Optimal. Leaf size=199 \[ -\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{2/3}}+\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{2/3}}-\frac{\left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{2/3}}-\frac{a e-b x (c+d x)}{3 a b \left (a+b x^3\right )} \]
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Rubi [A] time = 0.314682, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{2/3}}+\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{2/3}}-\frac{\left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{2/3}}-\frac{a e-b x (c+d x)}{3 a b \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x + e*x^2)/(a + b*x^3)^2,x]
[Out]
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Rubi in Sympy [A] time = 46.5138, size = 178, normalized size = 0.89 \[ - \frac{a e - b x \left (c + d x\right )}{3 a b \left (a + b x^{3}\right )} + \frac{\left (\frac{\sqrt [3]{a} d}{2} - \sqrt [3]{b} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{9 a^{\frac{5}{3}} b^{\frac{2}{3}}} - \frac{\left (\sqrt [3]{a} d - 2 \sqrt [3]{b} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{5}{3}} b^{\frac{2}{3}}} - \frac{\sqrt{3} \left (\sqrt [3]{a} d + 2 \sqrt [3]{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} b^{\frac{2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x**2+d*x+c)/(b*x**3+a)**2,x)
[Out]
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Mathematica [A] time = 0.382753, size = 189, normalized size = 0.95 \[ \frac{\sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a} d-2 \sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\left (4 \sqrt [3]{a} b^{2/3} c-2 a^{2/3} \sqrt [3]{b} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\frac{6 a (b x (c+d x)-a e)}{a+b x^3}-2 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{18 a^2 b} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x + e*x^2)/(a + b*x^3)^2,x]
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Maple [A] time = 0.006, size = 253, normalized size = 1.3 \[{\frac{cx}{3\,a \left ( b{x}^{3}+a \right ) }}+{\frac{2\,c}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c}{9\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,c\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{d{x}^{2}}{3\,a \left ( b{x}^{3}+a \right ) }}-{\frac{d}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d}{18\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{e}{3\,b \left ( b{x}^{3}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+d*x+c)/(b*x^3+a)^2,x)
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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)/(b*x^3 + a)^2,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)/(b*x^3 + a)^2,x, algorithm="fricas")
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Sympy [A] time = 2.61076, size = 116, normalized size = 0.58 \[ \operatorname{RootSum}{\left (729 t^{3} a^{5} b^{2} + 54 t a^{2} b c d + a d^{3} - 8 b c^{3}, \left ( t \mapsto t \log{\left (x + \frac{81 t^{2} a^{4} b d + 36 t a^{2} b c^{2} + 4 a c d^{2}}{a d^{3} + 8 b c^{3}} \right )} \right )\right )} + \frac{- a e + b c x + b d x^{2}}{3 a^{2} b + 3 a b^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x**2+d*x+c)/(b*x**3+a)**2,x)
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GIAC/XCAS [A] time = 0.216552, size = 266, normalized size = 1.34 \[ -\frac{{\left (d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2}} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b c - \left (-a b^{2}\right )^{\frac{2}{3}} d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{2}} + \frac{b d x^{2} + b c x - a e}{3 \,{\left (b x^{3} + a\right )} a b} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c + \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{3} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d*x + c)/(b*x^3 + a)^2,x, algorithm="giac")
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