Integrand size = 18, antiderivative size = 90 \[ \int \frac {(c+d x)^2}{x^4 (a+b x)} \, dx=-\frac {c^2}{3 a x^3}+\frac {c (b c-2 a d)}{2 a^2 x^2}-\frac {(b c-a d)^2}{a^3 x}-\frac {b (b c-a d)^2 \log (x)}{a^4}+\frac {b (b c-a d)^2 \log (a+b x)}{a^4} \]
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Time = 0.04 (sec) , antiderivative size = 90, 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} \[ \int \frac {(c+d x)^2}{x^4 (a+b x)} \, dx=-\frac {b \log (x) (b c-a d)^2}{a^4}+\frac {b (b c-a d)^2 \log (a+b x)}{a^4}-\frac {(b c-a d)^2}{a^3 x}+\frac {c (b c-2 a d)}{2 a^2 x^2}-\frac {c^2}{3 a x^3} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^2}{a x^4}+\frac {c (-b c+2 a d)}{a^2 x^3}+\frac {(-b c+a d)^2}{a^3 x^2}-\frac {b (-b c+a d)^2}{a^4 x}+\frac {b^2 (-b c+a d)^2}{a^4 (a+b x)}\right ) \, dx \\ & = -\frac {c^2}{3 a x^3}+\frac {c (b c-2 a d)}{2 a^2 x^2}-\frac {(b c-a d)^2}{a^3 x}-\frac {b (b c-a d)^2 \log (x)}{a^4}+\frac {b (b c-a d)^2 \log (a+b x)}{a^4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^2}{x^4 (a+b x)} \, dx=\frac {-6 a b^2 c^2 x^2+3 a^2 b c x (c+4 d x)-2 a^3 \left (c^2+3 c d x+3 d^2 x^2\right )-6 b (b c-a d)^2 x^3 \log (x)+6 b (b c-a d)^2 x^3 \log (a+b x)}{6 a^4 x^3} \]
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Time = 1.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.34
method | result | size |
default | \(-\frac {c^{2}}{3 a \,x^{3}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{a^{3} x}-\frac {c \left (2 a d -b c \right )}{2 a^{2} x^{2}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b \ln \left (x \right )}{a^{4}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b \ln \left (b x +a \right )}{a^{4}}\) | \(121\) |
norman | \(\frac {-\frac {c^{2}}{3 a}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{2}}{a^{3}}-\frac {c \left (2 a d -b c \right ) x}{2 a^{2}}}{x^{3}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b \ln \left (b x +a \right )}{a^{4}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b \ln \left (x \right )}{a^{4}}\) | \(121\) |
risch | \(\frac {-\frac {c^{2}}{3 a}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{2}}{a^{3}}-\frac {c \left (2 a d -b c \right ) x}{2 a^{2}}}{x^{3}}+\frac {b \ln \left (-b x -a \right ) d^{2}}{a^{2}}-\frac {2 b^{2} \ln \left (-b x -a \right ) c d}{a^{3}}+\frac {b^{3} \ln \left (-b x -a \right ) c^{2}}{a^{4}}-\frac {b \ln \left (x \right ) d^{2}}{a^{2}}+\frac {2 b^{2} \ln \left (x \right ) c d}{a^{3}}-\frac {b^{3} \ln \left (x \right ) c^{2}}{a^{4}}\) | \(151\) |
parallelrisch | \(-\frac {6 \ln \left (x \right ) x^{3} a^{2} b \,d^{2}-12 \ln \left (x \right ) x^{3} a \,b^{2} c d +6 \ln \left (x \right ) x^{3} b^{3} c^{2}-6 \ln \left (b x +a \right ) x^{3} a^{2} b \,d^{2}+12 \ln \left (b x +a \right ) x^{3} a \,b^{2} c d -6 \ln \left (b x +a \right ) x^{3} b^{3} c^{2}+6 a^{3} d^{2} x^{2}-12 a^{2} b c d \,x^{2}+6 a \,b^{2} c^{2} x^{2}+6 a^{3} c d x -3 a^{2} b \,c^{2} x +2 c^{2} a^{3}}{6 a^{4} x^{3}}\) | \(162\) |
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Time = 0.22 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.46 \[ \int \frac {(c+d x)^2}{x^4 (a+b x)} \, dx=-\frac {2 \, a^{3} c^{2} - 6 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} \log \left (b x + a\right ) + 6 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} \log \left (x\right ) + 6 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} - 3 \, {\left (a^{2} b c^{2} - 2 \, a^{3} c d\right )} x}{6 \, a^{4} x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (78) = 156\).
Time = 0.41 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.67 \[ \int \frac {(c+d x)^2}{x^4 (a+b x)} \, dx=\frac {- 2 a^{2} c^{2} + x^{2} \left (- 6 a^{2} d^{2} + 12 a b c d - 6 b^{2} c^{2}\right ) + x \left (- 6 a^{2} c d + 3 a b c^{2}\right )}{6 a^{3} x^{3}} - \frac {b \left (a d - b c\right )^{2} \log {\left (x + \frac {a^{3} b d^{2} - 2 a^{2} b^{2} c d + a b^{3} c^{2} - a b \left (a d - b c\right )^{2}}{2 a^{2} b^{2} d^{2} - 4 a b^{3} c d + 2 b^{4} c^{2}} \right )}}{a^{4}} + \frac {b \left (a d - b c\right )^{2} \log {\left (x + \frac {a^{3} b d^{2} - 2 a^{2} b^{2} c d + a b^{3} c^{2} + a b \left (a d - b c\right )^{2}}{2 a^{2} b^{2} d^{2} - 4 a b^{3} c d + 2 b^{4} c^{2}} \right )}}{a^{4}} \]
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Time = 0.20 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.40 \[ \int \frac {(c+d x)^2}{x^4 (a+b x)} \, dx=\frac {{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left (b x + a\right )}{a^{4}} - \frac {{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left (x\right )}{a^{4}} - \frac {2 \, a^{2} c^{2} + 6 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} - 3 \, {\left (a b c^{2} - 2 \, a^{2} c d\right )} x}{6 \, a^{3} x^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.53 \[ \int \frac {(c+d x)^2}{x^4 (a+b x)} \, dx=-\frac {{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac {2 \, a^{3} c^{2} + 6 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} - 3 \, {\left (a^{2} b c^{2} - 2 \, a^{3} c d\right )} x}{6 \, a^{4} x^{3}} \]
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Time = 0.17 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.37 \[ \int \frac {(c+d x)^2}{x^4 (a+b x)} \, dx=\frac {2\,b\,\mathrm {atanh}\left (\frac {b\,{\left (a\,d-b\,c\right )}^2\,\left (a+2\,b\,x\right )}{a\,\left (a^2\,b\,d^2-2\,a\,b^2\,c\,d+b^3\,c^2\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{a^4}-\frac {\frac {c^2}{3\,a}+\frac {x^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{a^3}+\frac {c\,x\,\left (2\,a\,d-b\,c\right )}{2\,a^2}}{x^3} \]
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