Optimal. Leaf size=161 \[ \frac{\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{2/3}}-\frac{\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} e^{2/3}}-\frac{\left (b \sqrt [3]{d}-a \sqrt [3]{e}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} e^{2/3}} \]
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Rubi [A] time = 0.215671, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{2/3}}-\frac{\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} e^{2/3}}-\frac{\left (b \sqrt [3]{d}-a \sqrt [3]{e}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} e^{2/3}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)/(d - e*x^3),x]
[Out]
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Rubi in Sympy [A] time = 31.7272, size = 150, normalized size = 0.93 \[ \frac{\sqrt{3} \left (a \sqrt [3]{e} - b \sqrt [3]{d}\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{d}}{3} + \frac{2 \sqrt [3]{e} x}{3}\right )}{\sqrt [3]{d}} \right )}}{3 d^{\frac{2}{3}} e^{\frac{2}{3}}} - \frac{\left (a \sqrt [3]{e} + b \sqrt [3]{d}\right ) \log{\left (\sqrt [3]{d} - \sqrt [3]{e} x \right )}}{3 d^{\frac{2}{3}} e^{\frac{2}{3}}} + \frac{\left (a \sqrt [3]{e} + b \sqrt [3]{d}\right ) \log{\left (d^{\frac{2}{3}} + \sqrt [3]{d} \sqrt [3]{e} x + e^{\frac{2}{3}} x^{2} \right )}}{6 d^{\frac{2}{3}} e^{\frac{2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)/(-e*x**3+d),x)
[Out]
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Mathematica [A] time = 0.0878635, size = 125, normalized size = 0.78 \[ \frac{-\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \left (2 \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )-\log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )\right )-2 \sqrt{3} \left (b \sqrt [3]{d}-a \sqrt [3]{e}\right ) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}+1}{\sqrt{3}}\right )}{6 d^{2/3} e^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)/(d - e*x^3),x]
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Maple [A] time = 0.007, size = 188, normalized size = 1.2 \[ -{\frac{a}{3\,e}\ln \left ( x-\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a}{6\,e}\ln \left ({x}^{2}+x\sqrt [3]{{\frac{d}{e}}}+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a\sqrt{3}}{3\,e}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+1 \right ) } \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{3\,e}\ln \left ( x-\sqrt [3]{{\frac{d}{e}}} \right ){\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+{\frac{b}{6\,e}\ln \left ({x}^{2}+x\sqrt [3]{{\frac{d}{e}}}+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}-{\frac{b\sqrt{3}}{3\,e}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)/(-e*x^3+d),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(-(b*x + a)/(e*x^3 - d),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(-(b*x + a)/(e*x^3 - d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.06266, size = 78, normalized size = 0.48 \[ - \operatorname{RootSum}{\left (27 t^{3} d^{2} e^{2} - 9 t a b d e - a^{3} e - b^{3} d, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} b d^{2} e - 3 t a^{2} d e - 2 a b^{2} d}{a^{3} e - b^{3} d} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)/(-e*x**3+d),x)
[Out]
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GIAC/XCAS [A] time = 0.213404, size = 155, normalized size = 0.96 \[ -\frac{\sqrt{3}{\left (b d^{\frac{2}{3}} e^{\frac{4}{3}} - a d^{\frac{1}{3}} e^{\frac{5}{3}}\right )} \arctan \left (\frac{\sqrt{3}{\left (d^{\frac{1}{3}} e^{\left (-\frac{1}{3}\right )} + 2 \, x\right )} e^{\frac{1}{3}}}{3 \, d^{\frac{1}{3}}}\right ) e^{\left (-2\right )}}{3 \, d} - \frac{{\left (b d^{\frac{1}{3}} e^{\left (-\frac{1}{3}\right )} + a\right )} e^{\left (-\frac{1}{3}\right )}{\rm ln}\left ({\left | -d^{\frac{1}{3}} e^{\left (-\frac{1}{3}\right )} + x \right |}\right )}{3 \, d^{\frac{2}{3}}} + \frac{{\left (b d^{\frac{2}{3}} e^{\frac{4}{3}} + a d^{\frac{1}{3}} e^{\frac{5}{3}}\right )} e^{\left (-2\right )}{\rm ln}\left (d^{\frac{1}{3}} x e^{\left (-\frac{1}{3}\right )} + x^{2} + d^{\frac{2}{3}} e^{\left (-\frac{2}{3}\right )}\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*x + a)/(e*x^3 - d),x, algorithm="giac")
[Out]