b^{4} e^{8}\right )} x^{4} + 420 \, {\left (b^{8} d^{5} e^{3} + 5 \, a b^{7} d^{4} e^{4} - 30 \, a^{2} b^{6} d^{3} e^{5} + 50 \, a^{3} b^{5} d^{2} e^{6} - 35 \, a^{4} b^{4} d e^{7} + 9 \, a^{5} b^{3} e^{8}\right )} x^{3} + 140 \, {\left (b^{8} d^{6} e^{2} + 3 \, a b^{7} d^{5} e^{3} + 15 \, a^{2} b^{6} d^{4} e^{4} - 110 \, a^{3} b^{5} d^{3} e^{5} + 195 \, a^{4} b^{4} d^{2} e^{6} - 141 \, a^{5} b^{3} d e^{7} + 37 \, a^{6} b^{2} e^{8}\right )} x^{2} + 10 \, {\left (3 \, b^{8} d^{7} e + 7 \, a b^{7} d^{6} e^{2} + 21 \, a^{2} b^{6} d^{5} e^{3} + 105 \, a^{3} b^{5} d^{4} e^{4} - 875 \, a^{4} b^{4} d^{3} e^{5} + 1617 \, a^{5} b^{3} d^{2} e^{6} - 1197 \, a^{6} b^{2} d e^{7} + 319 \, a^{7} b e^{8}\right )} x}{15 \, {\left (b x + a\right )}^{5} b^{9}} + \frac {b^{12} e^{8} x^{3} + 12 \, b^{12} d e^{7} x^{2} - 9 \, a b^{11} e^{8} x^{2} + 84 \, b^{12} d^{2} e^{6} x - 144 \, a b^{11} d e^{7} x + 63 \, a^{2} b^{10} e^{8} x}{3 \, b^{18}} \] Input:
integrate((e*x+d)^8/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
Output:
56*(b^3*d^3*e^5 - 3*a*b^2*d^2*e^6 + 3*a^2*b*d*e^7 - a^3*e^8)*log(abs(b*x + a))/b^9 - 1/15*(3*b^8*d^8 + 6*a*b^7*d^7*e + 14*a^2*b^6*d^6*e^2 + 42*a^3*b ^5*d^5*e^3 + 210*a^4*b^4*d^4*e^4 - 1918*a^5*b^3*d^3*e^5 + 3654*a^6*b^2*d^2 *e^6 - 2754*a^7*b*d*e^7 + 743*a^8*e^8 + 1050*(b^8*d^4*e^4 - 4*a*b^7*d^3*e^ 5 + 6*a^2*b^6*d^2*e^6 - 4*a^3*b^5*d*e^7 + a^4*b^4*e^8)*x^4 + 420*(b^8*d^5* e^3 + 5*a*b^7*d^4*e^4 - 30*a^2*b^6*d^3*e^5 + 50*a^3*b^5*d^2*e^6 - 35*a^4*b ^4*d*e^7 + 9*a^5*b^3*e^8)*x^3 + 140*(b^8*d^6*e^2 + 3*a*b^7*d^5*e^3 + 15*a^ 2*b^6*d^4*e^4 - 110*a^3*b^5*d^3*e^5 + 195*a^4*b^4*d^2*e^6 - 141*a^5*b^3*d* e^7 + 37*a^6*b^2*e^8)*x^2 + 10*(3*b^8*d^7*e + 7*a*b^7*d^6*e^2 + 21*a^2*b^6 *d^5*e^3 + 105*a^3*b^5*d^4*e^4 - 875*a^4*b^4*d^3*e^5 + 1617*a^5*b^3*d^2*e^ 6 - 1197*a^6*b^2*d*e^7 + 319*a^7*b*e^8)*x)/((b*x + a)^5*b^9) + 1/3*(b^12*e ^8*x^3 + 12*b^12*d*e^7*x^2 - 9*a*b^11*e^8*x^2 + 84*b^12*d^2*e^6*x - 144*a* b^11*d*e^7*x + 63*a^2*b^10*e^8*x)/b^18
Time = 0.20 (sec) , antiderivative size = 644, normalized size of antiderivative = 3.04 \[ \int \frac {(d+e x)^8}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=x\,\left (\frac {6\,a\,\left (\frac {6\,a\,e^8}{b^7}-\frac {8\,d\,e^7}{b^6}\right )}{b}-\frac {15\,a^2\,e^8}{b^8}+\frac {28\,d^2\,e^6}{b^6}\right )-\frac {x^4\,\left (70\,a^4\,b^3\,e^8-280\,a^3\,b^4\,d\,e^7+420\,a^2\,b^5\,d^2\,e^6-280\,a\,b^6\,d^3\,e^5+70\,b^7\,d^4\,e^4\right )+\frac {743\,a^8\,e^8-2754\,a^7\,b\,d\,e^7+3654\,a^6\,b^2\,d^2\,e^6-1918\,a^5\,b^3\,d^3\,e^5+210\,a^4\,b^4\,d^4\,e^4+42\,a^3\,b^5\,d^5\,e^3+14\,a^2\,b^6\,d^6\,e^2+6\,a\,b^7\,d^7\,e+3\,b^8\,d^8}{15\,b}+x\,\left (\frac {638\,a^7\,e^8}{3}-798\,a^6\,b\,d\,e^7+1078\,a^5\,b^2\,d^2\,e^6-\frac {1750\,a^4\,b^3\,d^3\,e^5}{3}+70\,a^3\,b^4\,d^4\,e^4+14\,a^2\,b^5\,d^5\,e^3+\frac {14\,a\,b^6\,d^6\,e^2}{3}+2\,b^7\,d^7\,e\right )+x^3\,\left (252\,a^5\,b^2\,e^8-980\,a^4\,b^3\,d\,e^7+1400\,a^3\,b^4\,d^2\,e^6-840\,a^2\,b^5\,d^3\,e^5+140\,a\,b^6\,d^4\,e^4+28\,b^7\,d^5\,e^3\right )+x^2\,\left (\frac {1036\,a^6\,b\,e^8}{3}-1316\,a^5\,b^2\,d\,e^7+1820\,a^4\,b^3\,d^2\,e^6-\frac {3080\,a^3\,b^4\,d^3\,e^5}{3}+140\,a^2\,b^5\,d^4\,e^4+28\,a\,b^6\,d^5\,e^3+\frac {28\,b^7\,d^6\,e^2}{3}\right )}{a^5\,b^8+5\,a^4\,b^9\,x+10\,a^3\,b^{10}\,x^2+10\,a^2\,b^{11}\,x^3+5\,a\,b^{12}\,x^4+b^{13}\,x^5}-x^2\,\left (\frac {3\,a\,e^8}{b^7}-\frac {4\,d\,e^7}{b^6}\right )-\frac {\ln \left (a+b\,x\right )\,\left (56\,a^3\,e^8-168\,a^2\,b\,d\,e^7+168\,a\,b^2\,d^2\,e^6-56\,b^3\,d^3\,e^5\right )}{b^9}+\frac {e^8\,x^3}{3\,b^6} \] Input:
int((d + e*x)^8/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)
Output:
x*((6*a*((6*a*e^8)/b^7 - (8*d*e^7)/b^6))/b - (15*a^2*e^8)/b^8 + (28*d^2*e^ 6)/b^6) - (x^4*(70*a^4*b^3*e^8 + 70*b^7*d^4*e^4 - 280*a*b^6*d^3*e^5 - 280* a^3*b^4*d*e^7 + 420*a^2*b^5*d^2*e^6) + (743*a^8*e^8 + 3*b^8*d^8 + 14*a^2*b ^6*d^6*e^2 + 42*a^3*b^5*d^5*e^3 + 210*a^4*b^4*d^4*e^4 - 1918*a^5*b^3*d^3*e ^5 + 3654*a^6*b^2*d^2*e^6 + 6*a*b^7*d^7*e - 2754*a^7*b*d*e^7)/(15*b) + x*( (638*a^7*e^8)/3 + 2*b^7*d^7*e + (14*a*b^6*d^6*e^2)/3 + 14*a^2*b^5*d^5*e^3 + 70*a^3*b^4*d^4*e^4 - (1750*a^4*b^3*d^3*e^5)/3 + 1078*a^5*b^2*d^2*e^6 - 7 98*a^6*b*d*e^7) + x^3*(252*a^5*b^2*e^8 + 28*b^7*d^5*e^3 + 140*a*b^6*d^4*e^ 4 - 980*a^4*b^3*d*e^7 - 840*a^2*b^5*d^3*e^5 + 1400*a^3*b^4*d^2*e^6) + x^2* ((1036*a^6*b*e^8)/3 + (28*b^7*d^6*e^2)/3 + 28*a*b^6*d^5*e^3 - 1316*a^5*b^2 *d*e^7 + 140*a^2*b^5*d^4*e^4 - (3080*a^3*b^4*d^3*e^5)/3 + 1820*a^4*b^3*d^2 *e^6))/(a^5*b^8 + b^13*x^5 + 5*a^4*b^9*x + 5*a*b^12*x^4 + 10*a^3*b^10*x^2 + 10*a^2*b^11*x^3) - x^2*((3*a*e^8)/b^7 - (4*d*e^7)/b^6) - (log(a + b*x)*( 56*a^3*e^8 - 56*b^3*d^3*e^5 + 168*a*b^2*d^2*e^6 - 168*a^2*b*d*e^7))/b^9 + (e^8*x^3)/(3*b^6)
Time = 0.25 (sec) , antiderivative size = 1073, normalized size of antiderivative = 5.06 \[ \int \frac {(d+e x)^8}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^8/(b^2*x^2+2*a*b*x+a^2)^3,x)
Output:
( - 840*log(a + b*x)*a**9*e**8 + 2520*log(a + b*x)*a**8*b*d*e**7 - 4200*lo g(a + b*x)*a**8*b*e**8*x - 2520*log(a + b*x)*a**7*b**2*d**2*e**6 + 12600*l og(a + b*x)*a**7*b**2*d*e**7*x - 8400*log(a + b*x)*a**7*b**2*e**8*x**2 + 8 40*log(a + b*x)*a**6*b**3*d**3*e**5 - 12600*log(a + b*x)*a**6*b**3*d**2*e* *6*x + 25200*log(a + b*x)*a**6*b**3*d*e**7*x**2 - 8400*log(a + b*x)*a**6*b **3*e**8*x**3 + 4200*log(a + b*x)*a**5*b**4*d**3*e**5*x - 25200*log(a + b* x)*a**5*b**4*d**2*e**6*x**2 + 25200*log(a + b*x)*a**5*b**4*d*e**7*x**3 - 4 200*log(a + b*x)*a**5*b**4*e**8*x**4 + 8400*log(a + b*x)*a**4*b**5*d**3*e* *5*x**2 - 25200*log(a + b*x)*a**4*b**5*d**2*e**6*x**3 + 12600*log(a + b*x) *a**4*b**5*d*e**7*x**4 - 840*log(a + b*x)*a**4*b**5*e**8*x**5 + 8400*log(a + b*x)*a**3*b**6*d**3*e**5*x**3 - 12600*log(a + b*x)*a**3*b**6*d**2*e**6* x**4 + 2520*log(a + b*x)*a**3*b**6*d*e**7*x**5 + 4200*log(a + b*x)*a**2*b* *7*d**3*e**5*x**4 - 2520*log(a + b*x)*a**2*b**7*d**2*e**6*x**5 + 840*log(a + b*x)*a*b**8*d**3*e**5*x**5 - 1078*a**9*e**8 + 3234*a**8*b*d*e**7 - 4550 *a**8*b*e**8*x - 3234*a**7*b**2*d**2*e**6 + 13650*a**7*b**2*d*e**7*x - 700 0*a**7*b**2*e**8*x**2 + 1078*a**6*b**3*d**3*e**5 - 13650*a**6*b**3*d**2*e* *6*x + 21000*a**6*b**3*d*e**7*x**2 - 4200*a**6*b**3*e**8*x**3 + 4550*a**5* b**4*d**3*e**5*x - 21000*a**5*b**4*d**2*e**6*x**2 + 12600*a**5*b**4*d*e**7 *x**3 - 42*a**4*b**5*d**5*e**3 + 7000*a**4*b**5*d**3*e**5*x**2 - 12600*a** 4*b**5*d**2*e**6*x**3 + 840*a**4*b**5*e**8*x**5 - 14*a**3*b**6*d**6*e**...