Optimal. Leaf size=347 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (a g+5 b d)-\sqrt [3]{a} (a h+2 b e)\right )}{54 a^{8/3} b^{5/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a g+5 b d)-\sqrt [3]{a} (a h+2 b e)\right )}{27 a^{8/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{9 \sqrt{3} a^{8/3} b^{5/3}}+\frac{x \left (-3 b x^2 (3 b c-a f)+a (a g+5 b d)+2 a x (a h+2 b e)\right )}{18 a^3 b \left (a+b x^3\right )}-\frac{c \log \left (a+b x^3\right )}{3 a^3}+\frac{c \log (x)}{a^3}+\frac{x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 1.40924, antiderivative size = 345, 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 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (a h+2 b e)}{\sqrt [3]{b}}+a g+5 b d\right )}{54 a^{8/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a g+5 b d)-\sqrt [3]{a} (a h+2 b e)\right )}{27 a^{8/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{9 \sqrt{3} a^{8/3} b^{5/3}}+\frac{x \left (-3 b x^2 (3 b c-a f)+a (a g+5 b d)+2 a x (a h+2 b e)\right )}{18 a^3 b \left (a+b x^3\right )}-\frac{c \log \left (a+b x^3\right )}{3 a^3}+\frac{c \log (x)}{a^3}+\frac{x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x*(a + b*x^3)^3),x]
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Rubi in Sympy [A] time = 136.863, size = 284, normalized size = 0.82 \[ - \frac{x \left (\frac{a f}{x} + a g + a h x - \frac{b c}{x} - b d - b e x\right )}{6 a b \left (a + b x^{3}\right )^{2}} + \frac{x \left (\frac{6 a f}{x} + a g + 2 a h x + 5 b d + 4 b e x\right )}{18 a^{2} b \left (a + b x^{3}\right )} - \frac{\left (\sqrt [3]{a} \left (a h + 2 b e\right ) - \sqrt [3]{b} \left (a g + 5 b d\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{8}{3}} b^{\frac{5}{3}}} + \frac{\left (\sqrt [3]{a} \left (a h + 2 b e\right ) - \sqrt [3]{b} \left (a g + 5 b d\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{8}{3}} b^{\frac{5}{3}}} - \frac{\sqrt{3} \left (a^{\frac{4}{3}} h + 2 \sqrt [3]{a} b e + a \sqrt [3]{b} g + 5 b^{\frac{4}{3}} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{8}{3}} b^{\frac{5}{3}}} \]
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)/x/(b*x**3+a)**3,x)
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Mathematica [A] time = 0.50999, size = 311, normalized size = 0.9 \[ \frac{\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e-a \sqrt [3]{b} g-5 b^{4/3} d\right )}{b^{5/3}}+\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} (-h)-2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{5/3}}-\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{5/3}}-\frac{9 a^2 (a (f+x (g+h x))-b (c+x (d+e x)))}{b \left (a+b x^3\right )^2}+\frac{3 a (a x (g+2 h x)+6 b c+b x (5 d+4 e x))}{b \left (a+b x^3\right )}-18 c \log \left (a+b x^3\right )+54 c \log (x)}{54 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x*(a + b*x^3)^3),x]
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Maple [B] time = 0.029, size = 620, normalized size = 1.8 \[ \text{result too large to display} \]
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/(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),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),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/(b*x**3+a)**3,x)
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GIAC/XCAS [A] time = 0.232004, size = 529, normalized size = 1.52 \[ -\frac{c{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac{c{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d + \left (-a b^{2}\right )^{\frac{1}{3}} a b g - \left (-a b^{2}\right )^{\frac{2}{3}} a h - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{3}} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d + \left (-a b^{2}\right )^{\frac{1}{3}} a b g + \left (-a b^{2}\right )^{\frac{2}{3}} a h + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{3} b^{3}} + \frac{6 \, a b^{2} c x^{3} + 2 \,{\left (a^{2} b h + 2 \, a b^{2} e\right )} x^{5} +{\left (5 \, a b^{2} d + a^{2} b g\right )} x^{4} + 9 \, a^{2} b c - 3 \, a^{3} f -{\left (a^{3} h - 7 \, a^{2} b e\right )} x^{2} + 2 \,{\left (4 \, a^{2} b d - a^{3} g\right )} x}{18 \,{\left (b x^{3} + a\right )}^{2} a^{3} b} - \frac{{\left (a^{5} b^{2} h \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, a^{4} b^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + 5 \, a^{4} b^{3} d + a^{5} b^{2} g\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{7} b^{3}} \]
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),x, algorithm="giac")
[Out]