Optimal. Leaf size=384 \[ \frac{x^5 \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac{x^2 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{2 b^6}+\frac{a x^2 \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{9 b^6 \left (a+b x^3\right )}-\frac{a^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )}{54 b^{20/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )}{27 b^{20/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )}{9 \sqrt{3} b^{20/3}}+\frac{x^8 (b e-3 a f)}{8 b^4}+\frac{f x^{11}}{11 b^3} \]
[Out]
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Rubi [A] time = 2.05217, antiderivative size = 384, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x^5 \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac{x^2 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{2 b^6}+\frac{a x^2 \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{9 b^6 \left (a+b x^3\right )}-\frac{a^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )}{54 b^{20/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )}{27 b^{20/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )}{9 \sqrt{3} b^{20/3}}+\frac{x^8 (b e-3 a f)}{8 b^4}+\frac{f x^{11}}{11 b^3} \]
Antiderivative was successfully verified.
[In] Int[(x^10*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**10*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)
[Out]
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Mathematica [A] time = 0.977661, size = 380, normalized size = 0.99 \[ \frac{x^5 \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac{x^2 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{2 b^6}+\frac{a x^2 \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{9 b^6 \left (a+b x^3\right )}+\frac{a^2 x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (119 a^3 f-77 a^2 b e+44 a b^2 d-20 b^3 c\right )}{54 b^{20/3}}-\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (119 a^3 f-77 a^2 b e+44 a b^2 d-20 b^3 c\right )}{27 b^{20/3}}-\frac{a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (119 a^3 f-77 a^2 b e+44 a b^2 d-20 b^3 c\right )}{9 \sqrt{3} b^{20/3}}+\frac{x^8 (b e-3 a f)}{8 b^4}+\frac{f x^{11}}{11 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^10*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]
[Out]
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Maple [B] time = 0.021, size = 668, normalized size = 1.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^10*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^10/(b*x^3 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.240825, size = 887, normalized size = 2.31 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^10/(b*x^3 + a)^3,x, algorithm="fricas")
[Out]
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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**10*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.223775, size = 663, normalized size = 1.73 \[ \frac{{\left (20 \, a b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 44 \, a^{2} b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 119 \, a^{4} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 77 \, a^{3} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a b^{6}} + \frac{\sqrt{3}{\left (20 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 119 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 77 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, b^{8}} - \frac{{\left (20 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 119 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 77 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, b^{8}} + \frac{14 \, a b^{4} c x^{5} - 20 \, a^{2} b^{3} d x^{5} - 32 \, a^{4} b f x^{5} + 26 \, a^{3} b^{2} x^{5} e + 11 \, a^{2} b^{3} c x^{2} - 17 \, a^{3} b^{2} d x^{2} - 29 \, a^{5} f x^{2} + 23 \, a^{4} b x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} b^{6}} + \frac{40 \, b^{30} f x^{11} - 165 \, a b^{29} f x^{8} + 55 \, b^{30} x^{8} e + 88 \, b^{30} d x^{5} + 528 \, a^{2} b^{28} f x^{5} - 264 \, a b^{29} x^{5} e + 220 \, b^{30} c x^{2} - 660 \, a b^{29} d x^{2} - 2200 \, a^{3} b^{27} f x^{2} + 1320 \, a^{2} b^{28} x^{2} e}{440 \, b^{33}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^10/(b*x^3 + a)^3,x, algorithm="giac")
[Out]