Optimal. Leaf size=81 \[ -\frac {\log (x) (b c-a d)}{a^2}+\frac {\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^2 b^2}-\frac {c}{3 a x^3}+\frac {f x^3}{3 b} \]
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Rubi [A] time = 0.12, antiderivative size = 81, 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} \[ \frac {\log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^2 b^2}-\frac {\log (x) (b c-a d)}{a^2}-\frac {c}{3 a x^3}+\frac {f x^3}{3 b} \]
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 )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {c+d x+e x^2+f x^3}{x^2 (a+b x)} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {f}{b}+\frac {c}{a x^2}+\frac {-b c+a d}{a^2 x}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{a^2 b (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {c}{3 a x^3}+\frac {f x^3}{3 b}-\frac {(b c-a d) \log (x)}{a^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a+b x^3\right )}{3 a^2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 77, normalized size = 0.95 \[ \frac {1}{3} \left (\frac {3 \log (x) (a d-b c)}{a^2}+\frac {\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^2 b^2}-\frac {c}{a x^3}+\frac {f x^3}{b}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 85, normalized size = 1.05 \[ \frac {a^{2} b f x^{6} + {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{3} \log \left (b x^{3} + a\right ) - 3 \, {\left (b^{3} c - a b^{2} d\right )} x^{3} \log \relax (x) - a b^{2} c}{3 \, a^{2} b^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 95, normalized size = 1.17 \[ \frac {f x^{3}}{3 \, b} - \frac {{\left (b c - a d\right )} \log \left ({\left | x \right |}\right )}{a^{2}} + \frac {{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2} b^{2}} + \frac {b c x^{3} - a d x^{3} - a c}{3 \, a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 94, normalized size = 1.16 \[ \frac {f \,x^{3}}{3 b}-\frac {a f \ln \left (b \,x^{3}+a \right )}{3 b^{2}}+\frac {d \ln \relax (x )}{a}-\frac {d \ln \left (b \,x^{3}+a \right )}{3 a}-\frac {b c \ln \relax (x )}{a^{2}}+\frac {b c \ln \left (b \,x^{3}+a \right )}{3 a^{2}}+\frac {e \ln \left (b \,x^{3}+a \right )}{3 b}-\frac {c}{3 a \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 77, normalized size = 0.95 \[ \frac {f x^{3}}{3 \, b} - \frac {{\left (b c - a d\right )} \log \left (x^{3}\right )}{3 \, a^{2}} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{2} b^{2}} - \frac {c}{3 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.97, size = 74, normalized size = 0.91 \[ \frac {f\,x^3}{3\,b}-\frac {c}{3\,a\,x^3}+\frac {\ln \relax (x)\,\left (a\,d-b\,c\right )}{a^2}+\frac {\ln \left (b\,x^3+a\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.56, size = 70, normalized size = 0.86 \[ \frac {f x^{3}}{3 b} - \frac {c}{3 a x^{3}} + \frac {\left (a d - b c\right ) \log {\relax (x )}}{a^{2}} - \frac {\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (\frac {a}{b} + x^{3} \right )}}{3 a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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