Optimal. Leaf size=274 \[ \frac{x^4 \left (a^2 f-a b e+b^2 d\right )}{4 b^3}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{13/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} b^{13/3}}+\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^4}+\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{13/3}}+\frac{x^7 (b e-a f)}{7 b^2}+\frac{f x^{10}}{10 b} \]
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Rubi [A] time = 0.592991, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{x^4 \left (a^2 f-a b e+b^2 d\right )}{4 b^3}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{13/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} b^{13/3}}+\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^4}+\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{13/3}}+\frac{x^7 (b e-a f)}{7 b^2}+\frac{f x^{10}}{10 b} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt [3]{a} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{13}{3}}} - \frac{\sqrt [3]{a} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{13}{3}}} - \frac{\sqrt{3} \sqrt [3]{a} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{13}{3}}} - \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \int \frac{1}{b^{4}}\, dx + \frac{f x^{10}}{10 b} - \frac{x^{7} \left (a f - b e\right )}{7 b^{2}} + \frac{x^{4} \left (a^{2} f - a b e + b^{2} d\right )}{4 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)
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Mathematica [A] time = 0.170181, size = 264, normalized size = 0.96 \[ \frac{105 b^{4/3} x^4 \left (a^2 f-a b e+b^2 d\right )+420 \sqrt [3]{b} x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )+140 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )-140 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )-70 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )+60 b^{7/3} x^7 (b e-a f)+42 b^{10/3} f x^{10}}{420 b^{13/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]
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Maple [B] time = 0.005, size = 492, normalized size = 1.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),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((f*x^9 + e*x^6 + d*x^3 + c)*x^3/(b*x^3 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.2231, size = 355, normalized size = 1.3 \[ \frac{\sqrt{3}{\left (70 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 140 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 420 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} x - \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (14 \, b^{3} f x^{10} + 20 \,{\left (b^{3} e - a b^{2} f\right )} x^{7} + 35 \,{\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{4} + 140 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x\right )}\right )}}{1260 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^3/(b*x^3 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.44902, size = 371, normalized size = 1.35 \[ \operatorname{RootSum}{\left (27 t^{3} b^{13} - a^{10} f^{3} + 3 a^{9} b e f^{2} - 3 a^{8} b^{2} d f^{2} - 3 a^{8} b^{2} e^{2} f + 3 a^{7} b^{3} c f^{2} + 6 a^{7} b^{3} d e f + a^{7} b^{3} e^{3} - 6 a^{6} b^{4} c e f - 3 a^{6} b^{4} d^{2} f - 3 a^{6} b^{4} d e^{2} + 6 a^{5} b^{5} c d f + 3 a^{5} b^{5} c e^{2} + 3 a^{5} b^{5} d^{2} e - 3 a^{4} b^{6} c^{2} f - 6 a^{4} b^{6} c d e - a^{4} b^{6} d^{3} + 3 a^{3} b^{7} c^{2} e + 3 a^{3} b^{7} c d^{2} - 3 a^{2} b^{8} c^{2} d + a b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{3 t b^{4}}{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c} + x \right )} \right )\right )} + \frac{f x^{10}}{10 b} - \frac{x^{7} \left (a f - b e\right )}{7 b^{2}} + \frac{x^{4} \left (a^{2} f - a b e + b^{2} d\right )}{4 b^{3}} - \frac{x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)
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GIAC/XCAS [A] time = 0.21903, size = 467, normalized size = 1.7 \[ -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{5}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{5}} + \frac{{\left (a b^{9} c - a^{2} b^{8} d - a^{4} b^{6} f + a^{3} b^{7} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{10}} + \frac{14 \, b^{9} f x^{10} - 20 \, a b^{8} f x^{7} + 20 \, b^{9} x^{7} e + 35 \, b^{9} d x^{4} + 35 \, a^{2} b^{7} f x^{4} - 35 \, a b^{8} x^{4} e + 140 \, b^{9} c x - 140 \, a b^{8} d x - 140 \, a^{3} b^{6} f x + 140 \, a^{2} b^{7} x e}{140 \, b^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^3/(b*x^3 + a),x, algorithm="giac")
[Out]