Integrand size = 18, antiderivative size = 103 \[ \int \frac {(c+d x)^3}{x^4 (a+b x)} \, dx=-\frac {c^3}{3 a x^3}+\frac {c^2 (b c-3 a d)}{2 a^2 x^2}-\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{a^3 x}-\frac {(b c-a d)^3 \log (x)}{a^4}+\frac {(b c-a d)^3 \log (a+b x)}{a^4} \]
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Time = 0.05 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {90} \[ \int \frac {(c+d x)^3}{x^4 (a+b x)} \, dx=-\frac {\log (x) (b c-a d)^3}{a^4}+\frac {(b c-a d)^3 \log (a+b x)}{a^4}+\frac {c^2 (b c-3 a d)}{2 a^2 x^2}-\frac {c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{a^3 x}-\frac {c^3}{3 a x^3} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^3}{a x^4}+\frac {c^2 (-b c+3 a d)}{a^2 x^3}+\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{a^3 x^2}+\frac {(-b c+a d)^3}{a^4 x}-\frac {b (-b c+a d)^3}{a^4 (a+b x)}\right ) \, dx \\ & = -\frac {c^3}{3 a x^3}+\frac {c^2 (b c-3 a d)}{2 a^2 x^2}-\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{a^3 x}-\frac {(b c-a d)^3 \log (x)}{a^4}+\frac {(b c-a d)^3 \log (a+b x)}{a^4} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.90 \[ \int \frac {(c+d x)^3}{x^4 (a+b x)} \, dx=-\frac {\frac {a c \left (6 b^2 c^2 x^2-3 a b c x (c+6 d x)+a^2 \left (2 c^2+9 c d x+18 d^2 x^2\right )\right )}{x^3}+6 (b c-a d)^3 \log (x)-6 (b c-a d)^3 \log (a+b x)}{6 a^4} \]
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Time = 1.20 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.49
method | result | size |
default | \(-\frac {c^{3}}{3 a \,x^{3}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (x \right )}{a^{4}}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right )}{a^{3} x}-\frac {c^{2} \left (3 a d -b c \right )}{2 a^{2} x^{2}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{4}}\) | \(153\) |
norman | \(\frac {-\frac {c^{3}}{3 a}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) x^{2}}{a^{3}}-\frac {c^{2} \left (3 a d -b c \right ) x}{2 a^{2}}}{x^{3}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (x \right )}{a^{4}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{4}}\) | \(153\) |
risch | \(\frac {-\frac {c^{3}}{3 a}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) x^{2}}{a^{3}}-\frac {c^{2} \left (3 a d -b c \right ) x}{2 a^{2}}}{x^{3}}+\frac {\ln \left (-x \right ) d^{3}}{a}-\frac {3 \ln \left (-x \right ) b c \,d^{2}}{a^{2}}+\frac {3 \ln \left (-x \right ) b^{2} c^{2} d}{a^{3}}-\frac {\ln \left (-x \right ) b^{3} c^{3}}{a^{4}}-\frac {\ln \left (b x +a \right ) d^{3}}{a}+\frac {3 \ln \left (b x +a \right ) b c \,d^{2}}{a^{2}}-\frac {3 \ln \left (b x +a \right ) b^{2} c^{2} d}{a^{3}}+\frac {\ln \left (b x +a \right ) b^{3} c^{3}}{a^{4}}\) | \(184\) |
parallelrisch | \(\frac {6 \ln \left (x \right ) x^{3} a^{3} d^{3}-18 \ln \left (x \right ) x^{3} a^{2} b c \,d^{2}+18 \ln \left (x \right ) x^{3} a \,b^{2} c^{2} d -6 \ln \left (x \right ) x^{3} b^{3} c^{3}-6 \ln \left (b x +a \right ) x^{3} a^{3} d^{3}+18 \ln \left (b x +a \right ) x^{3} a^{2} b c \,d^{2}-18 \ln \left (b x +a \right ) x^{3} a \,b^{2} c^{2} d +6 \ln \left (b x +a \right ) x^{3} b^{3} c^{3}-18 a^{3} c \,d^{2} x^{2}+18 a^{2} b \,c^{2} d \,x^{2}-6 a \,b^{2} c^{3} x^{2}-9 a^{3} c^{2} d x +3 a^{2} b \,c^{3} x -2 c^{3} a^{3}}{6 a^{4} x^{3}}\) | \(203\) |
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Time = 0.23 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.56 \[ \int \frac {(c+d x)^3}{x^4 (a+b x)} \, dx=-\frac {2 \, a^{3} c^{3} - 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \log \left (b x + a\right ) + 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \log \left (x\right ) + 6 \, {\left (a b^{2} c^{3} - 3 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} - 3 \, {\left (a^{2} b c^{3} - 3 \, a^{3} c^{2} d\right )} x}{6 \, a^{4} x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (94) = 188\).
Time = 0.60 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.81 \[ \int \frac {(c+d x)^3}{x^4 (a+b x)} \, dx=\frac {- 2 a^{2} c^{3} + x^{2} \left (- 18 a^{2} c d^{2} + 18 a b c^{2} d - 6 b^{2} c^{3}\right ) + x \left (- 9 a^{2} c^{2} d + 3 a b c^{3}\right )}{6 a^{3} x^{3}} + \frac {\left (a d - b c\right )^{3} \log {\left (x + \frac {a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3} - a \left (a d - b c\right )^{3}}{2 a^{3} b d^{3} - 6 a^{2} b^{2} c d^{2} + 6 a b^{3} c^{2} d - 2 b^{4} c^{3}} \right )}}{a^{4}} - \frac {\left (a d - b c\right )^{3} \log {\left (x + \frac {a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3} + a \left (a d - b c\right )^{3}}{2 a^{3} b d^{3} - 6 a^{2} b^{2} c d^{2} + 6 a b^{3} c^{2} d - 2 b^{4} c^{3}} \right )}}{a^{4}} \]
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Time = 0.24 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.51 \[ \int \frac {(c+d x)^3}{x^4 (a+b x)} \, dx=\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{4}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (x\right )}{a^{4}} - \frac {2 \, a^{2} c^{3} + 6 \, {\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{2} - 3 \, {\left (a b c^{3} - 3 \, a^{2} c^{2} d\right )} x}{6 \, a^{3} x^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.64 \[ \int \frac {(c+d x)^3}{x^4 (a+b x)} \, dx=-\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac {2 \, a^{3} c^{3} + 6 \, {\left (a b^{2} c^{3} - 3 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} - 3 \, {\left (a^{2} b c^{3} - 3 \, a^{3} c^{2} d\right )} x}{6 \, a^{4} x^{3}} \]
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Time = 0.54 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.33 \[ \int \frac {(c+d x)^3}{x^4 (a+b x)} \, dx=-\frac {\frac {c^3}{3\,a}+\frac {c^2\,x\,\left (3\,a\,d-b\,c\right )}{2\,a^2}+\frac {c\,x^2\,\left (3\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{a^3}}{x^3}-\frac {2\,\mathrm {atanh}\left (\frac {{\left (a\,d-b\,c\right )}^3\,\left (a+2\,b\,x\right )}{a\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{a^4} \]
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