Integrand size = 20, antiderivative size = 94 \[ \int \frac {x^6 (d+e x)}{\left (b x+c x^2\right )^3} \, dx=\frac {(c d-3 b e) x}{c^4}+\frac {e x^2}{2 c^3}+\frac {b^3 (c d-b e)}{2 c^5 (b+c x)^2}-\frac {b^2 (3 c d-4 b e)}{c^5 (b+c x)}-\frac {3 b (c d-2 b e) \log (b+c x)}{c^5} \] Output:
(-3*b*e+c*d)*x/c^4+1/2*e*x^2/c^3+1/2*b^3*(-b*e+c*d)/c^5/(c*x+b)^2-b^2*(-4* b*e+3*c*d)/c^5/(c*x+b)-3*b*(-2*b*e+c*d)*ln(c*x+b)/c^5
Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \frac {x^6 (d+e x)}{\left (b x+c x^2\right )^3} \, dx=\frac {2 c (c d-3 b e) x+c^2 e x^2+\frac {b^3 (c d-b e)}{(b+c x)^2}+\frac {2 b^2 (-3 c d+4 b e)}{b+c x}+6 b (-c d+2 b e) \log (b+c x)}{2 c^5} \] Input:
Integrate[(x^6*(d + e*x))/(b*x + c*x^2)^3,x]
Output:
(2*c*(c*d - 3*b*e)*x + c^2*e*x^2 + (b^3*(c*d - b*e))/(b + c*x)^2 + (2*b^2* (-3*c*d + 4*b*e))/(b + c*x) + 6*b*(-(c*d) + 2*b*e)*Log[b + c*x])/(2*c^5)
Time = 0.45 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {9, 86, 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.
\(\displaystyle \int \frac {x^6 (d+e x)}{\left (b x+c x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \frac {x^3 (d+e x)}{(b+c x)^3}dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {b^3 (b e-c d)}{c^4 (b+c x)^3}-\frac {b^2 (4 b e-3 c d)}{c^4 (b+c x)^2}+\frac {3 b (2 b e-c d)}{c^4 (b+c x)}+\frac {c d-3 b e}{c^4}+\frac {e x}{c^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b^3 (c d-b e)}{2 c^5 (b+c x)^2}-\frac {b^2 (3 c d-4 b e)}{c^5 (b+c x)}-\frac {3 b (c d-2 b e) \log (b+c x)}{c^5}+\frac {x (c d-3 b e)}{c^4}+\frac {e x^2}{2 c^3}\) |
Input:
Int[(x^6*(d + e*x))/(b*x + c*x^2)^3,x]
Output:
((c*d - 3*b*e)*x)/c^4 + (e*x^2)/(2*c^3) + (b^3*(c*d - b*e))/(2*c^5*(b + c* x)^2) - (b^2*(3*c*d - 4*b*e))/(c^5*(b + c*x)) - (3*b*(c*d - 2*b*e)*Log[b + c*x])/c^5
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Time = 0.77 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {-\frac {1}{2} c e \,x^{2}+3 b e x -c d x}{c^{4}}+\frac {3 b \left (2 b e -c d \right ) \ln \left (c x +b \right )}{c^{5}}+\frac {b^{2} \left (4 b e -3 c d \right )}{c^{5} \left (c x +b \right )}-\frac {b^{3} \left (b e -c d \right )}{2 c^{5} \left (c x +b \right )^{2}}\) | \(92\) |
risch | \(\frac {e \,x^{2}}{2 c^{3}}-\frac {3 b e x}{c^{4}}+\frac {d x}{c^{3}}+\frac {\left (4 e \,b^{3}-3 c d \,b^{2}\right ) x +\frac {b^{3} \left (7 b e -5 c d \right )}{2 c}}{c^{4} \left (c x +b \right )^{2}}+\frac {6 b^{2} \ln \left (c x +b \right ) e}{c^{5}}-\frac {3 b \ln \left (c x +b \right ) d}{c^{4}}\) | \(98\) |
norman | \(\frac {-\frac {\left (2 b e -c d \right ) x^{5}}{c^{2}}+\frac {e \,x^{6}}{2 c}+\frac {2 b \left (6 e \,b^{2}-3 d b c \right ) x^{3}}{c^{4}}+\frac {b^{2} \left (18 e \,b^{2}-9 d b c \right ) x^{2}}{2 c^{5}}}{x^{2} \left (c x +b \right )^{2}}+\frac {3 b \left (2 b e -c d \right ) \ln \left (c x +b \right )}{c^{5}}\) | \(105\) |
parallelrisch | \(\frac {e \,x^{4} c^{4}+12 \ln \left (c x +b \right ) x^{2} b^{2} c^{2} e -6 \ln \left (c x +b \right ) x^{2} b \,c^{3} d -4 x^{3} b \,c^{3} e +2 c^{4} d \,x^{3}+24 \ln \left (c x +b \right ) x \,b^{3} c e -12 \ln \left (c x +b \right ) x \,b^{2} c^{2} d +12 \ln \left (c x +b \right ) b^{4} e -6 \ln \left (c x +b \right ) b^{3} c d +24 x \,b^{3} c e -12 b^{2} c^{2} d x +18 b^{4} e -9 b^{3} c d}{2 c^{5} \left (c x +b \right )^{2}}\) | \(161\) |
Input:
int(x^6*(e*x+d)/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)
Output:
-1/c^4*(-1/2*c*e*x^2+3*b*e*x-c*d*x)+3*b/c^5*(2*b*e-c*d)*ln(c*x+b)+b^2/c^5* (4*b*e-3*c*d)/(c*x+b)-1/2*b^3*(b*e-c*d)/c^5/(c*x+b)^2
Time = 0.08 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.78 \[ \int \frac {x^6 (d+e x)}{\left (b x+c x^2\right )^3} \, dx=\frac {c^{4} e x^{4} - 5 \, b^{3} c d + 7 \, b^{4} e + 2 \, {\left (c^{4} d - 2 \, b c^{3} e\right )} x^{3} + {\left (4 \, b c^{3} d - 11 \, b^{2} c^{2} e\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{2} d - b^{3} c e\right )} x - 6 \, {\left (b^{3} c d - 2 \, b^{4} e + {\left (b c^{3} d - 2 \, b^{2} c^{2} e\right )} x^{2} + 2 \, {\left (b^{2} c^{2} d - 2 \, b^{3} c e\right )} x\right )} \log \left (c x + b\right )}{2 \, {\left (c^{7} x^{2} + 2 \, b c^{6} x + b^{2} c^{5}\right )}} \] Input:
integrate(x^6*(e*x+d)/(c*x^2+b*x)^3,x, algorithm="fricas")
Output:
1/2*(c^4*e*x^4 - 5*b^3*c*d + 7*b^4*e + 2*(c^4*d - 2*b*c^3*e)*x^3 + (4*b*c^ 3*d - 11*b^2*c^2*e)*x^2 - 2*(2*b^2*c^2*d - b^3*c*e)*x - 6*(b^3*c*d - 2*b^4 *e + (b*c^3*d - 2*b^2*c^2*e)*x^2 + 2*(b^2*c^2*d - 2*b^3*c*e)*x)*log(c*x + b))/(c^7*x^2 + 2*b*c^6*x + b^2*c^5)
Time = 0.39 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.14 \[ \int \frac {x^6 (d+e x)}{\left (b x+c x^2\right )^3} \, dx=\frac {3 b \left (2 b e - c d\right ) \log {\left (b + c x \right )}}{c^{5}} + x \left (- \frac {3 b e}{c^{4}} + \frac {d}{c^{3}}\right ) + \frac {7 b^{4} e - 5 b^{3} c d + x \left (8 b^{3} c e - 6 b^{2} c^{2} d\right )}{2 b^{2} c^{5} + 4 b c^{6} x + 2 c^{7} x^{2}} + \frac {e x^{2}}{2 c^{3}} \] Input:
integrate(x**6*(e*x+d)/(c*x**2+b*x)**3,x)
Output:
3*b*(2*b*e - c*d)*log(b + c*x)/c**5 + x*(-3*b*e/c**4 + d/c**3) + (7*b**4*e - 5*b**3*c*d + x*(8*b**3*c*e - 6*b**2*c**2*d))/(2*b**2*c**5 + 4*b*c**6*x + 2*c**7*x**2) + e*x**2/(2*c**3)
Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.13 \[ \int \frac {x^6 (d+e x)}{\left (b x+c x^2\right )^3} \, dx=-\frac {5 \, b^{3} c d - 7 \, b^{4} e + 2 \, {\left (3 \, b^{2} c^{2} d - 4 \, b^{3} c e\right )} x}{2 \, {\left (c^{7} x^{2} + 2 \, b c^{6} x + b^{2} c^{5}\right )}} + \frac {c e x^{2} + 2 \, {\left (c d - 3 \, b e\right )} x}{2 \, c^{4}} - \frac {3 \, {\left (b c d - 2 \, b^{2} e\right )} \log \left (c x + b\right )}{c^{5}} \] Input:
integrate(x^6*(e*x+d)/(c*x^2+b*x)^3,x, algorithm="maxima")
Output:
-1/2*(5*b^3*c*d - 7*b^4*e + 2*(3*b^2*c^2*d - 4*b^3*c*e)*x)/(c^7*x^2 + 2*b* c^6*x + b^2*c^5) + 1/2*(c*e*x^2 + 2*(c*d - 3*b*e)*x)/c^4 - 3*(b*c*d - 2*b^ 2*e)*log(c*x + b)/c^5
Time = 0.14 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05 \[ \int \frac {x^6 (d+e x)}{\left (b x+c x^2\right )^3} \, dx=-\frac {3 \, {\left (b c d - 2 \, b^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{5}} + \frac {c^{3} e x^{2} + 2 \, c^{3} d x - 6 \, b c^{2} e x}{2 \, c^{6}} - \frac {5 \, b^{3} c d - 7 \, b^{4} e + 2 \, {\left (3 \, b^{2} c^{2} d - 4 \, b^{3} c e\right )} x}{2 \, {\left (c x + b\right )}^{2} c^{5}} \] Input:
integrate(x^6*(e*x+d)/(c*x^2+b*x)^3,x, algorithm="giac")
Output:
-3*(b*c*d - 2*b^2*e)*log(abs(c*x + b))/c^5 + 1/2*(c^3*e*x^2 + 2*c^3*d*x - 6*b*c^2*e*x)/c^6 - 1/2*(5*b^3*c*d - 7*b^4*e + 2*(3*b^2*c^2*d - 4*b^3*c*e)* x)/((c*x + b)^2*c^5)
Time = 5.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.15 \[ \int \frac {x^6 (d+e x)}{\left (b x+c x^2\right )^3} \, dx=x\,\left (\frac {d}{c^3}-\frac {3\,b\,e}{c^4}\right )+\frac {x\,\left (4\,b^3\,e-3\,b^2\,c\,d\right )+\frac {7\,b^4\,e-5\,b^3\,c\,d}{2\,c}}{b^2\,c^4+2\,b\,c^5\,x+c^6\,x^2}+\frac {e\,x^2}{2\,c^3}+\frac {\ln \left (b+c\,x\right )\,\left (6\,b^2\,e-3\,b\,c\,d\right )}{c^5} \] Input:
int((x^6*(d + e*x))/(b*x + c*x^2)^3,x)
Output:
x*(d/c^3 - (3*b*e)/c^4) + (x*(4*b^3*e - 3*b^2*c*d) + (7*b^4*e - 5*b^3*c*d) /(2*c))/(b^2*c^4 + c^6*x^2 + 2*b*c^5*x) + (e*x^2)/(2*c^3) + (log(b + c*x)* (6*b^2*e - 3*b*c*d))/c^5
Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.86 \[ \int \frac {x^6 (d+e x)}{\left (b x+c x^2\right )^3} \, dx=\frac {12 \,\mathrm {log}\left (c x +b \right ) b^{4} e -6 \,\mathrm {log}\left (c x +b \right ) b^{3} c d +24 \,\mathrm {log}\left (c x +b \right ) b^{3} c e x -12 \,\mathrm {log}\left (c x +b \right ) b^{2} c^{2} d x +12 \,\mathrm {log}\left (c x +b \right ) b^{2} c^{2} e \,x^{2}-6 \,\mathrm {log}\left (c x +b \right ) b \,c^{3} d \,x^{2}+6 b^{4} e -3 b^{3} c d -12 b^{2} c^{2} e \,x^{2}+6 b \,c^{3} d \,x^{2}-4 b \,c^{3} e \,x^{3}+2 c^{4} d \,x^{3}+c^{4} e \,x^{4}}{2 c^{5} \left (c^{2} x^{2}+2 b c x +b^{2}\right )} \] Input:
int(x^6*(e*x+d)/(c*x^2+b*x)^3,x)
Output:
(12*log(b + c*x)*b**4*e - 6*log(b + c*x)*b**3*c*d + 24*log(b + c*x)*b**3*c *e*x - 12*log(b + c*x)*b**2*c**2*d*x + 12*log(b + c*x)*b**2*c**2*e*x**2 - 6*log(b + c*x)*b*c**3*d*x**2 + 6*b**4*e - 3*b**3*c*d - 12*b**2*c**2*e*x**2 + 6*b*c**3*d*x**2 - 4*b*c**3*e*x**3 + 2*c**4*d*x**3 + c**4*e*x**4)/(2*c** 5*(b**2 + 2*b*c*x + c**2*x**2))