Integrand size = 19, antiderivative size = 173 \[ \int \frac {\left (c+d x^3\right )^2}{a+b x^3} \, dx=\frac {d (2 b c-a d) x}{b^2}+\frac {d^2 x^4}{4 b}-\frac {(b c-a d)^2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{7/3}}+\frac {(b c-a d)^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{7/3}}-\frac {(b c-a d)^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{7/3}} \]
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Time = 0.08 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {398, 206, 31, 648, 631, 210, 642} \[ \int \frac {\left (c+d x^3\right )^2}{a+b x^3} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) (b c-a d)^2}{\sqrt {3} a^{2/3} b^{7/3}}-\frac {(b c-a d)^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{7/3}}+\frac {(b c-a d)^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{7/3}}+\frac {d x (2 b c-a d)}{b^2}+\frac {d^2 x^4}{4 b} \]
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Rule 31
Rule 206
Rule 210
Rule 398
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d (2 b c-a d)}{b^2}+\frac {d^2 x^3}{b}+\frac {b^2 c^2-2 a b c d+a^2 d^2}{b^2 \left (a+b x^3\right )}\right ) \, dx \\ & = \frac {d (2 b c-a d) x}{b^2}+\frac {d^2 x^4}{4 b}+\frac {(b c-a d)^2 \int \frac {1}{a+b x^3} \, dx}{b^2} \\ & = \frac {d (2 b c-a d) x}{b^2}+\frac {d^2 x^4}{4 b}+\frac {(b c-a d)^2 \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{2/3} b^2}+\frac {(b c-a d)^2 \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}{3 a^{2/3} b^2} \\ & = \frac {d (2 b c-a d) x}{b^2}+\frac {d^2 x^4}{4 b}+\frac {(b c-a d)^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{7/3}}-\frac {(b c-a d)^2 \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^{2/3} b^{7/3}}+\frac {(b c-a d)^2 \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 \sqrt [3]{a} b^2} \\ & = \frac {d (2 b c-a d) x}{b^2}+\frac {d^2 x^4}{4 b}+\frac {(b c-a d)^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{7/3}}-\frac {(b c-a d)^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{7/3}}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{2/3} b^{7/3}} \\ & = \frac {d (2 b c-a d) x}{b^2}+\frac {d^2 x^4}{4 b}-\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{7/3}}+\frac {(b c-a d)^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{7/3}}-\frac {(b c-a d)^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{7/3}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.97 \[ \int \frac {\left (c+d x^3\right )^2}{a+b x^3} \, dx=\frac {-12 a^{2/3} \sqrt [3]{b} d (-2 b c+a d) x+3 a^{2/3} b^{4/3} d^2 x^4+4 \sqrt {3} (b c-a d)^2 \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )+4 (b c-a d)^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 (b c-a d)^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{12 a^{2/3} b^{7/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.91 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.45
method | result | size |
risch | \(\frac {d^{2} x^{4}}{4 b}-\frac {d^{2} a x}{b^{2}}+\frac {2 d c x}{b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b^{3}}\) | \(78\) |
default | \(-\frac {d \left (-\frac {1}{4} b d \,x^{4}+a d x -2 b c x \right )}{b^{2}}+\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{2}}\) | \(140\) |
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Time = 0.32 (sec) , antiderivative size = 507, normalized size of antiderivative = 2.93 \[ \int \frac {\left (c+d x^3\right )^2}{a+b x^3} \, dx=\left [\frac {3 \, a^{2} b^{2} d^{2} x^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) - 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 12 \, {\left (2 \, a^{2} b^{2} c d - a^{3} b d^{2}\right )} x}{12 \, a^{2} b^{3}}, \frac {3 \, a^{2} b^{2} d^{2} x^{4} + 12 \, \sqrt {\frac {1}{3}} {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 12 \, {\left (2 \, a^{2} b^{2} c d - a^{3} b d^{2}\right )} x}{12 \, a^{2} b^{3}}\right ] \]
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Time = 0.34 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.90 \[ \int \frac {\left (c+d x^3\right )^2}{a+b x^3} \, dx=x \left (- \frac {a d^{2}}{b^{2}} + \frac {2 c d}{b}\right ) + \operatorname {RootSum} {\left (27 t^{3} a^{2} b^{7} - a^{6} d^{6} + 6 a^{5} b c d^{5} - 15 a^{4} b^{2} c^{2} d^{4} + 20 a^{3} b^{3} c^{3} d^{3} - 15 a^{2} b^{4} c^{4} d^{2} + 6 a b^{5} c^{5} d - b^{6} c^{6}, \left ( t \mapsto t \log {\left (\frac {3 t a b^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )} \right )\right )} + \frac {d^{2} x^{4}}{4 b} \]
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Time = 0.27 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.10 \[ \int \frac {\left (c+d x^3\right )^2}{a+b x^3} \, dx=\frac {b d^{2} x^{4} + 4 \, {\left (2 \, b c d - a d^{2}\right )} x}{4 \, b^{2}} + \frac {\sqrt {3} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.31 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.22 \[ \int \frac {\left (c+d x^3\right )^2}{a+b x^3} \, dx=-\frac {\sqrt {3} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}} b} - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}} b} - \frac {{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{4}} + \frac {b^{3} d^{2} x^{4} + 8 \, b^{3} c d x - 4 \, a b^{2} d^{2} x}{4 \, b^{4}} \]
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Time = 0.25 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c+d x^3\right )^2}{a+b x^3} \, dx=\frac {d^2\,x^4}{4\,b}-x\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,{\left (a\,d-b\,c\right )}^2}{3\,a^{2/3}\,b^{7/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,{\left (a\,d-b\,c\right )}^2}{a^{2/3}\,b^{7/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^2}{3\,a^{2/3}\,b^{7/3}} \]
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