Optimal. Leaf size=280 \[ \frac {b c-a d}{8 a^2 x^8}-\frac {a^2 e-a b d+b^2 c}{5 a^3 x^5}-\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{14/3}}+\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{14/3}}-\frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} a^{14/3}}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{2 a^4 x^2}-\frac {c}{11 a x^{11}} \]
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Rubi [A] time = 0.20, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {1834, 200, 31, 634, 617, 204, 628} \[ \frac {a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{2 a^4 x^2}-\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{14/3}}+\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{14/3}}-\frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt {3} a^{14/3}}-\frac {a^2 e-a b d+b^2 c}{5 a^3 x^5}+\frac {b c-a d}{8 a^2 x^8}-\frac {c}{11 a x^{11}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 617
Rule 628
Rule 634
Rule 1834
Rubi steps
\begin {align*} \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )} \, dx &=\int \left (\frac {c}{a x^{12}}+\frac {-b c+a d}{a^2 x^9}+\frac {b^2 c-a b d+a^2 e}{a^3 x^6}+\frac {-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^3}-\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^4 \left (a+b x^3\right )}\right ) \, dx\\ &=-\frac {c}{11 a x^{11}}+\frac {b c-a d}{8 a^2 x^8}-\frac {b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{2 a^4 x^2}+\frac {\left (b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{a+b x^3} \, dx}{a^4}\\ &=-\frac {c}{11 a x^{11}}+\frac {b c-a d}{8 a^2 x^8}-\frac {b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{2 a^4 x^2}+\frac {\left (b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{14/3}}+\frac {\left (b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{14/3}}\\ &=-\frac {c}{11 a x^{11}}+\frac {b c-a d}{8 a^2 x^8}-\frac {b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{2 a^4 x^2}+\frac {b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{14/3}}-\frac {\left (b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{14/3}}+\frac {\left (b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^{13/3}}\\ &=-\frac {c}{11 a x^{11}}+\frac {b c-a d}{8 a^2 x^8}-\frac {b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{2 a^4 x^2}+\frac {b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{14/3}}-\frac {b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{14/3}}+\frac {\left (b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{14/3}}\\ &=-\frac {c}{11 a x^{11}}+\frac {b c-a d}{8 a^2 x^8}-\frac {b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{2 a^4 x^2}-\frac {b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{14/3}}+\frac {b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{14/3}}-\frac {b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{14/3}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 266, normalized size = 0.95 \[ \frac {\frac {165 a^{8/3} (b c-a d)}{x^8}-\frac {120 a^{11/3} c}{x^{11}}-\frac {264 a^{5/3} \left (a^2 e-a b d+b^2 c\right )}{x^5}+440 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )-440 \sqrt {3} b^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )+\frac {660 a^{2/3} \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{x^2}+220 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{1320 a^{14/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 295, normalized size = 1.05 \[ -\frac {440 \, \sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{11} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - 220 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{11} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) + 440 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{11} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 660 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{9} + 264 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{6} + 120 \, a^{3} c - 165 \, {\left (a^{2} b c - a^{3} d\right )} x^{3}}{1320 \, a^{4} x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 338, normalized size = 1.21 \[ \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{5}} - \frac {{\left (b^{4} c - a b^{3} d - a^{3} b f + a^{2} b^{2} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{5}} + \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{5}} + \frac {220 \, b^{3} c x^{9} - 220 \, a b^{2} d x^{9} - 220 \, a^{3} f x^{9} + 220 \, a^{2} b x^{9} e - 88 \, a b^{2} c x^{6} + 88 \, a^{2} b d x^{6} - 88 \, a^{3} x^{6} e + 55 \, a^{2} b c x^{3} - 55 \, a^{3} d x^{3} - 40 \, a^{3} c}{440 \, a^{4} x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 493, normalized size = 1.76 \[ -\frac {\sqrt {3}\, f \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a}-\frac {f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a}+\frac {f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} a}+\frac {\sqrt {3}\, b e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}+\frac {b e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}-\frac {b e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}-\frac {\sqrt {3}\, b^{2} d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}-\frac {b^{2} d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}+\frac {b^{2} d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}+\frac {\sqrt {3}\, b^{3} c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}+\frac {b^{3} c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}-\frac {b^{3} c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}-\frac {f}{2 a \,x^{2}}+\frac {b e}{2 a^{2} x^{2}}-\frac {b^{2} d}{2 a^{3} x^{2}}+\frac {b^{3} c}{2 a^{4} x^{2}}-\frac {e}{5 a \,x^{5}}+\frac {b d}{5 a^{2} x^{5}}-\frac {b^{2} c}{5 a^{3} x^{5}}-\frac {d}{8 a \,x^{8}}+\frac {b c}{8 a^{2} x^{8}}-\frac {c}{11 a \,x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 260, normalized size = 0.93 \[ \frac {\sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {220 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{9} - 88 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{6} - 40 \, a^{3} c + 55 \, {\left (a^{2} b c - a^{3} d\right )} x^{3}}{440 \, a^{4} x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.15, size = 253, normalized size = 0.90 \[ \frac {b^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{14/3}}-\frac {\frac {c}{11\,a}-\frac {x^9\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{2\,a^4}+\frac {x^3\,\left (a\,d-b\,c\right )}{8\,a^2}+\frac {x^6\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{5\,a^3}}{x^{11}}+\frac {b^{2/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{14/3}}-\frac {b^{2/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{14/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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