Integrand size = 23, antiderivative size = 267 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^3} \, dx=-\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\left (14 b^{2/3} c-5 a^{2/3} e\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} \sqrt [3]{b}}+\frac {d \log (x)}{a^3}+\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}-\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^3} \]
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Time = 0.31 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1843, 1848, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^3} \, dx=\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (14 b^{2/3} c-5 a^{2/3} e\right )}{9 \sqrt {3} a^{10/3} \sqrt [3]{b}}-\frac {\left (5 a^{2/3} e+14 b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} \sqrt [3]{b}}+\frac {\left (5 a^{2/3} e+14 b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac {d \log \left (a+b x^3\right )}{3 a^3}-\frac {c}{a^3 x}+\frac {d \log (x)}{a^3}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2} \]
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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 (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {\int \frac {-6 b c-6 b d x-5 b e x^2+\frac {4 b^2 c x^3}{a}+\frac {3 b^2 d x^4}{a}}{x^2 \left (a+b x^3\right )^2} \, dx}{6 a b} \\ & = \frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\int \frac {18 b^3 c+18 b^3 d x+10 b^3 e x^2-\frac {10 b^4 c x^3}{a}}{x^2 \left (a+b x^3\right )} \, dx}{18 a^2 b^3} \\ & = \frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\int \left (\frac {18 b^3 c}{a x^2}+\frac {18 b^3 d}{a x}+\frac {2 b^3 \left (5 a e-14 b c x-9 b d x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^3} \\ & = -\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}+\frac {\int \frac {5 a e-14 b c x-9 b d x^2}{a+b x^3} \, dx}{9 a^3} \\ & = -\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}+\frac {\int \frac {5 a e-14 b c x}{a+b x^3} \, dx}{9 a^3}-\frac {(b d) \int \frac {x^2}{a+b x^3} \, dx}{a^3} \\ & = -\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}-\frac {d \log \left (a+b x^3\right )}{3 a^3}+\frac {\int \frac {\sqrt [3]{a} \left (-14 \sqrt [3]{a} b c+10 a \sqrt [3]{b} e\right )+\sqrt [3]{b} \left (-14 \sqrt [3]{a} b c-5 a \sqrt [3]{b} e\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{11/3} \sqrt [3]{b}}+\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{10/3}} \\ & = -\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}+\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^3}-\frac {\left (14 b^{2/3} c-5 a^{2/3} e\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^3}-\frac {\left (14 b^{2/3} c+5 a^{2/3} e\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^{10/3} \sqrt [3]{b}} \\ & = -\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}+\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}-\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^3}-\frac {\left (14 b^{2/3} c-5 a^{2/3} e\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^{10/3} \sqrt [3]{b}} \\ & = -\frac {c}{a^3 x}+\frac {x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\left (14 b^{2/3} c-5 a^{2/3} e\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} \sqrt [3]{b}}+\frac {d \log (x)}{a^3}+\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}-\frac {\left (14 b^{2/3} c+5 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^3} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^3} \, dx=\frac {-\frac {54 a c}{x}+\frac {3 a \left (6 a d+5 a e x-10 b c x^2\right )}{a+b x^3}+\frac {9 a^2 \left (-b c x^2+a (d+e x)\right )}{\left (a+b x^3\right )^2}-\frac {2 \sqrt {3} a^{2/3} \left (-14 b^{2/3} c+5 a^{2/3} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+54 a d \log (x)+\frac {2 \left (14 a^{2/3} b^{2/3} c+5 a^{4/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {\left (14 a^{2/3} b^{2/3} c+5 a^{4/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}-18 a d \log \left (a+b x^3\right )}{54 a^4} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.55 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {-\frac {14 c \,b^{2} x^{6}}{9 a^{3}}+\frac {5 b e \,x^{5}}{18 a^{2}}+\frac {b d \,x^{4}}{3 a^{2}}-\frac {49 b c \,x^{3}}{18 a^{2}}+\frac {4 e \,x^{2}}{9 a}+\frac {x d}{2 a}-\frac {c}{a}}{x \left (b \,x^{3}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{10} b \,\textit {\_Z}^{3}+27 a^{7} b d \,\textit {\_Z}^{2}+\left (-210 a^{4} b c e +243 a^{4} b \,d^{2}\right ) \textit {\_Z} -125 a^{2} e^{3}-1890 a b c d e +729 a b \,d^{3}-2744 b^{2} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{10} b -72 \textit {\_R}^{2} a^{7} b d +\left (700 a^{4} b c e -324 a^{4} b \,d^{2}\right ) \textit {\_R} +375 a^{2} e^{3}+3780 a b c d e +8232 b^{2} c^{3}\right ) x -14 a^{7} b c \,\textit {\_R}^{2}+\left (-25 a^{5} e^{2}+252 a^{4} b c d \right ) \textit {\_R} +675 a^{2} d \,e^{2}+3402 a b c \,d^{2}\right )\right )}{27}+\frac {d \ln \left (x \right )}{a^{3}}\) | \(277\) |
default | \(-\frac {c}{a^{3} x}+\frac {d \ln \left (x \right )}{a^{3}}+\frac {\frac {-\frac {5}{9} b^{2} c \,x^{5}+\frac {5}{18} a b e \,x^{4}+\frac {1}{3} x^{3} a b d -\frac {13}{18} a b c \,x^{2}+\frac {4}{9} a^{2} e x +\frac {1}{2} a^{2} d}{\left (b \,x^{3}+a \right )^{2}}+\frac {5 a e \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9}-\frac {14 b c \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9}-\frac {d \ln \left (b \,x^{3}+a \right )}{3}}{a^{3}}\) | \(279\) |
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Result contains complex when optimal does not.
Time = 1.14 (sec) , antiderivative size = 5112, normalized size of antiderivative = 19.15 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^3} \, dx=-\frac {28 \, b^{2} c x^{6} - 5 \, a b e x^{5} - 6 \, a b d x^{4} + 49 \, a b c x^{3} - 8 \, a^{2} e x^{2} - 9 \, a^{2} d x + 18 \, a^{2} c}{18 \, {\left (a^{3} b^{2} x^{7} + 2 \, a^{4} b x^{4} + a^{5} x\right )}} + \frac {d \log \left (x\right )}{a^{3}} - \frac {\sqrt {3} {\left (14 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 5 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4}} - \frac {{\left (18 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} + 14 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a e\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (9 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 14 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a e\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.29 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^3} \, dx=-\frac {d \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac {d \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {\sqrt {3} {\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a e + 14 \, \left (-a b^{2}\right )^{\frac {2}{3}} c\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} b} + \frac {{\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a e - 14 \, \left (-a b^{2}\right )^{\frac {2}{3}} c\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{4} b} - \frac {28 \, b^{2} c x^{6} - 5 \, a b e x^{5} - 6 \, a b d x^{4} + 49 \, a b c x^{3} - 8 \, a^{2} e x^{2} - 9 \, a^{2} d x + 18 \, a^{2} c}{18 \, {\left (b x^{3} + a\right )}^{2} a^{3} x} + \frac {{\left (14 \, a^{3} b^{2} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a^{4} b e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{7} b} \]
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Time = 9.35 (sec) , antiderivative size = 793, normalized size of antiderivative = 2.97 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^3} \, dx=\frac {\frac {4\,e\,x^2}{9\,a}-\frac {c}{a}+\frac {d\,x}{2\,a}-\frac {14\,b^2\,c\,x^6}{9\,a^3}-\frac {49\,b\,c\,x^3}{18\,a^2}+\frac {b\,d\,x^4}{3\,a^2}+\frac {5\,b\,e\,x^5}{18\,a^2}}{a^2\,x+2\,a\,b\,x^4+b^2\,x^7}+\left (\sum _{k=1}^3\ln \left (\frac {b^2\,\left (-\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )\,a^5\,e^2\,225+225\,a^2\,d\,e^2+2744\,b^2\,c^3\,x+125\,a^2\,e^3\,x+1134\,a\,b\,c\,d^2-{\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )}^2\,a^7\,b\,c\,3402-{\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )}^3\,a^{10}\,b\,x\,26244-\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )\,a^4\,b\,d^2\,x\,2916-{\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )}^2\,a^7\,b\,d\,x\,17496+\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )\,a^4\,b\,c\,d\,2268+\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )\,a^4\,b\,c\,e\,x\,6300+1260\,a\,b\,c\,d\,e\,x\right )}{a^8\,729}\right )\,\mathrm {root}\left (19683\,a^{10}\,b\,z^3+19683\,a^7\,b\,d\,z^2-5670\,a^4\,b\,c\,e\,z+6561\,a^4\,b\,d^2\,z-1890\,a\,b\,c\,d\,e+729\,a\,b\,d^3-125\,a^2\,e^3-2744\,b^2\,c^3,z,k\right )\right )+\frac {d\,\ln \left (x\right )}{a^3} \]
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