Optimal. Leaf size=301 \[ \frac{x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{9 a^2 b^3 \left (a+b x^3\right )}+\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{54 a^{7/3} b^{11/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{27 a^{7/3} b^{11/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{9 \sqrt{3} a^{7/3} b^{11/3}}+\frac{f x^2}{2 b^3} \]
[Out]
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Rubi [A] time = 0.805486, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{9 a^2 b^3 \left (a+b x^3\right )}+\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{54 a^{7/3} b^{11/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{27 a^{7/3} b^{11/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{9 \sqrt{3} a^{7/3} b^{11/3}}+\frac{f x^2}{2 b^3} \]
Antiderivative was successfully verified.
[In] Int[(x*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]
[Out]
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Rubi in Sympy [A] time = 128.405, size = 291, normalized size = 0.97 \[ \frac{f x^{2}}{2 b^{3}} - \frac{x^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{6 a b^{3} \left (a + b x^{3}\right )^{2}} + \frac{x^{2} \left (7 a^{3} f - 4 a^{2} b e + a b^{2} d + 2 b^{3} c\right )}{9 a^{2} b^{3} \left (a + b x^{3}\right )} + \frac{\left (a \left (20 a^{2} f - 5 a b e - b^{2} d\right ) - 2 b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{7}{3}} b^{\frac{11}{3}}} - \frac{\left (a \left (20 a^{2} f - 5 a b e - b^{2} d\right ) - 2 b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{7}{3}} b^{\frac{11}{3}}} + \frac{\sqrt{3} \left (a \left (20 a^{2} f - 5 a b e - b^{2} d\right ) - 2 b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{7}{3}} b^{\frac{11}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)
[Out]
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Mathematica [A] time = 0.379933, size = 284, normalized size = 0.94 \[ \frac{\frac{6 b^{2/3} x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{a^2 \left (a+b x^3\right )}+\frac{9 b^{2/3} x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a \left (a+b x^3\right )^2}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{a^{7/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{a^{7/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{a^{7/3}}+27 b^{2/3} f x^2}{54 b^{11/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]
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Maple [B] time = 0.017, size = 550, normalized size = 1.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(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/(b*x^3 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222284, size = 711, normalized size = 2.36 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (2 \, b^{5} c + a b^{4} d + 5 \, a^{2} b^{3} e - 20 \, a^{3} b^{2} f\right )} x^{6} + 2 \, a^{2} b^{3} c + a^{3} b^{2} d + 5 \, a^{4} b e - 20 \, a^{5} f + 2 \,{\left (2 \, a b^{4} c + a^{2} b^{3} d + 5 \, a^{3} b^{2} e - 20 \, a^{4} b f\right )} x^{3}\right )} \log \left (\left (a b^{2}\right )^{\frac{1}{3}} b x^{2} + a b - \left (a b^{2}\right )^{\frac{2}{3}} x\right ) - 2 \, \sqrt{3}{\left ({\left (2 \, b^{5} c + a b^{4} d + 5 \, a^{2} b^{3} e - 20 \, a^{3} b^{2} f\right )} x^{6} + 2 \, a^{2} b^{3} c + a^{3} b^{2} d + 5 \, a^{4} b e - 20 \, a^{5} f + 2 \,{\left (2 \, a b^{4} c + a^{2} b^{3} d + 5 \, a^{3} b^{2} e - 20 \, a^{4} b f\right )} x^{3}\right )} \log \left (a b + \left (a b^{2}\right )^{\frac{2}{3}} x\right ) + 6 \,{\left ({\left (2 \, b^{5} c + a b^{4} d + 5 \, a^{2} b^{3} e - 20 \, a^{3} b^{2} f\right )} x^{6} + 2 \, a^{2} b^{3} c + a^{3} b^{2} d + 5 \, a^{4} b e - 20 \, a^{5} f + 2 \,{\left (2 \, a b^{4} c + a^{2} b^{3} d + 5 \, a^{3} b^{2} e - 20 \, a^{4} b f\right )} x^{3}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) + 3 \, \sqrt{3}{\left (9 \, a^{2} b^{2} f x^{8} + 2 \,{\left (2 \, b^{4} c + a b^{3} d - 4 \, a^{2} b^{2} e + 16 \, a^{3} b f\right )} x^{5} +{\left (7 \, a b^{3} c - a^{2} b^{2} d - 5 \, a^{3} b e + 20 \, a^{4} f\right )} x^{2}\right )} \left (a b^{2}\right )^{\frac{1}{3}}\right )}}{162 \,{\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )} \left (a b^{2}\right )^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x/(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*(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 = 0.220603, size = 522, normalized size = 1.73 \[ \frac{f x^{2}}{2 \, b^{3}} - \frac{{\left (2 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 20 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, a^{2} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3} b^{3}} - \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c + \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 20 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{5}} + \frac{4 \, b^{4} c x^{5} + 2 \, a b^{3} d x^{5} + 14 \, a^{3} b f x^{5} - 8 \, a^{2} b^{2} x^{5} e + 7 \, a b^{3} c x^{2} - a^{2} b^{2} d x^{2} + 11 \, a^{4} f x^{2} - 5 \, a^{3} b x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2} b^{3}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c + \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 20 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{3} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x/(b*x^3 + a)^3,x, algorithm="giac")
[Out]