Integrand size = 31, antiderivative size = 122 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{d+e x} \, dx=\frac {b (b d-a e)^4 x}{e^5}-\frac {(b d-a e)^3 (a+b x)^2}{2 e^4}+\frac {(b d-a e)^2 (a+b x)^3}{3 e^3}-\frac {(b d-a e) (a+b x)^4}{4 e^2}+\frac {(a+b x)^5}{5 e}-\frac {(b d-a e)^5 \log (d+e x)}{e^6} \]
b*(-a*e+b*d)^4*x/e^5-1/2*(-a*e+b*d)^3*(b*x+a)^2/e^4+1/3*(-a*e+b*d)^2*(b*x+ a)^3/e^3-1/4*(-a*e+b*d)*(b*x+a)^4/e^2+1/5*(b*x+a)^5/e-(-a*e+b*d)^5*ln(e*x+ d)/e^6
Time = 0.04 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.37 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{d+e x} \, dx=\frac {b e x \left (300 a^4 e^4+300 a^3 b e^3 (-2 d+e x)+100 a^2 b^2 e^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )+25 a b^3 e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+b^4 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )-60 (b d-a e)^5 \log (d+e x)}{60 e^6} \]
(b*e*x*(300*a^4*e^4 + 300*a^3*b*e^3*(-2*d + e*x) + 100*a^2*b^2*e^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 25*a*b^3*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3 *e^3*x^3) + b^4*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12* e^4*x^4)) - 60*(b*d - a*e)^5*Log[d + e*x])/(60*e^6)
Time = 0.29 (sec) , antiderivative size = 122, 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.
\(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{d+e x} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \frac {\int \frac {b^4 (a+b x)^5}{d+e x}dx}{b^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {(a+b x)^5}{d+e x}dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (\frac {(a e-b d)^5}{e^5 (d+e x)}+\frac {b (b d-a e)^4}{e^5}-\frac {b (a+b x) (b d-a e)^3}{e^4}+\frac {b (a+b x)^2 (b d-a e)^2}{e^3}-\frac {b (a+b x)^3 (b d-a e)}{e^2}+\frac {b (a+b x)^4}{e}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {(b d-a e)^5 \log (d+e x)}{e^6}+\frac {b x (b d-a e)^4}{e^5}-\frac {(a+b x)^2 (b d-a e)^3}{2 e^4}+\frac {(a+b x)^3 (b d-a e)^2}{3 e^3}-\frac {(a+b x)^4 (b d-a e)}{4 e^2}+\frac {(a+b x)^5}{5 e}\) |
(b*(b*d - a*e)^4*x)/e^5 - ((b*d - a*e)^3*(a + b*x)^2)/(2*e^4) + ((b*d - a* e)^2*(a + b*x)^3)/(3*e^3) - ((b*d - a*e)*(a + b*x)^4)/(4*e^2) + (a + b*x)^ 5/(5*e) - ((b*d - a*e)^5*Log[d + e*x])/e^6
3.20.12.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(243\) vs. \(2(114)=228\).
Time = 0.27 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.00
method | result | size |
norman | \(\frac {b \left (5 e^{4} a^{4}-10 b d \,e^{3} a^{3}+10 b^{2} d^{2} e^{2} a^{2}-5 b^{3} d^{3} e a +b^{4} d^{4}\right ) x}{e^{5}}+\frac {b^{5} x^{5}}{5 e}+\frac {b^{2} \left (10 a^{3} e^{3}-10 a^{2} b d \,e^{2}+5 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) x^{2}}{2 e^{4}}+\frac {b^{3} \left (10 e^{2} a^{2}-5 a b d e +b^{2} d^{2}\right ) x^{3}}{3 e^{3}}+\frac {b^{4} \left (5 a e -b d \right ) x^{4}}{4 e^{2}}+\frac {\left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right ) \ln \left (e x +d \right )}{e^{6}}\) | \(244\) |
default | \(\frac {b \left (\frac {1}{5} b^{4} x^{5} e^{4}+\frac {5}{4} a \,b^{3} e^{4} x^{4}-\frac {1}{4} b^{4} d \,e^{3} x^{4}+\frac {10}{3} a^{2} b^{2} e^{4} x^{3}-\frac {5}{3} a \,b^{3} d \,e^{3} x^{3}+\frac {1}{3} b^{4} d^{2} e^{2} x^{3}+5 a^{3} b \,e^{4} x^{2}-5 a^{2} b^{2} d \,e^{3} x^{2}+\frac {5}{2} a \,b^{3} d^{2} e^{2} x^{2}-\frac {1}{2} b^{4} d^{3} e \,x^{2}+5 e^{4} a^{4} x -10 b d \,e^{3} a^{3} x +10 b^{2} d^{2} e^{2} a^{2} x -5 b^{3} d^{3} e a x +b^{4} d^{4} x \right )}{e^{5}}+\frac {\left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right ) \ln \left (e x +d \right )}{e^{6}}\) | \(266\) |
risch | \(\frac {b^{5} x^{5}}{5 e}+\frac {5 b^{4} a \,x^{4}}{4 e}-\frac {b^{5} d \,x^{4}}{4 e^{2}}+\frac {10 b^{3} a^{2} x^{3}}{3 e}-\frac {5 b^{4} a d \,x^{3}}{3 e^{2}}+\frac {b^{5} d^{2} x^{3}}{3 e^{3}}+\frac {5 b^{2} a^{3} x^{2}}{e}-\frac {5 b^{3} a^{2} d \,x^{2}}{e^{2}}+\frac {5 b^{4} a \,d^{2} x^{2}}{2 e^{3}}-\frac {b^{5} d^{3} x^{2}}{2 e^{4}}+\frac {5 b \,a^{4} x}{e}-\frac {10 b^{2} d \,a^{3} x}{e^{2}}+\frac {10 b^{3} d^{2} a^{2} x}{e^{3}}-\frac {5 b^{4} d^{3} a x}{e^{4}}+\frac {b^{5} d^{4} x}{e^{5}}+\frac {\ln \left (e x +d \right ) a^{5}}{e}-\frac {5 \ln \left (e x +d \right ) b d \,a^{4}}{e^{2}}+\frac {10 \ln \left (e x +d \right ) b^{2} d^{2} a^{3}}{e^{3}}-\frac {10 \ln \left (e x +d \right ) b^{3} d^{3} a^{2}}{e^{4}}+\frac {5 \ln \left (e x +d \right ) b^{4} d^{4} a}{e^{5}}-\frac {\ln \left (e x +d \right ) b^{5} d^{5}}{e^{6}}\) | \(302\) |
parallelrisch | \(\frac {12 x^{5} b^{5} e^{5}+75 x^{4} a \,b^{4} e^{5}-15 x^{4} b^{5} d \,e^{4}+200 x^{3} a^{2} b^{3} e^{5}-100 x^{3} a \,b^{4} d \,e^{4}+20 x^{3} b^{5} d^{2} e^{3}+300 x^{2} a^{3} b^{2} e^{5}-300 x^{2} a^{2} b^{3} d \,e^{4}+150 x^{2} a \,b^{4} d^{2} e^{3}-30 x^{2} b^{5} d^{3} e^{2}+60 \ln \left (e x +d \right ) a^{5} e^{5}-300 \ln \left (e x +d \right ) a^{4} b d \,e^{4}+600 \ln \left (e x +d \right ) a^{3} b^{2} d^{2} e^{3}-600 \ln \left (e x +d \right ) a^{2} b^{3} d^{3} e^{2}+300 \ln \left (e x +d \right ) a \,b^{4} d^{4} e -60 \ln \left (e x +d \right ) b^{5} d^{5}+300 x \,a^{4} b \,e^{5}-600 x \,a^{3} b^{2} d \,e^{4}+600 x \,a^{2} b^{3} d^{2} e^{3}-300 x a \,b^{4} d^{3} e^{2}+60 x \,b^{5} d^{4} e}{60 e^{6}}\) | \(302\) |
b*(5*a^4*e^4-10*a^3*b*d*e^3+10*a^2*b^2*d^2*e^2-5*a*b^3*d^3*e+b^4*d^4)/e^5* x+1/5*b^5/e*x^5+1/2*b^2/e^4*(10*a^3*e^3-10*a^2*b*d*e^2+5*a*b^2*d^2*e-b^3*d ^3)*x^2+1/3*b^3/e^3*(10*a^2*e^2-5*a*b*d*e+b^2*d^2)*x^3+1/4*b^4/e^2*(5*a*e- b*d)*x^4+(a^5*e^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)/e^6*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (114) = 228\).
Time = 0.33 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.12 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{d+e x} \, dx=\frac {12 \, b^{5} e^{5} x^{5} - 15 \, {\left (b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 20 \, {\left (b^{5} d^{2} e^{3} - 5 \, a b^{4} d e^{4} + 10 \, a^{2} b^{3} e^{5}\right )} x^{3} - 30 \, {\left (b^{5} d^{3} e^{2} - 5 \, a b^{4} d^{2} e^{3} + 10 \, a^{2} b^{3} d e^{4} - 10 \, a^{3} b^{2} e^{5}\right )} x^{2} + 60 \, {\left (b^{5} d^{4} e - 5 \, a b^{4} d^{3} e^{2} + 10 \, a^{2} b^{3} d^{2} e^{3} - 10 \, a^{3} b^{2} d e^{4} + 5 \, a^{4} b e^{5}\right )} x - 60 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \log \left (e x + d\right )}{60 \, e^{6}} \]
1/60*(12*b^5*e^5*x^5 - 15*(b^5*d*e^4 - 5*a*b^4*e^5)*x^4 + 20*(b^5*d^2*e^3 - 5*a*b^4*d*e^4 + 10*a^2*b^3*e^5)*x^3 - 30*(b^5*d^3*e^2 - 5*a*b^4*d^2*e^3 + 10*a^2*b^3*d*e^4 - 10*a^3*b^2*e^5)*x^2 + 60*(b^5*d^4*e - 5*a*b^4*d^3*e^2 + 10*a^2*b^3*d^2*e^3 - 10*a^3*b^2*d*e^4 + 5*a^4*b*e^5)*x - 60*(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)*log(e*x + d))/e^6
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (104) = 208\).
Time = 0.27 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.71 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{d+e x} \, dx=\frac {b^{5} x^{5}}{5 e} + x^{4} \cdot \left (\frac {5 a b^{4}}{4 e} - \frac {b^{5} d}{4 e^{2}}\right ) + x^{3} \cdot \left (\frac {10 a^{2} b^{3}}{3 e} - \frac {5 a b^{4} d}{3 e^{2}} + \frac {b^{5} d^{2}}{3 e^{3}}\right ) + x^{2} \cdot \left (\frac {5 a^{3} b^{2}}{e} - \frac {5 a^{2} b^{3} d}{e^{2}} + \frac {5 a b^{4} d^{2}}{2 e^{3}} - \frac {b^{5} d^{3}}{2 e^{4}}\right ) + x \left (\frac {5 a^{4} b}{e} - \frac {10 a^{3} b^{2} d}{e^{2}} + \frac {10 a^{2} b^{3} d^{2}}{e^{3}} - \frac {5 a b^{4} d^{3}}{e^{4}} + \frac {b^{5} d^{4}}{e^{5}}\right ) + \frac {\left (a e - b d\right )^{5} \log {\left (d + e x \right )}}{e^{6}} \]
b**5*x**5/(5*e) + x**4*(5*a*b**4/(4*e) - b**5*d/(4*e**2)) + x**3*(10*a**2* b**3/(3*e) - 5*a*b**4*d/(3*e**2) + b**5*d**2/(3*e**3)) + x**2*(5*a**3*b**2 /e - 5*a**2*b**3*d/e**2 + 5*a*b**4*d**2/(2*e**3) - b**5*d**3/(2*e**4)) + x *(5*a**4*b/e - 10*a**3*b**2*d/e**2 + 10*a**2*b**3*d**2/e**3 - 5*a*b**4*d** 3/e**4 + b**5*d**4/e**5) + (a*e - b*d)**5*log(d + e*x)/e**6
Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (114) = 228\).
Time = 0.19 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.11 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{d+e x} \, dx=\frac {12 \, b^{5} e^{4} x^{5} - 15 \, {\left (b^{5} d e^{3} - 5 \, a b^{4} e^{4}\right )} x^{4} + 20 \, {\left (b^{5} d^{2} e^{2} - 5 \, a b^{4} d e^{3} + 10 \, a^{2} b^{3} e^{4}\right )} x^{3} - 30 \, {\left (b^{5} d^{3} e - 5 \, a b^{4} d^{2} e^{2} + 10 \, a^{2} b^{3} d e^{3} - 10 \, a^{3} b^{2} e^{4}\right )} x^{2} + 60 \, {\left (b^{5} d^{4} - 5 \, a b^{4} d^{3} e + 10 \, a^{2} b^{3} d^{2} e^{2} - 10 \, a^{3} b^{2} d e^{3} + 5 \, a^{4} b e^{4}\right )} x}{60 \, e^{5}} - \frac {{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \log \left (e x + d\right )}{e^{6}} \]
1/60*(12*b^5*e^4*x^5 - 15*(b^5*d*e^3 - 5*a*b^4*e^4)*x^4 + 20*(b^5*d^2*e^2 - 5*a*b^4*d*e^3 + 10*a^2*b^3*e^4)*x^3 - 30*(b^5*d^3*e - 5*a*b^4*d^2*e^2 + 10*a^2*b^3*d*e^3 - 10*a^3*b^2*e^4)*x^2 + 60*(b^5*d^4 - 5*a*b^4*d^3*e + 10* a^2*b^3*d^2*e^2 - 10*a^3*b^2*d*e^3 + 5*a^4*b*e^4)*x)/e^5 - (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)*log(e*x + d)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (114) = 228\).
Time = 0.26 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.24 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{d+e x} \, dx=\frac {12 \, b^{5} e^{4} x^{5} - 15 \, b^{5} d e^{3} x^{4} + 75 \, a b^{4} e^{4} x^{4} + 20 \, b^{5} d^{2} e^{2} x^{3} - 100 \, a b^{4} d e^{3} x^{3} + 200 \, a^{2} b^{3} e^{4} x^{3} - 30 \, b^{5} d^{3} e x^{2} + 150 \, a b^{4} d^{2} e^{2} x^{2} - 300 \, a^{2} b^{3} d e^{3} x^{2} + 300 \, a^{3} b^{2} e^{4} x^{2} + 60 \, b^{5} d^{4} x - 300 \, a b^{4} d^{3} e x + 600 \, a^{2} b^{3} d^{2} e^{2} x - 600 \, a^{3} b^{2} d e^{3} x + 300 \, a^{4} b e^{4} x}{60 \, e^{5}} - \frac {{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{6}} \]
1/60*(12*b^5*e^4*x^5 - 15*b^5*d*e^3*x^4 + 75*a*b^4*e^4*x^4 + 20*b^5*d^2*e^ 2*x^3 - 100*a*b^4*d*e^3*x^3 + 200*a^2*b^3*e^4*x^3 - 30*b^5*d^3*e*x^2 + 150 *a*b^4*d^2*e^2*x^2 - 300*a^2*b^3*d*e^3*x^2 + 300*a^3*b^2*e^4*x^2 + 60*b^5* d^4*x - 300*a*b^4*d^3*e*x + 600*a^2*b^3*d^2*e^2*x - 600*a^3*b^2*d*e^3*x + 300*a^4*b*e^4*x)/e^5 - (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)*log(abs(e*x + d))/e^6
Time = 10.71 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.30 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{d+e x} \, dx=x\,\left (\frac {5\,a^4\,b}{e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{e}-\frac {b^5\,d}{e^2}\right )}{e}-\frac {10\,a^2\,b^3}{e}\right )}{e}+\frac {10\,a^3\,b^2}{e}\right )}{e}\right )+x^4\,\left (\frac {5\,a\,b^4}{4\,e}-\frac {b^5\,d}{4\,e^2}\right )+x^2\,\left (\frac {d\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{e}-\frac {b^5\,d}{e^2}\right )}{e}-\frac {10\,a^2\,b^3}{e}\right )}{2\,e}+\frac {5\,a^3\,b^2}{e}\right )-x^3\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{e}-\frac {b^5\,d}{e^2}\right )}{3\,e}-\frac {10\,a^2\,b^3}{3\,e}\right )+\frac {b^5\,x^5}{5\,e}+\frac {\ln \left (d+e\,x\right )\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}{e^6} \]
x*((5*a^4*b)/e - (d*((d*((d*((5*a*b^4)/e - (b^5*d)/e^2))/e - (10*a^2*b^3)/ e))/e + (10*a^3*b^2)/e))/e) + x^4*((5*a*b^4)/(4*e) - (b^5*d)/(4*e^2)) + x^ 2*((d*((d*((5*a*b^4)/e - (b^5*d)/e^2))/e - (10*a^2*b^3)/e))/(2*e) + (5*a^3 *b^2)/e) - x^3*((d*((5*a*b^4)/e - (b^5*d)/e^2))/(3*e) - (10*a^2*b^3)/(3*e) ) + (b^5*x^5)/(5*e) + (log(d + e*x)*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^ 2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4))/e^6