Integrand size = 18, antiderivative size = 85 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)} \, dx=-\frac {c^3}{2 a x^2}+\frac {c^2 (b c-3 a d)}{a^2 x}+\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) \log (x)}{a^3}-\frac {(b c-a d)^3 \log (a+b x)}{a^3 b} \]
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Time = 0.04 (sec) , antiderivative size = 85, 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} \[ \int \frac {(c+d x)^3}{x^3 (a+b x)} \, dx=-\frac {(b c-a d)^3 \log (a+b x)}{a^3 b}+\frac {c^2 (b c-3 a d)}{a^2 x}+\frac {c \log (x) \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{a^3}-\frac {c^3}{2 a x^2} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^3}{a x^3}+\frac {c^2 (-b c+3 a d)}{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}{a^3 (a+b x)}\right ) \, dx \\ & = -\frac {c^3}{2 a x^2}+\frac {c^2 (b c-3 a d)}{a^2 x}+\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) \log (x)}{a^3}-\frac {(b c-a d)^3 \log (a+b x)}{a^3 b} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)} \, dx=-\frac {\frac {a c^2 (-2 b c x+a (c+6 d x))}{x^2}-2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) \log (x)+\frac {2 (b c-a d)^3 \log (a+b x)}{b}}{2 a^3} \]
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Time = 0.44 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.31
method | result | size |
default | \(-\frac {c^{3}}{2 a \,x^{2}}+\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) \ln \left (x \right )}{a^{3}}-\frac {c^{2} \left (3 a d -b c \right )}{x \,a^{2}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{3} b}\) | \(111\) |
norman | \(\frac {-\frac {c^{3}}{2 a}-\frac {c^{2} \left (3 a d -b c \right ) x}{a^{2}}}{x^{2}}+\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) \ln \left (x \right )}{a^{3}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{3} b}\) | \(111\) |
risch | \(\frac {-\frac {c^{3}}{2 a}-\frac {c^{2} \left (3 a d -b c \right ) x}{a^{2}}}{x^{2}}+\frac {3 c \ln \left (x \right ) d^{2}}{a}-\frac {3 c^{2} \ln \left (x \right ) b d}{a^{2}}+\frac {c^{3} \ln \left (x \right ) b^{2}}{a^{3}}+\frac {\ln \left (-b x -a \right ) d^{3}}{b}-\frac {3 \ln \left (-b x -a \right ) c \,d^{2}}{a}+\frac {3 b \ln \left (-b x -a \right ) c^{2} d}{a^{2}}-\frac {b^{2} \ln \left (-b x -a \right ) c^{3}}{a^{3}}\) | \(141\) |
parallelrisch | \(\frac {6 \ln \left (x \right ) x^{2} a^{2} b c \,d^{2}-6 \ln \left (x \right ) x^{2} a \,b^{2} c^{2} d +2 \ln \left (x \right ) x^{2} b^{3} c^{3}+2 \ln \left (b x +a \right ) x^{2} a^{3} d^{3}-6 \ln \left (b x +a \right ) x^{2} a^{2} b c \,d^{2}+6 \ln \left (b x +a \right ) x^{2} a \,b^{2} c^{2} d -2 \ln \left (b x +a \right ) x^{2} b^{3} c^{3}-6 x \,a^{2} b \,c^{2} d +2 x a \,b^{2} c^{3}-a^{2} b \,c^{3}}{2 a^{3} x^{2} b}\) | \(158\) |
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Time = 0.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.46 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)} \, dx=-\frac {a^{2} b c^{3} + 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2} \log \left (b x + a\right ) - 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2}\right )} x^{2} \log \left (x\right ) - 2 \, {\left (a b^{2} c^{3} - 3 \, a^{2} b c^{2} d\right )} x}{2 \, a^{3} b x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (78) = 156\).
Time = 0.71 (sec) , antiderivative size = 257, normalized size of antiderivative = 3.02 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)} \, dx=\frac {- a c^{3} + x \left (- 6 a c^{2} d + 2 b c^{3}\right )}{2 a^{2} x^{2}} + \frac {c \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {- 3 a^{3} c d^{2} + 3 a^{2} b c^{2} d - a b^{2} c^{3} + a c \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{a^{3} d^{3} - 6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{3}} + \frac {\left (a d - b c\right )^{3} \log {\left (x + \frac {- 3 a^{3} c d^{2} + 3 a^{2} b c^{2} d - a b^{2} c^{3} + \frac {a \left (a d - b c\right )^{3}}{b}}{a^{3} d^{3} - 6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{3} b} \]
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Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.32 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)} \, dx=\frac {{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \log \left (x\right )}{a^{3}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{3} b} - \frac {a c^{3} - 2 \, {\left (b c^{3} - 3 \, a c^{2} d\right )} x}{2 \, a^{2} x^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.40 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)} \, dx=\frac {{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{3} b} - \frac {a^{2} c^{3} - 2 \, {\left (a b c^{3} - 3 \, a^{2} c^{2} d\right )} x}{2 \, a^{3} x^{2}} \]
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Time = 0.47 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.99 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)} \, dx=\frac {\ln \left (a+b\,x\right )\,{\left (a\,d-b\,c\right )}^3}{a^3\,b}-\frac {\frac {c^3}{2\,a}+\frac {c^2\,x\,\left (3\,a\,d-b\,c\right )}{a^2}}{x^2}+\frac {c\,\ln \left (x\right )\,\left (3\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{a^3} \]
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