Integrand size = 38, antiderivative size = 395 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left (a+b x^3\right )^3} \, dx=-\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac {5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\left (20 b^{4/3} d+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g-2 a^{4/3} h\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} b^{2/3}}-\frac {(3 b c-a f) \log (x)}{a^4}-\frac {\left (5 \sqrt [3]{b} (4 b d-a g)-2 \sqrt [3]{a} (7 b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} b^{2/3}}+\frac {\left (5 \sqrt [3]{b} (4 b d-a g)-2 \sqrt [3]{a} (7 b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} b^{2/3}}+\frac {(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4} \]
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Time = 0.66 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1843, 1848, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left (a+b x^3\right )^3} \, dx=\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-2 a^{4/3} h+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+20 b^{4/3} d\right )}{9 \sqrt {3} a^{11/3} b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac {2 \sqrt [3]{a} (7 b e-a h)}{\sqrt [3]{b}}-5 a g+20 b d\right )}{54 a^{11/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 \sqrt [3]{b} (4 b d-a g)-2 \sqrt [3]{a} (7 b e-a h)\right )}{27 a^{11/3} b^{2/3}}+\frac {(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4}-\frac {\log (x) (3 b c-a f)}{a^4}-\frac {x \left (-3 b x^2 \left (\frac {5 b c}{a}-3 f\right )+2 x (5 b e-2 a h)-5 a g+11 b d\right )}{18 a^3 \left (a+b x^3\right )}-\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (-b x^2 \left (\frac {b c}{a}-f\right )+x (b e-a h)-a g+b d\right )}{6 a^2 \left (a+b x^3\right )^2} \]
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Rule 31
Rule 210
Rule 266
Rule 631
Rule 642
Rule 648
Rule 1843
Rule 1848
Rule 1874
Rule 1885
Rubi steps \begin{align*} \text {integral}& = -\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {\int \frac {-6 b^2 c-6 b^2 d x-6 b^2 e x^2+6 b^2 \left (\frac {b c}{a}-f\right ) x^3+5 b^2 \left (\frac {b d}{a}-g\right ) x^4+4 b^2 \left (\frac {b e}{a}-h\right ) x^5-\frac {3 b^3 (b c-a f) x^6}{a^2}}{x^4 \left (a+b x^3\right )^2} \, dx}{6 a b^2} \\ & = -\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac {5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\int \frac {18 b^4 c+18 b^4 d x+18 b^4 e x^2-18 b^4 \left (\frac {2 b c}{a}-f\right ) x^3-2 b^4 \left (\frac {11 b d}{a}-5 g\right ) x^4-2 b^4 \left (\frac {5 b e}{a}-2 h\right ) x^5}{x^4 \left (a+b x^3\right )} \, dx}{18 a^2 b^4} \\ & = -\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac {5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\int \left (\frac {18 b^4 c}{a x^4}+\frac {18 b^4 d}{a x^3}+\frac {18 b^4 e}{a x^2}+\frac {18 b^4 (-3 b c+a f)}{a^2 x}+\frac {2 b^4 \left (-5 a (4 b d-a g)-2 a (7 b e-a h) x+9 b (3 b c-a f) x^2\right )}{a^2 \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^4} \\ & = -\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac {5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac {(3 b c-a f) \log (x)}{a^4}+\frac {\int \frac {-5 a (4 b d-a g)-2 a (7 b e-a h) x+9 b (3 b c-a f) x^2}{a+b x^3} \, dx}{9 a^4} \\ & = -\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac {5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac {(3 b c-a f) \log (x)}{a^4}+\frac {\int \frac {-5 a (4 b d-a g)-2 a (7 b e-a h) x}{a+b x^3} \, dx}{9 a^4}+\frac {(b (3 b c-a f)) \int \frac {x^2}{a+b x^3} \, dx}{a^4} \\ & = -\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac {5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac {(3 b c-a f) \log (x)}{a^4}+\frac {(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4}+\frac {\int \frac {\sqrt [3]{a} \left (-10 a \sqrt [3]{b} (4 b d-a g)-2 a^{4/3} (7 b e-a h)\right )+\sqrt [3]{b} \left (5 a \sqrt [3]{b} (4 b d-a g)-2 a^{4/3} (7 b e-a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{14/3} \sqrt [3]{b}}-\frac {\left (20 b d-5 a g-\frac {2 \sqrt [3]{a} (7 b e-a h)}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{11/3}} \\ & = -\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac {5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac {(3 b c-a f) \log (x)}{a^4}-\frac {\left (20 b d-5 a g-\frac {2 \sqrt [3]{a} (7 b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac {(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4}-\frac {\left (20 b^{4/3} d+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g-2 a^{4/3} h\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{10/3} \sqrt [3]{b}}+\frac {\left (20 b d-5 a g-\frac {2 \sqrt [3]{a} (7 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}{54 a^{11/3} \sqrt [3]{b}} \\ & = -\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac {5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac {(3 b c-a f) \log (x)}{a^4}-\frac {\left (20 b d-5 a g-\frac {2 \sqrt [3]{a} (7 b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac {\left (20 b d-5 a g-\frac {2 \sqrt [3]{a} (7 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 )}{54 a^{11/3} \sqrt [3]{b}}+\frac {(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4}-\frac {\left (20 b^{4/3} d+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g-2 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 )}{9 a^{11/3} b^{2/3}} \\ & = -\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (b d-a g+(b e-a h) x-b \left (\frac {b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac {5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\left (20 b^{4/3} d+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g-2 a^{4/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} b^{2/3}}-\frac {(3 b c-a f) \log (x)}{a^4}-\frac {\left (20 b d-5 a g-\frac {2 \sqrt [3]{a} (7 b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac {\left (20 b d-5 a g-\frac {2 \sqrt [3]{a} (7 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 )}{54 a^{11/3} \sqrt [3]{b}}+\frac {(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.89 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left (a+b x^3\right )^3} \, dx=\frac {-\frac {18 a c}{x^3}-\frac {27 a d}{x^2}-\frac {54 a e}{x}+\frac {3 a (-12 b c+6 a f-b x (11 d+10 e x)+a x (5 g+4 h x))}{a+b x^3}+\frac {a^2 (-9 b (c+x (d+e x))+9 a (f+x (g+h x)))}{\left (a+b x^3\right )^2}+\frac {2 \sqrt {3} \sqrt [3]{a} \left (20 b^{4/3} d+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g-2 a^{4/3} h\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}+54 (-3 b c+a f) \log (x)-\frac {2 \sqrt [3]{a} \left (20 b^{4/3} d-14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+2 a^{4/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {\sqrt [3]{a} \left (20 b^{4/3} d-14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+2 a^{4/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+18 (3 b c-a f) \log \left (a+b x^3\right )}{54 a^4} \]
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Time = 1.59 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {c}{3 a^{3} x^{3}}-\frac {d}{2 a^{3} x^{2}}-\frac {e}{a^{3} x}+\frac {\left (a f -3 b c \right ) \ln \left (x \right )}{a^{4}}+\frac {\frac {\left (\frac {2}{9} a^{2} b h -\frac {5}{9} a \,b^{2} e \right ) x^{5}+\left (\frac {5}{18} a^{2} b g -\frac {11}{18} a \,b^{2} d \right ) x^{4}+\left (\frac {1}{3} f \,a^{2} b -\frac {2}{3} a \,b^{2} c \right ) x^{3}+\frac {a^{2} \left (7 a h -13 b e \right ) x^{2}}{18}+\frac {a^{2} \left (4 a g -7 b d \right ) x}{9}+\frac {f \,a^{3}}{2}-\frac {5 a^{2} b c}{6}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (5 a^{2} g -20 a b d \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9}+\frac {\left (2 a^{2} h -14 a e b \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9}+\frac {\left (-9 a f b +27 b^{2} c \right ) \ln \left (b \,x^{3}+a \right )}{27 b}}{a^{4}}\) | \(394\) |
risch | \(\frac {\frac {2 b \left (a h -7 b e \right ) x^{8}}{9 a^{3}}+\frac {5 b \left (a g -4 b d \right ) x^{7}}{18 a^{3}}+\frac {b \left (a f -3 b c \right ) x^{6}}{3 a^{3}}+\frac {7 \left (a h -7 b e \right ) x^{5}}{18 a^{2}}+\frac {4 \left (a g -4 b d \right ) x^{4}}{9 a^{2}}+\frac {\left (a f -3 b c \right ) x^{3}}{2 a^{2}}-\frac {e \,x^{2}}{a}-\frac {x d}{2 a}-\frac {c}{3 a}}{x^{3} \left (b \,x^{3}+a \right )^{2}}+\frac {\ln \left (-x \right ) f}{a^{3}}-\frac {3 \ln \left (-x \right ) b c}{a^{4}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{12} b^{2} \textit {\_Z}^{3}+\left (27 a^{9} b^{2} f -81 a^{8} b^{3} c \right ) \textit {\_Z}^{2}+\left (30 a^{7} b g h -120 a^{6} b^{2} d h -210 a^{6} b^{2} e g +243 a^{6} b^{2} f^{2}-1458 a^{5} b^{3} c f +840 a^{5} b^{3} d e +2187 a^{4} b^{4} c^{2}\right ) \textit {\_Z} +8 a^{5} h^{3}-168 a^{4} b e \,h^{2}+270 a^{4} b f g h -125 a^{4} b \,g^{3}-810 a^{3} b^{2} c g h -1080 a^{3} b^{2} d f h +1500 a^{3} b^{2} d \,g^{2}+1176 a^{3} b^{2} e^{2} h -1890 a^{3} b^{2} e f g +729 a^{3} b^{2} f^{3}+3240 a^{2} b^{3} c d h +5670 a^{2} b^{3} c e g -6561 a^{2} b^{3} c \,f^{2}-6000 a^{2} b^{3} d^{2} g +7560 a^{2} b^{3} d e f -2744 a^{2} b^{3} e^{3}+19683 a \,b^{4} c^{2} f -22680 a \,b^{4} c d e +8000 a \,b^{4} d^{3}-19683 b^{5} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{11} b^{2}+\left (-72 a^{8} b^{2} f +216 a^{7} b^{3} c \right ) \textit {\_R}^{2}+\left (-100 a^{6} b g h +400 a^{5} b^{2} d h +700 a^{5} b^{2} e g -324 a^{5} b^{2} f^{2}+1944 a^{4} b^{3} c f -2800 a^{4} b^{3} d e -2916 a^{3} b^{4} c^{2}\right ) \textit {\_R} -24 a^{4} h^{3}+504 a^{3} b e \,h^{2}-540 a^{3} b f g h +375 a^{3} b \,g^{3}+1620 a^{2} b^{2} c g h +2160 a^{2} b^{2} d f h -4500 a^{2} b^{2} d \,g^{2}-3528 a^{2} b^{2} e^{2} h +3780 a^{2} b^{2} e f g -6480 a \,b^{3} c d h -11340 a \,b^{3} c e g +18000 a \,b^{3} d^{2} g -15120 a \,b^{3} d e f +8232 a \,b^{3} e^{3}+45360 b^{4} c d e -24000 b^{4} d^{3}\right ) x +\left (2 a^{9} b h -14 a^{8} b^{2} e \right ) \textit {\_R}^{2}+\left (-36 a^{6} b f h -25 a^{6} b \,g^{2}+108 a^{5} b^{2} c h +200 a^{5} b^{2} d g +252 a^{5} b^{2} e f -756 a^{4} b^{3} c e -400 a^{4} b^{3} d^{2}\right ) \textit {\_R} -486 a^{3} b \,f^{2} h +675 a^{3} b f \,g^{2}+2916 a^{2} b^{2} c f h -2025 a^{2} b^{2} c \,g^{2}-5400 a^{2} b^{2} d f g +3402 a^{2} b^{2} e \,f^{2}-4374 a \,b^{3} c^{2} h +16200 a \,b^{3} c d g -20412 a \,b^{3} c e f +10800 a \,b^{3} d^{2} f +30618 b^{4} c^{2} e -32400 b^{4} c \,d^{2}\right )\right )}{27}\) | \(961\) |
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Result contains complex when optimal does not.
Time = 101.30 (sec) , antiderivative size = 16697, normalized size of antiderivative = 42.27 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.12 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left (a+b x^3\right )^3} \, dx=-\frac {4 \, {\left (7 \, b^{2} e - a b h\right )} x^{8} + 5 \, {\left (4 \, b^{2} d - a b g\right )} x^{7} + 6 \, {\left (3 \, b^{2} c - a b f\right )} x^{6} + 7 \, {\left (7 \, a b e - a^{2} h\right )} x^{5} + 18 \, a^{2} e x^{2} + 8 \, {\left (4 \, a b d - a^{2} g\right )} x^{4} + 9 \, a^{2} d x + 9 \, {\left (3 \, a b c - a^{2} f\right )} x^{3} + 6 \, a^{2} c}{18 \, {\left (a^{3} b^{2} x^{9} + 2 \, a^{4} b x^{6} + a^{5} x^{3}\right )}} - \frac {{\left (3 \, b c - a f\right )} \log \left (x\right )}{a^{4}} - \frac {\sqrt {3} {\left (14 \, a b e \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} + 20 \, a b d \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a^{2} g \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{5}} + \frac {{\left (54 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 18 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} - 14 \, a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} + 20 \, a b d - 5 \, a^{2} g\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (27 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 9 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} + 14 \, a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - 20 \, a b d + 5 \, a^{2} g\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.28 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.08 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left (a+b x^3\right )^3} \, dx=\frac {\sqrt {3} {\left (20 \, b^{2} d - 5 \, a b g - 14 \, \left (-a b^{2}\right )^{\frac {1}{3}} b e + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} a h\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3}} + \frac {{\left (20 \, b^{2} d - 5 \, a b g + 14 \, \left (-a b^{2}\right )^{\frac {1}{3}} b e - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} a h\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3}} + \frac {{\left (3 \, b c - a f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4}} - \frac {{\left (3 \, b c - a f\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {{\left (14 \, a^{5} b^{2} e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{6} b h \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 20 \, a^{5} b^{2} d - 5 \, a^{6} b g\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{9} b} - \frac {4 \, {\left (7 \, a b^{2} e - a^{2} b h\right )} x^{8} + 5 \, {\left (4 \, a b^{2} d - a^{2} b g\right )} x^{7} + 6 \, {\left (3 \, a b^{2} c - a^{2} b f\right )} x^{6} + 18 \, a^{3} e x^{2} + 7 \, {\left (7 \, a^{2} b e - a^{3} h\right )} x^{5} + 9 \, a^{3} d x + 8 \, {\left (4 \, a^{2} b d - a^{3} g\right )} x^{4} + 6 \, a^{3} c + 9 \, {\left (3 \, a^{2} b c - a^{3} f\right )} x^{3}}{18 \, {\left (b x^{3} + a\right )}^{2} a^{4} x^{3}} \]
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Time = 10.35 (sec) , antiderivative size = 1994, normalized size of antiderivative = 5.05 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
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