Optimal. Leaf size=258 \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{6 a^{2/3} b^{5/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 a^{2/3} b^{5/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt {3} a^{2/3} b^{5/3}}-\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a b}+\frac {c \log (x)}{a}+\frac {g x}{b}+\frac {h x^2}{2 b} \]
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Rubi [A] time = 0.47, antiderivative size = 256, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.237, Rules used = {1834, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}-a g+b d\right )}{6 a^{2/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 a^{2/3} b^{5/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt {3} a^{2/3} b^{5/3}}-\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a b}+\frac {c \log (x)}{a}+\frac {g x}{b}+\frac {h x^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 260
Rule 617
Rule 628
Rule 634
Rule 1834
Rule 1860
Rule 1871
Rubi steps
\begin {align*} \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )} \, dx &=\int \left (\frac {g}{b}+\frac {c}{a x}+\frac {h x}{b}+\frac {a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2}{a b \left (a+b x^3\right )}\right ) \, dx\\ &=\frac {g x}{b}+\frac {h x^2}{2 b}+\frac {c \log (x)}{a}+\frac {\int \frac {a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2}{a+b x^3} \, dx}{a b}\\ &=\frac {g x}{b}+\frac {h x^2}{2 b}+\frac {c \log (x)}{a}+\frac {\int \frac {a (b d-a g)+a (b e-a h) x}{a+b x^3} \, dx}{a b}-\frac {(b c-a f) \int \frac {x^2}{a+b x^3} \, dx}{a}\\ &=\frac {g x}{b}+\frac {h x^2}{2 b}+\frac {c \log (x)}{a}-\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a b}+\frac {\int \frac {\sqrt [3]{a} \left (2 a \sqrt [3]{b} (b d-a g)+a^{4/3} (b e-a h)\right )+\sqrt [3]{b} \left (-a \sqrt [3]{b} (b d-a g)+a^{4/3} (b e-a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{5/3} b^{4/3}}+\frac {\left (b d-a g-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{2/3} b}\\ &=\frac {g x}{b}+\frac {h x^2}{2 b}+\frac {c \log (x)}{a}+\frac {\left (b d-a g-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a b}+\frac {\left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 \sqrt [3]{a} b^{4/3}}-\frac {\left (b d-a g-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\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^{2/3} b^{4/3}}\\ &=\frac {g x}{b}+\frac {h x^2}{2 b}+\frac {c \log (x)}{a}+\frac {\left (b d-a g-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\frac {\left (b d-a g-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{4/3}}-\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a b}+\frac {\left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{2/3} b^{5/3}}\\ &=\frac {g x}{b}+\frac {h x^2}{2 b}-\frac {\left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{5/3}}+\frac {c \log (x)}{a}+\frac {\left (b d-a g-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\frac {\left (b d-a g-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{4/3}}-\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a b}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 258, normalized size = 1.00 \[ \frac {-\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} h-\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )+2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} h-\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )+2 \sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^{4/3} h-\sqrt [3]{a} b e+a \sqrt [3]{b} g-b^{4/3} d\right )-2 b^{2/3} (b c-a f) \log \left (a+b x^3\right )+6 a b^{2/3} g x+3 a b^{2/3} h x^2+6 b^{5/3} c \log (x)}{6 a b^{5/3}} \]
Antiderivative was successfully verified.
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fricas [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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giac [A] time = 0.19, size = 281, normalized size = 1.09 \[ \frac {c \log \left ({\left | x \right |}\right )}{a} - \frac {\sqrt {3} {\left (b^{2} d - a b g + \left (-a b^{2}\right )^{\frac {1}{3}} a h - \left (-a b^{2}\right )^{\frac {1}{3}} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}} b} - \frac {{\left (b^{2} d - a b g - \left (-a b^{2}\right )^{\frac {1}{3}} a h + \left (-a b^{2}\right )^{\frac {1}{3}} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}} b} - \frac {{\left (b c - a f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a b} + \frac {b h x^{2} + 2 \, b g x}{2 \, b^{2}} + \frac {{\left (a^{3} b^{2} h \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b^{3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e - a^{2} b^{3} d + a^{3} b^{2} g\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{3} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 426, normalized size = 1.65 \[ \frac {h \,x^{2}}{2 b}-\frac {\sqrt {3}\, a g \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}} b^{2}}-\frac {a g \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}+\frac {a g \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}} b^{2}}-\frac {\sqrt {3}\, a h \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 h \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 h \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 {c \ln \relax (x )}{a}-\frac {c \ln \left (b \,x^{3}+a \right )}{3 a}+\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 {2}{3}} b}+\frac {d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}-\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 {2}{3}} b}+\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 {f \ln \left (b \,x^{3}+a \right )}{3 b}+\frac {g x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 290, normalized size = 1.12 \[ \frac {c \log \relax (x)}{a} + \frac {h x^{2} + 2 \, g x}{2 \, b} + \frac {\sqrt {3} {\left (a b e \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b d \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} g \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} b} - \frac {{\left (2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b d - a^{2} g\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b d + a^{2} g\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.10, size = 1731, normalized size = 6.71 \[ \text {result too large to display} \]
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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