Integrand size = 18, antiderivative size = 87 \[ \int \frac {(c+d x)^3}{x^2 (a+b x)^2} \, dx=-\frac {c^3}{a^2 x}-\frac {(b c-a d)^3}{a^2 b^2 (a+b x)}-\frac {c^2 (2 b c-3 a d) \log (x)}{a^3}+\frac {(b c-a d)^2 (2 b c+a d) \log (a+b x)}{a^3 b^2} \]
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Time = 0.05 (sec) , antiderivative size = 87, 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} \[ \int \frac {(c+d x)^3}{x^2 (a+b x)^2} \, dx=\frac {(b c-a d)^2 (a d+2 b c) \log (a+b x)}{a^3 b^2}-\frac {c^2 \log (x) (2 b c-3 a d)}{a^3}-\frac {(b c-a d)^3}{a^2 b^2 (a+b x)}-\frac {c^3}{a^2 x} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^3}{a^2 x^2}+\frac {c^2 (-2 b c+3 a d)}{a^3 x}-\frac {(-b c+a d)^3}{a^2 b (a+b x)^2}+\frac {(-b c+a d)^2 (2 b c+a d)}{a^3 b (a+b x)}\right ) \, dx \\ & = -\frac {c^3}{a^2 x}-\frac {(b c-a d)^3}{a^2 b^2 (a+b x)}-\frac {c^2 (2 b c-3 a d) \log (x)}{a^3}+\frac {(b c-a d)^2 (2 b c+a d) \log (a+b x)}{a^3 b^2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.91 \[ \int \frac {(c+d x)^3}{x^2 (a+b x)^2} \, dx=\frac {-\frac {a c^3}{x}+\frac {a (-b c+a d)^3}{b^2 (a+b x)}+c^2 (-2 b c+3 a d) \log (x)+\frac {(b c-a d)^2 (2 b c+a d) \log (a+b x)}{b^2}}{a^3} \]
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Time = 0.47 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.39
method | result | size |
default | \(-\frac {c^{3}}{a^{2} x}+\frac {c^{2} \left (3 a d -2 b c \right ) \ln \left (x \right )}{a^{3}}+\frac {\left (a^{3} d^{3}-3 a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{3} b^{2}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{a^{2} b^{2} \left (b x +a \right )}\) | \(121\) |
norman | \(\frac {-\frac {c^{3}}{a}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{2}}{a^{3} b}}{x \left (b x +a \right )}+\frac {c^{2} \left (3 a d -2 b c \right ) \ln \left (x \right )}{a^{3}}+\frac {\left (a^{3} d^{3}-3 a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{3} b^{2}}\) | \(126\) |
risch | \(\frac {\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x}{b^{2} a^{2}}-\frac {c^{3}}{a}}{x \left (b x +a \right )}+\frac {\ln \left (-b x -a \right ) d^{3}}{b^{2}}-\frac {3 \ln \left (-b x -a \right ) c^{2} d}{a^{2}}+\frac {2 b \ln \left (-b x -a \right ) c^{3}}{a^{3}}+\frac {3 c^{2} \ln \left (x \right ) d}{a^{2}}-\frac {2 c^{3} \ln \left (x \right ) b}{a^{3}}\) | \(140\) |
parallelrisch | \(\frac {3 \ln \left (x \right ) x^{2} a \,b^{3} c^{2} d -2 \ln \left (x \right ) x^{2} b^{4} c^{3}+\ln \left (b x +a \right ) x^{2} a^{3} b \,d^{3}-3 \ln \left (b x +a \right ) x^{2} a \,b^{3} c^{2} d +2 \ln \left (b x +a \right ) x^{2} b^{4} c^{3}+3 \ln \left (x \right ) x \,a^{2} b^{2} c^{2} d -2 \ln \left (x \right ) x a \,b^{3} c^{3}+\ln \left (b x +a \right ) x \,a^{4} d^{3}-3 \ln \left (b x +a \right ) x \,a^{2} b^{2} c^{2} d +2 \ln \left (b x +a \right ) x a \,b^{3} c^{3}+a^{4} d^{3} x -3 a^{3} b c \,d^{2} x +3 a^{2} b^{2} c^{2} d x -2 b^{3} c^{3} a x -a^{2} b^{2} c^{3}}{b^{2} a^{3} x \left (b x +a \right )}\) | \(229\) |
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Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (87) = 174\).
Time = 0.23 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.28 \[ \int \frac {(c+d x)^3}{x^2 (a+b x)^2} \, dx=-\frac {a^{2} b^{2} c^{3} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x - {\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d + a^{3} b d^{3}\right )} x^{2} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + a^{4} d^{3}\right )} x\right )} \log \left (b x + a\right ) + {\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d\right )} x^{2} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d\right )} x\right )} \log \left (x\right )}{a^{3} b^{3} x^{2} + a^{4} b^{2} x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (78) = 156\).
Time = 0.74 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.87 \[ \int \frac {(c+d x)^3}{x^2 (a+b x)^2} \, dx=\frac {- a b^{2} c^{3} + x \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}\right )}{a^{3} b^{2} x + a^{2} b^{3} x^{2}} + \frac {c^{2} \cdot \left (3 a d - 2 b c\right ) \log {\left (x + \frac {- 3 a^{2} b c^{2} d + 2 a b^{2} c^{3} + a b c^{2} \cdot \left (3 a d - 2 b c\right )}{a^{3} d^{3} - 6 a b^{2} c^{2} d + 4 b^{3} c^{3}} \right )}}{a^{3}} + \frac {\left (a d - b c\right )^{2} \left (a d + 2 b c\right ) \log {\left (x + \frac {- 3 a^{2} b c^{2} d + 2 a b^{2} c^{3} + \frac {a \left (a d - b c\right )^{2} \left (a d + 2 b c\right )}{b}}{a^{3} d^{3} - 6 a b^{2} c^{2} d + 4 b^{3} c^{3}} \right )}}{a^{3} b^{2}} \]
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Time = 0.21 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.52 \[ \int \frac {(c+d x)^3}{x^2 (a+b x)^2} \, dx=-\frac {a b^{2} c^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{a^{2} b^{3} x^{2} + a^{3} b^{2} x} - \frac {{\left (2 \, b c^{3} - 3 \, a c^{2} d\right )} \log \left (x\right )}{a^{3}} + \frac {{\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{3} b^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.90 \[ \int \frac {(c+d x)^3}{x^2 (a+b x)^2} \, dx=-\frac {d^{3} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{2}} + \frac {b c^{3}}{a^{3} {\left (\frac {a}{b x + a} - 1\right )}} - \frac {{\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{3} b} - \frac {\frac {b^{5} c^{3}}{b x + a} - \frac {3 \, a b^{4} c^{2} d}{b x + a} + \frac {3 \, a^{2} b^{3} c d^{2}}{b x + a} - \frac {a^{3} b^{2} d^{3}}{b x + a}}{a^{2} b^{4}} \]
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Time = 0.50 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.36 \[ \int \frac {(c+d x)^3}{x^2 (a+b x)^2} \, dx=\ln \left (a+b\,x\right )\,\left (\frac {d^3}{b^2}+\frac {2\,b\,c^3}{a^3}-\frac {3\,c^2\,d}{a^2}\right )-\frac {\frac {c^3}{a}-\frac {x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}{a^2\,b^2}}{b\,x^2+a\,x}+\frac {c^2\,\ln \left (x\right )\,\left (3\,a\,d-2\,b\,c\right )}{a^3} \]
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