Optimal. Leaf size=381 \[ \frac{3 b c-a d}{7 a^4 x^7}-\frac{c}{10 a^3 x^{10}}-\frac{a^2 e-3 a b d+6 b^2 c}{4 a^5 x^4}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{27 a^{19/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{9 \sqrt{3} a^{19/3}}+\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{54 a^{19/3}}+\frac{b x^2 \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{9 a^6 \left (a+b x^3\right )}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{a^6 x}+\frac{b x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2} \]
[Out]
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Rubi [A] time = 1.40591, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{3 b c-a d}{7 a^4 x^7}-\frac{c}{10 a^3 x^{10}}-\frac{a^2 e-3 a b d+6 b^2 c}{4 a^5 x^4}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{27 a^{19/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{9 \sqrt{3} a^{19/3}}+\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{54 a^{19/3}}+\frac{b x^2 \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{9 a^6 \left (a+b x^3\right )}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{a^6 x}+\frac{b x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)^3),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**11/(b*x**3+a)**3,x)
[Out]
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Mathematica [A] time = 0.850349, size = 366, normalized size = 0.96 \[ \frac{-\frac{540 a^{7/3} (a d-3 b c)}{x^7}-\frac{378 a^{10/3} c}{x^{10}}-\frac{945 a^{4/3} \left (a^2 e-3 a b d+6 b^2 c\right )}{x^4}-\frac{420 \sqrt [3]{a} b x^2 \left (5 a^3 f-8 a^2 b e+11 a b^2 d-14 b^3 c\right )}{a+b x^3}-\frac{3780 \sqrt [3]{a} \left (a^3 f-3 a^2 b e+6 a b^2 d-10 b^3 c\right )}{x}+140 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (14 a^3 f-35 a^2 b e+65 a b^2 d-104 b^3 c\right )-140 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )-\frac{630 a^{4/3} b x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+70 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{3780 a^{19/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)^3),x]
[Out]
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Maple [A] time = 0.025, size = 659, normalized size = 1.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^9+e*x^6+d*x^3+c)/x^11/(b*x^3+a)^3,x)
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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)/((b*x^3 + a)^3*x^11),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.230684, size = 873, normalized size = 2.29 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)^3*x^11),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((f*x**9+e*x**6+d*x**3+c)/x**11/(b*x**3+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.219558, size = 656, normalized size = 1.72 \[ -\frac{{\left (104 \, b^{4} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 65 \, a b^{3} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 14 \, a^{3} b f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 35 \, a^{2} b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{7}} - \frac{\sqrt{3}{\left (104 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 35 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{7} b} + \frac{{\left (104 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 35 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{7} b} + \frac{28 \, b^{5} c x^{5} - 22 \, a b^{4} d x^{5} - 10 \, a^{3} b^{2} f x^{5} + 16 \, a^{2} b^{3} x^{5} e + 31 \, a b^{4} c x^{2} - 25 \, a^{2} b^{3} d x^{2} - 13 \, a^{4} b f x^{2} + 19 \, a^{3} b^{2} x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{6}} + \frac{1400 \, b^{3} c x^{9} - 840 \, a b^{2} d x^{9} - 140 \, a^{3} f x^{9} + 420 \, a^{2} b x^{9} e - 210 \, a b^{2} c x^{6} + 105 \, a^{2} b d x^{6} - 35 \, a^{3} x^{6} e + 60 \, a^{2} b c x^{3} - 20 \, a^{3} d x^{3} - 14 \, a^{3} c}{140 \, a^{6} x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)^3*x^11),x, algorithm="giac")
[Out]