Optimal. Leaf size=203 \[ \frac{b^{2/3} \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3}}-\frac{\sqrt [3]{b} \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3}}+\frac{\sqrt [3]{b} \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3}}-\frac{e \log \left (a+b x^3\right )}{3 a}-\frac{c}{2 a x^2}-\frac{d}{a x}+\frac{e \log (x)}{a} \]
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Rubi [A] time = 0.426781, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ \frac{b^{2/3} \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3}}-\frac{\sqrt [3]{b} \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3}}+\frac{\sqrt [3]{b} \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3}}-\frac{e \log \left (a+b x^3\right )}{3 a}-\frac{c}{2 a x^2}-\frac{d}{a x}+\frac{e \log (x)}{a} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x + e*x^2)/(x^3*(a + b*x^3)),x]
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Rubi in Sympy [A] time = 67.2907, size = 184, normalized size = 0.91 \[ - \frac{c}{2 a x^{2}} - \frac{d}{a x} + \frac{e \log{\left (x \right )}}{a} - \frac{e \log{\left (a + b x^{3} \right )}}{3 a} + \frac{\sqrt [3]{b} \left (\sqrt [3]{a} d - \sqrt [3]{b} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{5}{3}}} - \frac{\sqrt [3]{b} \left (\sqrt [3]{a} d - \sqrt [3]{b} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{5}{3}}} + \frac{\sqrt{3} \sqrt [3]{b} \left (\sqrt [3]{a} d + \sqrt [3]{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{5}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x**2+d*x+c)/x**3/(b*x**3+a),x)
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Mathematica [A] time = 0.373687, size = 192, normalized size = 0.95 \[ \frac{\sqrt [3]{b} \left (\sqrt [3]{a} \sqrt [3]{b} c-a^{2/3} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt [3]{b} \left (a^{2/3} d-\sqrt [3]{a} \sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )-2 a e \log \left (a+b x^3\right )-\frac{3 a c}{x^2}-\frac{6 a d}{x}+6 a e \log (x)}{6 a^2} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x + e*x^2)/(x^3*(a + b*x^3)),x]
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Maple [A] time = 0.01, size = 225, normalized size = 1.1 \[ -{\frac{d}{ax}}+{\frac{e\ln \left ( x \right ) }{a}}-{\frac{c}{2\,a{x}^{2}}}-{\frac{c}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{c}{6\,a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c\sqrt{3}}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{d}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{d}{6\,a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{d\sqrt{3}}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{e\ln \left ( b{x}^{3}+a \right ) }{3\,a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+d*x+c)/x^3/(b*x^3+a),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)*x^3),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)*x^3),x, algorithm="fricas")
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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((e*x**2+d*x+c)/x**3/(b*x**3+a),x)
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GIAC/XCAS [A] time = 0.217112, size = 285, normalized size = 1.4 \[ -\frac{e{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a} + \frac{e{\rm ln}\left ({\left | x \right |}\right )}{a} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b c + \left (-a b^{2}\right )^{\frac{2}{3}} d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b} + \frac{{\left (a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a b^{2} c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{3} b} - \frac{2 \, d x + c}{2 \, a x^{2}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{3} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d*x + c)/((b*x^3 + a)*x^3),x, algorithm="giac")
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