Optimal. Leaf size=289 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (2 a f+b c)-a^{2/3} (b e-4 a h)\right )}{18 a^{4/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (2 a f+b c)-a^{2/3} (b e-4 a h)\right )}{9 a^{4/3} b^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{2/3} b e-4 a^{5/3} h+2 a b^{2/3} f+b^{5/3} c\right )}{3 \sqrt{3} a^{4/3} b^{7/3}}-\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 a b^2 \left (a+b x^3\right )}+\frac{g \log \left (a+b x^3\right )}{3 b^2}+\frac{h x}{b^2} \]
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Rubi [A] time = 1.04753, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (2 a f+b c)-a^{2/3} (b e-4 a h)\right )}{18 a^{4/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (2 a f+b c)-a^{2/3} (b e-4 a h)\right )}{9 a^{4/3} b^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{2/3} b e-4 a^{5/3} h+2 a b^{2/3} f+b^{5/3} c\right )}{3 \sqrt{3} a^{4/3} b^{7/3}}-\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 a b^2 \left (a+b x^3\right )}+\frac{g \log \left (a+b x^3\right )}{3 b^2}+\frac{h x}{b^2} \]
Antiderivative was successfully verified.
[In] Int[(x*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^2,x]
[Out]
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Rubi in Sympy [A] time = 160.671, size = 265, normalized size = 0.92 \[ \frac{g \log{\left (a + b x^{3} \right )}}{3 b^{2}} + \frac{h x}{b^{2}} + \frac{x \left (a \left (a h - b e\right ) - b x^{2} \left (a g - b d\right ) - b x \left (a f - b c\right )\right )}{3 a b^{2} \left (a + b x^{3}\right )} - \frac{\left (a^{\frac{2}{3}} \left (4 a h - b e\right ) + b^{\frac{2}{3}} \left (2 a f + b c\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{4}{3}} b^{\frac{7}{3}}} + \frac{\left (a^{\frac{2}{3}} \left (4 a h - b e\right ) + b^{\frac{2}{3}} \left (2 a f + b c\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{4}{3}} b^{\frac{7}{3}}} + \frac{\sqrt{3} \left (4 a^{\frac{5}{3}} h - a^{\frac{2}{3}} b e - 2 a b^{\frac{2}{3}} f - b^{\frac{5}{3}} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{4}{3}} b^{\frac{7}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a)**2,x)
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Mathematica [A] time = 0.385777, size = 285, normalized size = 0.99 \[ \frac{\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^{2/3} b^{4/3} e+4 a^{5/3} \sqrt [3]{b} h+2 a b f+b^2 c\right )}{a^{4/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^{2/3} b^{4/3} e+4 a^{5/3} \sqrt [3]{b} h+2 a b f+b^2 c\right )}{a^{4/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{2/3} b^{4/3} e-4 a^{5/3} \sqrt [3]{b} h+2 a b f+b^2 c\right )}{a^{4/3}}+\frac{6 b^{2/3} \left (a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right )}{a \left (a+b x^3\right )}+6 b^{2/3} g \log \left (a+b x^3\right )+18 b^{2/3} h x}{18 b^{8/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^2,x]
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Maple [B] time = 0.014, size = 504, normalized size = 1.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^2,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/(b*x^3 + a)^2,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)*x/(b*x^3 + a)^2,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(x*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.227123, size = 451, normalized size = 1.56 \[ \frac{h x}{b^{2}} + \frac{g{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} - \frac{\sqrt{3}{\left (4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - \left (-a b^{2}\right )^{\frac{1}{3}} a b e + \left (-a b^{2}\right )^{\frac{2}{3}} b c + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{3}} - \frac{a b d - a^{2} g -{\left (b^{2} c - a b f\right )} x^{2} -{\left (a^{2} h - a b e\right )} x}{3 \,{\left (b x^{3} + a\right )} a b^{2}} - \frac{{\left (4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - \left (-a b^{2}\right )^{\frac{1}{3}} a b e - \left (-a b^{2}\right )^{\frac{2}{3}} b c - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{3}} - \frac{{\left (a b^{5} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, a^{2} b^{4} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 4 \, a^{3} b^{3} h + a^{2} b^{4} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{3} b^{5}} \]
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)^2,x, algorithm="giac")
[Out]