Optimal. Leaf size=205 \[ \frac{\sqrt [3]{a} \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3}}-\frac{\sqrt [3]{a} \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3}}+\frac{\sqrt [3]{a} \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3}}-\frac{a e \log \left (a+b x^3\right )}{3 b^2}+\frac{c x}{b}+\frac{d x^2}{2 b}+\frac{e x^3}{3 b} \]
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Rubi [A] time = 0.529306, antiderivative size = 205, 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 \[ \frac{\sqrt [3]{a} \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3}}-\frac{\sqrt [3]{a} \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3}}+\frac{\sqrt [3]{a} \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3}}-\frac{a e \log \left (a+b x^3\right )}{3 b^2}+\frac{c x}{b}+\frac{d x^2}{2 b}+\frac{e x^3}{3 b} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(c + d*x + e*x^2))/(a + b*x^3),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt [3]{a} \left (\sqrt [3]{a} d - \sqrt [3]{b} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{5}{3}}} - \frac{\sqrt [3]{a} \left (\sqrt [3]{a} d - \sqrt [3]{b} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{5}{3}}} + \frac{\sqrt{3} \sqrt [3]{a} \left (\sqrt [3]{a} d + \sqrt [3]{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{5}{3}}} - \frac{a e \log{\left (a + b x^{3} \right )}}{3 b^{2}} + \frac{d \int x\, dx}{b} + \frac{e x^{3}}{3 b} + \frac{\int c\, dx}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(e*x**2+d*x+c)/(b*x**3+a),x)
[Out]
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Mathematica [A] time = 0.226753, size = 191, normalized size = 0.93 \[ \frac{\sqrt [3]{b} \left (\sqrt [3]{a} \sqrt [3]{b} c-a^{2/3} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt [3]{b} \left (a^{2/3} d-\sqrt [3]{a} \sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )-2 a e \log \left (a+b x^3\right )+6 b c x+3 b d x^2+2 b e x^3}{6 b^2} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(c + d*x + e*x^2))/(a + b*x^3),x]
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Maple [A] time = 0.005, size = 231, normalized size = 1.1 \[{\frac{e{x}^{3}}{3\,b}}+{\frac{d{x}^{2}}{2\,b}}+{\frac{cx}{b}}-{\frac{ac}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{ac}{6\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a\sqrt{3}c}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{ad}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{ad}{6\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{a\sqrt{3}d}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{ae\ln \left ( b{x}^{3}+a \right ) }{3\,{b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(e*x^2+d*x+c)/(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)*x^3/(b*x^3 + a),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)*x^3/(b*x^3 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.88853, size = 178, normalized size = 0.87 \[ \operatorname{RootSum}{\left (27 t^{3} b^{6} + 27 t^{2} a b^{4} e + t \left (9 a^{2} b^{2} e^{2} + 9 a b^{3} c d\right ) + a^{3} e^{3} + 3 a^{2} b c d e - a^{2} b d^{3} + a b^{2} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} b^{4} d + 6 t a b^{2} d e - 3 t b^{3} c^{2} + a^{2} d e^{2} - a b c^{2} e + 2 a b c d^{2}}{a b d^{3} + b^{2} c^{3}} \right )} \right )\right )} + \frac{c x}{b} + \frac{d x^{2}}{2 b} + \frac{e x^{3}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(e*x**2+d*x+c)/(b*x**3+a),x)
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GIAC/XCAS [A] time = 0.215968, size = 302, normalized size = 1.47 \[ -\frac{a e{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} + \frac{2 \, b^{2} x^{3} e + 3 \, b^{2} d x^{2} + 6 \, b^{2} c x}{6 \, b^{3}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{4}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} c + \left (-a b^{2}\right )^{\frac{2}{3}} a b d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{4}} + \frac{{\left (a b^{6} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a b^{6} c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{7}} \]
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, algorithm="giac")
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