Integrand size = 52, antiderivative size = 94 \[ \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{(c+d x)^2} \, dx=-\frac {b c \left (b c^2+2 a d^2\right ) x}{d^4}+\frac {b \left (b c^2+2 a d^2\right ) x^2}{2 d^3}-\frac {b^2 c x^3}{3 d^2}+\frac {b^2 x^4}{4 d}+\frac {\left (b c^2+a d^2\right )^2 \log (c+d x)}{d^5} \] Output:
-b*c*(2*a*d^2+b*c^2)*x/d^4+1/2*b*(2*a*d^2+b*c^2)*x^2/d^3-1/3*b^2*c*x^3/d^2 +1/4*b^2*x^4/d+(a*d^2+b*c^2)^2*ln(d*x+c)/d^5
Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.84 \[ \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{(c+d x)^2} \, dx=\frac {b d x \left (12 a d^2 (-2 c+d x)+b \left (-12 c^3+6 c^2 d x-4 c d^2 x^2+3 d^3 x^3\right )\right )+12 \left (b c^2+a d^2\right )^2 \log (c+d x)}{12 d^5} \] Input:
Integrate[(a^2*c + a^2*d*x + 2*a*b*c*x^2 + 2*a*b*d*x^3 + b^2*c*x^4 + b^2*d *x^5)/(c + d*x)^2,x]
Output:
(b*d*x*(12*a*d^2*(-2*c + d*x) + b*(-12*c^3 + 6*c^2*d*x - 4*c*d^2*x^2 + 3*d ^3*x^3)) + 12*(b*c^2 + a*d^2)^2*Log[c + d*x])/(12*d^5)
Time = 0.49 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.096, Rules used = {2019, 1380, 27, 476, 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.
\(\displaystyle \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{(c+d x)^2} \, dx\) |
\(\Big \downarrow \) 2019 |
\(\displaystyle \int \frac {a^2+2 a b x^2+b^2 x^4}{c+d x}dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \frac {\int \frac {b^2 \left (b x^2+a\right )^2}{c+d x}dx}{b^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {\left (a+b x^2\right )^2}{c+d x}dx\) |
\(\Big \downarrow \) 476 |
\(\displaystyle \int \left (\frac {\left (a d^2+b c^2\right )^2}{d^4 (c+d x)}-\frac {b c \left (2 a d^2+b c^2\right )}{d^4}+\frac {b x \left (2 a d^2+b c^2\right )}{d^3}-\frac {b^2 c x^2}{d^2}+\frac {b^2 x^3}{d}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (a d^2+b c^2\right )^2 \log (c+d x)}{d^5}-\frac {b c x \left (2 a d^2+b c^2\right )}{d^4}+\frac {b x^2 \left (2 a d^2+b c^2\right )}{2 d^3}-\frac {b^2 c x^3}{3 d^2}+\frac {b^2 x^4}{4 d}\) |
Input:
Int[(a^2*c + a^2*d*x + 2*a*b*c*x^2 + 2*a*b*d*x^3 + b^2*c*x^4 + b^2*d*x^5)/ (c + d*x)^2,x]
Output:
-((b*c*(b*c^2 + 2*a*d^2)*x)/d^4) + (b*(b*c^2 + 2*a*d^2)*x^2)/(2*d^3) - (b^ 2*c*x^3)/(3*d^2) + (b^2*x^4)/(4*d) + ((b*c^2 + a*d^2)^2*Log[c + d*x])/d^5
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Time = 0.18 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.02
method | result | size |
default | \(-\frac {b \left (-\frac {b \,d^{3} x^{4}}{4}+\frac {b c \,x^{3} d^{2}}{3}-\frac {\left (2 a \,d^{2}+b \,c^{2}\right ) x^{2} d}{2}+x c \left (2 a \,d^{2}+b \,c^{2}\right )\right )}{d^{4}}+\frac {\left (a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}\right ) \ln \left (d x +c \right )}{d^{5}}\) | \(96\) |
risch | \(\frac {b^{2} x^{4}}{4 d}-\frac {b^{2} c \,x^{3}}{3 d^{2}}+\frac {b a \,x^{2}}{d}+\frac {b^{2} c^{2} x^{2}}{2 d^{3}}-\frac {2 b a c x}{d^{2}}-\frac {b^{2} c^{3} x}{d^{4}}+\frac {\ln \left (d x +c \right ) a^{2}}{d}+\frac {2 \ln \left (d x +c \right ) a b \,c^{2}}{d^{3}}+\frac {\ln \left (d x +c \right ) b^{2} c^{4}}{d^{5}}\) | \(114\) |
parallelrisch | \(\frac {3 x^{4} b^{2} d^{4}-4 b^{2} c \,x^{3} d^{3}+12 x^{2} a b \,d^{4}+6 x^{2} b^{2} c^{2} d^{2}+12 \ln \left (d x +c \right ) a^{2} d^{4}+24 \ln \left (d x +c \right ) a b \,c^{2} d^{2}+12 \ln \left (d x +c \right ) b^{2} c^{4}-24 x a b c \,d^{3}-12 x \,b^{2} c^{3} d}{12 d^{5}}\) | \(117\) |
norman | \(\frac {\frac {c \left (2 a b \,c^{2} d^{2}+b^{2} c^{4}\right )}{d^{5}}+\frac {b^{2} x^{5}}{4}+\frac {b \left (6 a \,d^{2}+b \,c^{2}\right ) x^{3}}{6 d^{2}}-\frac {c \,b^{2} x^{4}}{12 d}-\frac {b c \left (2 a \,d^{2}+b \,c^{2}\right ) x^{2}}{2 d^{3}}}{d x +c}+\frac {\left (a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}\right ) \ln \left (d x +c \right )}{d^{5}}\) | \(132\) |
Input:
int((b^2*d*x^5+b^2*c*x^4+2*a*b*d*x^3+2*a*b*c*x^2+a^2*d*x+a^2*c)/(d*x+c)^2, x,method=_RETURNVERBOSE)
Output:
-b/d^4*(-1/4*b*d^3*x^4+1/3*b*c*x^3*d^2-1/2*(2*a*d^2+b*c^2)*x^2*d+x*c*(2*a* d^2+b*c^2))+(a^2*d^4+2*a*b*c^2*d^2+b^2*c^4)/d^5*ln(d*x+c)
Time = 0.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.12 \[ \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{(c+d x)^2} \, dx=\frac {3 \, b^{2} d^{4} x^{4} - 4 \, b^{2} c d^{3} x^{3} + 6 \, {\left (b^{2} c^{2} d^{2} + 2 \, a b d^{4}\right )} x^{2} - 12 \, {\left (b^{2} c^{3} d + 2 \, a b c d^{3}\right )} x + 12 \, {\left (b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} \log \left (d x + c\right )}{12 \, d^{5}} \] Input:
integrate((b^2*d*x^5+b^2*c*x^4+2*a*b*d*x^3+2*a*b*c*x^2+a^2*d*x+a^2*c)/(d*x +c)^2,x, algorithm="fricas")
Output:
1/12*(3*b^2*d^4*x^4 - 4*b^2*c*d^3*x^3 + 6*(b^2*c^2*d^2 + 2*a*b*d^4)*x^2 - 12*(b^2*c^3*d + 2*a*b*c*d^3)*x + 12*(b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)*lo g(d*x + c))/d^5
Time = 0.16 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.94 \[ \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{(c+d x)^2} \, dx=- \frac {b^{2} c x^{3}}{3 d^{2}} + \frac {b^{2} x^{4}}{4 d} + x^{2} \left (\frac {a b}{d} + \frac {b^{2} c^{2}}{2 d^{3}}\right ) + x \left (- \frac {2 a b c}{d^{2}} - \frac {b^{2} c^{3}}{d^{4}}\right ) + \frac {\left (a d^{2} + b c^{2}\right )^{2} \log {\left (c + d x \right )}}{d^{5}} \] Input:
integrate((b**2*d*x**5+b**2*c*x**4+2*a*b*d*x**3+2*a*b*c*x**2+a**2*d*x+a**2 *c)/(d*x+c)**2,x)
Output:
-b**2*c*x**3/(3*d**2) + b**2*x**4/(4*d) + x**2*(a*b/d + b**2*c**2/(2*d**3) ) + x*(-2*a*b*c/d**2 - b**2*c**3/d**4) + (a*d**2 + b*c**2)**2*log(c + d*x) /d**5
Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.12 \[ \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{(c+d x)^2} \, dx=\frac {3 \, b^{2} d^{3} x^{4} - 4 \, b^{2} c d^{2} x^{3} + 6 \, {\left (b^{2} c^{2} d + 2 \, a b d^{3}\right )} x^{2} - 12 \, {\left (b^{2} c^{3} + 2 \, a b c d^{2}\right )} x}{12 \, d^{4}} + \frac {{\left (b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} \log \left (d x + c\right )}{d^{5}} \] Input:
integrate((b^2*d*x^5+b^2*c*x^4+2*a*b*d*x^3+2*a*b*c*x^2+a^2*d*x+a^2*c)/(d*x +c)^2,x, algorithm="maxima")
Output:
1/12*(3*b^2*d^3*x^4 - 4*b^2*c*d^2*x^3 + 6*(b^2*c^2*d + 2*a*b*d^3)*x^2 - 12 *(b^2*c^3 + 2*a*b*c*d^2)*x)/d^4 + (b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)*log( d*x + c)/d^5
Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (88) = 176\).
Time = 0.13 (sec) , antiderivative size = 365, normalized size of antiderivative = 3.88 \[ \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{(c+d x)^2} \, dx=-\frac {1}{12} \, b^{2} d {\left (\frac {{\left (d x + c\right )}^{4} {\left (\frac {20 \, c}{d x + c} - \frac {60 \, c^{2}}{{\left (d x + c\right )}^{2}} + \frac {120 \, c^{3}}{{\left (d x + c\right )}^{3}} - 3\right )}}{d^{6}} + \frac {60 \, c^{4} \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{d^{6}} - \frac {12 \, c^{5}}{{\left (d x + c\right )} d^{6}}\right )} - \frac {1}{3} \, b^{2} c {\left (\frac {{\left (d x + c\right )}^{3} {\left (\frac {6 \, c}{d x + c} - \frac {18 \, c^{2}}{{\left (d x + c\right )}^{2}} - 1\right )}}{d^{5}} - \frac {12 \, c^{3} \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{d^{5}} + \frac {3 \, c^{4}}{{\left (d x + c\right )} d^{5}}\right )} - a b d {\left (\frac {{\left (d x + c\right )}^{2} {\left (\frac {6 \, c}{d x + c} - 1\right )}}{d^{4}} + \frac {6 \, c^{2} \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{d^{4}} - \frac {2 \, c^{3}}{{\left (d x + c\right )} d^{4}}\right )} + 2 \, a b c {\left (\frac {2 \, c \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{d^{3}} + \frac {d x + c}{d^{3}} - \frac {c^{2}}{{\left (d x + c\right )} d^{3}}\right )} - a^{2} {\left (\frac {\log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{d} - \frac {c}{{\left (d x + c\right )} d}\right )} - \frac {a^{2} c}{{\left (d x + c\right )} d} \] Input:
integrate((b^2*d*x^5+b^2*c*x^4+2*a*b*d*x^3+2*a*b*c*x^2+a^2*d*x+a^2*c)/(d*x +c)^2,x, algorithm="giac")
Output:
-1/12*b^2*d*((d*x + c)^4*(20*c/(d*x + c) - 60*c^2/(d*x + c)^2 + 120*c^3/(d *x + c)^3 - 3)/d^6 + 60*c^4*log(abs(d*x + c)/((d*x + c)^2*abs(d)))/d^6 - 1 2*c^5/((d*x + c)*d^6)) - 1/3*b^2*c*((d*x + c)^3*(6*c/(d*x + c) - 18*c^2/(d *x + c)^2 - 1)/d^5 - 12*c^3*log(abs(d*x + c)/((d*x + c)^2*abs(d)))/d^5 + 3 *c^4/((d*x + c)*d^5)) - a*b*d*((d*x + c)^2*(6*c/(d*x + c) - 1)/d^4 + 6*c^2 *log(abs(d*x + c)/((d*x + c)^2*abs(d)))/d^4 - 2*c^3/((d*x + c)*d^4)) + 2*a *b*c*(2*c*log(abs(d*x + c)/((d*x + c)^2*abs(d)))/d^3 + (d*x + c)/d^3 - c^2 /((d*x + c)*d^3)) - a^2*(log(abs(d*x + c)/((d*x + c)^2*abs(d)))/d - c/((d* x + c)*d)) - a^2*c/((d*x + c)*d)
Time = 22.51 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.13 \[ \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{(c+d x)^2} \, dx=x^2\,\left (\frac {b^2\,c^2}{2\,d^3}+\frac {a\,b}{d}\right )+\frac {\ln \left (c+d\,x\right )\,\left (a^2\,d^4+2\,a\,b\,c^2\,d^2+b^2\,c^4\right )}{d^5}+\frac {b^2\,x^4}{4\,d}-\frac {b^2\,c\,x^3}{3\,d^2}-\frac {c\,x\,\left (\frac {b^2\,c^2}{d^3}+\frac {2\,a\,b}{d}\right )}{d} \] Input:
int((a^2*c + b^2*c*x^4 + b^2*d*x^5 + a^2*d*x + 2*a*b*c*x^2 + 2*a*b*d*x^3)/ (c + d*x)^2,x)
Output:
x^2*((b^2*c^2)/(2*d^3) + (a*b)/d) + (log(c + d*x)*(a^2*d^4 + b^2*c^4 + 2*a *b*c^2*d^2))/d^5 + (b^2*x^4)/(4*d) - (b^2*c*x^3)/(3*d^2) - (c*x*((b^2*c^2) /d^3 + (2*a*b)/d))/d
Time = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.23 \[ \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{(c+d x)^2} \, dx=\frac {12 \,\mathrm {log}\left (d x +c \right ) a^{2} d^{4}+24 \,\mathrm {log}\left (d x +c \right ) a b \,c^{2} d^{2}+12 \,\mathrm {log}\left (d x +c \right ) b^{2} c^{4}-24 a b c \,d^{3} x +12 a b \,d^{4} x^{2}-12 b^{2} c^{3} d x +6 b^{2} c^{2} d^{2} x^{2}-4 b^{2} c \,d^{3} x^{3}+3 b^{2} d^{4} x^{4}}{12 d^{5}} \] Input:
int((b^2*d*x^5+b^2*c*x^4+2*a*b*d*x^3+2*a*b*c*x^2+a^2*d*x+a^2*c)/(d*x+c)^2, x)
Output:
(12*log(c + d*x)*a**2*d**4 + 24*log(c + d*x)*a*b*c**2*d**2 + 12*log(c + d* x)*b**2*c**4 - 24*a*b*c*d**3*x + 12*a*b*d**4*x**2 - 12*b**2*c**3*d*x + 6*b **2*c**2*d**2*x**2 - 4*b**2*c*d**3*x**3 + 3*b**2*d**4*x**4)/(12*d**5)