Optimal. Leaf size=208 \[ \frac {x^{12} \left (a^2 f-a b e+b^2 d\right )}{12 b^3}-\frac {a^3 \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^7}+\frac {a^2 x^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^6}-\frac {a x^6 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5}+\frac {x^9 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{9 b^4}+\frac {x^{15} (b e-a f)}{15 b^2}+\frac {f x^{18}}{18 b} \]
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Rubi [A] time = 0.32, antiderivative size = 208, 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} \[ \frac {x^9 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{9 b^4}-\frac {a x^6 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^5}+\frac {a^2 x^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^6}-\frac {a^3 \log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^7}+\frac {x^{12} \left (a^2 f-a b e+b^2 d\right )}{12 b^3}+\frac {x^{15} (b e-a f)}{15 b^2}+\frac {f x^{18}}{18 b} \]
Antiderivative was successfully verified.
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Rule 1620
Rule 1821
Rubi steps
\begin {align*} \int \frac {x^{11} \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{a+b x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {a^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^6}+\frac {a \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{b^5}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^3}{b^3}+\frac {(b e-a f) x^4}{b^2}+\frac {f x^5}{b}+\frac {a^3 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^6 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3}{3 b^6}-\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6}{6 b^5}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{9 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^{12}}{12 b^3}+\frac {(b e-a f) x^{15}}{15 b^2}+\frac {f x^{18}}{18 b}-\frac {a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a+b x^3\right )}{3 b^7}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 187, normalized size = 0.90 \[ \frac {60 a^3 \log \left (a+b x^3\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )+b x^3 \left (-60 a^5 f+30 a^4 b \left (2 e+f x^3\right )-10 a^3 b^2 \left (6 d+3 e x^3+2 f x^6\right )+5 a^2 b^3 \left (12 c+6 d x^3+4 e x^6+3 f x^9\right )-a b^4 x^3 \left (30 c+20 d x^3+15 e x^6+12 f x^9\right )+b^5 x^6 \left (20 c+15 d x^3+12 e x^6+10 f x^9\right )\right )}{180 b^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 210, normalized size = 1.01 \[ \frac {10 \, b^{6} f x^{18} + 12 \, {\left (b^{6} e - a b^{5} f\right )} x^{15} + 15 \, {\left (b^{6} d - a b^{5} e + a^{2} b^{4} f\right )} x^{12} + 20 \, {\left (b^{6} c - a b^{5} d + a^{2} b^{4} e - a^{3} b^{3} f\right )} x^{9} - 30 \, {\left (a b^{5} c - a^{2} b^{4} d + a^{3} b^{3} e - a^{4} b^{2} f\right )} x^{6} + 60 \, {\left (a^{2} b^{4} c - a^{3} b^{3} d + a^{4} b^{2} e - a^{5} b f\right )} x^{3} - 60 \, {\left (a^{3} b^{3} c - a^{4} b^{2} d + a^{5} b e - a^{6} f\right )} \log \left (b x^{3} + a\right )}{180 \, b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 246, normalized size = 1.18 \[ \frac {10 \, b^{5} f x^{18} - 12 \, a b^{4} f x^{15} + 12 \, b^{5} x^{15} e + 15 \, b^{5} d x^{12} + 15 \, a^{2} b^{3} f x^{12} - 15 \, a b^{4} x^{12} e + 20 \, b^{5} c x^{9} - 20 \, a b^{4} d x^{9} - 20 \, a^{3} b^{2} f x^{9} + 20 \, a^{2} b^{3} x^{9} e - 30 \, a b^{4} c x^{6} + 30 \, a^{2} b^{3} d x^{6} + 30 \, a^{4} b f x^{6} - 30 \, a^{3} b^{2} x^{6} e + 60 \, a^{2} b^{3} c x^{3} - 60 \, a^{3} b^{2} d x^{3} - 60 \, a^{5} f x^{3} + 60 \, a^{4} b x^{3} e}{180 \, b^{6}} - \frac {{\left (a^{3} b^{3} c - a^{4} b^{2} d - a^{6} f + a^{5} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 266, normalized size = 1.28 \[ \frac {f \,x^{18}}{18 b}-\frac {a f \,x^{15}}{15 b^{2}}+\frac {e \,x^{15}}{15 b}+\frac {a^{2} f \,x^{12}}{12 b^{3}}-\frac {a e \,x^{12}}{12 b^{2}}+\frac {d \,x^{12}}{12 b}-\frac {a^{3} f \,x^{9}}{9 b^{4}}+\frac {a^{2} e \,x^{9}}{9 b^{3}}-\frac {a d \,x^{9}}{9 b^{2}}+\frac {c \,x^{9}}{9 b}+\frac {a^{4} f \,x^{6}}{6 b^{5}}-\frac {a^{3} e \,x^{6}}{6 b^{4}}+\frac {a^{2} d \,x^{6}}{6 b^{3}}-\frac {a c \,x^{6}}{6 b^{2}}-\frac {a^{5} f \,x^{3}}{3 b^{6}}+\frac {a^{4} e \,x^{3}}{3 b^{5}}-\frac {a^{3} d \,x^{3}}{3 b^{4}}+\frac {a^{2} c \,x^{3}}{3 b^{3}}+\frac {a^{6} f \ln \left (b \,x^{3}+a \right )}{3 b^{7}}-\frac {a^{5} e \ln \left (b \,x^{3}+a \right )}{3 b^{6}}+\frac {a^{4} d \ln \left (b \,x^{3}+a \right )}{3 b^{5}}-\frac {a^{3} c \ln \left (b \,x^{3}+a \right )}{3 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 209, normalized size = 1.00 \[ \frac {10 \, b^{5} f x^{18} + 12 \, {\left (b^{5} e - a b^{4} f\right )} x^{15} + 15 \, {\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{12} + 20 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{9} - 30 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{6} + 60 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x^{3}}{180 \, b^{6}} - \frac {{\left (a^{3} b^{3} c - a^{4} b^{2} d + a^{5} b e - a^{6} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.92, size = 237, normalized size = 1.14 \[ x^{15}\,\left (\frac {e}{15\,b}-\frac {a\,f}{15\,b^2}\right )+x^{12}\,\left (\frac {d}{12\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{12\,b}\right )+x^9\,\left (\frac {c}{9\,b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{9\,b}\right )+\frac {\ln \left (b\,x^3+a\right )\,\left (f\,a^6-e\,a^5\,b+d\,a^4\,b^2-c\,a^3\,b^3\right )}{3\,b^7}+\frac {f\,x^{18}}{18\,b}+\frac {a^2\,x^3\,\left (\frac {c}{b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{b}\right )}{3\,b^2}-\frac {a\,x^6\,\left (\frac {c}{b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{b}\right )}{6\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.32, size = 216, normalized size = 1.04 \[ \frac {a^{3} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (a + b x^{3} \right )}}{3 b^{7}} + x^{15} \left (- \frac {a f}{15 b^{2}} + \frac {e}{15 b}\right ) + x^{12} \left (\frac {a^{2} f}{12 b^{3}} - \frac {a e}{12 b^{2}} + \frac {d}{12 b}\right ) + x^{9} \left (- \frac {a^{3} f}{9 b^{4}} + \frac {a^{2} e}{9 b^{3}} - \frac {a d}{9 b^{2}} + \frac {c}{9 b}\right ) + x^{6} \left (\frac {a^{4} f}{6 b^{5}} - \frac {a^{3} e}{6 b^{4}} + \frac {a^{2} d}{6 b^{3}} - \frac {a c}{6 b^{2}}\right ) + x^{3} \left (- \frac {a^{5} f}{3 b^{6}} + \frac {a^{4} e}{3 b^{5}} - \frac {a^{3} d}{3 b^{4}} + \frac {a^{2} c}{3 b^{3}}\right ) + \frac {f x^{18}}{18 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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