Optimal. Leaf size=313 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (a f+5 b c)-\sqrt [3]{a} (a g+2 b d)\right )}{54 a^{8/3} b^{5/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+5 b c)-\sqrt [3]{a} (a g+2 b d)\right )}{27 a^{8/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} g+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right )}{9 \sqrt{3} a^{8/3} b^{5/3}}-\frac{3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{18 a^2 b^2 \left (a+b x^3\right )}+\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.916225, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (a f+5 b c)-\sqrt [3]{a} (a g+2 b d)\right )}{54 a^{8/3} b^{5/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+5 b c)-\sqrt [3]{a} (a g+2 b d)\right )}{27 a^{8/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} g+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right )}{9 \sqrt{3} a^{8/3} b^{5/3}}-\frac{3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{18 a^2 b^2 \left (a+b x^3\right )}+\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a + b*x^3)^3,x]
[Out]
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Rubi in Sympy [A] time = 120.495, size = 287, normalized size = 0.92 \[ - \frac{x \left (a f - b c + x^{2} \left (a h - b e\right ) + x \left (a g - b d\right )\right )}{6 a b \left (a + b x^{3}\right )^{2}} - \frac{3 a \left (a h + b e\right ) - b x \left (a f + 5 b c + x \left (2 a g + 4 b d\right )\right )}{18 a^{2} b^{2} \left (a + b x^{3}\right )} - \frac{\left (\sqrt [3]{a} \left (a g + 2 b d\right ) - \sqrt [3]{b} \left (a f + 5 b c\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{8}{3}} b^{\frac{5}{3}}} + \frac{\left (\sqrt [3]{a} \left (a g + 2 b d\right ) - \sqrt [3]{b} \left (a f + 5 b c\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{8}{3}} b^{\frac{5}{3}}} - \frac{\sqrt{3} \left (a^{\frac{4}{3}} g + 2 \sqrt [3]{a} b d + a \sqrt [3]{b} f + 5 b^{\frac{4}{3}} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{8}{3}} b^{\frac{5}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a)**3,x)
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Mathematica [A] time = 0.45834, size = 295, normalized size = 0.94 \[ \frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} g+2 \sqrt [3]{a} b d-a \sqrt [3]{b} f-5 b^{4/3} c\right )+2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} (-g)-2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right )-2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} g+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right )+\frac{9 a^{5/3} \left (a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right )}{\left (a+b x^3\right )^2}+\frac{3 a^{2/3} \left (-6 a^2 h+a b x (f+2 g x)+b^2 x (5 c+4 d x)\right )}{a+b x^3}}{54 a^{8/3} b^2} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a + b*x^3)^3,x]
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Maple [A] time = 0.014, size = 506, normalized size = 1.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^3,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)/(b*x^3 + a)^3,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((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^3 + a)^3,x, algorithm="fricas")
[Out]
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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((h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.225924, size = 467, normalized size = 1.49 \[ -\frac{{\left (2 \, b d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a g \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, b c + a f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3} b} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c + \left (-a b^{2}\right )^{\frac{1}{3}} a b f - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d - \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{3}} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c + \left (-a b^{2}\right )^{\frac{1}{3}} a b f + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d + \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{3} b^{3}} + \frac{4 \, b^{3} d x^{5} + 2 \, a b^{2} g x^{5} + 5 \, b^{3} c x^{4} + a b^{2} f x^{4} - 6 \, a^{2} b h x^{3} + 7 \, a b^{2} d x^{2} - a^{2} b g x^{2} + 8 \, a b^{2} c x - 2 \, a^{2} b f x - 3 \, a^{3} h - 3 \, a^{2} b e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2} b^{2}} \]
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)/(b*x^3 + a)^3,x, algorithm="giac")
[Out]