Optimal. Leaf size=103 \[ \frac {\log \left (a+b x^3\right ) \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}-\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 b^4 \left (a+b x^3\right )}+\frac {x^3 (b e-2 a f)}{3 b^3}+\frac {f x^6}{6 b^2} \]
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Rubi [A] time = 0.15, antiderivative size = 103, 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} \[ -\frac {a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 b^4 \left (a+b x^3\right )}+\frac {\log \left (a+b x^3\right ) \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}+\frac {x^3 (b e-2 a f)}{3 b^3}+\frac {f x^6}{6 b^2} \]
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 )}{\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}{(a+b x)^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {b e-2 a f}{b^3}+\frac {f x}{b^2}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{b^3 (a+b x)^2}+\frac {b^2 d-2 a b e+3 a^2 f}{b^3 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac {(b e-2 a f) x^3}{3 b^3}+\frac {f x^6}{6 b^2}-\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{3 b^4 \left (a+b x^3\right )}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) \log \left (a+b x^3\right )}{3 b^4}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 93, normalized size = 0.90 \[ \frac {2 \log \left (a+b x^3\right ) \left (3 a^2 f-2 a b e+b^2 d\right )+\frac {2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a+b x^3}+2 b x^3 (b e-2 a f)+b^2 f x^6}{6 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 143, normalized size = 1.39 \[ \frac {b^{3} f x^{9} + {\left (2 \, b^{3} e - 3 \, a b^{2} f\right )} x^{6} - 2 \, b^{3} c + 2 \, a b^{2} d - 2 \, a^{2} b e + 2 \, a^{3} f + 2 \, {\left (a b^{2} e - 2 \, a^{2} b f\right )} x^{3} + 2 \, {\left (a b^{2} d - 2 \, a^{2} b e + 3 \, a^{3} f + {\left (b^{3} d - 2 \, a b^{2} e + 3 \, a^{2} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \, {\left (b^{5} x^{3} + a b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 206, normalized size = 2.00 \[ -\frac {1}{6} \, f {\left (\frac {{\left (b x^{3} + a\right )}^{2} {\left (\frac {6 \, a}{b x^{3} + a} - 1\right )}}{b^{4}} + \frac {6 \, a^{2} \log \left (\frac {{\left | b x^{3} + a \right |}}{{\left (b x^{3} + a\right )}^{2} {\left | b \right |}}\right )}{b^{4}} - \frac {2 \, a^{3}}{{\left (b x^{3} + a\right )} b^{4}}\right )} + \frac {1}{3} \, {\left (\frac {2 \, a \log \left (\frac {{\left | b x^{3} + a \right |}}{{\left (b x^{3} + a\right )}^{2} {\left | b \right |}}\right )}{b^{3}} + \frac {b x^{3} + a}{b^{3}} - \frac {a^{2}}{{\left (b x^{3} + a\right )} b^{3}}\right )} e - \frac {d {\left (\frac {\log \left (\frac {{\left | b x^{3} + a \right |}}{{\left (b x^{3} + a\right )}^{2} {\left | b \right |}}\right )}{b} - \frac {a}{{\left (b x^{3} + a\right )} b}\right )}}{3 \, b} - \frac {c}{3 \, {\left (b x^{3} + a\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 142, normalized size = 1.38 \[ \frac {f \,x^{6}}{6 b^{2}}-\frac {2 a f \,x^{3}}{3 b^{3}}+\frac {e \,x^{3}}{3 b^{2}}+\frac {a^{3} f}{3 \left (b \,x^{3}+a \right ) b^{4}}-\frac {a^{2} e}{3 \left (b \,x^{3}+a \right ) b^{3}}+\frac {a^{2} f \ln \left (b \,x^{3}+a \right )}{b^{4}}+\frac {a d}{3 \left (b \,x^{3}+a \right ) b^{2}}-\frac {2 a e \ln \left (b \,x^{3}+a \right )}{3 b^{3}}-\frac {c}{3 \left (b \,x^{3}+a \right ) b}+\frac {d \ln \left (b \,x^{3}+a \right )}{3 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 98, normalized size = 0.95 \[ -\frac {b^{3} c - a b^{2} d + a^{2} b e - a^{3} f}{3 \, {\left (b^{5} x^{3} + a b^{4}\right )}} + \frac {b f x^{6} + 2 \, {\left (b e - 2 \, a f\right )} x^{3}}{6 \, b^{3}} + \frac {{\left (b^{2} d - 2 \, a b e + 3 \, a^{2} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 103, normalized size = 1.00 \[ x^3\,\left (\frac {e}{3\,b^2}-\frac {2\,a\,f}{3\,b^3}\right )+\frac {f\,x^6}{6\,b^2}-\frac {-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3}{3\,b\,\left (b^4\,x^3+a\,b^3\right )}+\frac {\ln \left (b\,x^3+a\right )\,\left (3\,f\,a^2-2\,e\,a\,b+d\,b^2\right )}{3\,b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.61, size = 100, normalized size = 0.97 \[ x^{3} \left (- \frac {2 a f}{3 b^{3}} + \frac {e}{3 b^{2}}\right ) + \frac {a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{3 a b^{4} + 3 b^{5} x^{3}} + \frac {f x^{6}}{6 b^{2}} + \frac {\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log {\left (a + b x^{3} \right )}}{3 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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