Optimal. Leaf size=205 \[ \frac {b c-a d}{12 a^2 x^{12}}-\frac {a^2 e-a b d+b^2 c}{9 a^3 x^9}+\frac {b^2 \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^6}-\frac {b^2 \log (x) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^6}-\frac {b \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 x^3}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 a^4 x^6}-\frac {c}{15 a x^{15}} \]
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Rubi [A] time = 0.21, antiderivative size = 205, 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} \[ -\frac {b \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5 x^3}+\frac {a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{6 a^4 x^6}+\frac {b^2 \log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^6}-\frac {b^2 \log (x) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^6}-\frac {a^2 e-a b d+b^2 c}{9 a^3 x^9}+\frac {b c-a d}{12 a^2 x^{12}}-\frac {c}{15 a x^{15}} \]
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^{16} \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^6 (a+b x)} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {c}{a x^6}+\frac {-b c+a d}{a^2 x^5}+\frac {b^2 c-a b d+a^2 e}{a^3 x^4}+\frac {-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^3}-\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 x^2}+\frac {b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^6 x}-\frac {b^3 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^6 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {c}{15 a x^{15}}+\frac {b c-a d}{12 a^2 x^{12}}-\frac {b^2 c-a b d+a^2 e}{9 a^3 x^9}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{6 a^4 x^6}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 a^5 x^3}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log (x)}{a^6}+\frac {b^2 \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^6}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 194, normalized size = 0.95 \[ -\frac {-60 b^2 \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )+180 b^2 \log (x) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )+\frac {a \left (a^4 \left (12 c+15 d x^3+20 e x^6+30 f x^9\right )-5 a^3 b x^3 \left (3 c+4 d x^3+6 e x^6+12 f x^9\right )+10 a^2 b^2 x^6 \left (2 c+3 d x^3+6 e x^6\right )-30 a b^3 x^9 \left (c+2 d x^3\right )+60 b^4 c x^{12}\right )}{x^{15}}}{180 a^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 210, normalized size = 1.02 \[ \frac {60 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{15} \log \left (b x^{3} + a\right ) - 180 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{15} \log \relax (x) - 60 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{12} + 30 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x^{9} - 20 \, {\left (a^{3} b^{2} c - a^{4} b d + a^{5} e\right )} x^{6} - 12 \, a^{5} c + 15 \, {\left (a^{4} b c - a^{5} d\right )} x^{3}}{180 \, a^{6} x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 287, normalized size = 1.40 \[ -\frac {{\left (b^{5} c - a b^{4} d - a^{3} b^{2} f + a^{2} b^{3} e\right )} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac {{\left (b^{6} c - a b^{5} d - a^{3} b^{3} f + a^{2} b^{4} e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{6} b} + \frac {137 \, b^{5} c x^{15} - 137 \, a b^{4} d x^{15} - 137 \, a^{3} b^{2} f x^{15} + 137 \, a^{2} b^{3} x^{15} e - 60 \, a b^{4} c x^{12} + 60 \, a^{2} b^{3} d x^{12} + 60 \, a^{4} b f x^{12} - 60 \, a^{3} b^{2} x^{12} e + 30 \, a^{2} b^{3} c x^{9} - 30 \, a^{3} b^{2} d x^{9} - 30 \, a^{5} f x^{9} + 30 \, a^{4} b x^{9} e - 20 \, a^{3} b^{2} c x^{6} + 20 \, a^{4} b d x^{6} - 20 \, a^{5} x^{6} e + 15 \, a^{4} b c x^{3} - 15 \, a^{5} d x^{3} - 12 \, a^{5} c}{180 \, a^{6} x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 260, normalized size = 1.27 \[ \frac {b^{2} f \ln \relax (x )}{a^{3}}-\frac {b^{2} f \ln \left (b \,x^{3}+a \right )}{3 a^{3}}-\frac {b^{3} e \ln \relax (x )}{a^{4}}+\frac {b^{3} e \ln \left (b \,x^{3}+a \right )}{3 a^{4}}+\frac {b^{4} d \ln \relax (x )}{a^{5}}-\frac {b^{4} d \ln \left (b \,x^{3}+a \right )}{3 a^{5}}-\frac {b^{5} c \ln \relax (x )}{a^{6}}+\frac {b^{5} c \ln \left (b \,x^{3}+a \right )}{3 a^{6}}+\frac {b f}{3 a^{2} x^{3}}-\frac {b^{2} e}{3 a^{3} x^{3}}+\frac {b^{3} d}{3 a^{4} x^{3}}-\frac {b^{4} c}{3 a^{5} x^{3}}-\frac {f}{6 a \,x^{6}}+\frac {b e}{6 a^{2} x^{6}}-\frac {b^{2} d}{6 a^{3} x^{6}}+\frac {b^{3} c}{6 a^{4} x^{6}}-\frac {e}{9 a \,x^{9}}+\frac {b d}{9 a^{2} x^{9}}-\frac {b^{2} c}{9 a^{3} x^{9}}-\frac {d}{12 a \,x^{12}}+\frac {b c}{12 a^{2} x^{12}}-\frac {c}{15 a \,x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 208, normalized size = 1.01 \[ \frac {{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{6}} - \frac {{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} \log \left (x^{3}\right )}{3 \, a^{6}} - \frac {60 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{12} - 30 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{9} + 20 \, {\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{6} + 12 \, a^{4} c - 15 \, {\left (a^{3} b c - a^{4} d\right )} x^{3}}{180 \, a^{5} x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 200, normalized size = 0.98 \[ \frac {\ln \left (b\,x^3+a\right )\,\left (-f\,a^3\,b^2+e\,a^2\,b^3-d\,a\,b^4+c\,b^5\right )}{3\,a^6}-\frac {\frac {c}{15\,a}-\frac {x^9\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{6\,a^4}+\frac {x^3\,\left (a\,d-b\,c\right )}{12\,a^2}+\frac {x^6\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{9\,a^3}+\frac {b\,x^{12}\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^5}}{x^{15}}-\frac {\ln \relax (x)\,\left (-f\,a^3\,b^2+e\,a^2\,b^3-d\,a\,b^4+c\,b^5\right )}{a^6} \]
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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