Optimal. Leaf size=266 \[ \frac{x^9 \left (6 a^2 f-3 a b e+b^2 d\right )}{9 b^5}+\frac{a^3 \left (-7 a^3 f+6 a^2 b e-5 a b^2 d+4 b^3 c\right )}{3 b^8 \left (a+b x^3\right )}+\frac{a^2 \log \left (a+b x^3\right ) \left (-21 a^3 f+15 a^2 b e-10 a b^2 d+6 b^3 c\right )}{3 b^8}-\frac{a x^3 \left (-15 a^3 f+10 a^2 b e-6 a b^2 d+3 b^3 c\right )}{3 b^7}+\frac{x^6 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{6 b^6}-\frac{a^4 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^8 \left (a+b x^3\right )^2}+\frac{x^{12} (b e-3 a f)}{12 b^4}+\frac{f x^{15}}{15 b^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.866896, antiderivative size = 266, 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 \[ \frac{x^9 \left (6 a^2 f-3 a b e+b^2 d\right )}{9 b^5}+\frac{a^3 \left (-7 a^3 f+6 a^2 b e-5 a b^2 d+4 b^3 c\right )}{3 b^8 \left (a+b x^3\right )}+\frac{a^2 \log \left (a+b x^3\right ) \left (-21 a^3 f+15 a^2 b e-10 a b^2 d+6 b^3 c\right )}{3 b^8}-\frac{a x^3 \left (-15 a^3 f+10 a^2 b e-6 a b^2 d+3 b^3 c\right )}{3 b^7}+\frac{x^6 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{6 b^6}-\frac{a^4 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^8 \left (a+b x^3\right )^2}+\frac{x^{12} (b e-3 a f)}{12 b^4}+\frac{f x^{15}}{15 b^3} \]
Antiderivative was successfully verified.
[In] Int[(x^14*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a^{4} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{6 b^{8} \left (a + b x^{3}\right )^{2}} - \frac{a^{3} \left (7 a^{3} f - 6 a^{2} b e + 5 a b^{2} d - 4 b^{3} c\right )}{3 b^{8} \left (a + b x^{3}\right )} - \frac{a^{2} \left (21 a^{3} f - 15 a^{2} b e + 10 a b^{2} d - 6 b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 b^{8}} + \frac{f x^{15}}{15 b^{3}} - \frac{x^{12} \left (3 a f - b e\right )}{12 b^{4}} + \frac{x^{9} \left (6 a^{2} f - 3 a b e + b^{2} d\right )}{9 b^{5}} - \frac{\left (10 a^{3} f - 6 a^{2} b e + 3 a b^{2} d - b^{3} c\right ) \int ^{x^{3}} x\, dx}{3 b^{6}} + \frac{\left (15 a^{3} f - 10 a^{2} b e + 6 a b^{2} d - 3 b^{3} c\right ) \int ^{x^{3}} a\, dx}{3 b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**14*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.271543, size = 246, normalized size = 0.92 \[ \frac{20 b^3 x^9 \left (6 a^2 f-3 a b e+b^2 d\right )+30 b^2 x^6 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )+60 a b x^3 \left (15 a^3 f-10 a^2 b e+6 a b^2 d-3 b^3 c\right )-\frac{60 a^3 \left (7 a^3 f-6 a^2 b e+5 a b^2 d-4 b^3 c\right )}{a+b x^3}+60 a^2 \log \left (a+b x^3\right ) \left (-21 a^3 f+15 a^2 b e-10 a b^2 d+6 b^3 c\right )+\frac{30 a^4 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+15 b^4 x^{12} (b e-3 a f)+12 b^5 f x^{15}}{180 b^8} \]
Antiderivative was successfully verified.
[In] Integrate[(x^14*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.021, size = 361, normalized size = 1.4 \[{\frac{{a}^{5}d}{6\,{b}^{6} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{4}c}{6\,{b}^{5} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{7\,{a}^{6}f}{3\,{b}^{8} \left ( b{x}^{3}+a \right ) }}+2\,{\frac{{a}^{5}e}{{b}^{7} \left ( b{x}^{3}+a \right ) }}-{\frac{5\,{a}^{4}d}{3\,{b}^{6} \left ( b{x}^{3}+a \right ) }}+{\frac{4\,{a}^{3}c}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }}-7\,{\frac{{a}^{5}\ln \left ( b{x}^{3}+a \right ) f}{{b}^{8}}}+5\,{\frac{{a}^{4}\ln \left ( b{x}^{3}+a \right ) e}{{b}^{7}}}-{\frac{10\,{a}^{3}\ln \left ( b{x}^{3}+a \right ) d}{3\,{b}^{6}}}+2\,{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) c}{{b}^{5}}}-{\frac{{x}^{12}af}{4\,{b}^{4}}}+{\frac{2\,{x}^{9}{a}^{2}f}{3\,{b}^{5}}}-{\frac{{x}^{9}ae}{3\,{b}^{4}}}-{\frac{5\,{a}^{3}f{x}^{6}}{3\,{b}^{6}}}+{\frac{{a}^{2}e{x}^{6}}{{b}^{5}}}-{\frac{ad{x}^{6}}{2\,{b}^{4}}}+5\,{\frac{{a}^{4}f{x}^{3}}{{b}^{7}}}-{\frac{10\,{a}^{3}e{x}^{3}}{3\,{b}^{6}}}+2\,{\frac{{a}^{2}d{x}^{3}}{{b}^{5}}}-{\frac{ac{x}^{3}}{{b}^{4}}}+{\frac{{x}^{12}e}{12\,{b}^{3}}}+{\frac{{x}^{9}d}{9\,{b}^{3}}}+{\frac{{x}^{6}c}{6\,{b}^{3}}}-{\frac{{a}^{6}e}{6\,{b}^{7} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{{a}^{7}f}{6\,{b}^{8} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{f{x}^{15}}{15\,{b}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^14*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.37891, size = 371, normalized size = 1.39 \[ \frac{7 \, a^{4} b^{3} c - 9 \, a^{5} b^{2} d + 11 \, a^{6} b e - 13 \, a^{7} f + 2 \,{\left (4 \, a^{3} b^{4} c - 5 \, a^{4} b^{3} d + 6 \, a^{5} b^{2} e - 7 \, a^{6} b f\right )} x^{3}}{6 \,{\left (b^{10} x^{6} + 2 \, a b^{9} x^{3} + a^{2} b^{8}\right )}} + \frac{12 \, b^{4} f x^{15} + 15 \,{\left (b^{4} e - 3 \, a b^{3} f\right )} x^{12} + 20 \,{\left (b^{4} d - 3 \, a b^{3} e + 6 \, a^{2} b^{2} f\right )} x^{9} + 30 \,{\left (b^{4} c - 3 \, a b^{3} d + 6 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{6} - 60 \,{\left (3 \, a b^{3} c - 6 \, a^{2} b^{2} d + 10 \, a^{3} b e - 15 \, a^{4} f\right )} x^{3}}{180 \, b^{7}} + \frac{{\left (6 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 15 \, a^{4} b e - 21 \, a^{5} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^14/(b*x^3 + a)^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.205264, size = 535, normalized size = 2.01 \[ \frac{12 \, b^{7} f x^{21} + 3 \,{\left (5 \, b^{7} e - 7 \, a b^{6} f\right )} x^{18} + 2 \,{\left (10 \, b^{7} d - 15 \, a b^{6} e + 21 \, a^{2} b^{5} f\right )} x^{15} + 5 \,{\left (6 \, b^{7} c - 10 \, a b^{6} d + 15 \, a^{2} b^{5} e - 21 \, a^{3} b^{4} f\right )} x^{12} - 20 \,{\left (6 \, a b^{6} c - 10 \, a^{2} b^{5} d + 15 \, a^{3} b^{4} e - 21 \, a^{4} b^{3} f\right )} x^{9} + 210 \, a^{4} b^{3} c - 270 \, a^{5} b^{2} d + 330 \, a^{6} b e - 390 \, a^{7} f - 30 \,{\left (11 \, a^{2} b^{5} c - 21 \, a^{3} b^{4} d + 34 \, a^{4} b^{3} e - 50 \, a^{5} b^{2} f\right )} x^{6} + 60 \,{\left (a^{3} b^{4} c + a^{4} b^{3} d - 4 \, a^{5} b^{2} e + 8 \, a^{6} b f\right )} x^{3} + 60 \,{\left (6 \, a^{4} b^{3} c - 10 \, a^{5} b^{2} d + 15 \, a^{6} b e - 21 \, a^{7} f +{\left (6 \, a^{2} b^{5} c - 10 \, a^{3} b^{4} d + 15 \, a^{4} b^{3} e - 21 \, a^{5} b^{2} f\right )} x^{6} + 2 \,{\left (6 \, a^{3} b^{4} c - 10 \, a^{4} b^{3} d + 15 \, a^{5} b^{2} e - 21 \, a^{6} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{180 \,{\left (b^{10} x^{6} + 2 \, a b^{9} x^{3} + a^{2} b^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^14/(b*x^3 + a)^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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**14*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.218528, size = 471, normalized size = 1.77 \[ \frac{{\left (6 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d - 21 \, a^{5} f + 15 \, a^{4} b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{8}} - \frac{18 \, a^{2} b^{5} c x^{6} - 30 \, a^{3} b^{4} d x^{6} - 63 \, a^{5} b^{2} f x^{6} + 45 \, a^{4} b^{3} x^{6} e + 28 \, a^{3} b^{4} c x^{3} - 50 \, a^{4} b^{3} d x^{3} - 112 \, a^{6} b f x^{3} + 78 \, a^{5} b^{2} x^{3} e + 11 \, a^{4} b^{3} c - 21 \, a^{5} b^{2} d - 50 \, a^{7} f + 34 \, a^{6} b e}{6 \,{\left (b x^{3} + a\right )}^{2} b^{8}} + \frac{12 \, b^{12} f x^{15} - 45 \, a b^{11} f x^{12} + 15 \, b^{12} x^{12} e + 20 \, b^{12} d x^{9} + 120 \, a^{2} b^{10} f x^{9} - 60 \, a b^{11} x^{9} e + 30 \, b^{12} c x^{6} - 90 \, a b^{11} d x^{6} - 300 \, a^{3} b^{9} f x^{6} + 180 \, a^{2} b^{10} x^{6} e - 180 \, a b^{11} c x^{3} + 360 \, a^{2} b^{10} d x^{3} + 900 \, a^{4} b^{8} f x^{3} - 600 \, a^{3} b^{9} x^{3} e}{180 \, b^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^14/(b*x^3 + a)^3,x, algorithm="giac")
[Out]