Optimal. Leaf size=360 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 \sqrt [3]{b} (4 b c-a f)-2 \sqrt [3]{a} (7 b d-a g)\right )}{54 a^{11/3} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 \sqrt [3]{b} (4 b c-a f)-2 \sqrt [3]{a} (7 b d-a g)\right )}{27 a^{11/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^{4/3} g+14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f+20 b^{4/3} c\right )}{9 \sqrt{3} a^{11/3} b^{2/3}}-\frac{x \left (2 x (5 b d-2 a g)+3 x^2 (3 b e-a h)-5 a f+11 b c\right )}{18 a^3 \left (a+b x^3\right )}-\frac{e \log \left (a+b x^3\right )}{3 a^3}-\frac{c}{2 a^3 x^2}-\frac{d}{a^3 x}+\frac{e \log (x)}{a^3}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a^2 \left (a+b x^3\right )^2} \]
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Rubi [A] time = 1.56156, antiderivative size = 357, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 9, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.237 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{2 \sqrt [3]{a} (7 b d-a g)}{\sqrt [3]{b}}-5 a f+20 b c\right )}{54 a^{11/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 \sqrt [3]{b} (4 b c-a f)-2 \sqrt [3]{a} (7 b d-a g)\right )}{27 a^{11/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^{4/3} g+14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f+20 b^{4/3} c\right )}{9 \sqrt{3} a^{11/3} b^{2/3}}-\frac{x \left (2 x (5 b d-2 a g)+3 x^2 (3 b e-a h)-5 a f+11 b c\right )}{18 a^3 \left (a+b x^3\right )}-\frac{e \log \left (a+b x^3\right )}{3 a^3}-\frac{c}{2 a^3 x^2}-\frac{d}{a^3 x}+\frac{e \log (x)}{a^3}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a^2 \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^3*(a + b*x^3)^3),x]
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Rubi in Sympy [A] time = 51.3888, size = 83, normalized size = 0.23 \[ \frac{x \left (\frac{6 f}{x^{3}} + \frac{6 g}{x^{2}} + \frac{6 h}{x}\right )}{18 a b \left (a + b x^{3}\right )} - \frac{x \left (\frac{a f}{x^{3}} + \frac{a g}{x^{2}} + \frac{a h}{x} - \frac{b c}{x^{3}} - \frac{b d}{x^{2}} - \frac{b e}{x}\right )}{6 a b \left (a + b x^{3}\right )^{2}} \]
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)/x**3/(b*x**3+a)**3,x)
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Mathematica [A] time = 1.29957, size = 337, normalized size = 0.94 \[ -\frac{-\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^{4/3} g-14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f+20 b^{4/3} c\right )}{b^{2/3}}+\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^{4/3} g-14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f+20 b^{4/3} c\right )}{b^{2/3}}+\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (2 a^{4/3} g-14 \sqrt [3]{a} b d+5 a \sqrt [3]{b} f-20 b^{4/3} c\right )}{b^{2/3}}+\frac{9 a^2 \left (a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right )}{b \left (a+b x^3\right )^2}-\frac{3 a (6 a e+a x (5 f+4 g x)-b x (11 c+10 d x))}{a+b x^3}+18 a e \log \left (a+b x^3\right )+\frac{27 a c}{x^2}+\frac{54 a d}{x}-54 a e \log (x)}{54 a^4} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^3*(a + b*x^3)^3),x]
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Maple [B] time = 0.027, size = 626, normalized size = 1.7 \[ \text{result too large to display} \]
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)/x^3/(b*x^3+a)^3,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((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/((b*x^3 + a)^3*x^3),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)/((b*x^3 + a)^3*x^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)/x**3/(b*x**3+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.228878, size = 568, normalized size = 1.58 \[ -\frac{e{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac{e{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{\sqrt{3}{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{4} b^{2}} - \frac{{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f + 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{4} b^{2}} - \frac{28 \, b^{3} d x^{7} - 4 \, a b^{2} g x^{7} + 20 \, b^{3} c x^{6} - 5 \, a b^{2} f x^{6} - 6 \, a b^{2} x^{5} e + 49 \, a b^{2} d x^{4} - 7 \, a^{2} b g x^{4} + 32 \, a b^{2} c x^{3} - 8 \, a^{2} b f x^{3} + 3 \, a^{3} h x^{2} - 9 \, a^{2} b x^{2} e + 18 \, a^{2} b d x + 9 \, a^{2} b c}{18 \,{\left (b x^{4} + a x\right )}^{2} a^{3} b} + \frac{{\left (14 \, a^{3} b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 2 \, a^{4} b g \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 20 \, a^{3} b^{2} c - 5 \, a^{4} b f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{7} b} \]
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^3),x, algorithm="giac")
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