Optimal. Leaf size=226 \[ \frac{x^6 \left (6 a^2 f-3 a b e+b^2 d\right )}{6 b^5}-\frac{a^2 \left (-6 a^3 f+5 a^2 b e-4 a b^2 d+3 b^3 c\right )}{3 b^7 \left (a+b x^3\right )}+\frac{a^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}-\frac{a \log \left (a+b x^3\right ) \left (-15 a^3 f+10 a^2 b e-6 a b^2 d+3 b^3 c\right )}{3 b^7}+\frac{x^3 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{3 b^6}+\frac{x^9 (b e-3 a f)}{9 b^4}+\frac{f x^{12}}{12 b^3} \]
[Out]
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Rubi [A] time = 0.698882, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{x^6 \left (6 a^2 f-3 a b e+b^2 d\right )}{6 b^5}-\frac{a^2 \left (-6 a^3 f+5 a^2 b e-4 a b^2 d+3 b^3 c\right )}{3 b^7 \left (a+b x^3\right )}+\frac{a^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}-\frac{a \log \left (a+b x^3\right ) \left (-15 a^3 f+10 a^2 b e-6 a b^2 d+3 b^3 c\right )}{3 b^7}+\frac{x^3 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{3 b^6}+\frac{x^9 (b e-3 a f)}{9 b^4}+\frac{f x^{12}}{12 b^3} \]
Antiderivative was successfully verified.
[In] Int[(x^11*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{3} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{6 b^{7} \left (a + b x^{3}\right )^{2}} + \frac{a^{2} \left (6 a^{3} f - 5 a^{2} b e + 4 a b^{2} d - 3 b^{3} c\right )}{3 b^{7} \left (a + b x^{3}\right )} + \frac{a \left (15 a^{3} f - 10 a^{2} b e + 6 a b^{2} d - 3 b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 b^{7}} - \left (\frac{10 a^{3} f}{3} - 2 a^{2} b e + a b^{2} d - \frac{b^{3} c}{3}\right ) \int ^{x^{3}} \frac{1}{b^{6}}\, dx + \frac{f x^{12}}{12 b^{3}} - \frac{x^{9} \left (3 a f - b e\right )}{9 b^{4}} + \frac{\left (6 a^{2} f - 3 a b e + b^{2} d\right ) \int ^{x^{3}} x\, dx}{3 b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**11*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)
[Out]
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Mathematica [A] time = 0.230815, 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.
[In] Integrate[(x^11*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]
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Maple [A] time = 0.022, size = 313, normalized size = 1.4 \[{\frac{f{x}^{12}}{12\,{b}^{3}}}-{\frac{{x}^{9}af}{3\,{b}^{4}}}+{\frac{{x}^{9}e}{9\,{b}^{3}}}+{\frac{{a}^{2}f{x}^{6}}{{b}^{5}}}-{\frac{ae{x}^{6}}{2\,{b}^{4}}}+{\frac{d{x}^{6}}{6\,{b}^{3}}}-{\frac{10\,{a}^{3}f{x}^{3}}{3\,{b}^{6}}}+2\,{\frac{{a}^{2}e{x}^{3}}{{b}^{5}}}-{\frac{ad{x}^{3}}{{b}^{4}}}+{\frac{c{x}^{3}}{3\,{b}^{3}}}+5\,{\frac{{a}^{4}\ln \left ( b{x}^{3}+a \right ) f}{{b}^{7}}}-{\frac{10\,{a}^{3}\ln \left ( b{x}^{3}+a \right ) e}{3\,{b}^{6}}}+2\,{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) d}{{b}^{5}}}-{\frac{a\ln \left ( b{x}^{3}+a \right ) c}{{b}^{4}}}-{\frac{{a}^{6}f}{6\,{b}^{7} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{{a}^{5}e}{6\,{b}^{6} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{4}d}{6\,{b}^{5} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{{a}^{3}c}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}+2\,{\frac{{a}^{5}f}{{b}^{7} \left ( b{x}^{3}+a \right ) }}-{\frac{5\,{a}^{4}e}{3\,{b}^{6} \left ( b{x}^{3}+a \right ) }}+{\frac{4\,{a}^{3}d}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}c}{{b}^{4} \left ( b{x}^{3}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^11*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)
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Maxima [A] time = 1.44419, size = 315, normalized size = 1.39 \[ -\frac{5 \, a^{3} b^{3} c - 7 \, a^{4} b^{2} d + 9 \, a^{5} b e - 11 \, a^{6} f + 2 \,{\left (3 \, a^{2} b^{4} c - 4 \, a^{3} b^{3} d + 5 \, a^{4} b^{2} e - 6 \, a^{5} b f\right )} x^{3}}{6 \,{\left (b^{9} x^{6} + 2 \, a b^{8} x^{3} + a^{2} b^{7}\right )}} + \frac{3 \, b^{3} f x^{12} + 4 \,{\left (b^{3} e - 3 \, a b^{2} f\right )} x^{9} + 6 \,{\left (b^{3} d - 3 \, a b^{2} e + 6 \, a^{2} b f\right )} x^{6} + 12 \,{\left (b^{3} c - 3 \, a b^{2} d + 6 \, a^{2} b e - 10 \, a^{3} f\right )} x^{3}}{36 \, b^{6}} - \frac{{\left (3 \, a b^{3} c - 6 \, a^{2} b^{2} d + 10 \, a^{3} b e - 15 \, a^{4} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^11/(b*x^3 + a)^3,x, algorithm="maxima")
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Fricas [A] time = 0.206411, size = 477, normalized size = 2.11 \[ \frac{3 \, b^{6} f x^{18} + 2 \,{\left (2 \, b^{6} e - 3 \, a b^{5} f\right )} x^{15} +{\left (6 \, b^{6} d - 10 \, a b^{5} e + 15 \, a^{2} b^{4} f\right )} x^{12} + 4 \,{\left (3 \, b^{6} c - 6 \, a b^{5} d + 10 \, a^{2} b^{4} e - 15 \, a^{3} b^{3} f\right )} x^{9} - 30 \, a^{3} b^{3} c + 42 \, a^{4} b^{2} d - 54 \, a^{5} b e + 66 \, a^{6} f + 6 \,{\left (4 \, a b^{5} c - 11 \, a^{2} b^{4} d + 21 \, a^{3} b^{3} e - 34 \, a^{4} b^{2} f\right )} x^{6} - 12 \,{\left (2 \, a^{2} b^{4} c - a^{3} b^{3} d - a^{4} b^{2} e + 4 \, a^{5} b f\right )} x^{3} - 12 \,{\left (3 \, a^{3} b^{3} c - 6 \, a^{4} b^{2} d + 10 \, a^{5} b e - 15 \, a^{6} f +{\left (3 \, a b^{5} c - 6 \, a^{2} b^{4} d + 10 \, a^{3} b^{3} e - 15 \, a^{4} b^{2} f\right )} x^{6} + 2 \,{\left (3 \, a^{2} b^{4} c - 6 \, a^{3} b^{3} d + 10 \, a^{4} b^{2} e - 15 \, a^{5} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{36 \,{\left (b^{9} x^{6} + 2 \, a b^{8} x^{3} + a^{2} b^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^11/(b*x^3 + a)^3,x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**11*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)
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GIAC/XCAS [A] time = 0.215971, size = 402, normalized size = 1.78 \[ -\frac{{\left (3 \, a b^{3} c - 6 \, a^{2} b^{2} d - 15 \, a^{4} f + 10 \, a^{3} b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{7}} + \frac{9 \, a b^{5} c x^{6} - 18 \, a^{2} b^{4} d x^{6} - 45 \, a^{4} b^{2} f x^{6} + 30 \, a^{3} b^{3} x^{6} e + 12 \, a^{2} b^{4} c x^{3} - 28 \, a^{3} b^{3} d x^{3} - 78 \, a^{5} b f x^{3} + 50 \, a^{4} b^{2} x^{3} e + 4 \, a^{3} b^{3} c - 11 \, a^{4} b^{2} d - 34 \, a^{6} f + 21 \, a^{5} b e}{6 \,{\left (b x^{3} + a\right )}^{2} b^{7}} + \frac{3 \, b^{9} f x^{12} - 12 \, a b^{8} f x^{9} + 4 \, b^{9} x^{9} e + 6 \, b^{9} d x^{6} + 36 \, a^{2} b^{7} f x^{6} - 18 \, a b^{8} x^{6} e + 12 \, b^{9} c x^{3} - 36 \, a b^{8} d x^{3} - 120 \, a^{3} b^{6} f x^{3} + 72 \, a^{2} b^{7} x^{3} e}{36 \, b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^11/(b*x^3 + a)^3,x, algorithm="giac")
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