Optimal. Leaf size=109 \[ \frac {\log \left (a+b x^3\right ) \left (a^3 f-a b^2 d+2 b^3 c\right )}{3 a^3 b^2}-\frac {\log (x) (2 b c-a d)}{a^3}-\frac {c}{3 a^2 x^3}-\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a^2 b^2 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.14, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1821, 1620} \[ -\frac {a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 a^2 b^2 \left (a+b x^3\right )}+\frac {\log \left (a+b x^3\right ) \left (a^3 f-a b^2 d+2 b^3 c\right )}{3 a^3 b^2}-\frac {\log (x) (2 b c-a d)}{a^3}-\frac {c}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Rule 1620
Rule 1821
Rubi steps
\begin {align*} \int \frac {c+d x^3+e x^6+f x^9}{x^4 \left (a+b x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {c+d x+e x^2+f x^3}{x^2 (a+b x)^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {c}{a^2 x^2}+\frac {-2 b c+a d}{a^3 x}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{a^2 b (a+b x)^2}+\frac {2 b^3 c-a b^2 d+a^3 f}{a^3 b (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {c}{3 a^2 x^3}-\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^2 b^2 \left (a+b x^3\right )}-\frac {(2 b c-a d) \log (x)}{a^3}+\frac {\left (2 b^3 c-a b^2 d+a^3 f\right ) \log \left (a+b x^3\right )}{3 a^3 b^2}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 97, normalized size = 0.89 \[ \frac {\frac {\log \left (a+b x^3\right ) \left (a^3 f-a b^2 d+2 b^3 c\right )}{b^2}+\frac {a \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b^2 \left (a+b x^3\right )}+3 \log (x) (a d-2 b c)-\frac {a c}{x^3}}{3 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 172, normalized size = 1.58 \[ -\frac {a^{2} b^{2} c + {\left (2 \, a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{3} - {\left ({\left (2 \, b^{4} c - a b^{3} d + a^{3} b f\right )} x^{6} + {\left (2 \, a b^{3} c - a^{2} b^{2} d + a^{4} f\right )} x^{3}\right )} \log \left (b x^{3} + a\right ) + 3 \, {\left ({\left (2 \, b^{4} c - a b^{3} d\right )} x^{6} + {\left (2 \, a b^{3} c - a^{2} b^{2} d\right )} x^{3}\right )} \log \relax (x)}{3 \, {\left (a^{3} b^{3} x^{6} + a^{4} b^{2} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 131, normalized size = 1.20 \[ -\frac {{\left (2 \, b c - a d\right )} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {{\left (2 \, b^{3} c - a b^{2} d + a^{3} f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3} b^{2}} - \frac {a^{2} b f x^{6} + 4 \, b^{3} c x^{3} - 2 \, a b^{2} d x^{3} - a^{3} f x^{3} + 2 \, a^{2} b x^{3} e + 2 \, a b^{2} c}{6 \, {\left (b x^{6} + a x^{3}\right )} a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 132, normalized size = 1.21 \[ \frac {a f}{3 \left (b \,x^{3}+a \right ) b^{2}}+\frac {d}{3 \left (b \,x^{3}+a \right ) a}-\frac {b c}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {d \ln \relax (x )}{a^{2}}-\frac {d \ln \left (b \,x^{3}+a \right )}{3 a^{2}}-\frac {2 b c \ln \relax (x )}{a^{3}}+\frac {2 b c \ln \left (b \,x^{3}+a \right )}{3 a^{3}}-\frac {e}{3 \left (b \,x^{3}+a \right ) b}+\frac {f \ln \left (b \,x^{3}+a \right )}{3 b^{2}}-\frac {c}{3 a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 116, normalized size = 1.06 \[ -\frac {a b^{2} c + {\left (2 \, b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{3}}{3 \, {\left (a^{2} b^{3} x^{6} + a^{3} b^{2} x^{3}\right )}} - \frac {{\left (2 \, b c - a d\right )} \log \left (x^{3}\right )}{3 \, a^{3}} + \frac {{\left (2 \, b^{3} c - a b^{2} d + a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{3} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.05, size = 109, normalized size = 1.00 \[ \frac {\ln \relax (x)\,\left (a\,d-2\,b\,c\right )}{a^3}-\frac {\frac {c}{3\,a}+\frac {x^3\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+2\,c\,b^3\right )}{3\,a^2\,b^2}}{b\,x^6+a\,x^3}+\frac {\ln \left (b\,x^3+a\right )\,\left (f\,a^3-d\,a\,b^2+2\,c\,b^3\right )}{3\,a^3\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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