Optimal. Leaf size=226 \[ \frac {x^6 \left (6 a^2 f-3 a b e+b^2 d\right )}{6 b^5}-\frac {a^2 \left (-6 a^3 f+5 a^2 b e-4 a b^2 d+3 b^3 c\right )}{3 b^7 \left (a+b x^3\right )}+\frac {a^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}-\frac {a \log \left (a+b x^3\right ) \left (-15 a^3 f+10 a^2 b e-6 a b^2 d+3 b^3 c\right )}{3 b^7}+\frac {x^3 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{3 b^6}+\frac {x^9 (b e-3 a f)}{9 b^4}+\frac {f x^{12}}{12 b^3} \]
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Rubi [A] time = 0.33, antiderivative size = 226, 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} \[ \frac {x^3 \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{3 b^6}-\frac {a^2 \left (5 a^2 b e-6 a^3 f-4 a b^2 d+3 b^3 c\right )}{3 b^7 \left (a+b x^3\right )}+\frac {a^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}-\frac {a \log \left (a+b x^3\right ) \left (10 a^2 b e-15 a^3 f-6 a b^2 d+3 b^3 c\right )}{3 b^7}+\frac {x^6 \left (6 a^2 f-3 a b e+b^2 d\right )}{6 b^5}+\frac {x^9 (b e-3 a f)}{9 b^4}+\frac {f x^{12}}{12 b^3} \]
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 )}{\left (a+b x^3\right )^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)^3} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f}{b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x}{b^5}+\frac {(b e-3 a f) x^2}{b^4}+\frac {f x^3}{b^3}+\frac {a^3 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^6 (a+b x)^3}-\frac {a^2 \left (-3 b^3 c+4 a b^2 d-5 a^2 b e+6 a^3 f\right )}{b^6 (a+b x)^2}+\frac {a \left (-3 b^3 c+6 a b^2 d-10 a^2 b e+15 a^3 f\right )}{b^6 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^3}{3 b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^6}{6 b^5}+\frac {(b e-3 a f) x^9}{9 b^4}+\frac {f x^{12}}{12 b^3}+\frac {a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{6 b^7 \left (a+b x^3\right )^2}-\frac {a^2 \left (3 b^3 c-4 a b^2 d+5 a^2 b e-6 a^3 f\right )}{3 b^7 \left (a+b x^3\right )}-\frac {a \left (3 b^3 c-6 a b^2 d+10 a^2 b e-15 a^3 f\right ) \log \left (a+b x^3\right )}{3 b^7}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 208, normalized size = 0.92 \[ \frac {6 b^2 x^6 \left (6 a^2 f-3 a b e+b^2 d\right )+12 b x^3 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )+\frac {12 a^2 \left (6 a^3 f-5 a^2 b e+4 a b^2 d-3 b^3 c\right )}{a+b x^3}+\frac {6 a^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\left (a+b x^3\right )^2}+12 a \log \left (a+b x^3\right ) \left (15 a^3 f-10 a^2 b e+6 a b^2 d-3 b^3 c\right )+4 b^3 x^9 (b e-3 a f)+3 b^4 f x^{12}}{36 b^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 353, normalized size = 1.56 \[ \frac {3 \, b^{6} f x^{18} + 2 \, {\left (2 \, b^{6} e - 3 \, a b^{5} f\right )} x^{15} + {\left (6 \, b^{6} d - 10 \, a b^{5} e + 15 \, a^{2} b^{4} f\right )} x^{12} + 4 \, {\left (3 \, b^{6} c - 6 \, a b^{5} d + 10 \, a^{2} b^{4} e - 15 \, a^{3} b^{3} f\right )} x^{9} - 30 \, a^{3} b^{3} c + 42 \, a^{4} b^{2} d - 54 \, a^{5} b e + 66 \, a^{6} f + 6 \, {\left (4 \, a b^{5} c - 11 \, a^{2} b^{4} d + 21 \, a^{3} b^{3} e - 34 \, a^{4} b^{2} f\right )} x^{6} - 12 \, {\left (2 \, a^{2} b^{4} c - a^{3} b^{3} d - a^{4} b^{2} e + 4 \, a^{5} b f\right )} x^{3} - 12 \, {\left (3 \, a^{3} b^{3} c - 6 \, a^{4} b^{2} d + 10 \, a^{5} b e - 15 \, a^{6} f + {\left (3 \, a b^{5} c - 6 \, a^{2} b^{4} d + 10 \, a^{3} b^{3} e - 15 \, a^{4} b^{2} f\right )} x^{6} + 2 \, {\left (3 \, a^{2} b^{4} c - 6 \, a^{3} b^{3} d + 10 \, a^{4} b^{2} e - 15 \, a^{5} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{36 \, {\left (b^{9} x^{6} + 2 \, a b^{8} x^{3} + a^{2} b^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 298, normalized size = 1.32 \[ -\frac {{\left (3 \, a b^{3} c - 6 \, a^{2} b^{2} d - 15 \, a^{4} f + 10 \, a^{3} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{7}} + \frac {9 \, a b^{5} c x^{6} - 18 \, a^{2} b^{4} d x^{6} - 45 \, a^{4} b^{2} f x^{6} + 30 \, a^{3} b^{3} x^{6} e + 12 \, a^{2} b^{4} c x^{3} - 28 \, a^{3} b^{3} d x^{3} - 78 \, a^{5} b f x^{3} + 50 \, a^{4} b^{2} x^{3} e + 4 \, a^{3} b^{3} c - 11 \, a^{4} b^{2} d - 34 \, a^{6} f + 21 \, a^{5} b e}{6 \, {\left (b x^{3} + a\right )}^{2} b^{7}} + \frac {3 \, b^{9} f x^{12} - 12 \, a b^{8} f x^{9} + 4 \, b^{9} x^{9} e + 6 \, b^{9} d x^{6} + 36 \, a^{2} b^{7} f x^{6} - 18 \, a b^{8} x^{6} e + 12 \, b^{9} c x^{3} - 36 \, a b^{8} d x^{3} - 120 \, a^{3} b^{6} f x^{3} + 72 \, a^{2} b^{7} x^{3} e}{36 \, b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 313, normalized size = 1.38 \[ \frac {f \,x^{12}}{12 b^{3}}-\frac {a f \,x^{9}}{3 b^{4}}+\frac {e \,x^{9}}{9 b^{3}}+\frac {a^{2} f \,x^{6}}{b^{5}}-\frac {a e \,x^{6}}{2 b^{4}}+\frac {d \,x^{6}}{6 b^{3}}-\frac {10 a^{3} f \,x^{3}}{3 b^{6}}+\frac {2 a^{2} e \,x^{3}}{b^{5}}-\frac {a d \,x^{3}}{b^{4}}+\frac {c \,x^{3}}{3 b^{3}}-\frac {a^{6} f}{6 \left (b \,x^{3}+a \right )^{2} b^{7}}+\frac {a^{5} e}{6 \left (b \,x^{3}+a \right )^{2} b^{6}}-\frac {a^{4} d}{6 \left (b \,x^{3}+a \right )^{2} b^{5}}+\frac {a^{3} c}{6 \left (b \,x^{3}+a \right )^{2} b^{4}}+\frac {2 a^{5} f}{\left (b \,x^{3}+a \right ) b^{7}}-\frac {5 a^{4} e}{3 \left (b \,x^{3}+a \right ) b^{6}}+\frac {5 a^{4} f \ln \left (b \,x^{3}+a \right )}{b^{7}}+\frac {4 a^{3} d}{3 \left (b \,x^{3}+a \right ) b^{5}}-\frac {10 a^{3} e \ln \left (b \,x^{3}+a \right )}{3 b^{6}}-\frac {a^{2} c}{\left (b \,x^{3}+a \right ) b^{4}}+\frac {2 a^{2} d \ln \left (b \,x^{3}+a \right )}{b^{5}}-\frac {a c \ln \left (b \,x^{3}+a \right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 233, normalized size = 1.03 \[ -\frac {5 \, a^{3} b^{3} c - 7 \, a^{4} b^{2} d + 9 \, a^{5} b e - 11 \, a^{6} f + 2 \, {\left (3 \, a^{2} b^{4} c - 4 \, a^{3} b^{3} d + 5 \, a^{4} b^{2} e - 6 \, a^{5} b f\right )} x^{3}}{6 \, {\left (b^{9} x^{6} + 2 \, a b^{8} x^{3} + a^{2} b^{7}\right )}} + \frac {3 \, b^{3} f x^{12} + 4 \, {\left (b^{3} e - 3 \, a b^{2} f\right )} x^{9} + 6 \, {\left (b^{3} d - 3 \, a b^{2} e + 6 \, a^{2} b f\right )} x^{6} + 12 \, {\left (b^{3} c - 3 \, a b^{2} d + 6 \, a^{2} b e - 10 \, a^{3} f\right )} x^{3}}{36 \, b^{6}} - \frac {{\left (3 \, a b^{3} c - 6 \, a^{2} b^{2} d + 10 \, a^{3} b e - 15 \, a^{4} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.97, size = 293, normalized size = 1.30 \[ x^9\,\left (\frac {e}{9\,b^3}-\frac {a\,f}{3\,b^4}\right )+x^3\,\left (\frac {c}{3\,b^3}-\frac {a^3\,f}{3\,b^6}-\frac {a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b^2}+\frac {a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b}\right )-x^6\,\left (\frac {a^2\,f}{2\,b^5}-\frac {d}{6\,b^3}+\frac {a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{2\,b}\right )+\frac {\frac {11\,f\,a^6-9\,e\,a^5\,b+7\,d\,a^4\,b^2-5\,c\,a^3\,b^3}{6\,b}+x^3\,\left (2\,f\,a^5-\frac {5\,e\,a^4\,b}{3}+\frac {4\,d\,a^3\,b^2}{3}-c\,a^2\,b^3\right )}{a^2\,b^6+2\,a\,b^7\,x^3+b^8\,x^6}+\frac {f\,x^{12}}{12\,b^3}+\frac {\ln \left (b\,x^3+a\right )\,\left (15\,f\,a^4-10\,e\,a^3\,b+6\,d\,a^2\,b^2-3\,c\,a\,b^3\right )}{3\,b^7} \]
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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