Optimal. Leaf size=298 \[ \frac{\sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3}}-\frac{2 \sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3}}+\frac{2 \sqrt [3]{b} \left (7 \sqrt [3]{a} e+10 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3}}+\frac{b c \log \left (a+b x^3\right )}{a^4}-\frac{3 b c \log (x)}{a^4}-\frac{x \left (-\frac{15 b^2 c x^2}{a}+11 b d+10 b e x\right )}{18 a^3 \left (a+b x^3\right )}-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (-\frac{b^2 c x^2}{a}+b d+b e x\right )}{6 a^2 \left (a+b x^3\right )^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 1.11201, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391 \[ \frac{\sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3}}-\frac{2 \sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3}}+\frac{2 \sqrt [3]{b} \left (7 \sqrt [3]{a} e+10 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3}}+\frac{b c \log \left (a+b x^3\right )}{a^4}-\frac{3 b c \log (x)}{a^4}-\frac{x \left (-\frac{15 b^2 c x^2}{a}+11 b d+10 b e x\right )}{18 a^3 \left (a+b x^3\right )}-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (-\frac{b^2 c x^2}{a}+b d+b e x\right )}{6 a^2 \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x + e*x^2)/(x^4*(a + b*x^3)^3),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.1222, size = 27, normalized size = 0.09 \[ \frac{x \left (\frac{c}{x^{4}} + \frac{d}{x^{3}} + \frac{e}{x^{2}}\right )}{6 a \left (a + b x^{3}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x**2+d*x+c)/x**4/(b*x**3+a)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.698342, size = 255, normalized size = 0.86 \[ -\frac{-2 \sqrt [3]{b} \left (10 \sqrt [3]{a} \sqrt [3]{b} d-7 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+4 \sqrt [3]{b} \left (10 \sqrt [3]{a} \sqrt [3]{b} d-7 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\frac{9 a^2 b (c+x (d+e x))}{\left (a+b x^3\right )^2}+\frac{3 a b (12 c+x (11 d+10 e x))}{a+b x^3}-54 b c \log \left (a+b x^3\right )-4 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (7 \sqrt [3]{a} e+10 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+\frac{18 a c}{x^3}+\frac{27 a d}{x^2}+\frac{54 a e}{x}+162 b c \log (x)}{54 a^4} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x + e*x^2)/(x^4*(a + b*x^3)^3),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.025, size = 351, normalized size = 1.2 \[ -{\frac{d}{2\,{a}^{3}{x}^{2}}}-{\frac{e}{{a}^{3}x}}-{\frac{c}{3\,{a}^{3}{x}^{3}}}-3\,{\frac{bc\ln \left ( x \right ) }{{a}^{4}}}-{\frac{5\,{x}^{5}e{b}^{2}}{9\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{11\,{x}^{4}{b}^{2}d}{18\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{2\,{b}^{2}{x}^{3}c}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{13\,be{x}^{2}}{18\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{7\,bxd}{9\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{5\,bc}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{20\,d}{27\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{10\,d}{27\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{20\,d\sqrt{3}}{27\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{14\,e}{27\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{7\,e}{27\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{14\,e\sqrt{3}}{27\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{bc\ln \left ( b{x}^{3}+a \right ) }{{a}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+d*x+c)/x^4/(b*x^3+a)^3,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((e*x^2 + d*x + c)/((b*x^3 + a)^3*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((e*x^2 + d*x + c)/((b*x^3 + a)^3*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((e*x**2+d*x+c)/x**4/(b*x**3+a)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.222242, size = 421, normalized size = 1.41 \[ \frac{b c{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{a^{4}} - \frac{3 \, b c{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{{\left (10 \, \left (-a b^{2}\right )^{\frac{1}{3}} b d + 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{27 \, a^{4} b} - \frac{2 \, \sqrt{3}{\left (10 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} d - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{5} b^{3}} + \frac{2 \,{\left (7 \, a^{5} b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + 10 \, a^{5} b^{2} d\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{9} b} - \frac{28 \, a b^{2} x^{8} e + 20 \, a b^{2} d x^{7} + 18 \, a b^{2} c x^{6} + 49 \, a^{2} b x^{5} e + 32 \, a^{2} b d x^{4} + 27 \, a^{2} b c x^{3} + 18 \, a^{3} x^{2} e + 9 \, a^{3} d x + 6 \, a^{3} c}{18 \,{\left (b x^{3} + a\right )}^{2} a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d*x + c)/((b*x^3 + a)^3*x^4),x, algorithm="giac")
[Out]