Optimal. Leaf size=175 \[ \frac {2 b c-a d}{6 a^3 x^6}-\frac {c}{9 a^2 x^9}-\frac {a^2 e-2 a b d+3 b^2 c}{3 a^4 x^3}+\frac {\log \left (a+b x^3\right ) \left (a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c\right )}{3 a^5}-\frac {\log (x) \left (a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c\right )}{a^5}-\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a^4 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.20, antiderivative size = 175, 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} \[ -\frac {a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 a^4 \left (a+b x^3\right )}+\frac {\log \left (a+b x^3\right ) \left (2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c\right )}{3 a^5}-\frac {\log (x) \left (2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c\right )}{a^5}-\frac {a^2 e-2 a b d+3 b^2 c}{3 a^4 x^3}+\frac {2 b c-a d}{6 a^3 x^6}-\frac {c}{9 a^2 x^9} \]
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^{10} \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^4 (a+b x)^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {c}{a^2 x^4}+\frac {-2 b c+a d}{a^3 x^3}+\frac {3 b^2 c-2 a b d+a^2 e}{a^4 x^2}+\frac {-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f}{a^5 x}-\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^4 (a+b x)^2}-\frac {b \left (-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f\right )}{a^5 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {c}{9 a^2 x^9}+\frac {2 b c-a d}{6 a^3 x^6}-\frac {3 b^2 c-2 a b d+a^2 e}{3 a^4 x^3}-\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^4 \left (a+b x^3\right )}-\frac {\left (4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f\right ) \log (x)}{a^5}+\frac {\left (4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f\right ) \log \left (a+b x^3\right )}{3 a^5}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 160, normalized size = 0.91 \[ \frac {-\frac {2 a^3 c}{x^9}-\frac {6 a \left (a^2 e-2 a b d+3 b^2 c\right )}{x^3}-\frac {3 a^2 (a d-2 b c)}{x^6}+\frac {6 a \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a+b x^3}+6 \log \left (a+b x^3\right ) \left (a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c\right )+18 \log (x) \left (a^3 f-2 a^2 b e+3 a b^2 d-4 b^3 c\right )}{18 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 261, normalized size = 1.49 \[ -\frac {6 \, {\left (4 \, a b^{3} c - 3 \, a^{2} b^{2} d + 2 \, a^{3} b e - a^{4} f\right )} x^{9} + 3 \, {\left (4 \, a^{2} b^{2} c - 3 \, a^{3} b d + 2 \, a^{4} e\right )} x^{6} + 2 \, a^{4} c - {\left (4 \, a^{3} b c - 3 \, a^{4} d\right )} x^{3} - 6 \, {\left ({\left (4 \, b^{4} c - 3 \, a b^{3} d + 2 \, a^{2} b^{2} e - a^{3} b f\right )} x^{12} + {\left (4 \, a b^{3} c - 3 \, a^{2} b^{2} d + 2 \, a^{3} b e - a^{4} f\right )} x^{9}\right )} \log \left (b x^{3} + a\right ) + 18 \, {\left ({\left (4 \, b^{4} c - 3 \, a b^{3} d + 2 \, a^{2} b^{2} e - a^{3} b f\right )} x^{12} + {\left (4 \, a b^{3} c - 3 \, a^{2} b^{2} d + 2 \, a^{3} b e - a^{4} f\right )} x^{9}\right )} \log \relax (x)}{18 \, {\left (a^{5} b x^{12} + a^{6} x^{9}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 275, normalized size = 1.57 \[ -\frac {{\left (4 \, b^{3} c - 3 \, a b^{2} d - a^{3} f + 2 \, a^{2} b e\right )} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac {{\left (4 \, b^{4} c - 3 \, a b^{3} d - a^{3} b f + 2 \, a^{2} b^{2} e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{5} b} - \frac {4 \, b^{4} c x^{3} - 3 \, a b^{3} d x^{3} - a^{3} b f x^{3} + 2 \, a^{2} b^{2} x^{3} e + 5 \, a b^{3} c - 4 \, a^{2} b^{2} d - 2 \, a^{4} f + 3 \, a^{3} b e}{3 \, {\left (b x^{3} + a\right )} a^{5}} + \frac {44 \, b^{3} c x^{9} - 33 \, a b^{2} d x^{9} - 11 \, a^{3} f x^{9} + 22 \, a^{2} b x^{9} e - 18 \, a b^{2} c x^{6} + 12 \, a^{2} b d x^{6} - 6 \, a^{3} x^{6} e + 6 \, a^{2} b c x^{3} - 3 \, a^{3} d x^{3} - 2 \, a^{3} c}{18 \, a^{5} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 229, normalized size = 1.31 \[ \frac {f}{3 \left (b \,x^{3}+a \right ) a}-\frac {b e}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {f \ln \relax (x )}{a^{2}}-\frac {f \ln \left (b \,x^{3}+a \right )}{3 a^{2}}+\frac {b^{2} d}{3 \left (b \,x^{3}+a \right ) a^{3}}-\frac {2 b e \ln \relax (x )}{a^{3}}+\frac {2 b e \ln \left (b \,x^{3}+a \right )}{3 a^{3}}-\frac {b^{3} c}{3 \left (b \,x^{3}+a \right ) a^{4}}+\frac {3 b^{2} d \ln \relax (x )}{a^{4}}-\frac {b^{2} d \ln \left (b \,x^{3}+a \right )}{a^{4}}-\frac {4 b^{3} c \ln \relax (x )}{a^{5}}+\frac {4 b^{3} c \ln \left (b \,x^{3}+a \right )}{3 a^{5}}-\frac {e}{3 a^{2} x^{3}}+\frac {2 b d}{3 a^{3} x^{3}}-\frac {b^{2} c}{a^{4} x^{3}}-\frac {d}{6 a^{2} x^{6}}+\frac {b c}{3 a^{3} x^{6}}-\frac {c}{9 a^{2} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 181, normalized size = 1.03 \[ -\frac {6 \, {\left (4 \, b^{3} c - 3 \, a b^{2} d + 2 \, a^{2} b e - a^{3} f\right )} x^{9} + 3 \, {\left (4 \, a b^{2} c - 3 \, a^{2} b d + 2 \, a^{3} e\right )} x^{6} + 2 \, a^{3} c - {\left (4 \, a^{2} b c - 3 \, a^{3} d\right )} x^{3}}{18 \, {\left (a^{4} b x^{12} + a^{5} x^{9}\right )}} + \frac {{\left (4 \, b^{3} c - 3 \, a b^{2} d + 2 \, a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{5}} - \frac {{\left (4 \, b^{3} c - 3 \, a b^{2} d + 2 \, a^{2} b e - a^{3} f\right )} \log \left (x^{3}\right )}{3 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.08, size = 175, normalized size = 1.00 \[ \frac {\ln \left (b\,x^3+a\right )\,\left (-f\,a^3+2\,e\,a^2\,b-3\,d\,a\,b^2+4\,c\,b^3\right )}{3\,a^5}-\frac {\frac {c}{9\,a}+\frac {x^9\,\left (-f\,a^3+2\,e\,a^2\,b-3\,d\,a\,b^2+4\,c\,b^3\right )}{3\,a^4}+\frac {x^3\,\left (3\,a\,d-4\,b\,c\right )}{18\,a^2}+\frac {x^6\,\left (2\,e\,a^2-3\,d\,a\,b+4\,c\,b^2\right )}{6\,a^3}}{b\,x^{12}+a\,x^9}-\frac {\ln \relax (x)\,\left (-f\,a^3+2\,e\,a^2\,b-3\,d\,a\,b^2+4\,c\,b^3\right )}{a^5} \]
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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