Optimal. Leaf size=96 \[ \frac {x^3 \left (a^2 f-a b e+b^2 d\right )}{3 b^3}+\frac {\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^4}+\frac {x^6 (b e-a f)}{6 b^2}+\frac {f x^9}{9 b} \]
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Rubi [A] time = 0.14, antiderivative size = 96, 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 = {1819, 1850} \[ \frac {\log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^4}+\frac {x^3 \left (a^2 f-a b e+b^2 d\right )}{3 b^3}+\frac {x^6 (b e-a f)}{6 b^2}+\frac {f x^9}{9 b} \]
Antiderivative was successfully verified.
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Rule 1819
Rule 1850
Rubi steps
\begin {align*} \int \frac {x^2 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {c+d x+e x^2+f x^3}{a+b x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {b^2 d-a b e+a^2 f}{b^3}+\frac {(b e-a f) x}{b^2}+\frac {f x^2}{b}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{b^3 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac {\left (b^2 d-a b e+a^2 f\right ) x^3}{3 b^3}+\frac {(b e-a f) x^6}{6 b^2}+\frac {f x^9}{9 b}+\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 b^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 88, normalized size = 0.92 \[ \frac {b x^3 \left (6 a^2 f-3 a b \left (2 e+f x^3\right )+b^2 \left (6 d+3 e x^3+2 f x^6\right )\right )+6 \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{18 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 92, normalized size = 0.96 \[ \frac {2 \, b^{3} f x^{9} + 3 \, {\left (b^{3} e - a b^{2} f\right )} x^{6} + 6 \, {\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{3} + 6 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{18 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 101, normalized size = 1.05 \[ \frac {2 \, b^{2} f x^{9} - 3 \, a b f x^{6} + 3 \, b^{2} x^{6} e + 6 \, b^{2} d x^{3} + 6 \, a^{2} f x^{3} - 6 \, a b x^{3} e}{18 \, b^{3}} + \frac {{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 124, normalized size = 1.29 \[ \frac {f \,x^{9}}{9 b}-\frac {a f \,x^{6}}{6 b^{2}}+\frac {e \,x^{6}}{6 b}+\frac {a^{2} f \,x^{3}}{3 b^{3}}-\frac {a e \,x^{3}}{3 b^{2}}+\frac {d \,x^{3}}{3 b}-\frac {a^{3} f \ln \left (b \,x^{3}+a \right )}{3 b^{4}}+\frac {a^{2} e \ln \left (b \,x^{3}+a \right )}{3 b^{3}}-\frac {a d \ln \left (b \,x^{3}+a \right )}{3 b^{2}}+\frac {c \ln \left (b \,x^{3}+a \right )}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 91, normalized size = 0.95 \[ \frac {2 \, b^{2} f x^{9} + 3 \, {\left (b^{2} e - a b f\right )} x^{6} + 6 \, {\left (b^{2} d - a b e + a^{2} f\right )} x^{3}}{18 \, b^{3}} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.83, size = 96, normalized size = 1.00 \[ x^6\,\left (\frac {e}{6\,b}-\frac {a\,f}{6\,b^2}\right )+x^3\,\left (\frac {d}{3\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{3\,b}\right )+\frac {\ln \left (b\,x^3+a\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^4}+\frac {f\,x^9}{9\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.13, size = 88, normalized size = 0.92 \[ x^{6} \left (- \frac {a f}{6 b^{2}} + \frac {e}{6 b}\right ) + x^{3} \left (\frac {a^{2} f}{3 b^{3}} - \frac {a e}{3 b^{2}} + \frac {d}{3 b}\right ) + \frac {f x^{9}}{9 b} - \frac {\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (a + b x^{3} \right )}}{3 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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