Integrand size = 38, antiderivative size = 347 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx=\frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}-\frac {\left (5 b^{4/3} d+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+a^{4/3} h\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} b^{5/3}}+\frac {c \log (x)}{a^3}+\frac {\left (\sqrt [3]{b} (5 b d+a g)-\sqrt [3]{a} (2 b e+a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{5/3}}-\frac {\left (\sqrt [3]{b} (5 b d+a g)-\sqrt [3]{a} (2 b e+a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{5/3}}-\frac {c \log \left (a+b x^3\right )}{3 a^3} \]
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Time = 0.49 (sec) , antiderivative size = 345, 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} \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{9 \sqrt {3} a^{8/3} b^{5/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac {\sqrt [3]{a} (a h+2 b e)}{\sqrt [3]{b}}+a g+5 b d\right )}{54 a^{8/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a g+5 b d)-\sqrt [3]{a} (a h+2 b e)\right )}{27 a^{8/3} b^{5/3}}+\frac {x \left (-3 b x^2 (3 b c-a f)+a (a g+5 b d)+2 a x (a h+2 b e)\right )}{18 a^3 b \left (a+b x^3\right )}-\frac {c \log \left (a+b x^3\right )}{3 a^3}+\frac {c \log (x)}{a^3}+\frac {x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2} \]
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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 (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}-\frac {\int \frac {-6 b^2 c-b (5 b d+a g) x-2 b (2 b e+a h) x^2+3 b^2 \left (\frac {b c}{a}-f\right ) x^3}{x \left (a+b x^3\right )^2} \, dx}{6 a b^2} \\ & = \frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {\int \frac {18 b^3 c+2 b^2 (5 b d+a g) x+2 b^2 (2 b e+a h) x^2}{x \left (a+b x^3\right )} \, dx}{18 a^2 b^3} \\ & = \frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {\int \left (\frac {18 b^3 c}{a x}+\frac {2 b^2 \left (a (5 b d+a g)+a (2 b e+a h) x-9 b^2 c x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^3} \\ & = \frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {c \log (x)}{a^3}+\frac {\int \frac {a (5 b d+a g)+a (2 b e+a h) x-9 b^2 c x^2}{a+b x^3} \, dx}{9 a^3 b} \\ & = \frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {c \log (x)}{a^3}+\frac {\int \frac {a (5 b d+a g)+a (2 b e+a h) x}{a+b x^3} \, dx}{9 a^3 b}-\frac {(b c) \int \frac {x^2}{a+b x^3} \, dx}{a^3} \\ & = \frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {c \log (x)}{a^3}-\frac {c \log \left (a+b x^3\right )}{3 a^3}+\frac {\int \frac {\sqrt [3]{a} \left (2 a \sqrt [3]{b} (5 b d+a g)+a^{4/3} (2 b e+a h)\right )+\sqrt [3]{b} \left (-a \sqrt [3]{b} (5 b d+a g)+a^{4/3} (2 b e+a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{11/3} b^{4/3}}+\frac {\left (5 b d+a g-\frac {\sqrt [3]{a} (2 b e+a h)}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{8/3} b} \\ & = \frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {c \log (x)}{a^3}+\frac {\left (5 b d+a g-\frac {\sqrt [3]{a} (2 b e+a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{4/3}}-\frac {c \log \left (a+b x^3\right )}{3 a^3}+\frac {\left (5 b^{4/3} d+2 \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}{18 a^{7/3} b^{4/3}}-\frac {\left (5 b d+a g-\frac {\sqrt [3]{a} (2 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^{8/3} b^{4/3}} \\ & = \frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {c \log (x)}{a^3}+\frac {\left (5 b d+a g-\frac {\sqrt [3]{a} (2 b e+a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{4/3}}-\frac {\left (5 b d+a g-\frac {\sqrt [3]{a} (2 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^{8/3} b^{4/3}}-\frac {c \log \left (a+b x^3\right )}{3 a^3}+\frac {\left (5 b^{4/3} d+2 \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 )}{9 a^{8/3} b^{5/3}} \\ & = \frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}-\frac {\left (5 b^{4/3} d+2 \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 )}{9 \sqrt {3} a^{8/3} b^{5/3}}+\frac {c \log (x)}{a^3}+\frac {\left (5 b d+a g-\frac {\sqrt [3]{a} (2 b e+a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{4/3}}-\frac {\left (5 b d+a g-\frac {\sqrt [3]{a} (2 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^{8/3} b^{4/3}}-\frac {c \log \left (a+b x^3\right )}{3 a^3} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx=\frac {\frac {3 a (6 b c+b x (5 d+4 e x)+a x (g+2 h x))}{b \left (a+b x^3\right )}-\frac {9 a^2 (-b (c+x (d+e x))+a (f+x (g+h x)))}{b \left (a+b x^3\right )^2}-\frac {2 \sqrt {3} \sqrt [3]{a} \left (5 b^{4/3} d+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+a^{4/3} h\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{5/3}}+54 c \log (x)+\frac {2 \sqrt [3]{a} \left (5 b^{4/3} d-2 \sqrt [3]{a} b e+a \sqrt [3]{b} g-a^{4/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{5/3}}+\frac {\sqrt [3]{a} \left (-5 b^{4/3} d+2 \sqrt [3]{a} b e-a \sqrt [3]{b} g+a^{4/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{5/3}}-18 c \log \left (a+b x^3\right )}{54 a^3} \]
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Time = 1.58 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.98
method | result | size |
default | \(\frac {c \ln \left (x \right )}{a^{3}}+\frac {\frac {\left (\frac {1}{9} a^{2} h +\frac {2}{9} a e b \right ) x^{5}+\left (\frac {1}{18} a^{2} g +\frac {5}{18} a b d \right ) x^{4}+\frac {a b c \,x^{3}}{3}-\frac {a^{2} \left (a h -7 b e \right ) x^{2}}{18 b}-\frac {a^{2} \left (a g -4 b d \right ) x}{9 b}-\frac {a^{2} \left (a f -3 b c \right )}{6 b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (a^{2} g +5 a b d \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (a^{2} h +2 a e b \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-3 b c \ln \left (b \,x^{3}+a \right )}{9 b}}{a^{3}}\) | \(339\) |
risch | \(\frac {\frac {\left (a h +2 b e \right ) x^{5}}{9 a^{2}}+\frac {\left (a g +5 b d \right ) x^{4}}{18 a^{2}}+\frac {b c \,x^{3}}{3 a^{2}}-\frac {\left (a h -7 b e \right ) x^{2}}{18 a b}-\frac {\left (a g -4 b d \right ) x}{9 a b}-\frac {a f -3 b c}{6 a b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {c \ln \left (-x \right )}{a^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{9} b^{5} \textit {\_Z}^{3}+27 a^{6} b^{5} c \,\textit {\_Z}^{2}+\left (3 a^{6} b^{2} g h +15 a^{5} b^{3} d h +6 a^{5} b^{3} e g +30 a^{4} b^{4} d e +243 a^{3} b^{5} c^{2}\right ) \textit {\_Z} +a^{5} h^{3}+6 a^{4} b e \,h^{2}-a^{4} b \,g^{3}+27 a^{3} b^{2} c g h -15 a^{3} b^{2} d \,g^{2}+12 a^{3} b^{2} e^{2} h +135 a^{2} b^{3} c d h +54 a^{2} b^{3} c e g -75 a^{2} b^{3} d^{2} g +8 a^{2} b^{3} e^{3}+270 a \,b^{4} c d e -125 a \,b^{4} d^{3}+729 b^{5} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{8} b^{5}-72 \textit {\_R}^{2} a^{5} b^{5} c +\left (-10 a^{5} b^{2} g h -50 a^{4} b^{3} d h -20 a^{4} b^{3} e g -100 a^{3} b^{4} d e -324 a^{2} b^{5} c^{2}\right ) \textit {\_R} -3 a^{4} h^{3}-18 a^{3} b e \,h^{2}+3 a^{3} b \,g^{3}-54 a^{2} b^{2} c g h +45 a^{2} b^{2} d \,g^{2}-36 a^{2} b^{2} e^{2} h -270 a \,b^{3} c d h -108 a \,b^{3} c e g +225 a \,b^{3} d^{2} g -24 a \,b^{3} e^{3}-540 b^{4} c d e +375 b^{4} d^{3}\right ) x +\left (a^{7} b^{3} h +2 b^{4} e \,a^{6}\right ) \textit {\_R}^{2}+\left (-a^{5} b^{2} g^{2}-18 a^{4} b^{3} c h -10 a^{4} b^{3} d g -36 a^{3} b^{4} c e -25 a^{3} b^{4} d^{2}\right ) \textit {\_R} +27 a^{2} b^{2} c \,g^{2}-243 a \,b^{3} c^{2} h +270 a \,b^{3} c d g -486 b^{4} c^{2} e +675 b^{4} c \,d^{2}\right )\right )}{27}\) | \(656\) |
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Result contains complex when optimal does not.
Time = 21.99 (sec) , antiderivative size = 12815, normalized size of antiderivative = 36.93 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.06 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx=\frac {6 \, b^{2} c x^{3} + 2 \, {\left (2 \, b^{2} e + a b h\right )} x^{5} + {\left (5 \, b^{2} d + a b g\right )} x^{4} + 9 \, a b c - 3 \, a^{2} f + {\left (7 \, a b e - a^{2} h\right )} x^{2} + 2 \, {\left (4 \, a b d - a^{2} g\right )} x}{18 \, {\left (a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{3} + a^{4} b\right )}} + \frac {c \log \left (x\right )}{a^{3}} + \frac {\sqrt {3} {\left (2 \, a b e \left (\frac {a}{b}\right )^{\frac {2}{3}} + a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} + 5 \, a b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} g \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} b} - \frac {{\left (18 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a b d + a^{2} g\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (9 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a b d - a^{2} g\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.28 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.07 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx=-\frac {c \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac {c \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {\sqrt {3} {\left (5 \, b^{2} d + a b g - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b e - \left (-a b^{2}\right )^{\frac {1}{3}} a h\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b} - \frac {{\left (5 \, b^{2} d + a b g + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b e + \left (-a b^{2}\right )^{\frac {1}{3}} a h\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b} + \frac {6 \, a b^{2} c x^{3} + 2 \, {\left (2 \, a b^{2} e + a^{2} b h\right )} x^{5} + {\left (5 \, a b^{2} d + a^{2} b g\right )} x^{4} + 9 \, a^{2} b c - 3 \, a^{3} f + {\left (7 \, a^{2} b e - a^{3} h\right )} x^{2} + 2 \, {\left (4 \, a^{2} b d - a^{3} g\right )} x}{18 \, {\left (b x^{3} + a\right )}^{2} a^{3} b} - \frac {{\left (2 \, a^{4} b^{3} e \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{5} b^{2} h \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a^{4} b^{3} d + a^{5} b^{2} g\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{7} b^{3}} \]
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Time = 9.75 (sec) , antiderivative size = 1716, normalized size of antiderivative = 4.95 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
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