Optimal. Leaf size=245 \[ \frac{x^2 \left (a^2 f-a b e+b^2 d\right )}{2 b^3}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 \sqrt [3]{a} b^{11/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} \sqrt [3]{a} b^{11/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 \sqrt [3]{a} b^{11/3}}+\frac{x^5 (b e-a f)}{5 b^2}+\frac{f x^8}{8 b} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.480416, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{x^2 \left (a^2 f-a b e+b^2 d\right )}{2 b^3}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 \sqrt [3]{a} b^{11/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} \sqrt [3]{a} b^{11/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 \sqrt [3]{a} b^{11/3}}+\frac{x^5 (b e-a f)}{5 b^2}+\frac{f x^8}{8 b} \]
Antiderivative was successfully verified.
[In] Int[(x*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{f x^{8}}{8 b} - \frac{x^{5} \left (a f - b e\right )}{5 b^{2}} + \frac{\left (a^{2} f - a b e + b^{2} d\right ) \int x\, dx}{b^{3}} + \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 \sqrt [3]{a} b^{\frac{11}{3}}} - \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 \sqrt [3]{a} b^{\frac{11}{3}}} + \frac{\sqrt{3} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 \sqrt [3]{a} b^{\frac{11}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.2958, size = 231, normalized size = 0.94 \[ \frac{60 b^{2/3} x^2 \left (a^2 f-a b e+b^2 d\right )+\frac{40 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\sqrt [3]{a}}+\frac{40 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\sqrt [3]{a}}+\frac{20 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt [3]{a}}+24 b^{5/3} x^5 (b e-a f)+15 b^{8/3} f x^8}{120 b^{11/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.005, size = 450, normalized size = 1.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),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((f*x^9 + e*x^6 + d*x^3 + c)*x/(b*x^3 + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.222158, size = 313, normalized size = 1.28 \[ \frac{\sqrt{3}{\left (20 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (\left (a b^{2}\right )^{\frac{1}{3}} b x^{2} + a b - \left (a b^{2}\right )^{\frac{2}{3}} x\right ) - 40 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (a b + \left (a b^{2}\right )^{\frac{2}{3}} x\right ) + 120 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) + 3 \, \sqrt{3}{\left (5 \, b^{2} f x^{8} + 8 \,{\left (b^{2} e - a b f\right )} x^{5} + 20 \,{\left (b^{2} d - a b e + a^{2} f\right )} x^{2}\right )} \left (a b^{2}\right )^{\frac{1}{3}}\right )}}{360 \, \left (a b^{2}\right )^{\frac{1}{3}} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x/(b*x^3 + a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 2.89422, size = 422, normalized size = 1.72 \[ \operatorname{RootSum}{\left (27 t^{3} a b^{11} - a^{9} f^{3} + 3 a^{8} b e f^{2} - 3 a^{7} b^{2} d f^{2} - 3 a^{7} b^{2} e^{2} f + 3 a^{6} b^{3} c f^{2} + 6 a^{6} b^{3} d e f + a^{6} b^{3} e^{3} - 6 a^{5} b^{4} c e f - 3 a^{5} b^{4} d^{2} f - 3 a^{5} b^{4} d e^{2} + 6 a^{4} b^{5} c d f + 3 a^{4} b^{5} c e^{2} + 3 a^{4} b^{5} d^{2} e - 3 a^{3} b^{6} c^{2} f - 6 a^{3} b^{6} c d e - a^{3} b^{6} d^{3} + 3 a^{2} b^{7} c^{2} e + 3 a^{2} b^{7} c d^{2} - 3 a b^{8} c^{2} d + b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{9 t^{2} a b^{7}}{a^{6} f^{2} - 2 a^{5} b e f + 2 a^{4} b^{2} d f + a^{4} b^{2} e^{2} - 2 a^{3} b^{3} c f - 2 a^{3} b^{3} d e + 2 a^{2} b^{4} c e + a^{2} b^{4} d^{2} - 2 a b^{5} c d + b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{8}}{8 b} - \frac{x^{5} \left (a f - b e\right )}{5 b^{2}} + \frac{x^{2} \left (a^{2} f - a b e + b^{2} d\right )}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.219204, size = 466, normalized size = 1.9 \[ -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{5}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{5}} - \frac{{\left (b^{8} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{7} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} b^{5} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{2} b^{6} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{8}} + \frac{5 \, b^{7} f x^{8} - 8 \, a b^{6} f x^{5} + 8 \, b^{7} x^{5} e + 20 \, b^{7} d x^{2} + 20 \, a^{2} b^{5} f x^{2} - 20 \, a b^{6} x^{2} e}{40 \, b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x/(b*x^3 + a),x, algorithm="giac")
[Out]