Optimal. Leaf size=164 \[ \frac {b c-a d}{9 a^2 x^9}-\frac {a^2 e-a b d+b^2 c}{6 a^3 x^6}-\frac {b \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5}+\frac {b \log (x) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^5}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a^4 x^3}-\frac {c}{12 a x^{12}} \]
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Rubi [A] time = 0.18, antiderivative size = 164, 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} \[ \frac {a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 a^4 x^3}-\frac {b \log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5}+\frac {b \log (x) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^5}-\frac {a^2 e-a b d+b^2 c}{6 a^3 x^6}+\frac {b c-a d}{9 a^2 x^9}-\frac {c}{12 a x^{12}} \]
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^{13} \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^5 (a+b x)} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {c}{a x^5}+\frac {-b c+a d}{a^2 x^4}+\frac {b^2 c-a b d+a^2 e}{a^3 x^3}+\frac {-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^2}-\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 x}+\frac {b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {c}{12 a x^{12}}+\frac {b c-a d}{9 a^2 x^9}-\frac {b^2 c-a b d+a^2 e}{6 a^3 x^6}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^4 x^3}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log (x)}{a^5}-\frac {b \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^5}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 164, normalized size = 1.00 \[ \frac {-a^4 \left (3 c+4 d x^3+6 e x^6+12 f x^9\right )+2 a^3 b x^3 \left (2 c+3 d x^3+6 e x^6\right )-6 a^2 b^2 x^6 \left (c+2 d x^3\right )+36 b x^{12} \log (x) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )-12 b x^{12} \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )+12 a b^3 c x^9}{36 a^5 x^{12}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 168, normalized size = 1.02 \[ -\frac {12 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{12} \log \left (b x^{3} + a\right ) - 36 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{12} \log \relax (x) - 12 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{9} + 6 \, {\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{6} + 3 \, a^{4} c - 4 \, {\left (a^{3} b c - a^{4} d\right )} x^{3}}{36 \, a^{5} x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 235, normalized size = 1.43 \[ \frac {{\left (b^{4} c - a b^{3} d - a^{3} b f + a^{2} b^{2} e\right )} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {{\left (b^{5} c - a b^{4} d - a^{3} b^{2} f + a^{2} b^{3} e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{5} b} - \frac {25 \, b^{4} c x^{12} - 25 \, a b^{3} d x^{12} - 25 \, a^{3} b f x^{12} + 25 \, a^{2} b^{2} x^{12} e - 12 \, a b^{3} c x^{9} + 12 \, a^{2} b^{2} d x^{9} + 12 \, a^{4} f x^{9} - 12 \, a^{3} b x^{9} e + 6 \, a^{2} b^{2} c x^{6} - 6 \, a^{3} b d x^{6} + 6 \, a^{4} x^{6} e - 4 \, a^{3} b c x^{3} + 4 \, a^{4} d x^{3} + 3 \, a^{4} c}{36 \, a^{5} x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 210, normalized size = 1.28 \[ -\frac {b f \ln \relax (x )}{a^{2}}+\frac {b f \ln \left (b \,x^{3}+a \right )}{3 a^{2}}+\frac {b^{2} e \ln \relax (x )}{a^{3}}-\frac {b^{2} e \ln \left (b \,x^{3}+a \right )}{3 a^{3}}-\frac {b^{3} d \ln \relax (x )}{a^{4}}+\frac {b^{3} d \ln \left (b \,x^{3}+a \right )}{3 a^{4}}+\frac {b^{4} c \ln \relax (x )}{a^{5}}-\frac {b^{4} c \ln \left (b \,x^{3}+a \right )}{3 a^{5}}-\frac {f}{3 a \,x^{3}}+\frac {b e}{3 a^{2} x^{3}}-\frac {b^{2} d}{3 a^{3} x^{3}}+\frac {b^{3} c}{3 a^{4} x^{3}}-\frac {e}{6 a \,x^{6}}+\frac {b d}{6 a^{2} x^{6}}-\frac {b^{2} c}{6 a^{3} x^{6}}-\frac {d}{9 a \,x^{9}}+\frac {b c}{9 a^{2} x^{9}}-\frac {c}{12 a \,x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 166, normalized size = 1.01 \[ -\frac {{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{5}} + \frac {{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \log \left (x^{3}\right )}{3 \, a^{5}} + \frac {12 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{9} - 6 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{6} - 3 \, a^{3} c + 4 \, {\left (a^{2} b c - a^{3} d\right )} x^{3}}{36 \, a^{4} x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.07, size = 161, normalized size = 0.98 \[ \frac {\ln \relax (x)\,\left (-f\,a^3\,b+e\,a^2\,b^2-d\,a\,b^3+c\,b^4\right )}{a^5}-\frac {\ln \left (b\,x^3+a\right )\,\left (-f\,a^3\,b+e\,a^2\,b^2-d\,a\,b^3+c\,b^4\right )}{3\,a^5}-\frac {\frac {c}{12\,a}-\frac {x^9\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^4}+\frac {x^3\,\left (a\,d-b\,c\right )}{9\,a^2}+\frac {x^6\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{6\,a^3}}{x^{12}} \]
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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