Integrand size = 18, antiderivative size = 110 \[ \int \frac {x^4}{(a+b x) (c+d x)^2} \, dx=-\frac {(2 b c+a d) x}{b^2 d^3}+\frac {x^2}{2 b d^2}+\frac {c^4}{d^4 (b c-a d) (c+d x)}+\frac {a^4 \log (a+b x)}{b^3 (b c-a d)^2}+\frac {c^3 (3 b c-4 a d) \log (c+d x)}{d^4 (b c-a d)^2} \] Output:
-(a*d+2*b*c)*x/b^2/d^3+1/2*x^2/b/d^2+c^4/d^4/(-a*d+b*c)/(d*x+c)+a^4*ln(b*x +a)/b^3/(-a*d+b*c)^2+c^3*(-4*a*d+3*b*c)*ln(d*x+c)/d^4/(-a*d+b*c)^2
Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.97 \[ \int \frac {x^4}{(a+b x) (c+d x)^2} \, dx=\frac {a^4 \log (a+b x)}{b^3 (b c-a d)^2}+\frac {-\frac {2 a d^2 x}{b^2}+\frac {d x (-4 c+d x)}{b}+\frac {2 c^4}{(b c-a d) (c+d x)}+\frac {2 c^3 (3 b c-4 a d) \log (c+d x)}{(b c-a d)^2}}{2 d^4} \] Input:
Integrate[x^4/((a + b*x)*(c + d*x)^2),x]
Output:
(a^4*Log[a + b*x])/(b^3*(b*c - a*d)^2) + ((-2*a*d^2*x)/b^2 + (d*x*(-4*c + d*x))/b + (2*c^4)/((b*c - a*d)*(c + d*x)) + (2*c^3*(3*b*c - 4*a*d)*Log[c + d*x])/(b*c - a*d)^2)/(2*d^4)
Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {99, 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.
\(\displaystyle \int \frac {x^4}{(a+b x) (c+d x)^2} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {a^4}{b^2 (a+b x) (b c-a d)^2}+\frac {-a d-2 b c}{b^2 d^3}+\frac {c^4}{d^3 (c+d x)^2 (a d-b c)}+\frac {c^3 (3 b c-4 a d)}{d^3 (c+d x) (a d-b c)^2}+\frac {x}{b d^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^4 \log (a+b x)}{b^3 (b c-a d)^2}-\frac {x (a d+2 b c)}{b^2 d^3}+\frac {c^4}{d^4 (c+d x) (b c-a d)}+\frac {c^3 (3 b c-4 a d) \log (c+d x)}{d^4 (b c-a d)^2}+\frac {x^2}{2 b d^2}\) |
Input:
Int[x^4/((a + b*x)*(c + d*x)^2),x]
Output:
-(((2*b*c + a*d)*x)/(b^2*d^3)) + x^2/(2*b*d^2) + c^4/(d^4*(b*c - a*d)*(c + d*x)) + (a^4*Log[a + b*x])/(b^3*(b*c - a*d)^2) + (c^3*(3*b*c - 4*a*d)*Log [c + d*x])/(d^4*(b*c - a*d)^2)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 0.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {-\frac {1}{2} b d \,x^{2}+a d x +2 b c x}{b^{2} d^{3}}-\frac {c^{4}}{d^{4} \left (a d -b c \right ) \left (x d +c \right )}-\frac {c^{3} \left (4 a d -3 b c \right ) \ln \left (x d +c \right )}{d^{4} \left (a d -b c \right )^{2}}+\frac {a^{4} \ln \left (b x +a \right )}{b^{3} \left (a d -b c \right )^{2}}\) | \(108\) |
norman | \(\frac {\frac {\left (a^{2} d^{2} c +a b \,c^{2} d -3 b^{2} c^{3}\right ) c}{d^{4} b^{2} \left (a d -b c \right )}+\frac {x^{3}}{2 b d}-\frac {\left (2 a d +3 b c \right ) x^{2}}{2 b^{2} d^{2}}}{x d +c}+\frac {a^{4} \ln \left (b x +a \right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{3}}-\frac {c^{3} \left (4 a d -3 b c \right ) \ln \left (x d +c \right )}{d^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(166\) |
risch | \(\frac {x^{2}}{2 b \,d^{2}}-\frac {a x}{b^{2} d^{2}}-\frac {2 c x}{b \,d^{3}}-\frac {c^{4}}{d^{4} \left (a d -b c \right ) \left (x d +c \right )}+\frac {a^{4} \ln \left (-b x -a \right )}{b^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {4 c^{3} \ln \left (x d +c \right ) a}{d^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {3 c^{4} \ln \left (x d +c \right ) b}{d^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(173\) |
parallelrisch | \(\frac {x^{3} a^{2} b^{2} d^{5}-2 x^{3} a \,b^{3} c \,d^{4}+x^{3} b^{4} c^{2} d^{3}+2 \ln \left (b x +a \right ) x \,a^{4} d^{5}-8 \ln \left (x d +c \right ) x a \,b^{3} c^{3} d^{2}+6 \ln \left (x d +c \right ) x \,b^{4} c^{4} d -2 x^{2} a^{3} b \,d^{5}+x^{2} a^{2} b^{2} c \,d^{4}+4 x^{2} a \,b^{3} c^{2} d^{3}-3 x^{2} b^{4} c^{3} d^{2}+2 \ln \left (b x +a \right ) a^{4} c \,d^{4}-8 \ln \left (x d +c \right ) a \,b^{3} c^{4} d +6 \ln \left (x d +c \right ) b^{4} c^{5}+2 a^{3} b \,c^{2} d^{3}-8 a \,b^{3} c^{4} d +6 b^{4} c^{5}}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (x d +c \right ) b^{3} d^{4}}\) | \(259\) |
Input:
int(x^4/(b*x+a)/(d*x+c)^2,x,method=_RETURNVERBOSE)
Output:
-1/b^2/d^3*(-1/2*b*d*x^2+a*d*x+2*b*c*x)-1/d^4*c^4/(a*d-b*c)/(d*x+c)-1/d^4* c^3*(4*a*d-3*b*c)/(a*d-b*c)^2*ln(d*x+c)+1/b^3*a^4/(a*d-b*c)^2*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (108) = 216\).
Time = 0.09 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.59 \[ \int \frac {x^4}{(a+b x) (c+d x)^2} \, dx=\frac {2 \, b^{4} c^{5} - 2 \, a b^{3} c^{4} d + {\left (b^{4} c^{2} d^{3} - 2 \, a b^{3} c d^{4} + a^{2} b^{2} d^{5}\right )} x^{3} - {\left (3 \, b^{4} c^{3} d^{2} - 4 \, a b^{3} c^{2} d^{3} - a^{2} b^{2} c d^{4} + 2 \, a^{3} b d^{5}\right )} x^{2} - 2 \, {\left (2 \, b^{4} c^{4} d - 3 \, a b^{3} c^{3} d^{2} + a^{3} b c d^{4}\right )} x + 2 \, {\left (a^{4} d^{5} x + a^{4} c d^{4}\right )} \log \left (b x + a\right ) + 2 \, {\left (3 \, b^{4} c^{5} - 4 \, a b^{3} c^{4} d + {\left (3 \, b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (b^{5} c^{3} d^{4} - 2 \, a b^{4} c^{2} d^{5} + a^{2} b^{3} c d^{6} + {\left (b^{5} c^{2} d^{5} - 2 \, a b^{4} c d^{6} + a^{2} b^{3} d^{7}\right )} x\right )}} \] Input:
integrate(x^4/(b*x+a)/(d*x+c)^2,x, algorithm="fricas")
Output:
1/2*(2*b^4*c^5 - 2*a*b^3*c^4*d + (b^4*c^2*d^3 - 2*a*b^3*c*d^4 + a^2*b^2*d^ 5)*x^3 - (3*b^4*c^3*d^2 - 4*a*b^3*c^2*d^3 - a^2*b^2*c*d^4 + 2*a^3*b*d^5)*x ^2 - 2*(2*b^4*c^4*d - 3*a*b^3*c^3*d^2 + a^3*b*c*d^4)*x + 2*(a^4*d^5*x + a^ 4*c*d^4)*log(b*x + a) + 2*(3*b^4*c^5 - 4*a*b^3*c^4*d + (3*b^4*c^4*d - 4*a* b^3*c^3*d^2)*x)*log(d*x + c))/(b^5*c^3*d^4 - 2*a*b^4*c^2*d^5 + a^2*b^3*c*d ^6 + (b^5*c^2*d^5 - 2*a*b^4*c*d^6 + a^2*b^3*d^7)*x)
Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (100) = 200\).
Time = 1.99 (sec) , antiderivative size = 428, normalized size of antiderivative = 3.89 \[ \int \frac {x^4}{(a+b x) (c+d x)^2} \, dx=\frac {a^{4} \log {\left (x + \frac {\frac {a^{7} d^{6}}{b \left (a d - b c\right )^{2}} - \frac {3 a^{6} c d^{5}}{\left (a d - b c\right )^{2}} + \frac {3 a^{5} b c^{2} d^{4}}{\left (a d - b c\right )^{2}} - \frac {a^{4} b^{2} c^{3} d^{3}}{\left (a d - b c\right )^{2}} + a^{4} c d^{3} + 4 a^{2} b^{2} c^{3} d - 3 a b^{3} c^{4}}{a^{4} d^{4} + 4 a b^{3} c^{3} d - 3 b^{4} c^{4}} \right )}}{b^{3} \left (a d - b c\right )^{2}} - \frac {c^{4}}{a c d^{5} - b c^{2} d^{4} + x \left (a d^{6} - b c d^{5}\right )} - \frac {c^{3} \cdot \left (4 a d - 3 b c\right ) \log {\left (x + \frac {a^{4} c d^{3} - \frac {a^{3} b^{2} c^{3} d^{2} \cdot \left (4 a d - 3 b c\right )}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b^{3} c^{4} d \left (4 a d - 3 b c\right )}{\left (a d - b c\right )^{2}} + 4 a^{2} b^{2} c^{3} d - \frac {3 a b^{4} c^{5} \cdot \left (4 a d - 3 b c\right )}{\left (a d - b c\right )^{2}} - 3 a b^{3} c^{4} + \frac {b^{5} c^{6} \cdot \left (4 a d - 3 b c\right )}{d \left (a d - b c\right )^{2}}}{a^{4} d^{4} + 4 a b^{3} c^{3} d - 3 b^{4} c^{4}} \right )}}{d^{4} \left (a d - b c\right )^{2}} + x \left (- \frac {a}{b^{2} d^{2}} - \frac {2 c}{b d^{3}}\right ) + \frac {x^{2}}{2 b d^{2}} \] Input:
integrate(x**4/(b*x+a)/(d*x+c)**2,x)
Output:
a**4*log(x + (a**7*d**6/(b*(a*d - b*c)**2) - 3*a**6*c*d**5/(a*d - b*c)**2 + 3*a**5*b*c**2*d**4/(a*d - b*c)**2 - a**4*b**2*c**3*d**3/(a*d - b*c)**2 + a**4*c*d**3 + 4*a**2*b**2*c**3*d - 3*a*b**3*c**4)/(a**4*d**4 + 4*a*b**3*c **3*d - 3*b**4*c**4))/(b**3*(a*d - b*c)**2) - c**4/(a*c*d**5 - b*c**2*d**4 + x*(a*d**6 - b*c*d**5)) - c**3*(4*a*d - 3*b*c)*log(x + (a**4*c*d**3 - a* *3*b**2*c**3*d**2*(4*a*d - 3*b*c)/(a*d - b*c)**2 + 3*a**2*b**3*c**4*d*(4*a *d - 3*b*c)/(a*d - b*c)**2 + 4*a**2*b**2*c**3*d - 3*a*b**4*c**5*(4*a*d - 3 *b*c)/(a*d - b*c)**2 - 3*a*b**3*c**4 + b**5*c**6*(4*a*d - 3*b*c)/(d*(a*d - b*c)**2))/(a**4*d**4 + 4*a*b**3*c**3*d - 3*b**4*c**4))/(d**4*(a*d - b*c)* *2) + x*(-a/(b**2*d**2) - 2*c/(b*d**3)) + x**2/(2*b*d**2)
Time = 0.04 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.37 \[ \int \frac {x^4}{(a+b x) (c+d x)^2} \, dx=\frac {a^{4} \log \left (b x + a\right )}{b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}} + \frac {c^{4}}{b c^{2} d^{4} - a c d^{5} + {\left (b c d^{5} - a d^{6}\right )} x} + \frac {{\left (3 \, b c^{4} - 4 \, a c^{3} d\right )} \log \left (d x + c\right )}{b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}} + \frac {b d x^{2} - 2 \, {\left (2 \, b c + a d\right )} x}{2 \, b^{2} d^{3}} \] Input:
integrate(x^4/(b*x+a)/(d*x+c)^2,x, algorithm="maxima")
Output:
a^4*log(b*x + a)/(b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2) + c^4/(b*c^2*d^4 - a*c*d^5 + (b*c*d^5 - a*d^6)*x) + (3*b*c^4 - 4*a*c^3*d)*log(d*x + c)/(b^2*c ^2*d^4 - 2*a*b*c*d^5 + a^2*d^6) + 1/2*(b*d*x^2 - 2*(2*b*c + a*d)*x)/(b^2*d ^3)
Time = 0.12 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.68 \[ \int \frac {x^4}{(a+b x) (c+d x)^2} \, dx=\frac {c^{4} d^{3}}{{\left (b c d^{7} - a d^{8}\right )} {\left (d x + c\right )}} + \frac {a^{4} d \log \left ({\left | b - \frac {b c}{d x + c} + \frac {a d}{d x + c} \right |}\right )}{b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}} + \frac {{\left (b^{2} - \frac {2 \, {\left (3 \, b^{2} c d + a b d^{2}\right )}}{{\left (d x + c\right )} d}\right )} {\left (d x + c\right )}^{2}}{2 \, b^{3} d^{4}} - \frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{b^{3} d^{4}} \] Input:
integrate(x^4/(b*x+a)/(d*x+c)^2,x, algorithm="giac")
Output:
c^4*d^3/((b*c*d^7 - a*d^8)*(d*x + c)) + a^4*d*log(abs(b - b*c/(d*x + c) + a*d/(d*x + c)))/(b^5*c^2*d - 2*a*b^4*c*d^2 + a^2*b^3*d^3) + 1/2*(b^2 - 2*( 3*b^2*c*d + a*b*d^2)/((d*x + c)*d))*(d*x + c)^2/(b^3*d^4) - (3*b^2*c^2 + 2 *a*b*c*d + a^2*d^2)*log(abs(d*x + c)/((d*x + c)^2*abs(d)))/(b^3*d^4)
Time = 0.62 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.31 \[ \int \frac {x^4}{(a+b x) (c+d x)^2} \, dx=\frac {\ln \left (c+d\,x\right )\,\left (3\,b\,c^4-4\,a\,c^3\,d\right )}{a^2\,d^6-2\,a\,b\,c\,d^5+b^2\,c^2\,d^4}+\frac {x^2}{2\,b\,d^2}+\frac {a^4\,\ln \left (a+b\,x\right )}{b^3\,{\left (a\,d-b\,c\right )}^2}-\frac {x\,\left (a\,d^2+2\,b\,c\,d\right )}{b^2\,d^4}-\frac {b^2\,c^4}{d\,\left (x\,b^2\,d^4+c\,b^2\,d^3\right )\,\left (a\,d-b\,c\right )} \] Input:
int(x^4/((a + b*x)*(c + d*x)^2),x)
Output:
(log(c + d*x)*(3*b*c^4 - 4*a*c^3*d))/(a^2*d^6 + b^2*c^2*d^4 - 2*a*b*c*d^5) + x^2/(2*b*d^2) + (a^4*log(a + b*x))/(b^3*(a*d - b*c)^2) - (x*(a*d^2 + 2* b*c*d))/(b^2*d^4) - (b^2*c^4)/(d*(b^2*c*d^3 + b^2*d^4*x)*(a*d - b*c))
Time = 0.17 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.58 \[ \int \frac {x^4}{(a+b x) (c+d x)^2} \, dx=\frac {2 \,\mathrm {log}\left (b x +a \right ) a^{4} c \,d^{4}+2 \,\mathrm {log}\left (b x +a \right ) a^{4} d^{5} x -8 \,\mathrm {log}\left (d x +c \right ) a \,b^{3} c^{4} d -8 \,\mathrm {log}\left (d x +c \right ) a \,b^{3} c^{3} d^{2} x +6 \,\mathrm {log}\left (d x +c \right ) b^{4} c^{5}+6 \,\mathrm {log}\left (d x +c \right ) b^{4} c^{4} d x -2 a^{3} b c \,d^{4} x -2 a^{3} b \,d^{5} x^{2}+a^{2} b^{2} c \,d^{4} x^{2}+a^{2} b^{2} d^{5} x^{3}+8 a \,b^{3} c^{3} d^{2} x +4 a \,b^{3} c^{2} d^{3} x^{2}-2 a \,b^{3} c \,d^{4} x^{3}-6 b^{4} c^{4} d x -3 b^{4} c^{3} d^{2} x^{2}+b^{4} c^{2} d^{3} x^{3}}{2 b^{3} d^{4} \left (a^{2} d^{3} x -2 a b c \,d^{2} x +b^{2} c^{2} d x +a^{2} c \,d^{2}-2 a b \,c^{2} d +b^{2} c^{3}\right )} \] Input:
int(x^4/(b*x+a)/(d*x+c)^2,x)
Output:
(2*log(a + b*x)*a**4*c*d**4 + 2*log(a + b*x)*a**4*d**5*x - 8*log(c + d*x)* a*b**3*c**4*d - 8*log(c + d*x)*a*b**3*c**3*d**2*x + 6*log(c + d*x)*b**4*c* *5 + 6*log(c + d*x)*b**4*c**4*d*x - 2*a**3*b*c*d**4*x - 2*a**3*b*d**5*x**2 + a**2*b**2*c*d**4*x**2 + a**2*b**2*d**5*x**3 + 8*a*b**3*c**3*d**2*x + 4* a*b**3*c**2*d**3*x**2 - 2*a*b**3*c*d**4*x**3 - 6*b**4*c**4*d*x - 3*b**4*c* *3*d**2*x**2 + b**4*c**2*d**3*x**3)/(2*b**3*d**4*(a**2*c*d**2 + a**2*d**3* x - 2*a*b*c**2*d - 2*a*b*c*d**2*x + b**2*c**3 + b**2*c**2*d*x))