Optimal. Leaf size=183 \[ \frac{\left (a^{2/3} e+b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{4/3}}-\frac{\left (a^{2/3} e+b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{4/3}}-\frac{\left (b^{2/3} c-a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} b^{4/3}}+\frac{d \log \left (a+b x^3\right )}{3 b}+\frac{e x}{b} \]
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Rubi [A] time = 0.469453, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ \frac{\left (a^{2/3} e+b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{4/3}}-\frac{\left (a^{2/3} e+b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{4/3}}-\frac{\left (b^{2/3} c-a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} b^{4/3}}+\frac{d \log \left (a+b x^3\right )}{3 b}+\frac{e x}{b} \]
Antiderivative was successfully verified.
[In] Int[(x*(c + d*x + e*x^2))/(a + b*x^3),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{d \log{\left (a + b x^{3} \right )}}{3 b} + \frac{\int e\, dx}{b} + \frac{\sqrt{3} \left (a^{\frac{2}{3}} e - b^{\frac{2}{3}} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 \sqrt [3]{a} b^{\frac{4}{3}}} - \frac{\left (a^{\frac{2}{3}} e + b^{\frac{2}{3}} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 \sqrt [3]{a} b^{\frac{4}{3}}} + \frac{\left (a^{\frac{2}{3}} e + b^{\frac{2}{3}} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 \sqrt [3]{a} b^{\frac{4}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(e*x**2+d*x+c)/(b*x**3+a),x)
[Out]
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Mathematica [A] time = 0.126427, size = 200, normalized size = 1.09 \[ -\frac{\left (a^{4/3} \left (-\sqrt [3]{b}\right ) e-a^{2/3} b c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a b^{5/3}}+\frac{\left (a^{4/3} \left (-\sqrt [3]{b}\right ) e-a^{2/3} b c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a b^{5/3}}+\frac{\left (a^{2/3} b c-a^{4/3} \sqrt [3]{b} e\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a b^{5/3}}+\frac{d \log \left (a+b x^3\right )}{3 b}+\frac{e x}{b} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(c + d*x + e*x^2))/(a + b*x^3),x]
[Out]
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Maple [A] time = 0.004, size = 209, normalized size = 1.1 \[{\frac{ex}{b}}-{\frac{ae}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{ae}{6\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a\sqrt{3}e}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{c}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{c\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d\ln \left ( b{x}^{3}+a \right ) }{3\,b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(e*x^2+d*x+c)/(b*x^3+a),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),x, algorithm="maxima")
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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),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.95915, size = 160, normalized size = 0.87 \[ \operatorname{RootSum}{\left (27 t^{3} a b^{4} - 27 t^{2} a b^{3} d + t \left (- 9 a b^{2} c e + 9 a b^{2} d^{2}\right ) + a^{2} e^{3} + 3 a b c d e - a b d^{3} + b^{2} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{- 9 t^{2} a b^{3} c - 3 t a^{2} b e^{2} + 6 t a b^{2} c d + a^{2} d e^{2} + 2 a b c^{2} e - a b c d^{2}}{a^{2} e^{3} - b^{2} c^{3}} \right )} \right )\right )} + \frac{e x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(e*x**2+d*x+c)/(b*x**3+a),x)
[Out]
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GIAC/XCAS [A] time = 0.216157, size = 257, normalized size = 1.4 \[ \frac{x e}{b} + \frac{d{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a e + \left (-a b^{2}\right )^{\frac{2}{3}} c\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{2}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a e - \left (-a b^{2}\right )^{\frac{2}{3}} c\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{2}} - \frac{{\left (b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{2} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d*x + c)*x/(b*x^3 + a),x, algorithm="giac")
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