Integrand size = 30, antiderivative size = 130 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^7 \left (a+b x^3\right )^2} \, dx=-\frac {c}{6 a^2 x^6}+\frac {2 b c-a d}{3 a^3 x^3}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^3 b \left (a+b x^3\right )}+\frac {\left (3 b^2 c-2 a b d+a^2 e\right ) \log (x)}{a^4}-\frac {\left (3 b^2 c-2 a b d+a^2 e\right ) \log \left (a+b x^3\right )}{3 a^4} \]
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Time = 0.11 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1835, 1634} \[ \int \frac {c+d x^3+e x^6+f x^9}{x^7 \left (a+b x^3\right )^2} \, dx=\frac {2 b c-a d}{3 a^3 x^3}-\frac {c}{6 a^2 x^6}-\frac {\log \left (a+b x^3\right ) \left (a^2 e-2 a b d+3 b^2 c\right )}{3 a^4}+\frac {\log (x) \left (a^2 e-2 a b d+3 b^2 c\right )}{a^4}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a^3 b \left (a+b x^3\right )} \]
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Rule 1634
Rule 1835
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {c+d x+e x^2+f x^3}{x^3 (a+b x)^2} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\frac {c}{a^2 x^3}+\frac {-2 b c+a d}{a^3 x^2}+\frac {3 b^2 c-2 a b d+a^2 e}{a^4 x}+\frac {-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^3 (a+b x)^2}-\frac {b \left (3 b^2 c-2 a b d+a^2 e\right )}{a^4 (a+b x)}\right ) \, dx,x,x^3\right ) \\ & = -\frac {c}{6 a^2 x^6}+\frac {2 b c-a d}{3 a^3 x^3}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^3 b \left (a+b x^3\right )}+\frac {\left (3 b^2 c-2 a b d+a^2 e\right ) \log (x)}{a^4}-\frac {\left (3 b^2 c-2 a b d+a^2 e\right ) \log \left (a+b x^3\right )}{3 a^4} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.91 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^7 \left (a+b x^3\right )^2} \, dx=-\frac {\frac {a^2 c}{x^6}+\frac {2 a (-2 b c+a d)}{x^3}+\frac {2 a \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b \left (a+b x^3\right )}-6 \left (3 b^2 c-2 a b d+a^2 e\right ) \log (x)+2 \left (3 b^2 c-2 a b d+a^2 e\right ) \log \left (a+b x^3\right )}{6 a^4} \]
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Time = 1.50 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {c}{6 a^{2} x^{6}}-\frac {a d -2 b c}{3 a^{3} x^{3}}+\frac {\left (a^{2} e -2 a b d +3 b^{2} c \right ) \ln \left (x \right )}{a^{4}}+\frac {\left (-a^{2} e +2 a b d -3 b^{2} c \right ) \ln \left (b \,x^{3}+a \right )-\frac {a \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{b \left (b \,x^{3}+a \right )}}{3 a^{4}}\) | \(123\) |
norman | \(\frac {-\frac {c}{6 a}-\frac {\left (2 a d -3 b c \right ) x^{3}}{6 a^{2}}+\frac {\left (f \,a^{3}-a^{2} b e +2 a \,b^{2} d -3 b^{3} c \right ) x^{9}}{3 a^{4}}}{x^{6} \left (b \,x^{3}+a \right )}+\frac {\left (a^{2} e -2 a b d +3 b^{2} c \right ) \ln \left (x \right )}{a^{4}}-\frac {\left (a^{2} e -2 a b d +3 b^{2} c \right ) \ln \left (b \,x^{3}+a \right )}{3 a^{4}}\) | \(126\) |
risch | \(\frac {-\frac {\left (f \,a^{3}-a^{2} b e +2 a \,b^{2} d -3 b^{3} c \right ) x^{6}}{3 a^{3} b}-\frac {\left (2 a d -3 b c \right ) x^{3}}{6 a^{2}}-\frac {c}{6 a}}{x^{6} \left (b \,x^{3}+a \right )}+\frac {e \ln \left (x \right )}{a^{2}}-\frac {2 \ln \left (x \right ) b d}{a^{3}}+\frac {3 \ln \left (x \right ) b^{2} c}{a^{4}}-\frac {e \ln \left (b \,x^{3}+a \right )}{3 a^{2}}+\frac {2 \ln \left (b \,x^{3}+a \right ) b d}{3 a^{3}}-\frac {\ln \left (b \,x^{3}+a \right ) b^{2} c}{a^{4}}\) | \(149\) |
parallelrisch | \(\frac {6 \ln \left (x \right ) x^{9} a^{2} b e -12 \ln \left (x \right ) x^{9} a \,b^{2} d +18 \ln \left (x \right ) x^{9} b^{3} c -2 \ln \left (b \,x^{3}+a \right ) x^{9} a^{2} b e +4 \ln \left (b \,x^{3}+a \right ) x^{9} a \,b^{2} d -6 \ln \left (b \,x^{3}+a \right ) x^{9} b^{3} c +2 x^{9} a^{3} f -2 x^{9} a^{2} b e +4 x^{9} a \,b^{2} d -6 b^{3} c \,x^{9}+6 \ln \left (x \right ) x^{6} a^{3} e -12 \ln \left (x \right ) x^{6} a^{2} b d +18 \ln \left (x \right ) x^{6} a \,b^{2} c -2 \ln \left (b \,x^{3}+a \right ) x^{6} a^{3} e +4 \ln \left (b \,x^{3}+a \right ) x^{6} a^{2} b d -6 \ln \left (b \,x^{3}+a \right ) x^{6} a \,b^{2} c -2 a^{3} d \,x^{3}+3 a^{2} x^{3} b c -c \,a^{3}}{6 a^{4} x^{6} \left (b \,x^{3}+a \right )}\) | \(258\) |
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Time = 0.29 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.60 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^7 \left (a+b x^3\right )^2} \, dx=\frac {2 \, {\left (3 \, a b^{3} c - 2 \, a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{6} - a^{3} b c + {\left (3 \, a^{2} b^{2} c - 2 \, a^{3} b d\right )} x^{3} - 2 \, {\left ({\left (3 \, b^{4} c - 2 \, a b^{3} d + a^{2} b^{2} e\right )} x^{9} + {\left (3 \, a b^{3} c - 2 \, a^{2} b^{2} d + a^{3} b e\right )} x^{6}\right )} \log \left (b x^{3} + a\right ) + 6 \, {\left ({\left (3 \, b^{4} c - 2 \, a b^{3} d + a^{2} b^{2} e\right )} x^{9} + {\left (3 \, a b^{3} c - 2 \, a^{2} b^{2} d + a^{3} b e\right )} x^{6}\right )} \log \left (x\right )}{6 \, {\left (a^{4} b^{2} x^{9} + a^{5} b x^{6}\right )}} \]
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Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^7 \left (a+b x^3\right )^2} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.06 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^7 \left (a+b x^3\right )^2} \, dx=\frac {2 \, {\left (3 \, b^{3} c - 2 \, a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} - a^{2} b c + {\left (3 \, a b^{2} c - 2 \, a^{2} b d\right )} x^{3}}{6 \, {\left (a^{3} b^{2} x^{9} + a^{4} b x^{6}\right )}} - \frac {{\left (3 \, b^{2} c - 2 \, a b d + a^{2} e\right )} \log \left (b x^{3} + a\right )}{3 \, a^{4}} + \frac {{\left (3 \, b^{2} c - 2 \, a b d + a^{2} e\right )} \log \left (x^{3}\right )}{3 \, a^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.51 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^7 \left (a+b x^3\right )^2} \, dx=\frac {{\left (3 \, b^{2} c - 2 \, a b d + a^{2} e\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {{\left (3 \, b^{3} c - 2 \, a b^{2} d + a^{2} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4} b} + \frac {3 \, b^{4} c x^{3} - 2 \, a b^{3} d x^{3} + a^{2} b^{2} e x^{3} + 4 \, a b^{3} c - 3 \, a^{2} b^{2} d + 2 \, a^{3} b e - a^{4} f}{3 \, {\left (b x^{3} + a\right )} a^{4} b} - \frac {9 \, b^{2} c x^{6} - 6 \, a b d x^{6} + 3 \, a^{2} e x^{6} - 4 \, a b c x^{3} + 2 \, a^{2} d x^{3} + a^{2} c}{6 \, a^{4} x^{6}} \]
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Time = 9.75 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^7 \left (a+b x^3\right )^2} \, dx=\frac {\ln \left (x\right )\,\left (e\,a^2-2\,d\,a\,b+3\,c\,b^2\right )}{a^4}-\frac {\ln \left (b\,x^3+a\right )\,\left (e\,a^2-2\,d\,a\,b+3\,c\,b^2\right )}{3\,a^4}-\frac {\frac {c}{6\,a}+\frac {x^3\,\left (2\,a\,d-3\,b\,c\right )}{6\,a^2}-\frac {x^6\,\left (-f\,a^3+e\,a^2\,b-2\,d\,a\,b^2+3\,c\,b^3\right )}{3\,a^3\,b}}{b\,x^9+a\,x^6} \]
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