Integrand size = 38, antiderivative size = 276 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left (a+b x^3\right )} \, dx=-\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x}+\frac {\left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} b^{2/3}}-\frac {(b c-a f) \log (x)}{a^2}-\frac {\left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} b^{2/3}}+\frac {\left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} b^{2/3}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a^2} \]
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Time = 0.32 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.237, Rules used = {1848, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left (a+b x^3\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt {3} a^{5/3} b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}-a g+b d\right )}{6 a^{5/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 a^{5/3} b^{2/3}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a^2}-\frac {\log (x) (b c-a f)}{a^2}-\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x} \]
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Rule 31
Rule 210
Rule 266
Rule 631
Rule 642
Rule 648
Rule 1848
Rule 1874
Rule 1885
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c}{a x^4}+\frac {d}{a x^3}+\frac {e}{a x^2}+\frac {-b c+a f}{a^2 x}+\frac {-a (b d-a g)-a (b e-a h) x+b (b c-a f) x^2}{a^2 \left (a+b x^3\right )}\right ) \, dx \\ & = -\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {(b c-a f) \log (x)}{a^2}+\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^2} \\ & = -\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {(b c-a f) \log (x)}{a^2}+\frac {\int \frac {-a (b d-a g)-a (b e-a h) x}{a+b x^3} \, dx}{a^2}+\frac {(b (b c-a f)) \int \frac {x^2}{a+b x^3} \, dx}{a^2} \\ & = -\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {(b c-a f) \log (x)}{a^2}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a^2}+\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^{8/3} \sqrt [3]{b}}-\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^{5/3}} \\ & = -\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {(b c-a f) \log (x)}{a^2}-\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^{5/3} \sqrt [3]{b}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a^2}-\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 a^{4/3} \sqrt [3]{b}}+\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^{5/3} \sqrt [3]{b}} \\ & = -\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {(b c-a f) \log (x)}{a^2}-\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^{5/3} \sqrt [3]{b}}+\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^{5/3} \sqrt [3]{b}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a^2}-\frac {\left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{5/3} b^{2/3}} \\ & = -\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x}+\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^{5/3} b^{2/3}}-\frac {(b c-a f) \log (x)}{a^2}-\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^{5/3} \sqrt [3]{b}}+\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^{5/3} \sqrt [3]{b}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a^2} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left (a+b x^3\right )} \, dx=-\frac {\frac {2 a c}{x^3}+\frac {3 a d}{x^2}+\frac {6 a e}{x}+\frac {2 \sqrt {3} \sqrt [3]{a} \left (-b^{4/3} d-\sqrt [3]{a} b e+a \sqrt [3]{b} g+a^{4/3} h\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}+6 (b c-a f) \log (x)+\frac {2 \sqrt [3]{a} \left (b^{4/3} d-\sqrt [3]{a} b e-a \sqrt [3]{b} g+a^{4/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}-\frac {\sqrt [3]{a} \left (b^{4/3} d-\sqrt [3]{a} b e-a \sqrt [3]{b} g+a^{4/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}-2 (b c-a f) \log \left (a+b x^3\right )}{6 a^2} \]
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Time = 1.57 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {e}{a x}-\frac {c}{3 a \,x^{3}}-\frac {d}{2 a \,x^{2}}+\frac {\left (a f -b c \right ) \ln \left (x \right )}{a^{2}}+\frac {\left (a^{2} g -a b d \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (a^{2} h -a e b \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (-a f b +b^{2} c \right ) \ln \left (b \,x^{3}+a \right )}{3 b}}{a^{2}}\) | \(276\) |
risch | \(\frac {-\frac {e \,x^{2}}{a}-\frac {x d}{2 a}-\frac {c}{3 a}}{x^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{6} b^{2} \textit {\_Z}^{3}+\left (3 a^{5} b^{2} f -3 a^{4} b^{3} c \right ) \textit {\_Z}^{2}+\left (3 a^{5} b g h -3 a^{4} b^{2} d h -3 a^{4} b^{2} e g +3 a^{4} b^{2} f^{2}-6 a^{3} b^{3} c f +3 a^{3} b^{3} d e +3 a^{2} b^{4} c^{2}\right ) \textit {\_Z} +a^{5} h^{3}-3 a^{4} b e \,h^{2}+3 a^{4} b f g h -a^{4} b \,g^{3}-3 a^{3} b^{2} c g h -3 a^{3} b^{2} d f h +3 a^{3} b^{2} d \,g^{2}+3 a^{3} b^{2} e^{2} h -3 a^{3} b^{2} e f g +a^{3} b^{2} f^{3}+3 a^{2} b^{3} c d h +3 a^{2} b^{3} c e g -3 a^{2} b^{3} c \,f^{2}-3 a^{2} b^{3} d^{2} g +3 a^{2} b^{3} d e f -a^{2} b^{3} e^{3}+3 a \,b^{4} c^{2} f -3 a \,b^{4} c d e +a \,b^{4} d^{3}-b^{5} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{5} b^{2}+\left (-8 a^{4} b^{2} f +8 a^{3} b^{3} c \right ) \textit {\_R}^{2}+\left (-10 a^{4} b g h +10 a^{3} b^{2} d h +10 a^{3} b^{2} e g -4 a^{3} b^{2} f^{2}+8 a^{2} b^{3} c f -10 a^{2} b^{3} d e -4 c^{2} a \,b^{4}\right ) \textit {\_R} -3 a^{4} h^{3}+9 a^{3} b e \,h^{2}-6 a^{3} b f g h +3 a^{3} b \,g^{3}+6 a^{2} b^{2} c g h +6 a^{2} b^{2} d f h -9 a^{2} b^{2} d \,g^{2}-9 a^{2} b^{2} e^{2} h +6 a^{2} b^{2} e f g -6 a \,b^{3} c d h -6 a \,b^{3} c e g +9 a \,b^{3} d^{2} g -6 a \,b^{3} d e f +3 a \,b^{3} e^{3}+6 b^{4} c d e -3 b^{4} d^{3}\right ) x +\left (a^{5} b h -a^{4} e \,b^{2}\right ) \textit {\_R}^{2}+\left (-2 a^{4} b f h -a^{4} b \,g^{2}+2 a^{3} b^{2} c h +2 a^{3} b^{2} d g +2 a^{3} b^{2} e f -2 a^{2} b^{3} c e -a^{2} b^{3} d^{2}\right ) \textit {\_R} -3 a^{3} b \,f^{2} h +3 a^{3} b f \,g^{2}+6 a^{2} b^{2} c f h -3 a^{2} b^{2} c \,g^{2}-6 a^{2} b^{2} d f g +3 a^{2} b^{2} e \,f^{2}-3 a \,b^{3} c^{2} h +6 a \,b^{3} c d g -6 a \,b^{3} c e f +3 a \,b^{3} d^{2} f +3 b^{4} c^{2} e -3 b^{4} c \,d^{2}\right )\right )}{3}+\frac {\ln \left (-x \right ) f}{a}-\frac {\ln \left (-x \right ) b c}{a^{2}}\) | \(847\) |
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Result contains complex when optimal does not.
Time = 74.15 (sec) , antiderivative size = 15204, normalized size of antiderivative = 55.09 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left (a+b x^3\right )} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left (a+b x^3\right )} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.09 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left (a+b x^3\right )} \, dx=-\frac {{\left (b c - a f\right )} \log \left (x\right )}{a^{2}} - \frac {\sqrt {3} {\left (a b e \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b d \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} g \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{3}} + \frac {{\left (2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b d - a^{2} g\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b d + a^{2} g\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {6 \, e x^{2} + 3 \, d x + 2 \, c}{6 \, a x^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.04 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left (a+b x^3\right )} \, dx=\frac {\sqrt {3} {\left (b^{2} d - a b g - \left (-a b^{2}\right )^{\frac {1}{3}} b e + \left (-a b^{2}\right )^{\frac {1}{3}} a h\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}} a} + \frac {{\left (b^{2} d - a b g + \left (-a b^{2}\right )^{\frac {1}{3}} b e - \left (-a b^{2}\right )^{\frac {1}{3}} a h\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}} a} + \frac {{\left (b c - a f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} - \frac {{\left (b c - a f\right )} \log \left ({\left | x \right |}\right )}{a^{2}} + \frac {{\left (a^{3} b^{2} e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{4} b h \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{3} b^{2} d - a^{4} b g\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{5} b} - \frac {6 \, a e x^{2} + 3 \, a d x + 2 \, a c}{6 \, a^{2} x^{3}} \]
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Time = 10.00 (sec) , antiderivative size = 1842, normalized size of antiderivative = 6.67 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left (a+b x^3\right )} \, dx=\text {Too large to display} \]
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