Optimal. Leaf size=134 \[ \frac{a \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac{a \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac{a \tan ^{-1}\left (\frac{\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} \sqrt [3]{e}}-\frac{c \log \left (d-e x^3\right )}{3 e} \]
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Rubi [A] time = 0.181389, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444 \[ \frac{a \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac{a \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac{a \tan ^{-1}\left (\frac{\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} \sqrt [3]{e}}-\frac{c \log \left (d-e x^3\right )}{3 e} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)/(d - e*x^3),x]
[Out]
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Rubi in Sympy [A] time = 31.2776, size = 128, normalized size = 0.96 \[ - \frac{a \log{\left (\sqrt [3]{d} - \sqrt [3]{e} x \right )}}{3 d^{\frac{2}{3}} \sqrt [3]{e}} + \frac{a \log{\left (d^{\frac{2}{3}} + \sqrt [3]{d} \sqrt [3]{e} x + e^{\frac{2}{3}} x^{2} \right )}}{6 d^{\frac{2}{3}} \sqrt [3]{e}} + \frac{\sqrt{3} a \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{d}}{3} + \frac{2 \sqrt [3]{e} x}{3}\right )}{\sqrt [3]{d}} \right )}}{3 d^{\frac{2}{3}} \sqrt [3]{e}} - \frac{c \log{\left (d - e x^{3} \right )}}{3 e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)/(-e*x**3+d),x)
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Mathematica [A] time = 0.0612387, size = 123, normalized size = 0.92 \[ \frac{a e^{2/3} \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )-2 a e^{2/3} \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )+2 \sqrt{3} a e^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}+1}{\sqrt{3}}\right )-2 c d^{2/3} \log \left (d-e x^3\right )}{6 d^{2/3} e} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)/(d - e*x^3),x]
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Maple [A] time = 0.004, size = 111, normalized size = 0.8 \[ -{\frac{a}{3\,e}\ln \left ( x-\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a}{6\,e}\ln \left ({x}^{2}+x\sqrt [3]{{\frac{d}{e}}}+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a\sqrt{3}}{3\,e}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+1 \right ) } \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c\ln \left ( e{x}^{3}-d \right ) }{3\,e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)/(-e*x^3+d),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(-(c*x^2 + a)/(e*x^3 - d),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(-(c*x^2 + a)/(e*x^3 - d),x, algorithm="fricas")
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Sympy [A] time = 0.643426, size = 70, normalized size = 0.52 \[ - \operatorname{RootSum}{\left (27 t^{3} d^{2} e^{3} - 27 t^{2} c d^{2} e^{2} + 9 t c^{2} d^{2} e - a^{3} e^{2} - c^{3} d^{2}, \left ( t \mapsto t \log{\left (x + \frac{- 3 t d e + c d}{a e} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)/(-e*x**3+d),x)
[Out]
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GIAC/XCAS [A] time = 0.212121, size = 128, normalized size = 0.96 \[ -\frac{1}{3} \, c e^{\left (-1\right )}{\rm ln}\left ({\left | x^{3} e - d \right |}\right ) + \frac{\sqrt{3} a \arctan \left (\frac{\sqrt{3}{\left (d^{\frac{1}{3}} e^{\left (-\frac{1}{3}\right )} + 2 \, x\right )} e^{\frac{1}{3}}}{3 \, d^{\frac{1}{3}}}\right ) e^{\left (-\frac{1}{3}\right )}}{3 \, d^{\frac{2}{3}}} + \frac{a e^{\left (-\frac{1}{3}\right )}{\rm ln}\left (d^{\frac{1}{3}} x e^{\left (-\frac{1}{3}\right )} + x^{2} + d^{\frac{2}{3}} e^{\left (-\frac{2}{3}\right )}\right )}{6 \, d^{\frac{2}{3}}} - \frac{a e^{\left (-\frac{1}{3}\right )}{\rm ln}\left ({\left | -d^{\frac{1}{3}} e^{\left (-\frac{1}{3}\right )} + x \right |}\right )}{3 \, d^{\frac{2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(c*x^2 + a)/(e*x^3 - d),x, algorithm="giac")
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