Optimal. Leaf size=200 \[ \frac{\left (b^{2/3} c-a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{4/3}}-\frac{\left (b^{2/3} c-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{4/3}}-\frac{\left (a^{2/3} e+b^{2/3} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3} b^{4/3}}-\frac{x \left (a e-b c x-b d x^2\right )}{3 a b \left (a+b x^3\right )} \]
[Out]
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Rubi [A] time = 0.36624, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\left (b^{2/3} c-a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{4/3}}-\frac{\left (b^{2/3} c-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{4/3}}-\frac{\left (a^{2/3} e+b^{2/3} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3} b^{4/3}}-\frac{x \left (a e-b c x-b d x^2\right )}{3 a b \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Int[(x*(c + d*x + e*x^2))/(a + b*x^3)^2,x]
[Out]
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Rubi in Sympy [A] time = 53.9996, size = 178, normalized size = 0.89 \[ - \frac{x \left (a e - b c x - b d x^{2}\right )}{3 a b \left (a + b x^{3}\right )} + \frac{\left (a^{\frac{2}{3}} e - b^{\frac{2}{3}} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{4}{3}} b^{\frac{4}{3}}} - \frac{\left (a^{\frac{2}{3}} e - b^{\frac{2}{3}} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{4}{3}} b^{\frac{4}{3}}} - \frac{\sqrt{3} \left (a^{\frac{2}{3}} e + b^{\frac{2}{3}} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{4}{3}} b^{\frac{4}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(e*x**2+d*x+c)/(b*x**3+a)**2,x)
[Out]
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Mathematica [A] time = 0.45468, size = 186, normalized size = 0.93 \[ \frac{-\left (a^{4/3} \sqrt [3]{b} e-a^{2/3} b c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \left (a^{4/3} \sqrt [3]{b} e-a^{2/3} b c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} \left (a^{2/3} b c+a^{4/3} \sqrt [3]{b} e\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )-\frac{6 a b^{2/3} \left (a (d+e x)-b c x^2\right )}{a+b x^3}}{18 a^2 b^{5/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(c + d*x + e*x^2))/(a + b*x^3)^2,x]
[Out]
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Maple [A] time = 0.01, size = 228, normalized size = 1.1 \[{\frac{1}{b{x}^{3}+a} \left ({\frac{c{x}^{2}}{3\,a}}-{\frac{ex}{3\,b}}-{\frac{d}{3\,b}} \right ) }+{\frac{e}{9\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{e}{18\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{e\sqrt{3}}{9\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{c}{18\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{c\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(e*x^2+d*x+c)/(b*x^3+a)^2,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((e*x^2 + d*x + c)*x/(b*x^3 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d*x + c)*x/(b*x^3 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.5051, size = 124, normalized size = 0.62 \[ \operatorname{RootSum}{\left (729 t^{3} a^{4} b^{4} + 27 t a^{2} b^{2} c e - a^{2} e^{3} + b^{2} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{81 t^{2} a^{3} b^{3} c + 9 t a^{3} b e^{2} + 2 a b c^{2} e}{a^{2} e^{3} + b^{2} c^{3}} \right )} \right )\right )} + \frac{- a d - a e x + b c x^{2}}{3 a^{2} b + 3 a b^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(e*x**2+d*x+c)/(b*x**3+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.215455, size = 273, normalized size = 1.36 \[ -\frac{{\left (b c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2} b} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a e - \left (-a b^{2}\right )^{\frac{2}{3}} c\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{2}} + \frac{b c x^{2} - a x e - a d}{3 \,{\left (b x^{3} + a\right )} a b} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} e + \left (-a b^{2}\right )^{\frac{2}{3}} b^{2} c\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d*x + c)*x/(b*x^3 + a)^2,x, algorithm="giac")
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