Integrand size = 23, antiderivative size = 103 \[ \int \frac {(c+d x)^4}{(a-b x) (a+b x)} \, dx=-\frac {d^2 \left (6 b^2 c^2+a^2 d^2\right ) x}{b^4}-\frac {2 c d^3 x^2}{b^2}-\frac {d^4 x^3}{3 b^2}-\frac {(b c+a d)^4 \log (a-b x)}{2 a b^5}+\frac {(b c-a d)^4 \log (a+b x)}{2 a b^5} \]
-d^2*(a^2*d^2+6*b^2*c^2)*x/b^4-2*c*d^3*x^2/b^2-1/3*d^4*x^3/b^2-1/2*(a*d+b* c)^4*ln(-b*x+a)/a/b^5+1/2*(-a*d+b*c)^4*ln(b*x+a)/a/b^5
Time = 0.15 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.83 \[ \int \frac {(c+d x)^4}{(a-b x) (a+b x)} \, dx=\frac {-2 a b d^2 x \left (3 a^2 d^2+b^2 \left (18 c^2+6 c d x+d^2 x^2\right )\right )-3 (b c+a d)^4 \log (a-b x)+3 (b c-a d)^4 \log (a+b x)}{6 a b^5} \]
(-2*a*b*d^2*x*(3*a^2*d^2 + b^2*(18*c^2 + 6*c*d*x + d^2*x^2)) - 3*(b*c + a* d)^4*Log[a - b*x] + 3*(b*c - a*d)^4*Log[a + b*x])/(6*a*b^5)
Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {82, 477, 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.
\(\displaystyle \int \frac {(c+d x)^4}{(a-b x) (a+b x)} \, dx\) |
\(\Big \downarrow \) 82 |
\(\displaystyle \int \frac {(c+d x)^4}{a^2-b^2 x^2}dx\) |
\(\Big \downarrow \) 477 |
\(\displaystyle \frac {\int \left (-\frac {a^2 x^2 d^4}{b^2}-\frac {4 a^2 c x d^3}{b^2}-\frac {a^2 \left (6 b^2 c^2+a^2 d^2\right ) d^2}{b^4}+\frac {a (b c+a d)^4}{2 b^4 (a-b x)}+\frac {a (b c-a d)^4}{2 b^4 (a+b x)}\right )dx}{a^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {2 a^2 c d^3 x^2}{b^2}-\frac {a^2 d^4 x^3}{3 b^2}-\frac {a^2 d^2 x \left (a^2 d^2+6 b^2 c^2\right )}{b^4}-\frac {a (a d+b c)^4 \log (a-b x)}{2 b^5}+\frac {a (b c-a d)^4 \log (a+b x)}{2 b^5}}{a^2}\) |
(-((a^2*d^2*(6*b^2*c^2 + a^2*d^2)*x)/b^4) - (2*a^2*c*d^3*x^2)/b^2 - (a^2*d ^4*x^3)/(3*b^2) - (a*(b*c + a*d)^4*Log[a - b*x])/(2*b^5) + (a*(b*c - a*d)^ 4*Log[a + b*x])/(2*b^5))/a^2
3.16.30.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) )^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ a^p Int[ExpandIntegrand[(c + d*x)^n*(1 - Rt[-b/a, 2]*x)^p*(1 + Rt[-b/a, 2 ]*x)^p, x], x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[n] & & NiceSqrtQ[-b/a] && !FractionalPowerFactorQ[Rt[-b/a, 2]]
Time = 2.63 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.72
method | result | size |
norman | \(-\frac {d^{4} x^{3}}{3 b^{2}}-\frac {2 c \,d^{3} x^{2}}{b^{2}}-\frac {d^{2} \left (a^{2} d^{2}+6 b^{2} c^{2}\right ) x}{b^{4}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (b x +a \right )}{2 b^{5} a}-\frac {\left (a^{4} d^{4}+4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (-b x +a \right )}{2 a \,b^{5}}\) | \(177\) |
default | \(-\frac {d^{2} \left (\frac {1}{3} d^{2} x^{3} b^{2}+2 x^{2} b^{2} c d +a^{2} d^{2} x +6 b^{2} c^{2} x \right )}{b^{4}}+\frac {\left (-a^{4} d^{4}-4 a^{3} b c \,d^{3}-6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -b^{4} c^{4}\right ) \ln \left (-b x +a \right )}{2 b^{5} a}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (b x +a \right )}{2 b^{5} a}\) | \(178\) |
parallelrisch | \(-\frac {2 d^{4} x^{3} a \,b^{3}+12 c \,d^{3} x^{2} a \,b^{3}+3 \ln \left (b x -a \right ) a^{4} d^{4}+12 \ln \left (b x -a \right ) a^{3} b c \,d^{3}+18 \ln \left (b x -a \right ) a^{2} b^{2} c^{2} d^{2}+12 \ln \left (b x -a \right ) a \,b^{3} c^{3} d +3 \ln \left (b x -a \right ) b^{4} c^{4}-3 \ln \left (b x +a \right ) a^{4} d^{4}+12 \ln \left (b x +a \right ) a^{3} b c \,d^{3}-18 \ln \left (b x +a \right ) a^{2} b^{2} c^{2} d^{2}+12 \ln \left (b x +a \right ) a \,b^{3} c^{3} d -3 \ln \left (b x +a \right ) b^{4} c^{4}+6 x \,a^{3} b \,d^{4}+36 x a \,b^{3} c^{2} d^{2}}{6 a \,b^{5}}\) | \(228\) |
risch | \(-\frac {d^{4} x^{3}}{3 b^{2}}-\frac {2 c \,d^{3} x^{2}}{b^{2}}-\frac {d^{4} a^{2} x}{b^{4}}-\frac {6 d^{2} c^{2} x}{b^{2}}-\frac {a^{3} \ln \left (b x -a \right ) d^{4}}{2 b^{5}}-\frac {2 a^{2} \ln \left (b x -a \right ) c \,d^{3}}{b^{4}}-\frac {3 a \ln \left (b x -a \right ) c^{2} d^{2}}{b^{3}}-\frac {2 \ln \left (b x -a \right ) c^{3} d}{b^{2}}-\frac {\ln \left (b x -a \right ) c^{4}}{2 b a}+\frac {a^{3} \ln \left (-b x -a \right ) d^{4}}{2 b^{5}}-\frac {2 a^{2} \ln \left (-b x -a \right ) c \,d^{3}}{b^{4}}+\frac {3 a \ln \left (-b x -a \right ) c^{2} d^{2}}{b^{3}}-\frac {2 \ln \left (-b x -a \right ) c^{3} d}{b^{2}}+\frac {\ln \left (-b x -a \right ) c^{4}}{2 b a}\) | \(244\) |
-1/3*d^4*x^3/b^2-2*c*d^3*x^2/b^2-d^2*(a^2*d^2+6*b^2*c^2)*x/b^4+1/2/b^5*(a^ 4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/a*ln(b*x+a)-1 /2*(a^4*d^4+4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2+4*a*b^3*c^3*d+b^4*c^4)/a/b^5*l n(-b*x+a)
Time = 0.23 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.69 \[ \int \frac {(c+d x)^4}{(a-b x) (a+b x)} \, dx=-\frac {2 \, a b^{3} d^{4} x^{3} + 12 \, a b^{3} c d^{3} x^{2} + 6 \, {\left (6 \, a b^{3} c^{2} d^{2} + a^{3} b d^{4}\right )} x - 3 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x + a\right ) + 3 \, {\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x - a\right )}{6 \, a b^{5}} \]
-1/6*(2*a*b^3*d^4*x^3 + 12*a*b^3*c*d^3*x^2 + 6*(6*a*b^3*c^2*d^2 + a^3*b*d^ 4)*x - 3*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^ 4*d^4)*log(b*x + a) + 3*(b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 4*a ^3*b*c*d^3 + a^4*d^4)*log(b*x - a))/(a*b^5)
Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (92) = 184\).
Time = 0.70 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.08 \[ \int \frac {(c+d x)^4}{(a-b x) (a+b x)} \, dx=- x \left (\frac {a^{2} d^{4}}{b^{4}} + \frac {6 c^{2} d^{2}}{b^{2}}\right ) - \frac {2 c d^{3} x^{2}}{b^{2}} - \frac {d^{4} x^{3}}{3 b^{2}} + \frac {\left (a d - b c\right )^{4} \log {\left (x + \frac {4 a^{4} c d^{3} + 4 a^{2} b^{2} c^{3} d + \frac {a \left (a d - b c\right )^{4}}{b}}{a^{4} d^{4} + 6 a^{2} b^{2} c^{2} d^{2} + b^{4} c^{4}} \right )}}{2 a b^{5}} - \frac {\left (a d + b c\right )^{4} \log {\left (x + \frac {4 a^{4} c d^{3} + 4 a^{2} b^{2} c^{3} d - \frac {a \left (a d + b c\right )^{4}}{b}}{a^{4} d^{4} + 6 a^{2} b^{2} c^{2} d^{2} + b^{4} c^{4}} \right )}}{2 a b^{5}} \]
-x*(a**2*d**4/b**4 + 6*c**2*d**2/b**2) - 2*c*d**3*x**2/b**2 - d**4*x**3/(3 *b**2) + (a*d - b*c)**4*log(x + (4*a**4*c*d**3 + 4*a**2*b**2*c**3*d + a*(a *d - b*c)**4/b)/(a**4*d**4 + 6*a**2*b**2*c**2*d**2 + b**4*c**4))/(2*a*b**5 ) - (a*d + b*c)**4*log(x + (4*a**4*c*d**3 + 4*a**2*b**2*c**3*d - a*(a*d + b*c)**4/b)/(a**4*d**4 + 6*a**2*b**2*c**2*d**2 + b**4*c**4))/(2*a*b**5)
Time = 0.19 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.74 \[ \int \frac {(c+d x)^4}{(a-b x) (a+b x)} \, dx=-\frac {b^{2} d^{4} x^{3} + 6 \, b^{2} c d^{3} x^{2} + 3 \, {\left (6 \, b^{2} c^{2} d^{2} + a^{2} d^{4}\right )} x}{3 \, b^{4}} + \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x + a\right )}{2 \, a b^{5}} - \frac {{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x - a\right )}{2 \, a b^{5}} \]
-1/3*(b^2*d^4*x^3 + 6*b^2*c*d^3*x^2 + 3*(6*b^2*c^2*d^2 + a^2*d^4)*x)/b^4 + 1/2*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^ 4)*log(b*x + a)/(a*b^5) - 1/2*(b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + a^4*d^4)*log(b*x - a)/(a*b^5)
Time = 0.28 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.78 \[ \int \frac {(c+d x)^4}{(a-b x) (a+b x)} \, dx=\frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{2 \, a b^{5}} - \frac {{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left ({\left | b x - a \right |}\right )}{2 \, a b^{5}} - \frac {b^{4} d^{4} x^{3} + 6 \, b^{4} c d^{3} x^{2} + 18 \, b^{4} c^{2} d^{2} x + 3 \, a^{2} b^{2} d^{4} x}{3 \, b^{6}} \]
1/2*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4 )*log(abs(b*x + a))/(a*b^5) - 1/2*(b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2 *d^2 + 4*a^3*b*c*d^3 + a^4*d^4)*log(abs(b*x - a))/(a*b^5) - 1/3*(b^4*d^4*x ^3 + 6*b^4*c*d^3*x^2 + 18*b^4*c^2*d^2*x + 3*a^2*b^2*d^4*x)/b^6
Time = 1.56 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.71 \[ \int \frac {(c+d x)^4}{(a-b x) (a+b x)} \, dx=\frac {\ln \left (a+b\,x\right )\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{2\,a\,b^5}-\frac {d^4\,x^3}{3\,b^2}-\frac {2\,c\,d^3\,x^2}{b^2}-x\,\left (\frac {a^2\,d^4}{b^4}+\frac {6\,c^2\,d^2}{b^2}\right )-\frac {\ln \left (a-b\,x\right )\,\left (a^4\,d^4+4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2+4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{2\,a\,b^5} \]