Optimal. Leaf size=146 \[ \frac {\log \left (a+b x^3\right ) \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}-\frac {-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c}{3 b^5 \left (a+b x^3\right )}+\frac {a \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}+\frac {x^3 (b e-3 a f)}{3 b^4}+\frac {f x^6}{6 b^3} \]
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Rubi [A] time = 0.20, antiderivative size = 146, 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} \[ -\frac {3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c}{3 b^5 \left (a+b x^3\right )}+\frac {a \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}+\frac {\log \left (a+b x^3\right ) \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}+\frac {x^3 (b e-3 a f)}{3 b^4}+\frac {f x^6}{6 b^3} \]
Antiderivative was successfully verified.
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Rule 1620
Rule 1821
Rubi steps
\begin {align*} \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x \left (c+d x+e x^2+f x^3\right )}{(a+b x)^3} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {b e-3 a f}{b^4}+\frac {f x}{b^3}+\frac {a \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^4 (a+b x)^3}+\frac {b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f}{b^4 (a+b x)^2}+\frac {b^2 d-3 a b e+6 a^2 f}{b^4 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac {(b e-3 a f) x^3}{3 b^4}+\frac {f x^6}{6 b^3}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f}{3 b^5 \left (a+b x^3\right )}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) \log \left (a+b x^3\right )}{3 b^5}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 145, normalized size = 0.99 \[ \frac {7 a^4 f+a^3 b \left (2 f x^3-5 e\right )+2 \left (a+b x^3\right )^2 \log \left (a+b x^3\right ) \left (6 a^2 f-3 a b e+b^2 d\right )+a^2 b^2 \left (3 d-4 e x^3-11 f x^6\right )-a b^3 \left (c-4 x^3 \left (d+e x^3-f x^6\right )\right )+b^4 x^3 \left (-2 c+2 e x^6+f x^9\right )}{6 b^5 \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 225, normalized size = 1.54 \[ \frac {b^{4} f x^{12} + 2 \, {\left (b^{4} e - 2 \, a b^{3} f\right )} x^{9} + {\left (4 \, a b^{3} e - 11 \, a^{2} b^{2} f\right )} x^{6} - a b^{3} c + 3 \, a^{2} b^{2} d - 5 \, a^{3} b e + 7 \, a^{4} f - 2 \, {\left (b^{4} c - 2 \, a b^{3} d + 2 \, a^{2} b^{2} e - a^{3} b f\right )} x^{3} + 2 \, {\left ({\left (b^{4} d - 3 \, a b^{3} e + 6 \, a^{2} b^{2} f\right )} x^{6} + a^{2} b^{2} d - 3 \, a^{3} b e + 6 \, a^{4} f + 2 \, {\left (a b^{3} d - 3 \, a^{2} b^{2} e + 6 \, a^{3} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \, {\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 146, normalized size = 1.00 \[ \frac {{\left (b^{2} d + 6 \, a^{2} f - 3 \, a b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{5}} + \frac {b^{3} f x^{6} - 6 \, a b^{2} f x^{3} + 2 \, b^{3} x^{3} e}{6 \, b^{6}} - \frac {a b^{3} c - 3 \, a^{2} b^{2} d - 7 \, a^{4} f + 5 \, a^{3} b e + 2 \, {\left (b^{4} c - 2 \, a b^{3} d - 4 \, a^{3} b f + 3 \, a^{2} b^{2} e\right )} x^{3}}{6 \, {\left (b x^{3} + a\right )}^{2} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 213, normalized size = 1.46 \[ \frac {f \,x^{6}}{6 b^{3}}-\frac {a f \,x^{3}}{b^{4}}+\frac {e \,x^{3}}{3 b^{3}}-\frac {a^{4} f}{6 \left (b \,x^{3}+a \right )^{2} b^{5}}+\frac {a^{3} e}{6 \left (b \,x^{3}+a \right )^{2} b^{4}}-\frac {a^{2} d}{6 \left (b \,x^{3}+a \right )^{2} b^{3}}+\frac {a c}{6 \left (b \,x^{3}+a \right )^{2} b^{2}}+\frac {4 a^{3} f}{3 \left (b \,x^{3}+a \right ) b^{5}}-\frac {a^{2} e}{\left (b \,x^{3}+a \right ) b^{4}}+\frac {2 a^{2} f \ln \left (b \,x^{3}+a \right )}{b^{5}}+\frac {2 a d}{3 \left (b \,x^{3}+a \right ) b^{3}}-\frac {a e \ln \left (b \,x^{3}+a \right )}{b^{4}}-\frac {c}{3 \left (b \,x^{3}+a \right ) b^{2}}+\frac {d \ln \left (b \,x^{3}+a \right )}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 147, normalized size = 1.01 \[ -\frac {a b^{3} c - 3 \, a^{2} b^{2} d + 5 \, a^{3} b e - 7 \, a^{4} f + 2 \, {\left (b^{4} c - 2 \, a b^{3} d + 3 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{3}}{6 \, {\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} + \frac {b f x^{6} + 2 \, {\left (b e - 3 \, a f\right )} x^{3}}{6 \, b^{4}} + \frac {{\left (b^{2} d - 3 \, a b e + 6 \, a^{2} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 152, normalized size = 1.04 \[ x^3\,\left (\frac {e}{3\,b^3}-\frac {a\,f}{b^4}\right )+\frac {\frac {7\,f\,a^4-5\,e\,a^3\,b+3\,d\,a^2\,b^2-c\,a\,b^3}{6\,b}-x^3\,\left (-\frac {4\,f\,a^3}{3}+e\,a^2\,b-\frac {2\,d\,a\,b^2}{3}+\frac {c\,b^3}{3}\right )}{a^2\,b^4+2\,a\,b^5\,x^3+b^6\,x^6}+\frac {f\,x^6}{6\,b^3}+\frac {\ln \left (b\,x^3+a\right )\,\left (6\,f\,a^2-3\,e\,a\,b+d\,b^2\right )}{3\,b^5} \]
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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