Optimal. Leaf size=294 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{6 b^{8/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt{3} b^{8/3}}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 b^{8/3}}+\frac{(b c-a f) \log \left (a+b x^3\right )}{3 b^2}+\frac{x (b d-a g)}{b^2}+\frac{x^2 (b e-a h)}{2 b^2}+\frac{f x^3}{3 b}+\frac{g x^4}{4 b}+\frac{h x^5}{5 b} \]
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Rubi [A] time = 1.85241, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.237 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{6 b^{8/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt{3} b^{8/3}}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 b^{8/3}}+\frac{(b c-a f) \log \left (a+b x^3\right )}{3 b^2}+\frac{x (b d-a g)}{b^2}+\frac{x^2 (b e-a h)}{2 b^2}+\frac{f x^3}{3 b}+\frac{g x^4}{4 b}+\frac{h x^5}{5 b} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{\sqrt [3]{a} \left (\sqrt [3]{a} \left (a h - b e\right ) - \sqrt [3]{b} \left (a g - b d\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{8}{3}}} + \frac{\sqrt [3]{a} \left (\sqrt [3]{a} \left (a h - b e\right ) - \sqrt [3]{b} \left (a g - b d\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{8}{3}}} - \frac{\sqrt{3} \sqrt [3]{a} \left (\sqrt [3]{a} \left (a h - b e\right ) + \sqrt [3]{b} \left (a g - b d\right )\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{8}{3}}} - \left (a g - b d\right ) \int \frac{1}{b^{2}}\, dx + \frac{f x^{3}}{3 b} + \frac{g x^{4}}{4 b} + \frac{h x^{5}}{5 b} - \frac{\left (a f - b c\right ) \log{\left (a + b x^{3} \right )}}{3 b^{2}} - \frac{\left (a h - b e\right ) \int x\, dx}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a),x)
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Mathematica [A] time = 0.442584, size = 290, normalized size = 0.99 \[ \frac{10 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} h-\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )+20 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e+a \sqrt [3]{b} g-b^{4/3} d\right )-20 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} h-\sqrt [3]{a} b e+a \sqrt [3]{b} g-b^{4/3} d\right )+20 b^{2/3} (b c-a f) \log \left (a+b x^3\right )+60 b^{2/3} x (b d-a g)+30 b^{2/3} x^2 (b e-a h)+20 b^{5/3} f x^3+15 b^{5/3} g x^4+12 b^{5/3} h x^5}{60 b^{8/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3),x]
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Maple [B] time = 0.007, size = 483, normalized size = 1.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),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((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)*x^2/(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((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)*x^2/(b*x^3 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 95.2032, size = 789, normalized size = 2.68 \[ \operatorname{RootSum}{\left (27 t^{3} b^{8} + t^{2} \left (27 a b^{6} f - 27 b^{7} c\right ) + t \left (9 a^{3} b^{3} g h - 9 a^{2} b^{4} d h - 9 a^{2} b^{4} e g + 9 a^{2} b^{4} f^{2} - 18 a b^{5} c f + 9 a b^{5} d e + 9 b^{6} c^{2}\right ) + a^{5} h^{3} - 3 a^{4} b e h^{2} + 3 a^{4} b f g h - a^{4} b g^{3} - 3 a^{3} b^{2} c g h - 3 a^{3} b^{2} d f h + 3 a^{3} b^{2} d g^{2} + 3 a^{3} b^{2} e^{2} h - 3 a^{3} b^{2} e f g + a^{3} b^{2} f^{3} + 3 a^{2} b^{3} c d h + 3 a^{2} b^{3} c e g - 3 a^{2} b^{3} c f^{2} - 3 a^{2} b^{3} d^{2} g + 3 a^{2} b^{3} d e f - a^{2} b^{3} e^{3} + 3 a b^{4} c^{2} f - 3 a b^{4} c d e + a b^{4} d^{3} - b^{5} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} a b^{5} h - 9 t^{2} b^{6} e + 6 t a^{2} b^{3} f h + 3 t a^{2} b^{3} g^{2} - 6 t a b^{4} c h - 6 t a b^{4} d g - 6 t a b^{4} e f + 6 t b^{5} c e + 3 t b^{5} d^{2} + 2 a^{4} g h^{2} - 2 a^{3} b d h^{2} - 4 a^{3} b e g h + a^{3} b f^{2} h + a^{3} b f g^{2} - 2 a^{2} b^{2} c f h - a^{2} b^{2} c g^{2} + 4 a^{2} b^{2} d e h - 2 a^{2} b^{2} d f g + 2 a^{2} b^{2} e^{2} g - a^{2} b^{2} e f^{2} + a b^{3} c^{2} h + 2 a b^{3} c d g + 2 a b^{3} c e f + a b^{3} d^{2} f - 2 a b^{3} d e^{2} - b^{4} c^{2} e - b^{4} c d^{2}}{a^{4} h^{3} - 3 a^{3} b e h^{2} + a^{3} b g^{3} - 3 a^{2} b^{2} d g^{2} + 3 a^{2} b^{2} e^{2} h + 3 a b^{3} d^{2} g - a b^{3} e^{3} - b^{4} d^{3}} \right )} \right )\right )} + \frac{f x^{3}}{3 b} + \frac{g x^{4}}{4 b} + \frac{h x^{5}}{5 b} - \frac{x^{2} \left (a h - b e\right )}{2 b^{2}} - \frac{x \left (a g - b d\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a),x)
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GIAC/XCAS [A] time = 0.224048, size = 450, normalized size = 1.53 \[ \frac{{\left (b c - a f\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a b g + \left (-a b^{2}\right )^{\frac{2}{3}} a h - \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{4}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a b g - \left (-a b^{2}\right )^{\frac{2}{3}} a h + \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{4}} + \frac{12 \, b^{4} h x^{5} + 15 \, b^{4} g x^{4} + 20 \, b^{4} f x^{3} - 30 \, a b^{3} h x^{2} + 30 \, b^{4} x^{2} e + 60 \, b^{4} d x - 60 \, a b^{3} g x}{60 \, b^{5}} - \frac{{\left (a^{2} b^{9} h \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{10} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e - a b^{10} d + a^{2} b^{9} g\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)*x^2/(b*x^3 + a),x, algorithm="giac")
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