Optimal. Leaf size=227 \[ \frac {b c-a d}{a^2 x}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{7/3} b^{5/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{7/3} b^{5/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} a^{7/3} b^{5/3}}-\frac {c}{4 a x^4}+\frac {f x^2}{2 b} \]
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Rubi [A] time = 0.19, antiderivative size = 227, 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, 292, 31, 634, 617, 204, 628} \[ \frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{7/3} b^{5/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{7/3} b^{5/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt {3} a^{7/3} b^{5/3}}+\frac {b c-a d}{a^2 x}-\frac {c}{4 a x^4}+\frac {f x^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
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^5 \left (a+b x^3\right )} \, dx &=\int \left (\frac {c}{a x^5}+\frac {-b c+a d}{a^2 x^2}+\frac {f x}{b}-\frac {\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{a^2 b \left (a+b x^3\right )}\right ) \, dx\\ &=-\frac {c}{4 a x^4}+\frac {b c-a d}{a^2 x}+\frac {f x^2}{2 b}-\frac {\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \int \frac {x}{a+b x^3} \, dx}{a^2 b}\\ &=-\frac {c}{4 a x^4}+\frac {b c-a d}{a^2 x}+\frac {f x^2}{2 b}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{7/3} b^{4/3}}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \int \frac {\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^{7/3} b^{4/3}}\\ &=-\frac {c}{4 a x^4}+\frac {b c-a d}{a^2 x}+\frac {f x^2}{2 b}-\frac {\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^{7/3} b^{5/3}}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\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^{7/3} b^{5/3}}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^2 b^{4/3}}\\ &=-\frac {c}{4 a x^4}+\frac {b c-a d}{a^2 x}+\frac {f x^2}{2 b}-\frac {\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^{7/3} b^{5/3}}+\frac {\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^{7/3} b^{5/3}}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{7/3} b^{5/3}}\\ &=-\frac {c}{4 a x^4}+\frac {b c-a d}{a^2 x}+\frac {f x^2}{2 b}-\frac {\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^{7/3} b^{5/3}}-\frac {\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^{7/3} b^{5/3}}+\frac {\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^{7/3} b^{5/3}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 220, normalized size = 0.97 \[ \frac {1}{12} \left (\frac {12 (b c-a d)}{a^2 x}+\frac {2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{7/3} b^{5/3}}+\frac {4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^{7/3} b^{5/3}}+\frac {4 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^{7/3} b^{5/3}}-\frac {3 c}{a x^4}+\frac {6 f x^2}{b}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 556, normalized size = 2.45 \[ \left [\frac {6 \, a^{3} b^{2} f x^{6} - 6 \, \sqrt {\frac {1}{3}} {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{4} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b - 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (a b^{2}\right )^{\frac {2}{3}} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) + 2 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a b^{2}\right )^{\frac {2}{3}} x^{4} \log \left (b^{2} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} b x + \left (a b^{2}\right )^{\frac {2}{3}}\right ) - 4 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a b^{2}\right )^{\frac {2}{3}} x^{4} \log \left (b x + \left (a b^{2}\right )^{\frac {1}{3}}\right ) - 3 \, a^{2} b^{3} c + 12 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} x^{3}}{12 \, a^{3} b^{3} x^{4}}, \frac {6 \, a^{3} b^{2} f x^{6} - 12 \, \sqrt {\frac {1}{3}} {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{4} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x - \left (a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + 2 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a b^{2}\right )^{\frac {2}{3}} x^{4} \log \left (b^{2} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} b x + \left (a b^{2}\right )^{\frac {2}{3}}\right ) - 4 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a b^{2}\right )^{\frac {2}{3}} x^{4} \log \left (b x + \left (a b^{2}\right )^{\frac {1}{3}}\right ) - 3 \, a^{2} b^{3} c + 12 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} x^{3}}{12 \, a^{3} b^{3} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 261, normalized size = 1.15 \[ \frac {f x^{2}}{2 \, b} + \frac {\sqrt {3} {\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b} - \frac {{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b} - \frac {{\left (b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} b \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{3} b} + \frac {4 \, b c x^{3} - 4 \, a d x^{3} - a c}{4 \, a^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 412, normalized size = 1.81 \[ \frac {f \,x^{2}}{2 b}-\frac {\sqrt {3}\, a 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 {1}{3}} b^{2}}+\frac {a f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}-\frac {a 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 {1}{3}} b^{2}}-\frac {\sqrt {3}\, 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 {1}{3}} a}+\frac {d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} a}-\frac {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 {1}{3}} a}+\frac {\sqrt {3}\, b 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 {1}{3}} a^{2}}-\frac {b c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}+\frac {b 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 {1}{3}} a^{2}}+\frac {\sqrt {3}\, 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 {1}{3}} b}-\frac {e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b}+\frac {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 {1}{3}} b}-\frac {d}{a x}+\frac {b c}{a^{2} x}-\frac {c}{4 a \,x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.04, size = 217, normalized size = 0.96 \[ \frac {f x^{2}}{2 \, b} + \frac {\sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {4 \, {\left (b c - a d\right )} x^{3} - a c}{4 \, a^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.16, size = 209, normalized size = 0.92 \[ \frac {f\,x^2}{2\,b}-\frac {\frac {b\,c}{4\,a}+\frac {b\,x^3\,\left (a\,d-b\,c\right )}{a^2}}{b\,x^4}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{7/3}\,b^{5/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{7/3}\,b^{5/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{7/3}\,b^{5/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.53, size = 411, normalized size = 1.81 \[ \operatorname {RootSum} {\left (27 t^{3} a^{7} b^{5} - a^{9} f^{3} + 3 a^{8} b e f^{2} - 3 a^{7} b^{2} d f^{2} - 3 a^{7} b^{2} e^{2} f + 3 a^{6} b^{3} c f^{2} + 6 a^{6} b^{3} d e f + a^{6} b^{3} e^{3} - 6 a^{5} b^{4} c e f - 3 a^{5} b^{4} d^{2} f - 3 a^{5} b^{4} d e^{2} + 6 a^{4} b^{5} c d f + 3 a^{4} b^{5} c e^{2} + 3 a^{4} b^{5} d^{2} e - 3 a^{3} b^{6} c^{2} f - 6 a^{3} b^{6} c d e - a^{3} b^{6} d^{3} + 3 a^{2} b^{7} c^{2} e + 3 a^{2} b^{7} c d^{2} - 3 a b^{8} c^{2} d + b^{9} c^{3}, \left (t \mapsto t \log {\left (\frac {9 t^{2} a^{5} b^{3}}{a^{6} f^{2} - 2 a^{5} b e f + 2 a^{4} b^{2} d f + a^{4} b^{2} e^{2} - 2 a^{3} b^{3} c f - 2 a^{3} b^{3} d e + 2 a^{2} b^{4} c e + a^{2} b^{4} d^{2} - 2 a b^{5} c d + b^{6} c^{2}} + x \right )} \right )\right )} + \frac {f x^{2}}{2 b} + \frac {- a c + x^{3} \left (- 4 a d + 4 b c\right )}{4 a^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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