Integrand size = 31, antiderivative size = 130 \[ \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {1}{4 (b d-a e) (a+b x)^4}+\frac {e}{3 (b d-a e)^2 (a+b x)^3}-\frac {e^2}{2 (b d-a e)^3 (a+b x)^2}+\frac {e^3}{(b d-a e)^4 (a+b x)}+\frac {e^4 \log (a+b x)}{(b d-a e)^5}-\frac {e^4 \log (d+e x)}{(b d-a e)^5} \] Output:
-1/4/(-a*e+b*d)/(b*x+a)^4+1/3*e/(-a*e+b*d)^2/(b*x+a)^3-1/2*e^2/(-a*e+b*d)^ 3/(b*x+a)^2+e^3/(-a*e+b*d)^4/(b*x+a)+e^4*ln(b*x+a)/(-a*e+b*d)^5-e^4*ln(e*x +d)/(-a*e+b*d)^5
Time = 0.06 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {1}{4 (-b d+a e) (a+b x)^4}+\frac {e}{3 (b d-a e)^2 (a+b x)^3}-\frac {e^2}{2 (b d-a e)^3 (a+b x)^2}+\frac {e^3}{(b d-a e)^4 (a+b x)}+\frac {e^4 \log (a+b x)}{(b d-a e)^5}-\frac {e^4 \log (d+e x)}{(b d-a e)^5} \] Input:
Integrate[(a + b*x)/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
Output:
1/(4*(-(b*d) + a*e)*(a + b*x)^4) + e/(3*(b*d - a*e)^2*(a + b*x)^3) - e^2/( 2*(b*d - a*e)^3*(a + b*x)^2) + e^3/((b*d - a*e)^4*(a + b*x)) + (e^4*Log[a + b*x])/(b*d - a*e)^5 - (e^4*Log[d + e*x])/(b*d - a*e)^5
Time = 0.54 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1184, 27, 54, 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.
| \(\displaystyle \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^3 (d+e x)} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^6 \int \frac {1}{b^6 (a+b x)^5 (d+e x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{(a+b x)^5 (d+e x)}dx\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \int \left (-\frac {e^5}{(d+e x) (b d-a e)^5}+\frac {b e^4}{(a+b x) (b d-a e)^5}-\frac {b e^3}{(a+b x)^2 (b d-a e)^4}+\frac {b e^2}{(a+b x)^3 (b d-a e)^3}-\frac {b e}{(a+b x)^4 (b d-a e)^2}+\frac {b}{(a+b x)^5 (b d-a e)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \log (a+b x)}{(b d-a e)^5}-\frac {e^4 \log (d+e x)}{(b d-a e)^5}+\frac {e^3}{(a+b x) (b d-a e)^4}-\frac {e^2}{2 (a+b x)^2 (b d-a e)^3}+\frac {e}{3 (a+b x)^3 (b d-a e)^2}-\frac {1}{4 (a+b x)^4 (b d-a e)}\) |
Input:
Int[(a + b*x)/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
Output:
-1/4*1/((b*d - a*e)*(a + b*x)^4) + e/(3*(b*d - a*e)^2*(a + b*x)^3) - e^2/( 2*(b*d - a*e)^3*(a + b*x)^2) + e^3/((b*d - a*e)^4*(a + b*x)) + (e^4*Log[a + b*x])/(b*d - a*e)^5 - (e^4*Log[d + e*x])/(b*d - a*e)^5
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[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^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}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 1.45 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(\frac {1}{4 \left (a e -b d \right ) \left (b x +a \right )^{4}}-\frac {e^{4} \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}+\frac {e}{3 \left (a e -b d \right )^{2} \left (b x +a \right )^{3}}+\frac {e^{2}}{2 \left (a e -b d \right )^{3} \left (b x +a \right )^{2}}+\frac {e^{3}}{\left (a e -b d \right )^{4} \left (b x +a \right )}+\frac {e^{4} \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}\) | \(125\) |
| parallelrisch | \(-\frac {-25 a^{4} e^{4} b^{4}+48 \ln \left (b x +a \right ) x^{3} a \,b^{7} e^{4}-48 \ln \left (e x +d \right ) x^{3} a \,b^{7} e^{4}+72 \ln \left (b x +a \right ) x^{2} a^{2} b^{6} e^{4}-72 \ln \left (e x +d \right ) x^{2} a^{2} b^{6} e^{4}-3 b^{8} d^{4}-6 x^{2} b^{8} d^{2} e^{2}-52 x \,a^{3} b^{5} e^{4}+4 x \,b^{8} d^{3} e +12 \ln \left (b x +a \right ) x^{4} b^{8} e^{4}-12 \ln \left (e x +d \right ) x^{4} b^{8} e^{4}+12 \ln \left (b x +a \right ) a^{4} b^{4} e^{4}-12 \ln \left (e x +d \right ) a^{4} b^{4} e^{4}-12 x^{3} a \,b^{7} e^{4}+12 x^{3} b^{8} d \,e^{3}-42 x^{2} a^{2} b^{6} e^{4}+48 a^{3} b^{5} d \,e^{3}-36 a^{2} b^{6} d^{2} e^{2}+16 a \,b^{7} d^{3} e +48 \ln \left (b x +a \right ) x \,a^{3} b^{5} e^{4}-48 \ln \left (e x +d \right ) x \,a^{3} b^{5} e^{4}+48 x^{2} a \,b^{7} d \,e^{3}+72 x \,a^{2} b^{6} d \,e^{3}-24 x a \,b^{7} d^{2} e^{2}}{12 \left (e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right ) \left (b x +a \right )^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right ) b^{4}}\) | \(446\) |
| risch | \(\frac {\frac {b^{3} e^{3} x^{3}}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {\left (7 a e -b d \right ) b^{2} e^{2} x^{2}}{2 a^{4} e^{4}-8 a^{3} b d \,e^{3}+12 a^{2} b^{2} d^{2} e^{2}-8 a \,b^{3} d^{3} e +2 b^{4} d^{4}}+\frac {b e \left (13 e^{2} a^{2}-5 a b d e +b^{2} d^{2}\right ) x}{3 a^{4} e^{4}-12 a^{3} b d \,e^{3}+18 a^{2} b^{2} d^{2} e^{2}-12 a \,b^{3} d^{3} e +3 b^{4} d^{4}}+\frac {25 e^{3} a^{3}-23 a^{2} b d \,e^{2}+13 a \,b^{2} d^{2} e -3 b^{3} d^{3}}{12 a^{4} e^{4}-48 a^{3} b d \,e^{3}+72 a^{2} b^{2} d^{2} e^{2}-48 a \,b^{3} d^{3} e +12 b^{4} d^{4}}}{\left (b x +a \right )^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )}-\frac {e^{4} \ln \left (b x +a \right )}{e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}+\frac {e^{4} \ln \left (-e x -d \right )}{e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}\) | \(485\) |
| norman | \(\frac {\frac {b^{4} e^{3} x^{4}}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {\left (9 a \,b^{5} e^{3}-b^{6} d \,e^{2}\right ) x^{3}}{2 b^{2} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {\left (47 e^{3} a^{2} b^{5}-13 d \,e^{2} a \,b^{6}+2 d^{2} e \,b^{7}\right ) x^{2}}{6 b^{3} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {\left (25 a^{3} b^{5} e^{3}-23 a^{2} b^{6} d \,e^{2}+13 a \,b^{7} d^{2} e -3 b^{8} d^{3}\right ) a}{12 b^{5} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {\left (77 a^{3} b^{5} e^{3}-43 a^{2} b^{6} d \,e^{2}+17 a \,b^{7} d^{2} e -3 b^{8} d^{3}\right ) x}{12 \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) b^{4}}}{\left (b x +a \right )^{5}}+\frac {e^{4} \ln \left (e x +d \right )}{e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}-\frac {e^{4} \ln \left (b x +a \right )}{e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}\) | \(591\) |
Input:
int((b*x+a)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
Output:
1/4/(a*e-b*d)/(b*x+a)^4-e^4/(a*e-b*d)^5*ln(b*x+a)+1/3*e/(a*e-b*d)^2/(b*x+a )^3+1/2*e^2/(a*e-b*d)^3/(b*x+a)^2+e^3/(a*e-b*d)^4/(b*x+a)+e^4/(a*e-b*d)^5* ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 657 vs. \(2 (124) = 248\).
Time = 0.09 (sec) , antiderivative size = 657, normalized size of antiderivative = 5.05 \[ \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {3 \, b^{4} d^{4} - 16 \, a b^{3} d^{3} e + 36 \, a^{2} b^{2} d^{2} e^{2} - 48 \, a^{3} b d e^{3} + 25 \, a^{4} e^{4} - 12 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (b^{4} d^{2} e^{2} - 8 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \, {\left (b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} + 18 \, a^{2} b^{2} d e^{3} - 13 \, a^{3} b e^{4}\right )} x - 12 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \log \left (e x + d\right )}{12 \, {\left (a^{4} b^{5} d^{5} - 5 \, a^{5} b^{4} d^{4} e + 10 \, a^{6} b^{3} d^{3} e^{2} - 10 \, a^{7} b^{2} d^{2} e^{3} + 5 \, a^{8} b d e^{4} - a^{9} e^{5} + {\left (b^{9} d^{5} - 5 \, a b^{8} d^{4} e + 10 \, a^{2} b^{7} d^{3} e^{2} - 10 \, a^{3} b^{6} d^{2} e^{3} + 5 \, a^{4} b^{5} d e^{4} - a^{5} b^{4} e^{5}\right )} x^{4} + 4 \, {\left (a b^{8} d^{5} - 5 \, a^{2} b^{7} d^{4} e + 10 \, a^{3} b^{6} d^{3} e^{2} - 10 \, a^{4} b^{5} d^{2} e^{3} + 5 \, a^{5} b^{4} d e^{4} - a^{6} b^{3} e^{5}\right )} x^{3} + 6 \, {\left (a^{2} b^{7} d^{5} - 5 \, a^{3} b^{6} d^{4} e + 10 \, a^{4} b^{5} d^{3} e^{2} - 10 \, a^{5} b^{4} d^{2} e^{3} + 5 \, a^{6} b^{3} d e^{4} - a^{7} b^{2} e^{5}\right )} x^{2} + 4 \, {\left (a^{3} b^{6} d^{5} - 5 \, a^{4} b^{5} d^{4} e + 10 \, a^{5} b^{4} d^{3} e^{2} - 10 \, a^{6} b^{3} d^{2} e^{3} + 5 \, a^{7} b^{2} d e^{4} - a^{8} b e^{5}\right )} x\right )}} \] Input:
integrate((b*x+a)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
Output:
-1/12*(3*b^4*d^4 - 16*a*b^3*d^3*e + 36*a^2*b^2*d^2*e^2 - 48*a^3*b*d*e^3 + 25*a^4*e^4 - 12*(b^4*d*e^3 - a*b^3*e^4)*x^3 + 6*(b^4*d^2*e^2 - 8*a*b^3*d*e ^3 + 7*a^2*b^2*e^4)*x^2 - 4*(b^4*d^3*e - 6*a*b^3*d^2*e^2 + 18*a^2*b^2*d*e^ 3 - 13*a^3*b*e^4)*x - 12*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^ 2 + 4*a^3*b*e^4*x + a^4*e^4)*log(b*x + a) + 12*(b^4*e^4*x^4 + 4*a*b^3*e^4* x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*log(e*x + d))/(a^4*b^5* d^5 - 5*a^5*b^4*d^4*e + 10*a^6*b^3*d^3*e^2 - 10*a^7*b^2*d^2*e^3 + 5*a^8*b* d*e^4 - a^9*e^5 + (b^9*d^5 - 5*a*b^8*d^4*e + 10*a^2*b^7*d^3*e^2 - 10*a^3*b ^6*d^2*e^3 + 5*a^4*b^5*d*e^4 - a^5*b^4*e^5)*x^4 + 4*(a*b^8*d^5 - 5*a^2*b^7 *d^4*e + 10*a^3*b^6*d^3*e^2 - 10*a^4*b^5*d^2*e^3 + 5*a^5*b^4*d*e^4 - a^6*b ^3*e^5)*x^3 + 6*(a^2*b^7*d^5 - 5*a^3*b^6*d^4*e + 10*a^4*b^5*d^3*e^2 - 10*a ^5*b^4*d^2*e^3 + 5*a^6*b^3*d*e^4 - a^7*b^2*e^5)*x^2 + 4*(a^3*b^6*d^5 - 5*a ^4*b^5*d^4*e + 10*a^5*b^4*d^3*e^2 - 10*a^6*b^3*d^2*e^3 + 5*a^7*b^2*d*e^4 - a^8*b*e^5)*x)
Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (109) = 218\).
Time = 1.57 (sec) , antiderivative size = 802, normalized size of antiderivative = 6.17 \[ \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^{4} \log {\left (x + \frac {- \frac {a^{6} e^{10}}{\left (a e - b d\right )^{5}} + \frac {6 a^{5} b d e^{9}}{\left (a e - b d\right )^{5}} - \frac {15 a^{4} b^{2} d^{2} e^{8}}{\left (a e - b d\right )^{5}} + \frac {20 a^{3} b^{3} d^{3} e^{7}}{\left (a e - b d\right )^{5}} - \frac {15 a^{2} b^{4} d^{4} e^{6}}{\left (a e - b d\right )^{5}} + \frac {6 a b^{5} d^{5} e^{5}}{\left (a e - b d\right )^{5}} + a e^{5} - \frac {b^{6} d^{6} e^{4}}{\left (a e - b d\right )^{5}} + b d e^{4}}{2 b e^{5}} \right )}}{\left (a e - b d\right )^{5}} - \frac {e^{4} \log {\left (x + \frac {\frac {a^{6} e^{10}}{\left (a e - b d\right )^{5}} - \frac {6 a^{5} b d e^{9}}{\left (a e - b d\right )^{5}} + \frac {15 a^{4} b^{2} d^{2} e^{8}}{\left (a e - b d\right )^{5}} - \frac {20 a^{3} b^{3} d^{3} e^{7}}{\left (a e - b d\right )^{5}} + \frac {15 a^{2} b^{4} d^{4} e^{6}}{\left (a e - b d\right )^{5}} - \frac {6 a b^{5} d^{5} e^{5}}{\left (a e - b d\right )^{5}} + a e^{5} + \frac {b^{6} d^{6} e^{4}}{\left (a e - b d\right )^{5}} + b d e^{4}}{2 b e^{5}} \right )}}{\left (a e - b d\right )^{5}} + \frac {25 a^{3} e^{3} - 23 a^{2} b d e^{2} + 13 a b^{2} d^{2} e - 3 b^{3} d^{3} + 12 b^{3} e^{3} x^{3} + x^{2} \cdot \left (42 a b^{2} e^{3} - 6 b^{3} d e^{2}\right ) + x \left (52 a^{2} b e^{3} - 20 a b^{2} d e^{2} + 4 b^{3} d^{2} e\right )}{12 a^{8} e^{4} - 48 a^{7} b d e^{3} + 72 a^{6} b^{2} d^{2} e^{2} - 48 a^{5} b^{3} d^{3} e + 12 a^{4} b^{4} d^{4} + x^{4} \cdot \left (12 a^{4} b^{4} e^{4} - 48 a^{3} b^{5} d e^{3} + 72 a^{2} b^{6} d^{2} e^{2} - 48 a b^{7} d^{3} e + 12 b^{8} d^{4}\right ) + x^{3} \cdot \left (48 a^{5} b^{3} e^{4} - 192 a^{4} b^{4} d e^{3} + 288 a^{3} b^{5} d^{2} e^{2} - 192 a^{2} b^{6} d^{3} e + 48 a b^{7} d^{4}\right ) + x^{2} \cdot \left (72 a^{6} b^{2} e^{4} - 288 a^{5} b^{3} d e^{3} + 432 a^{4} b^{4} d^{2} e^{2} - 288 a^{3} b^{5} d^{3} e + 72 a^{2} b^{6} d^{4}\right ) + x \left (48 a^{7} b e^{4} - 192 a^{6} b^{2} d e^{3} + 288 a^{5} b^{3} d^{2} e^{2} - 192 a^{4} b^{4} d^{3} e + 48 a^{3} b^{5} d^{4}\right )} \] Input:
integrate((b*x+a)/(e*x+d)/(b**2*x**2+2*a*b*x+a**2)**3,x)
Output:
e**4*log(x + (-a**6*e**10/(a*e - b*d)**5 + 6*a**5*b*d*e**9/(a*e - b*d)**5 - 15*a**4*b**2*d**2*e**8/(a*e - b*d)**5 + 20*a**3*b**3*d**3*e**7/(a*e - b* d)**5 - 15*a**2*b**4*d**4*e**6/(a*e - b*d)**5 + 6*a*b**5*d**5*e**5/(a*e - b*d)**5 + a*e**5 - b**6*d**6*e**4/(a*e - b*d)**5 + b*d*e**4)/(2*b*e**5))/( a*e - b*d)**5 - e**4*log(x + (a**6*e**10/(a*e - b*d)**5 - 6*a**5*b*d*e**9/ (a*e - b*d)**5 + 15*a**4*b**2*d**2*e**8/(a*e - b*d)**5 - 20*a**3*b**3*d**3 *e**7/(a*e - b*d)**5 + 15*a**2*b**4*d**4*e**6/(a*e - b*d)**5 - 6*a*b**5*d* *5*e**5/(a*e - b*d)**5 + a*e**5 + b**6*d**6*e**4/(a*e - b*d)**5 + b*d*e**4 )/(2*b*e**5))/(a*e - b*d)**5 + (25*a**3*e**3 - 23*a**2*b*d*e**2 + 13*a*b** 2*d**2*e - 3*b**3*d**3 + 12*b**3*e**3*x**3 + x**2*(42*a*b**2*e**3 - 6*b**3 *d*e**2) + x*(52*a**2*b*e**3 - 20*a*b**2*d*e**2 + 4*b**3*d**2*e))/(12*a**8 *e**4 - 48*a**7*b*d*e**3 + 72*a**6*b**2*d**2*e**2 - 48*a**5*b**3*d**3*e + 12*a**4*b**4*d**4 + x**4*(12*a**4*b**4*e**4 - 48*a**3*b**5*d*e**3 + 72*a** 2*b**6*d**2*e**2 - 48*a*b**7*d**3*e + 12*b**8*d**4) + x**3*(48*a**5*b**3*e **4 - 192*a**4*b**4*d*e**3 + 288*a**3*b**5*d**2*e**2 - 192*a**2*b**6*d**3* e + 48*a*b**7*d**4) + x**2*(72*a**6*b**2*e**4 - 288*a**5*b**3*d*e**3 + 432 *a**4*b**4*d**2*e**2 - 288*a**3*b**5*d**3*e + 72*a**2*b**6*d**4) + x*(48*a **7*b*e**4 - 192*a**6*b**2*d*e**3 + 288*a**5*b**3*d**2*e**2 - 192*a**4*b** 4*d**3*e + 48*a**3*b**5*d**4))
Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (124) = 248\).
Time = 0.05 (sec) , antiderivative size = 558, normalized size of antiderivative = 4.29 \[ \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^{4} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac {e^{4} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac {12 \, b^{3} e^{3} x^{3} - 3 \, b^{3} d^{3} + 13 \, a b^{2} d^{2} e - 23 \, a^{2} b d e^{2} + 25 \, a^{3} e^{3} - 6 \, {\left (b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + 4 \, {\left (b^{3} d^{2} e - 5 \, a b^{2} d e^{2} + 13 \, a^{2} b e^{3}\right )} x}{12 \, {\left (a^{4} b^{4} d^{4} - 4 \, a^{5} b^{3} d^{3} e + 6 \, a^{6} b^{2} d^{2} e^{2} - 4 \, a^{7} b d e^{3} + a^{8} e^{4} + {\left (b^{8} d^{4} - 4 \, a b^{7} d^{3} e + 6 \, a^{2} b^{6} d^{2} e^{2} - 4 \, a^{3} b^{5} d e^{3} + a^{4} b^{4} e^{4}\right )} x^{4} + 4 \, {\left (a b^{7} d^{4} - 4 \, a^{2} b^{6} d^{3} e + 6 \, a^{3} b^{5} d^{2} e^{2} - 4 \, a^{4} b^{4} d e^{3} + a^{5} b^{3} e^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{6} d^{4} - 4 \, a^{3} b^{5} d^{3} e + 6 \, a^{4} b^{4} d^{2} e^{2} - 4 \, a^{5} b^{3} d e^{3} + a^{6} b^{2} e^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{5} d^{4} - 4 \, a^{4} b^{4} d^{3} e + 6 \, a^{5} b^{3} d^{2} e^{2} - 4 \, a^{6} b^{2} d e^{3} + a^{7} b e^{4}\right )} x\right )}} \] Input:
integrate((b*x+a)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
Output:
e^4*log(b*x + a)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^ 2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5) - e^4*log(e*x + d)/(b^5*d^5 - 5*a*b^4 *d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5 ) + 1/12*(12*b^3*e^3*x^3 - 3*b^3*d^3 + 13*a*b^2*d^2*e - 23*a^2*b*d*e^2 + 2 5*a^3*e^3 - 6*(b^3*d*e^2 - 7*a*b^2*e^3)*x^2 + 4*(b^3*d^2*e - 5*a*b^2*d*e^2 + 13*a^2*b*e^3)*x)/(a^4*b^4*d^4 - 4*a^5*b^3*d^3*e + 6*a^6*b^2*d^2*e^2 - 4 *a^7*b*d*e^3 + a^8*e^4 + (b^8*d^4 - 4*a*b^7*d^3*e + 6*a^2*b^6*d^2*e^2 - 4* a^3*b^5*d*e^3 + a^4*b^4*e^4)*x^4 + 4*(a*b^7*d^4 - 4*a^2*b^6*d^3*e + 6*a^3* b^5*d^2*e^2 - 4*a^4*b^4*d*e^3 + a^5*b^3*e^4)*x^3 + 6*(a^2*b^6*d^4 - 4*a^3* b^5*d^3*e + 6*a^4*b^4*d^2*e^2 - 4*a^5*b^3*d*e^3 + a^6*b^2*e^4)*x^2 + 4*(a^ 3*b^5*d^4 - 4*a^4*b^4*d^3*e + 6*a^5*b^3*d^2*e^2 - 4*a^6*b^2*d*e^3 + a^7*b* e^4)*x)
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (124) = 248\).
Time = 0.19 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.60 \[ \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {b e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}} - \frac {e^{5} \log \left ({\left | e x + d \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} - \frac {3 \, b^{4} d^{4} - 16 \, a b^{3} d^{3} e + 36 \, a^{2} b^{2} d^{2} e^{2} - 48 \, a^{3} b d e^{3} + 25 \, a^{4} e^{4} - 12 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (b^{4} d^{2} e^{2} - 8 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \, {\left (b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} + 18 \, a^{2} b^{2} d e^{3} - 13 \, a^{3} b e^{4}\right )} x}{12 \, {\left (b d - a e\right )}^{5} {\left (b x + a\right )}^{4}} \] Input:
integrate((b*x+a)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
Output:
b*e^4*log(abs(b*x + a))/(b^6*d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10 *a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5) - e^5*log(abs(e*x + d))/(b ^5*d^5*e - 5*a*b^4*d^4*e^2 + 10*a^2*b^3*d^3*e^3 - 10*a^3*b^2*d^2*e^4 + 5*a ^4*b*d*e^5 - a^5*e^6) - 1/12*(3*b^4*d^4 - 16*a*b^3*d^3*e + 36*a^2*b^2*d^2* e^2 - 48*a^3*b*d*e^3 + 25*a^4*e^4 - 12*(b^4*d*e^3 - a*b^3*e^4)*x^3 + 6*(b^ 4*d^2*e^2 - 8*a*b^3*d*e^3 + 7*a^2*b^2*e^4)*x^2 - 4*(b^4*d^3*e - 6*a*b^3*d^ 2*e^2 + 18*a^2*b^2*d*e^3 - 13*a^3*b*e^4)*x)/((b*d - a*e)^5*(b*x + a)^4)
Time = 11.19 (sec) , antiderivative size = 505, normalized size of antiderivative = 3.88 \[ \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {25\,a^3\,e^3-23\,a^2\,b\,d\,e^2+13\,a\,b^2\,d^2\,e-3\,b^3\,d^3}{12\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {b^3\,e^3\,x^3}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}-\frac {e^2\,x^2\,\left (b^3\,d-7\,a\,b^2\,e\right )}{2\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {e\,x\,\left (13\,a^2\,b\,e^2-5\,a\,b^2\,d\,e+b^3\,d^2\right )}{3\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4}-\frac {2\,e^4\,\mathrm {atanh}\left (\frac {a^5\,e^5-3\,a^4\,b\,d\,e^4+2\,a^3\,b^2\,d^2\,e^3+2\,a^2\,b^3\,d^3\,e^2-3\,a\,b^4\,d^4\,e+b^5\,d^5}{{\left (a\,e-b\,d\right )}^5}+\frac {2\,b\,e\,x\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^5}\right )}{{\left (a\,e-b\,d\right )}^5} \] Input:
int((a + b*x)/((d + e*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)
Output:
((25*a^3*e^3 - 3*b^3*d^3 + 13*a*b^2*d^2*e - 23*a^2*b*d*e^2)/(12*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3)) + (b^3*e^3* x^3)/(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^ 3) - (e^2*x^2*(b^3*d - 7*a*b^2*e))/(2*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e ^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3)) + (e*x*(b^3*d^2 + 13*a^2*b*e^2 - 5*a* b^2*d*e))/(3*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^ 3*b*d*e^3)))/(a^4 + b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x) - ( 2*e^4*atanh((a^5*e^5 + b^5*d^5 + 2*a^2*b^3*d^3*e^2 + 2*a^3*b^2*d^2*e^3 - 3 *a*b^4*d^4*e - 3*a^4*b*d*e^4)/(a*e - b*d)^5 + (2*b*e*x*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3))/(a*e - b*d)^5))/(a*e - b*d)^5
Time = 0.28 (sec) , antiderivative size = 769, normalized size of antiderivative = 5.92 \[ \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {-12 \,\mathrm {log}\left (b x +a \right ) a^{5} e^{4}-48 \,\mathrm {log}\left (b x +a \right ) a^{4} b \,e^{4} x -72 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{2} e^{4} x^{2}-48 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{3} e^{4} x^{3}-12 \,\mathrm {log}\left (b x +a \right ) a \,b^{4} e^{4} x^{4}+12 \,\mathrm {log}\left (e x +d \right ) a^{5} e^{4}+48 \,\mathrm {log}\left (e x +d \right ) a^{4} b \,e^{4} x +72 \,\mathrm {log}\left (e x +d \right ) a^{3} b^{2} e^{4} x^{2}+48 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{3} e^{4} x^{3}+12 \,\mathrm {log}\left (e x +d \right ) a \,b^{4} e^{4} x^{4}+22 a^{5} e^{4}-45 a^{4} b d \,e^{3}+40 a^{4} b \,e^{4} x +36 a^{3} b^{2} d^{2} e^{2}-60 a^{3} b^{2} d \,e^{3} x +24 a^{3} b^{2} e^{4} x^{2}-16 a^{2} b^{3} d^{3} e +24 a^{2} b^{3} d^{2} e^{2} x -30 a^{2} b^{3} d \,e^{3} x^{2}+3 a \,b^{4} d^{4}-4 a \,b^{4} d^{3} e x +6 a \,b^{4} d^{2} e^{2} x^{2}-3 a \,b^{4} e^{4} x^{4}+3 b^{5} d \,e^{3} x^{4}}{12 a \left (a^{5} b^{4} e^{5} x^{4}-5 a^{4} b^{5} d \,e^{4} x^{4}+10 a^{3} b^{6} d^{2} e^{3} x^{4}-10 a^{2} b^{7} d^{3} e^{2} x^{4}+5 a \,b^{8} d^{4} e \,x^{4}-b^{9} d^{5} x^{4}+4 a^{6} b^{3} e^{5} x^{3}-20 a^{5} b^{4} d \,e^{4} x^{3}+40 a^{4} b^{5} d^{2} e^{3} x^{3}-40 a^{3} b^{6} d^{3} e^{2} x^{3}+20 a^{2} b^{7} d^{4} e \,x^{3}-4 a \,b^{8} d^{5} x^{3}+6 a^{7} b^{2} e^{5} x^{2}-30 a^{6} b^{3} d \,e^{4} x^{2}+60 a^{5} b^{4} d^{2} e^{3} x^{2}-60 a^{4} b^{5} d^{3} e^{2} x^{2}+30 a^{3} b^{6} d^{4} e \,x^{2}-6 a^{2} b^{7} d^{5} x^{2}+4 a^{8} b \,e^{5} x -20 a^{7} b^{2} d \,e^{4} x +40 a^{6} b^{3} d^{2} e^{3} x -40 a^{5} b^{4} d^{3} e^{2} x +20 a^{4} b^{5} d^{4} e x -4 a^{3} b^{6} d^{5} x +a^{9} e^{5}-5 a^{8} b d \,e^{4}+10 a^{7} b^{2} d^{2} e^{3}-10 a^{6} b^{3} d^{3} e^{2}+5 a^{5} b^{4} d^{4} e -a^{4} b^{5} d^{5}\right )} \] Input:
int((b*x+a)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^3,x)
Output:
( - 12*log(a + b*x)*a**5*e**4 - 48*log(a + b*x)*a**4*b*e**4*x - 72*log(a + b*x)*a**3*b**2*e**4*x**2 - 48*log(a + b*x)*a**2*b**3*e**4*x**3 - 12*log(a + b*x)*a*b**4*e**4*x**4 + 12*log(d + e*x)*a**5*e**4 + 48*log(d + e*x)*a** 4*b*e**4*x + 72*log(d + e*x)*a**3*b**2*e**4*x**2 + 48*log(d + e*x)*a**2*b* *3*e**4*x**3 + 12*log(d + e*x)*a*b**4*e**4*x**4 + 22*a**5*e**4 - 45*a**4*b *d*e**3 + 40*a**4*b*e**4*x + 36*a**3*b**2*d**2*e**2 - 60*a**3*b**2*d*e**3* x + 24*a**3*b**2*e**4*x**2 - 16*a**2*b**3*d**3*e + 24*a**2*b**3*d**2*e**2* x - 30*a**2*b**3*d*e**3*x**2 + 3*a*b**4*d**4 - 4*a*b**4*d**3*e*x + 6*a*b** 4*d**2*e**2*x**2 - 3*a*b**4*e**4*x**4 + 3*b**5*d*e**3*x**4)/(12*a*(a**9*e* *5 - 5*a**8*b*d*e**4 + 4*a**8*b*e**5*x + 10*a**7*b**2*d**2*e**3 - 20*a**7* b**2*d*e**4*x + 6*a**7*b**2*e**5*x**2 - 10*a**6*b**3*d**3*e**2 + 40*a**6*b **3*d**2*e**3*x - 30*a**6*b**3*d*e**4*x**2 + 4*a**6*b**3*e**5*x**3 + 5*a** 5*b**4*d**4*e - 40*a**5*b**4*d**3*e**2*x + 60*a**5*b**4*d**2*e**3*x**2 - 2 0*a**5*b**4*d*e**4*x**3 + a**5*b**4*e**5*x**4 - a**4*b**5*d**5 + 20*a**4*b **5*d**4*e*x - 60*a**4*b**5*d**3*e**2*x**2 + 40*a**4*b**5*d**2*e**3*x**3 - 5*a**4*b**5*d*e**4*x**4 - 4*a**3*b**6*d**5*x + 30*a**3*b**6*d**4*e*x**2 - 40*a**3*b**6*d**3*e**2*x**3 + 10*a**3*b**6*d**2*e**3*x**4 - 6*a**2*b**7*d **5*x**2 + 20*a**2*b**7*d**4*e*x**3 - 10*a**2*b**7*d**3*e**2*x**4 - 4*a*b* *8*d**5*x**3 + 5*a*b**8*d**4*e*x**4 - b**9*d**5*x**4))