Integrand size = 21, antiderivative size = 331 \[ \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx=-\frac {b (6 b c-11 a d) x \left (a+b x^3\right )^{2/3}}{18 d^2}+\frac {b x \left (a+b x^3\right )^{5/3}}{6 d}+\frac {b^{2/3} \left (9 b^2 c^2-24 a b c d+20 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} d^3}-\frac {(b c-a d)^{8/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} d^3}-\frac {(b c-a d)^{8/3} \log \left (c+d x^3\right )}{6 c^{2/3} d^3}+\frac {(b c-a d)^{8/3} \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} d^3}-\frac {b^{2/3} \left (9 b^2 c^2-24 a b c d+20 a^2 d^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 d^3} \]
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Time = 0.31 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {427, 542, 544, 245, 384} \[ \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx=\frac {b^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right ) \left (20 a^2 d^2-24 a b c d+9 b^2 c^2\right )}{9 \sqrt {3} d^3}-\frac {b^{2/3} \left (20 a^2 d^2-24 a b c d+9 b^2 c^2\right ) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 d^3}-\frac {(b c-a d)^{8/3} \arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} d^3}-\frac {(b c-a d)^{8/3} \log \left (c+d x^3\right )}{6 c^{2/3} d^3}+\frac {(b c-a d)^{8/3} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} d^3}-\frac {b x \left (a+b x^3\right )^{2/3} (6 b c-11 a d)}{18 d^2}+\frac {b x \left (a+b x^3\right )^{5/3}}{6 d} \]
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Rule 245
Rule 384
Rule 427
Rule 542
Rule 544
Rubi steps \begin{align*} \text {integral}& = \frac {b x \left (a+b x^3\right )^{5/3}}{6 d}+\frac {\int \frac {\left (a+b x^3\right )^{2/3} \left (-a (b c-6 a d)-b (6 b c-11 a d) x^3\right )}{c+d x^3} \, dx}{6 d} \\ & = -\frac {b (6 b c-11 a d) x \left (a+b x^3\right )^{2/3}}{18 d^2}+\frac {b x \left (a+b x^3\right )^{5/3}}{6 d}+\frac {\int \frac {2 a \left (3 b^2 c^2-7 a b c d+9 a^2 d^2\right )+2 b \left (9 b^2 c^2-24 a b c d+20 a^2 d^2\right ) x^3}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{18 d^2} \\ & = -\frac {b (6 b c-11 a d) x \left (a+b x^3\right )^{2/3}}{18 d^2}+\frac {b x \left (a+b x^3\right )^{5/3}}{6 d}-\frac {(b c-a d)^3 \int \frac {1}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{d^3}+\frac {\left (b \left (9 b^2 c^2-24 a b c d+20 a^2 d^2\right )\right ) \int \frac {1}{\sqrt [3]{a+b x^3}} \, dx}{9 d^3} \\ & = -\frac {b (6 b c-11 a d) x \left (a+b x^3\right )^{2/3}}{18 d^2}+\frac {b x \left (a+b x^3\right )^{5/3}}{6 d}+\frac {b^{2/3} \left (9 b^2 c^2-24 a b c d+20 a^2 d^2\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} d^3}-\frac {(b c-a d)^{8/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} d^3}-\frac {(b c-a d)^{8/3} \log \left (c+d x^3\right )}{6 c^{2/3} d^3}+\frac {(b c-a d)^{8/3} \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} d^3}-\frac {b^{2/3} \left (9 b^2 c^2-24 a b c d+20 a^2 d^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 d^3} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.83 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.98 \[ \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx=\frac {3 b \sqrt [3]{b c-a d} \left (9 b^2 c^2-24 a b c d+20 a^2 d^2\right ) x^4 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 \sqrt [3]{c} \left (-18 a b^2 c^{5/3} \sqrt [3]{b c-a d} x+42 a^2 b c^{2/3} d \sqrt [3]{b c-a d} x-18 b^3 c^{5/3} \sqrt [3]{b c-a d} x^4+51 a b^2 c^{2/3} d \sqrt [3]{b c-a d} x^4+9 b^3 c^{2/3} d \sqrt [3]{b c-a d} x^7+2 \sqrt {3} a \left (3 b^2 c^2-7 a b c d+9 a^2 d^2\right ) \sqrt [3]{a+b x^3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 a \left (3 b^2 c^2-7 a b c d+9 a^2 d^2\right ) \sqrt [3]{a+b x^3} \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+3 a b^2 c^2 \sqrt [3]{a+b x^3} \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )-7 a^2 b c d \sqrt [3]{a+b x^3} \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+9 a^3 d^2 \sqrt [3]{a+b x^3} \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{108 c d^2 \sqrt [3]{b c-a d} \sqrt [3]{a+b x^3}} \]
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Time = 6.92 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.39
method | result | size |
pseudoelliptic | \(-\frac {\frac {\left (a d -b c \right )^{3} \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}-\frac {\left (\frac {20 a^{2} d^{2} b^{\frac {2}{3}}}{9}+b^{\frac {5}{3}} c \left (b c -\frac {8 a d}{3}\right )\right ) c \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}-\left (a d -b c \right )^{3} \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )+\left (\frac {20 a^{2} d^{2} b^{\frac {2}{3}}}{9}+b^{\frac {5}{3}} c \left (b c -\frac {8 a d}{3}\right )\right ) \sqrt {3}\, c \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )+\left (\frac {20 a^{2} d^{2} b^{\frac {2}{3}}}{9}+b^{\frac {5}{3}} c \left (b c -\frac {8 a d}{3}\right )\right ) c \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\frac {7 x b d \left (-\frac {3 b c}{7}+d \left (\frac {3 b \,x^{3}}{14}+a \right )\right ) c \left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}{3}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right ) \left (a d -b c \right )^{3}}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} c \,d^{3}}\) | \(461\) |
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Leaf count of result is larger than twice the leaf count of optimal. 643 vs. \(2 (273) = 546\).
Time = 6.48 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx=-\frac {18 \, \sqrt {3} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} {\left (b c - a d\right )} x + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} c \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}}}{3 \, {\left (b c - a d\right )} x}\right ) + 2 \, \sqrt {3} {\left (9 \, b^{2} c^{2} - 24 \, a b c d + 20 \, a^{2} d^{2}\right )} \left (-b^{2}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} b x - 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {1}{3}}}{3 \, b x}\right ) - 18 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (\frac {c x \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b c - a d\right )}}{x}\right ) - 2 \, {\left (9 \, b^{2} c^{2} - 24 \, a b c d + 20 \, a^{2} d^{2}\right )} \left (-b^{2}\right )^{\frac {1}{3}} \log \left (-\frac {\left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{x}\right ) + {\left (9 \, b^{2} c^{2} - 24 \, a b c d + 20 \, a^{2} d^{2}\right )} \left (-b^{2}\right )^{\frac {1}{3}} \log \left (-\frac {\left (-b^{2}\right )^{\frac {1}{3}} b x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{x^{2}}\right ) + 9 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (-\frac {{\left (b c - a d\right )} x^{2} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} c x \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b c - a d\right )}}{x^{2}}\right ) - 3 \, {\left (3 \, b^{2} d^{2} x^{4} - 2 \, {\left (3 \, b^{2} c d - 7 \, a b d^{2}\right )} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, d^{3}} \]
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\[ \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx=\int \frac {\left (a + b x^{3}\right )^{\frac {8}{3}}}{c + d x^{3}}\, dx \]
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\[ \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {8}{3}}}{d x^{3} + c} \,d x } \]
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\[ \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {8}{3}}}{d x^{3} + c} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{8/3}}{d\,x^3+c} \,d x \]
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