Integrand size = 31, antiderivative size = 111 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {(b d-a e)^4}{4 b^5 (a+b x)^4}-\frac {4 e (b d-a e)^3}{3 b^5 (a+b x)^3}-\frac {3 e^2 (b d-a e)^2}{b^5 (a+b x)^2}-\frac {4 e^3 (b d-a e)}{b^5 (a+b x)}+\frac {e^4 \log (a+b x)}{b^5} \]
-1/4*(-a*e+b*d)^4/b^5/(b*x+a)^4-4/3*e*(-a*e+b*d)^3/b^5/(b*x+a)^3-3*e^2*(-a *e+b*d)^2/b^5/(b*x+a)^2-4*e^3*(-a*e+b*d)/b^5/(b*x+a)+e^4*ln(b*x+a)/b^5
Time = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {(b d-a e) \left (25 a^3 e^3+a^2 b e^2 (13 d+88 e x)+a b^2 e \left (7 d^2+40 d e x+108 e^2 x^2\right )+b^3 \left (3 d^3+16 d^2 e x+36 d e^2 x^2+48 e^3 x^3\right )\right )}{12 b^5 (a+b x)^4}+\frac {e^4 \log (a+b x)}{b^5} \]
-1/12*((b*d - a*e)*(25*a^3*e^3 + a^2*b*e^2*(13*d + 88*e*x) + a*b^2*e*(7*d^ 2 + 40*d*e*x + 108*e^2*x^2) + b^3*(3*d^3 + 16*d^2*e*x + 36*d*e^2*x^2 + 48* e^3*x^3)))/(b^5*(a + b*x)^4) + (e^4*Log[a + b*x])/b^5
Time = 0.30 (sec) , antiderivative size = 111, 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, 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.
\(\displaystyle \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle b^6 \int \frac {(d+e x)^4}{b^6 (a+b x)^5}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {(d+e x)^4}{(a+b x)^5}dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (\frac {4 e^3 (b d-a e)}{b^4 (a+b x)^2}+\frac {6 e^2 (b d-a e)^2}{b^4 (a+b x)^3}+\frac {4 e (b d-a e)^3}{b^4 (a+b x)^4}+\frac {(b d-a e)^4}{b^4 (a+b x)^5}+\frac {e^4}{b^4 (a+b x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 e^3 (b d-a e)}{b^5 (a+b x)}-\frac {3 e^2 (b d-a e)^2}{b^5 (a+b x)^2}-\frac {4 e (b d-a e)^3}{3 b^5 (a+b x)^3}-\frac {(b d-a e)^4}{4 b^5 (a+b x)^4}+\frac {e^4 \log (a+b x)}{b^5}\) |
-1/4*(b*d - a*e)^4/(b^5*(a + b*x)^4) - (4*e*(b*d - a*e)^3)/(3*b^5*(a + b*x )^3) - (3*e^2*(b*d - a*e)^2)/(b^5*(a + b*x)^2) - (4*e^3*(b*d - a*e))/(b^5* (a + b*x)) + (e^4*Log[a + b*x])/b^5
3.20.45.3.1 Defintions of rubi rules used
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_.)*((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 = 0.28 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.66
method | result | size |
default | \(-\frac {e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}}{4 b^{5} \left (b x +a \right )^{4}}+\frac {4 e \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{3 b^{5} \left (b x +a \right )^{3}}+\frac {e^{4} \ln \left (b x +a \right )}{b^{5}}-\frac {3 e^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )}{b^{5} \left (b x +a \right )^{2}}+\frac {4 e^{3} \left (a e -b d \right )}{b^{5} \left (b x +a \right )}\) | \(184\) |
risch | \(\frac {\frac {4 e^{3} \left (a e -b d \right ) x^{3}}{b^{2}}+\frac {3 e^{2} \left (3 e^{2} a^{2}-2 a b d e -b^{2} d^{2}\right ) x^{2}}{b^{3}}+\frac {2 e \left (11 a^{3} e^{3}-6 a^{2} b d \,e^{2}-3 a \,b^{2} d^{2} e -2 b^{3} d^{3}\right ) x}{3 b^{4}}+\frac {25 e^{4} a^{4}-12 b d \,e^{3} a^{3}-6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a -3 b^{4} d^{4}}{12 b^{5}}}{\left (b x +a \right )^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )}+\frac {e^{4} \ln \left (b x +a \right )}{b^{5}}\) | \(195\) |
norman | \(\frac {\frac {\left (4 a \,e^{4}-4 b d \,e^{3}\right ) x^{4}}{b}+\frac {\left (13 a^{2} e^{4}-10 a b d \,e^{3}-3 b^{2} d^{2} e^{2}\right ) x^{3}}{b^{2}}+\frac {a \left (25 e^{4} a^{4}-12 b d \,e^{3} a^{3}-6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a -3 b^{4} d^{4}\right )}{12 b^{5}}+\frac {\left (49 a^{3} e^{4}-30 b \,a^{2} e^{3} d -15 b^{2} a \,d^{2} e^{2}-4 b^{3} d^{3} e \right ) x^{2}}{3 b^{3}}+\frac {\left (113 e^{4} a^{4}-60 b d \,e^{3} a^{3}-30 b^{2} d^{2} e^{2} a^{2}-20 b^{3} d^{3} e a -3 b^{4} d^{4}\right ) x}{12 b^{4}}}{\left (b x +a \right )^{5}}+\frac {e^{4} \ln \left (b x +a \right )}{b^{5}}\) | \(242\) |
parallelrisch | \(\frac {12 \ln \left (b x +a \right ) x^{4} b^{4} e^{4}+48 \ln \left (b x +a \right ) x^{3} a \,b^{3} e^{4}+72 \ln \left (b x +a \right ) x^{2} a^{2} b^{2} e^{4}+48 x^{3} a \,b^{3} e^{4}-48 x^{3} b^{4} d \,e^{3}+48 \ln \left (b x +a \right ) x \,a^{3} b \,e^{4}+108 x^{2} a^{2} b^{2} e^{4}-72 x^{2} a \,b^{3} d \,e^{3}-36 x^{2} b^{4} d^{2} e^{2}+12 \ln \left (b x +a \right ) a^{4} e^{4}+88 x \,a^{3} b \,e^{4}-48 x \,a^{2} b^{2} d \,e^{3}-24 x a \,b^{3} d^{2} e^{2}-16 x \,b^{4} d^{3} e +25 e^{4} a^{4}-12 b d \,e^{3} a^{3}-6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a -3 b^{4} d^{4}}{12 b^{5} \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )^{2}}\) | \(278\) |
-1/4*(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)/b^5/( b*x+a)^4+4/3*e*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)/b^5/(b*x+a)^3 +e^4*ln(b*x+a)/b^5-3/b^5*e^2*(a^2*e^2-2*a*b*d*e+b^2*d^2)/(b*x+a)^2+4/b^5*e ^3*(a*e-b*d)/(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (107) = 214\).
Time = 0.30 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.41 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {3 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 25 \, a^{4} e^{4} + 48 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 36 \, {\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} - 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \, {\left (2 \, b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} - 11 \, a^{3} b e^{4}\right )} x - 12 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \log \left (b x + a\right )}{12 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}} \]
-1/12*(3*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 12*a^3*b*d*e^3 - 25 *a^4*e^4 + 48*(b^4*d*e^3 - a*b^3*e^4)*x^3 + 36*(b^4*d^2*e^2 + 2*a*b^3*d*e^ 3 - 3*a^2*b^2*e^4)*x^2 + 8*(2*b^4*d^3*e + 3*a*b^3*d^2*e^2 + 6*a^2*b^2*d*e^ 3 - 11*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))/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2* b^7*x^2 + 4*a^3*b^6*x + a^4*b^5)
Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (100) = 200\).
Time = 10.66 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.07 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {25 a^{4} e^{4} - 12 a^{3} b d e^{3} - 6 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e - 3 b^{4} d^{4} + x^{3} \cdot \left (48 a b^{3} e^{4} - 48 b^{4} d e^{3}\right ) + x^{2} \cdot \left (108 a^{2} b^{2} e^{4} - 72 a b^{3} d e^{3} - 36 b^{4} d^{2} e^{2}\right ) + x \left (88 a^{3} b e^{4} - 48 a^{2} b^{2} d e^{3} - 24 a b^{3} d^{2} e^{2} - 16 b^{4} d^{3} e\right )}{12 a^{4} b^{5} + 48 a^{3} b^{6} x + 72 a^{2} b^{7} x^{2} + 48 a b^{8} x^{3} + 12 b^{9} x^{4}} + \frac {e^{4} \log {\left (a + b x \right )}}{b^{5}} \]
(25*a**4*e**4 - 12*a**3*b*d*e**3 - 6*a**2*b**2*d**2*e**2 - 4*a*b**3*d**3*e - 3*b**4*d**4 + x**3*(48*a*b**3*e**4 - 48*b**4*d*e**3) + x**2*(108*a**2*b **2*e**4 - 72*a*b**3*d*e**3 - 36*b**4*d**2*e**2) + x*(88*a**3*b*e**4 - 48* a**2*b**2*d*e**3 - 24*a*b**3*d**2*e**2 - 16*b**4*d**3*e))/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + e**4 *log(a + b*x)/b**5
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (107) = 214\).
Time = 0.21 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.97 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {3 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 25 \, a^{4} e^{4} + 48 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 36 \, {\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} - 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \, {\left (2 \, b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} - 11 \, a^{3} b e^{4}\right )} x}{12 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}} + \frac {e^{4} \log \left (b x + a\right )}{b^{5}} \]
-1/12*(3*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 12*a^3*b*d*e^3 - 25 *a^4*e^4 + 48*(b^4*d*e^3 - a*b^3*e^4)*x^3 + 36*(b^4*d^2*e^2 + 2*a*b^3*d*e^ 3 - 3*a^2*b^2*e^4)*x^2 + 8*(2*b^4*d^3*e + 3*a*b^3*d^2*e^2 + 6*a^2*b^2*d*e^ 3 - 11*a^3*b*e^4)*x)/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5) + e^4*log(b*x + a)/b^5
Time = 0.26 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.66 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{5}} - \frac {48 \, {\left (b^{3} d e^{3} - a b^{2} e^{4}\right )} x^{3} + 36 \, {\left (b^{3} d^{2} e^{2} + 2 \, a b^{2} d e^{3} - 3 \, a^{2} b e^{4}\right )} x^{2} + 8 \, {\left (2 \, b^{3} d^{3} e + 3 \, a b^{2} d^{2} e^{2} + 6 \, a^{2} b d e^{3} - 11 \, a^{3} e^{4}\right )} x + \frac {3 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 25 \, a^{4} e^{4}}{b}}{12 \, {\left (b x + a\right )}^{4} b^{4}} \]
e^4*log(abs(b*x + a))/b^5 - 1/12*(48*(b^3*d*e^3 - a*b^2*e^4)*x^3 + 36*(b^3 *d^2*e^2 + 2*a*b^2*d*e^3 - 3*a^2*b*e^4)*x^2 + 8*(2*b^3*d^3*e + 3*a*b^2*d^2 *e^2 + 6*a^2*b*d*e^3 - 11*a^3*e^4)*x + (3*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2* b^2*d^2*e^2 + 12*a^3*b*d*e^3 - 25*a^4*e^4)/b)/((b*x + a)^4*b^4)
Time = 0.13 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.92 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^4\,\ln \left (a+b\,x\right )}{b^5}-\frac {\frac {-25\,a^4\,e^4+12\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2+4\,a\,b^3\,d^3\,e+3\,b^4\,d^4}{12\,b^5}+\frac {3\,x^2\,\left (-3\,a^2\,e^4+2\,a\,b\,d\,e^3+b^2\,d^2\,e^2\right )}{b^3}+\frac {2\,x\,\left (-11\,a^3\,e^4+6\,a^2\,b\,d\,e^3+3\,a\,b^2\,d^2\,e^2+2\,b^3\,d^3\,e\right )}{3\,b^4}-\frac {4\,e^3\,x^3\,\left (a\,e-b\,d\right )}{b^2}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4} \]
(e^4*log(a + b*x))/b^5 - ((3*b^4*d^4 - 25*a^4*e^4 + 6*a^2*b^2*d^2*e^2 + 4* a*b^3*d^3*e + 12*a^3*b*d*e^3)/(12*b^5) + (3*x^2*(b^2*d^2*e^2 - 3*a^2*e^4 + 2*a*b*d*e^3))/b^3 + (2*x*(2*b^3*d^3*e - 11*a^3*e^4 + 3*a*b^2*d^2*e^2 + 6* a^2*b*d*e^3))/(3*b^4) - (4*e^3*x^3*(a*e - b*d))/b^2)/(a^4 + b^4*x^4 + 4*a* b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x)