Integrand size = 27, antiderivative size = 264 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^2} \, dx=\frac {(b e-2 a f) x}{b^3}+\frac {f x^4}{4 b^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a b^3 \left (a+b x^3\right )}-\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{10/3}}+\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{10/3}} \]
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Time = 0.18 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1872, 1425, 396, 206, 31, 648, 631, 210, 642} \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^2} \, dx=\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{3 \sqrt {3} a^{5/3} b^{10/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{18 a^{5/3} b^{10/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{9 a^{5/3} b^{10/3}}+\frac {x (b e-2 a f)}{b^3}+\frac {f x^4}{4 b^2} \]
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Rule 31
Rule 206
Rule 210
Rule 396
Rule 631
Rule 642
Rule 648
Rule 1425
Rule 1872
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a b^3 \left (a+b x^3\right )}-\frac {\int \frac {-2 b^3 c-a b^2 d+a^2 b e-a^3 f-3 a b (b e-a f) x^3-3 a b^2 f x^6}{a+b x^3} \, dx}{3 a b^3} \\ & = \frac {f x^4}{4 b^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a b^3 \left (a+b x^3\right )}-\frac {\int \frac {4 b \left (-2 b^3 c-a b^2 d+a^2 b e-a^3 f\right )-\left (-12 a^2 b^2 f+12 a b^2 (b e-a f)\right ) x^3}{a+b x^3} \, dx}{12 a b^4} \\ & = \frac {(b e-2 a f) x}{b^3}+\frac {f x^4}{4 b^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a b^3 \left (a+b x^3\right )}+\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \int \frac {1}{a+b x^3} \, dx}{3 a b^3} \\ & = \frac {(b e-2 a f) x}{b^3}+\frac {f x^4}{4 b^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a b^3 \left (a+b x^3\right )}+\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{5/3} b^3}+\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{5/3} b^3} \\ & = \frac {(b e-2 a f) x}{b^3}+\frac {f x^4}{4 b^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a b^3 \left (a+b x^3\right )}+\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\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}{18 a^{5/3} b^{10/3}}+\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{4/3} b^3} \\ & = \frac {(b e-2 a f) x}{b^3}+\frac {f x^4}{4 b^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a b^3 \left (a+b x^3\right )}+\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{10/3}}+\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{5/3} b^{10/3}} \\ & = \frac {(b e-2 a f) x}{b^3}+\frac {f x^4}{4 b^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a b^3 \left (a+b x^3\right )}-\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{10/3}}+\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac {\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{10/3}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^2} \, dx=\frac {36 \sqrt [3]{b} (b e-2 a f) x+9 b^{4/3} f x^4+\frac {12 \sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{a \left (a+b x^3\right )}-\frac {4 \sqrt {3} \left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3}}+\frac {4 \left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}-\frac {2 \left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}}{36 b^{10/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.53 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.47
method | result | size |
risch | \(\frac {f \,x^{4}}{4 b^{2}}-\frac {2 x a f}{b^{3}}+\frac {e x}{b^{2}}-\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) x}{3 a \,b^{3} \left (b \,x^{3}+a \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (7 f \,a^{3}-4 a^{2} b e +a \,b^{2} d +2 b^{3} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{9 b^{4} a}\) | \(123\) |
default | \(-\frac {-\frac {1}{4} b f \,x^{4}+2 a f x -b e x}{b^{3}}+\frac {-\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) x}{3 a \left (b \,x^{3}+a \right )}+\frac {\left (7 f \,a^{3}-4 a^{2} b e +a \,b^{2} d +2 b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3 a}}{b^{3}}\) | \(191\) |
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Time = 0.32 (sec) , antiderivative size = 861, normalized size of antiderivative = 3.26 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^2} \, dx=\left [\frac {9 \, a^{3} b^{3} f x^{7} + 9 \, {\left (4 \, a^{3} b^{3} e - 7 \, a^{4} b^{2} f\right )} x^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (2 \, a^{2} b^{4} c + a^{3} b^{3} d - 4 \, a^{4} b^{2} e + 7 \, a^{5} b f + {\left (2 \, a b^{5} c + a^{2} b^{4} d - 4 \, a^{3} b^{3} e + 7 \, a^{4} b^{2} f\right )} x^{3}\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) - 2 \, {\left (2 \, a b^{3} c + a^{2} b^{2} d - 4 \, a^{3} b e + 7 \, a^{4} f + {\left (2 \, b^{4} c + a b^{3} d - 4 \, a^{2} b^{2} e + 7 \, a^{3} b f\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 4 \, {\left (2 \, a b^{3} c + a^{2} b^{2} d - 4 \, a^{3} b e + 7 \, a^{4} f + {\left (2 \, b^{4} c + a b^{3} d - 4 \, a^{2} b^{2} e + 7 \, a^{3} b f\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 12 \, {\left (a^{2} b^{4} c - a^{3} b^{3} d + 4 \, a^{4} b^{2} e - 7 \, a^{5} b f\right )} x}{36 \, {\left (a^{3} b^{5} x^{3} + a^{4} b^{4}\right )}}, \frac {9 \, a^{3} b^{3} f x^{7} + 9 \, {\left (4 \, a^{3} b^{3} e - 7 \, a^{4} b^{2} f\right )} x^{4} + 12 \, \sqrt {\frac {1}{3}} {\left (2 \, a^{2} b^{4} c + a^{3} b^{3} d - 4 \, a^{4} b^{2} e + 7 \, a^{5} b f + {\left (2 \, a b^{5} c + a^{2} b^{4} d - 4 \, a^{3} b^{3} e + 7 \, a^{4} b^{2} f\right )} x^{3}\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - 2 \, {\left (2 \, a b^{3} c + a^{2} b^{2} d - 4 \, a^{3} b e + 7 \, a^{4} f + {\left (2 \, b^{4} c + a b^{3} d - 4 \, a^{2} b^{2} e + 7 \, a^{3} b f\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 4 \, {\left (2 \, a b^{3} c + a^{2} b^{2} d - 4 \, a^{3} b e + 7 \, a^{4} f + {\left (2 \, b^{4} c + a b^{3} d - 4 \, a^{2} b^{2} e + 7 \, a^{3} b f\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 12 \, {\left (a^{2} b^{4} c - a^{3} b^{3} d + 4 \, a^{4} b^{2} e - 7 \, a^{5} b f\right )} x}{36 \, {\left (a^{3} b^{5} x^{3} + a^{4} b^{4}\right )}}\right ] \]
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Time = 3.10 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.43 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^2} \, dx=x \left (- \frac {2 a f}{b^{3}} + \frac {e}{b^{2}}\right ) + \frac {x \left (- a^{3} f + a^{2} b e - a b^{2} d + b^{3} c\right )}{3 a^{2} b^{3} + 3 a b^{4} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} a^{5} b^{10} - 343 a^{9} f^{3} + 588 a^{8} b e f^{2} - 147 a^{7} b^{2} d f^{2} - 336 a^{7} b^{2} e^{2} f - 294 a^{6} b^{3} c f^{2} + 168 a^{6} b^{3} d e f + 64 a^{6} b^{3} e^{3} + 336 a^{5} b^{4} c e f - 21 a^{5} b^{4} d^{2} f - 48 a^{5} b^{4} d e^{2} - 84 a^{4} b^{5} c d f - 96 a^{4} b^{5} c e^{2} + 12 a^{4} b^{5} d^{2} e - 84 a^{3} b^{6} c^{2} f + 48 a^{3} b^{6} c d e - a^{3} b^{6} d^{3} + 48 a^{2} b^{7} c^{2} e - 6 a^{2} b^{7} c d^{2} - 12 a b^{8} c^{2} d - 8 b^{9} c^{3}, \left ( t \mapsto t \log {\left (\frac {9 t a^{2} b^{3}}{7 a^{3} f - 4 a^{2} b e + a b^{2} d + 2 b^{3} c} + x \right )} \right )\right )} + \frac {f x^{4}}{4 b^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^2} \, dx=\frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x}{3 \, {\left (a b^{4} x^{3} + a^{2} b^{3}\right )}} + \frac {b f x^{4} + 4 \, {\left (b e - 2 \, a f\right )} x}{4 \, b^{3}} + \frac {\sqrt {3} {\left (2 \, b^{3} c + a b^{2} d - 4 \, a^{2} b e + 7 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, b^{3} c + a b^{2} d - 4 \, a^{2} b e + 7 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, b^{3} c + a b^{2} d - 4 \, a^{2} b e + 7 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.27 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.02 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^2} \, dx=-\frac {\sqrt {3} {\left (2 \, b^{3} c + a b^{2} d - 4 \, a^{2} b e + 7 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2}} - \frac {{\left (2 \, b^{3} c + a b^{2} d - 4 \, a^{2} b e + 7 \, a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2}} - \frac {{\left (2 \, b^{3} c + a b^{2} d - 4 \, a^{2} b e + 7 \, a^{3} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2} b^{3}} + \frac {b^{3} c x - a b^{2} d x + a^{2} b e x - a^{3} f x}{3 \, {\left (b x^{3} + a\right )} a b^{3}} + \frac {b^{6} f x^{4} + 4 \, b^{6} e x - 8 \, a b^{5} f x}{4 \, b^{8}} \]
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Time = 9.77 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.91 \[ \int \frac {c+d x^3+e x^6+f x^9}{\left (a+b x^3\right )^2} \, dx=x\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )+\frac {f\,x^4}{4\,b^2}+\frac {x\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a\,\left (b^4\,x^3+a\,b^3\right )}+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (7\,f\,a^3-4\,e\,a^2\,b+d\,a\,b^2+2\,c\,b^3\right )}{9\,a^{5/3}\,b^{10/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (7\,f\,a^3-4\,e\,a^2\,b+d\,a\,b^2+2\,c\,b^3\right )}{9\,a^{5/3}\,b^{10/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (7\,f\,a^3-4\,e\,a^2\,b+d\,a\,b^2+2\,c\,b^3\right )}{9\,a^{5/3}\,b^{10/3}} \]
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