Optimal. Leaf size=239 \[ \frac{\left (2 b^{2/3} c-a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{4/3}}-\frac{\left (2 b^{2/3} c-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{4/3}}-\frac{\left (a^{2/3} e+2 b^{2/3} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{4/3}}-\frac{3 a d-x (a e+4 b c x)}{18 a^2 b \left (a+b x^3\right )}-\frac{x \left (a e-b c x-b d x^2\right )}{6 a b \left (a+b x^3\right )^2} \]
[Out]
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Rubi [A] time = 0.460489, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ \frac{\left (2 b^{2/3} c-a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{4/3}}-\frac{\left (2 b^{2/3} c-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{4/3}}-\frac{\left (a^{2/3} e+2 b^{2/3} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{4/3}}-\frac{3 a d-x (a e+4 b c x)}{18 a^2 b \left (a+b x^3\right )}-\frac{x \left (a e-b c x-b d x^2\right )}{6 a b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(x*(c + d*x + e*x^2))/(a + b*x^3)^3,x]
[Out]
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Rubi in Sympy [A] time = 73.7936, size = 216, normalized size = 0.9 \[ - \frac{x \left (a e - b c x - b d x^{2}\right )}{6 a b \left (a + b x^{3}\right )^{2}} - \frac{3 a d - x \left (a e + 4 b c x\right )}{18 a^{2} b \left (a + b x^{3}\right )} + \frac{\left (a^{\frac{2}{3}} e - 2 b^{\frac{2}{3}} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{7}{3}} b^{\frac{4}{3}}} - \frac{\left (a^{\frac{2}{3}} e - 2 b^{\frac{2}{3}} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{7}{3}} b^{\frac{4}{3}}} - \frac{\sqrt{3} \left (a^{\frac{2}{3}} e + 2 b^{\frac{2}{3}} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{7}{3}} b^{\frac{4}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(e*x**2+d*x+c)/(b*x**3+a)**3,x)
[Out]
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Mathematica [A] time = 0.556766, size = 214, normalized size = 0.9 \[ \frac{\left (2 a^{2/3} b c-a^{4/3} \sqrt [3]{b} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \sqrt{3} a^{2/3} \sqrt [3]{b} \left (a^{2/3} e+2 b^{2/3} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+2 \left (a^{4/3} \sqrt [3]{b} e-2 a^{2/3} b c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\frac{3 a b^{2/3} \left (-a^2 (3 d+2 e x)+a b x^2 \left (7 c+e x^2\right )+4 b^2 c x^5\right )}{\left (a+b x^3\right )^2}}{54 a^3 b^{5/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(c + d*x + e*x^2))/(a + b*x^3)^3,x]
[Out]
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Maple [A] time = 0.014, size = 256, normalized size = 1.1 \[{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{2}} \left ({\frac{2\,bc{x}^{5}}{9\,{a}^{2}}}+{\frac{e{x}^{4}}{18\,a}}+{\frac{7\,c{x}^{2}}{18\,a}}-{\frac{ex}{9\,b}}-{\frac{d}{6\,b}} \right ) }+{\frac{e}{27\,a{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{e}{54\,a{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}e}{27\,a{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,c}{27\,{a}^{2}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{c}{27\,{a}^{2}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,c\sqrt{3}}{27\,{a}^{2}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(e*x^2+d*x+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((e*x^2 + d*x + c)*x/(b*x^3 + a)^3,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)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.1414, size = 170, normalized size = 0.71 \[ \operatorname{RootSum}{\left (19683 t^{3} a^{7} b^{4} + 162 t a^{3} b^{2} c e - a^{2} e^{3} + 8 b^{2} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{1458 t^{2} a^{5} b^{3} c + 27 t a^{4} b e^{2} + 8 a b c^{2} e}{a^{2} e^{3} + 8 b^{2} c^{3}} \right )} \right )\right )} + \frac{- 3 a^{2} d - 2 a^{2} e x + 7 a b c x^{2} + a b e x^{4} + 4 b^{2} c x^{5}}{18 a^{4} b + 36 a^{3} b^{2} x^{3} + 18 a^{2} b^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(e*x**2+d*x+c)/(b*x**3+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.217957, size = 306, normalized size = 1.28 \[ -\frac{{\left (2 \, b c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3} b} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a e - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} c\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{2}} + \frac{4 \, b^{2} c x^{5} + a b x^{4} e + 7 \, a b c x^{2} - 2 \, a^{2} x e - 3 \, a^{2} d}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2} b} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} e + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{2} c\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{3} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d*x + c)*x/(b*x^3 + a)^3,x, algorithm="giac")
[Out]