Optimal. Leaf size=395 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 \sqrt [3]{b} (4 b d-a g)-2 \sqrt [3]{a} (7 b e-a h)\right )}{54 a^{11/3} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 \sqrt [3]{b} (4 b d-a g)-2 \sqrt [3]{a} (7 b e-a h)\right )}{27 a^{11/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^{4/3} h+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+20 b^{4/3} d\right )}{9 \sqrt{3} a^{11/3} b^{2/3}}+\frac{(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4}-\frac{\log (x) (3 b c-a f)}{a^4}-\frac{x \left (-3 b x^2 \left (\frac{5 b c}{a}-3 f\right )+2 x (5 b e-2 a h)-5 a g+11 b d\right )}{18 a^3 \left (a+b x^3\right )}-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (-b x^2 \left (\frac{b c}{a}-f\right )+x (b e-a h)-a g+b d\right )}{6 a^2 \left (a+b x^3\right )^2} \]
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Rubi [A] time = 1.97627, antiderivative size = 392, 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 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{2 \sqrt [3]{a} (7 b e-a h)}{\sqrt [3]{b}}-5 a g+20 b d\right )}{54 a^{11/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 \sqrt [3]{b} (4 b d-a g)-2 \sqrt [3]{a} (7 b e-a h)\right )}{27 a^{11/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^{4/3} h+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+20 b^{4/3} d\right )}{9 \sqrt{3} a^{11/3} b^{2/3}}+\frac{(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4}-\frac{\log (x) (3 b c-a f)}{a^4}-\frac{x \left (-3 b x^2 \left (\frac{5 b c}{a}-3 f\right )+2 x (5 b e-2 a h)-5 a g+11 b d\right )}{18 a^3 \left (a+b x^3\right )}-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (-b x^2 \left (\frac{b c}{a}-f\right )+x (b e-a h)-a g+b d\right )}{6 a^2 \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^4*(a + b*x^3)^3),x]
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Rubi in Sympy [A] time = 49.5221, size = 88, normalized size = 0.22 \[ \frac{x \left (\frac{6 f}{x^{4}} + \frac{6 g}{x^{3}} + \frac{6 h}{x^{2}}\right )}{18 a b \left (a + b x^{3}\right )} - \frac{x \left (\frac{a f}{x^{4}} + \frac{a g}{x^{3}} + \frac{a h}{x^{2}} - \frac{b c}{x^{4}} - \frac{b d}{x^{3}} - \frac{b e}{x^{2}}\right )}{6 a b \left (a + b x^{3}\right )^{2}} \]
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**4/(b*x**3+a)**3,x)
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Mathematica [A] time = 1.87989, size = 352, normalized size = 0.89 \[ \frac{\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^{4/3} h-14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+20 b^{4/3} d\right )}{b^{2/3}}-\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^{4/3} h-14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+20 b^{4/3} d\right )}{b^{2/3}}+\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-2 a^{4/3} h+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+20 b^{4/3} d\right )}{b^{2/3}}+\frac{a^2 (9 a (f+x (g+h x))-9 b (c+x (d+e x)))}{\left (a+b x^3\right )^2}+\frac{3 a (6 a f+a x (5 g+4 h x)-12 b c-b x (11 d+10 e x))}{a+b x^3}+18 (3 b c-a f) \log \left (a+b x^3\right )+54 \log (x) (a f-3 b c)-\frac{18 a c}{x^3}-\frac{27 a d}{x^2}-\frac{54 a e}{x}}{54 a^4} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^4*(a + b*x^3)^3),x]
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Maple [B] time = 0.028, size = 680, 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^4/(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^4),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^4),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**4/(b*x**3+a)**3,x)
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GIAC/XCAS [A] time = 0.229167, size = 612, normalized size = 1.55 \[ \frac{{\left (3 \, b c - a f\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4}} - \frac{{\left (3 \, b c - a f\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{\sqrt{3}{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a h - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{4} b^{2}} - \frac{{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a h + 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{4} b^{2}} - \frac{{\left (2 \, a^{6} b h \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 14 \, a^{5} b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e - 20 \, a^{5} b^{2} d + 5 \, a^{6} b g\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{9} b} + \frac{4 \,{\left (a^{2} b h - 7 \, a b^{2} e\right )} x^{8} - 5 \,{\left (4 \, a b^{2} d - a^{2} b g\right )} x^{7} - 6 \,{\left (3 \, a b^{2} c - a^{2} b f\right )} x^{6} + 7 \,{\left (a^{3} h - 7 \, a^{2} b e\right )} x^{5} - 18 \, a^{3} x^{2} e - 9 \, a^{3} d x - 8 \,{\left (4 \, a^{2} b d - a^{3} g\right )} x^{4} - 6 \, a^{3} c - 9 \,{\left (3 \, a^{2} b c - a^{3} f\right )} x^{3}}{18 \,{\left (b x^{3} + a\right )}^{2} a^{4} x^{3}} \]
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^4),x, algorithm="giac")
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