Optimal. Leaf size=338 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (5 b d-2 a g)-\sqrt [3]{a} (4 b e-a h)\right )}{18 a^{8/3} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (5 b d-2 a g)-\sqrt [3]{a} (4 b e-a h)\right )}{9 a^{8/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{3 \sqrt{3} a^{8/3} b^{2/3}}+\frac{(2 b c-a f) \log \left (a+b x^3\right )}{3 a^3}-\frac{\log (x) (2 b c-a f)}{a^3}-\frac{x \left (-b x^2 \left (\frac{b c}{a}-f\right )+x (b e-a h)-a g+b d\right )}{3 a^2 \left (a+b x^3\right )}-\frac{c}{3 a^2 x^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 1.45272, antiderivative size = 336, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 9, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.237 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (4 b e-a h)}{\sqrt [3]{b}}-2 a g+5 b d\right )}{18 a^{8/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (5 b d-2 a g)-\sqrt [3]{a} (4 b e-a h)\right )}{9 a^{8/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{3 \sqrt{3} a^{8/3} b^{2/3}}+\frac{(2 b c-a f) \log \left (a+b x^3\right )}{3 a^3}-\frac{\log (x) (2 b c-a f)}{a^3}-\frac{x \left (-b x^2 \left (\frac{b c}{a}-f\right )+x (b e-a h)-a g+b d\right )}{3 a^2 \left (a+b x^3\right )}-\frac{c}{3 a^2 x^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x} \]
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)^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 128.198, size = 255, normalized size = 0.75 \[ - \frac{f}{3 a b x^{3}} - \frac{g}{2 a b x^{2}} - \frac{h}{a b x} - \frac{x \left (\frac{a f}{x^{4}} + \frac{a g}{x^{3}} + \frac{a h}{x^{2}} - \frac{b c}{x^{4}} - \frac{b d}{x^{3}} - \frac{b e}{x^{2}}\right )}{3 a b \left (a + b x^{3}\right )} - \frac{f \log{\left (x \right )}}{a^{2}} + \frac{f \log{\left (a + b x^{3} \right )}}{3 a^{2}} + \frac{\left (\sqrt [3]{a} h - \sqrt [3]{b} g\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{5}{3}} b^{\frac{2}{3}}} - \frac{\left (\sqrt [3]{a} h - \sqrt [3]{b} g\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{5}{3}} b^{\frac{2}{3}}} + \frac{\sqrt{3} \left (\sqrt [3]{a} h + \sqrt [3]{b} g\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{5}{3}} b^{\frac{2}{3}}} \]
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**4/(b*x**3+a)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 1.1456, size = 303, normalized size = 0.9 \[ \frac{\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} h-4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{2/3}}-\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} h-4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{2/3}}-\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} h-4 \sqrt [3]{a} b e+2 a \sqrt [3]{b} g-5 b^{4/3} d\right )}{b^{2/3}}+\frac{a (6 a (f+x (g+h x))-6 b (c+x (d+e x)))}{a+b x^3}+6 (2 b c-a f) \log \left (a+b x^3\right )+18 \log (x) (a f-2 b c)-\frac{6 a c}{x^3}-\frac{9 a d}{x^2}-\frac{18 a e}{x}}{18 a^3} \]
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)^2),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.022, size = 561, 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)^2,x)
[Out]
_______________________________________________________________________________________
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)^2*x^4),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
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)^2*x^4),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.226904, size = 520, normalized size = 1.54 \[ \frac{{\left (2 \, b c - a f\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} - \frac{{\left (2 \, b c - a f\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g + \left (-a b^{2}\right )^{\frac{2}{3}} a h - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3} b^{2}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g - \left (-a b^{2}\right )^{\frac{2}{3}} a h + 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{3} b^{2}} - \frac{{\left (a^{5} b h \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 4 \, a^{4} b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e - 5 \, a^{4} b^{2} d + 2 \, a^{5} b g\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{7} b} + \frac{2 \,{\left (a^{2} h - 4 \, a b e\right )} x^{5} -{\left (5 \, a b d - 2 \, a^{2} g\right )} x^{4} - 6 \, a^{2} x^{2} e - 3 \, a^{2} d x - 2 \,{\left (2 \, a b c - a^{2} f\right )} x^{3} - 2 \, a^{2} c}{6 \,{\left (b x^{3} + a\right )} a^{3} x^{3}} \]
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)^2*x^4),x, algorithm="giac")
[Out]