Integrand size = 21, antiderivative size = 391 \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^3} \, dx=-\frac {(b c-a d) x \left (a+b x^3\right )^{5/3}}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (6 b c+5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 d^2 \left (c+d x^3\right )}+\frac {b^{8/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^3}-\frac {(b c-a d)^{2/3} \left (9 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} d^3}-\frac {(b c-a d)^{2/3} \left (9 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} d^3}+\frac {(b c-a d)^{2/3} \left (9 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} d^3}-\frac {b^{8/3} \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 d^3} \]
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Time = 0.30 (sec) , antiderivative size = 391, 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 = {424, 540, 544, 245, 384} \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^3} \, dx=-\frac {(b c-a d)^{2/3} \left (5 a^2 d^2+6 a b c d+9 b^2 c^2\right ) \arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} d^3}-\frac {(b c-a d)^{2/3} \left (5 a^2 d^2+6 a b c d+9 b^2 c^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} d^3}+\frac {(b c-a d)^{2/3} \left (5 a^2 d^2+6 a b c d+9 b^2 c^2\right ) \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} d^3}+\frac {b^{8/3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} d^3}-\frac {b^{8/3} \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 d^3}-\frac {x \left (a+b x^3\right )^{2/3} (b c-a d) (5 a d+6 b c)}{18 c^2 d^2 \left (c+d x^3\right )}-\frac {x \left (a+b x^3\right )^{5/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2} \]
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Rule 245
Rule 384
Rule 424
Rule 540
Rule 544
Rubi steps \begin{align*} \text {integral}& = -\frac {(b c-a d) x \left (a+b x^3\right )^{5/3}}{6 c d \left (c+d x^3\right )^2}+\frac {\int \frac {\left (a+b x^3\right )^{2/3} \left (a (b c+5 a d)+6 b^2 c x^3\right )}{\left (c+d x^3\right )^2} \, dx}{6 c d} \\ & = -\frac {(b c-a d) x \left (a+b x^3\right )^{5/3}}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (6 b c+5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 d^2 \left (c+d x^3\right )}-\frac {\int \frac {-2 a \left (3 b^2 c^2+a d (b c+5 a d)\right )-18 b^3 c^2 x^3}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{18 c^2 d^2} \\ & = -\frac {(b c-a d) x \left (a+b x^3\right )^{5/3}}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (6 b c+5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 d^2 \left (c+d x^3\right )}+\frac {b^3 \int \frac {1}{\sqrt [3]{a+b x^3}} \, dx}{d^3}-\frac {\left ((b c-a d) \left (9 b^2 c^2+6 a b c d+5 a^2 d^2\right )\right ) \int \frac {1}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{9 c^2 d^3} \\ & = -\frac {(b c-a d) x \left (a+b x^3\right )^{5/3}}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (6 b c+5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 d^2 \left (c+d x^3\right )}+\frac {b^{8/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^3}-\frac {(b c-a d)^{2/3} \left (9 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \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 )}{9 \sqrt {3} c^{8/3} d^3}-\frac {(b c-a d)^{2/3} \left (9 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} d^3}+\frac {(b c-a d)^{2/3} \left (9 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} d^3}-\frac {b^{8/3} \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 d^3} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 11.00 (sec) , antiderivative size = 651, normalized size of antiderivative = 1.66 \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^3} \, dx=\frac {\frac {6 c^{2/3} (-b c+a d) x \left (a+b x^3\right )^{2/3} \left (3 b c \left (2 c+3 d x^3\right )+a d \left (8 c+5 d x^3\right )\right )}{d^2 \left (c+d x^3\right )^2}+\frac {27 b^3 c^{5/3} x^4 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{d^2 \sqrt [3]{a+b x^3}}+\frac {10 a^3 \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{\sqrt [3]{b c-a d}}+\frac {6 a b^2 c^2 \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{d^2 \sqrt [3]{b c-a d}}+\frac {2 a^2 b c \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{d \sqrt [3]{b c-a d}}}{108 c^{8/3}} \]
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Time = 4.80 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.29
method | result | size |
pseudoelliptic | \(\frac {\frac {9 b^{\frac {8}{3}} c^{3} \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (d \,x^{3}+c \right )^{2} \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}+\left (5 a^{2} d^{2}+6 a b c d +9 b^{2} c^{2}\right ) \left (d \,x^{3}+c \right )^{2} \left (a d -b c \right ) \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-9 \sqrt {3}\, b^{\frac {8}{3}} c^{3} \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (d \,x^{3}+c \right )^{2} \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )-9 b^{\frac {8}{3}} c^{3} \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (d \,x^{3}+c \right )^{2} \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )+\frac {\left (3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} x d \left (5 a \,d^{2} x^{3}+9 b c d \,x^{3}+8 a c d +6 b \,c^{2}\right ) c \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}-\left (-2 \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right ) \sqrt {3}+\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )\right ) \left (5 a^{2} d^{2}+6 a b c d +9 b^{2} c^{2}\right ) \left (d \,x^{3}+c \right )^{2}\right ) \left (a d -b c \right )}{2}}{27 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (d \,x^{3}+c \right )^{2} d^{3} c^{3}}\) | \(505\) |
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Leaf count of result is larger than twice the leaf count of optimal. 954 vs. \(2 (334) = 668\).
Time = 1.95 (sec) , antiderivative size = 954, normalized size of antiderivative = 2.44 \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^3} \, dx=-\frac {2 \, \sqrt {3} {\left ({\left (9 \, b^{2} c^{2} d^{2} + 6 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{6} + 9 \, b^{2} c^{4} + 6 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2} + 2 \, {\left (9 \, b^{2} c^{3} d + 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{3}\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 ) + 18 \, \sqrt {3} {\left (b^{2} c^{2} d^{2} x^{6} + 2 \, b^{2} c^{3} d x^{3} + b^{2} c^{4}\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 ) - 2 \, {\left ({\left (9 \, b^{2} c^{2} d^{2} + 6 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{6} + 9 \, b^{2} c^{4} + 6 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2} + 2 \, {\left (9 \, b^{2} c^{3} d + 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{3}\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 ) - 18 \, {\left (b^{2} c^{2} d^{2} x^{6} + 2 \, b^{2} c^{3} d x^{3} + b^{2} c^{4}\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 ) + 9 \, {\left (b^{2} c^{2} d^{2} x^{6} + 2 \, b^{2} c^{3} d x^{3} + b^{2} c^{4}\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 ) + {\left ({\left (9 \, b^{2} c^{2} d^{2} + 6 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{6} + 9 \, b^{2} c^{4} + 6 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2} + 2 \, {\left (9 \, b^{2} c^{3} d + 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{3}\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 ({\left (9 \, b^{2} c^{2} d^{2} - 4 \, a b c d^{3} - 5 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{3} d + a b c^{2} d^{2} - 4 \, a^{2} c d^{3}\right )} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, {\left (c^{2} d^{5} x^{6} + 2 \, c^{3} d^{4} x^{3} + c^{4} d^{3}\right )}} \]
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Timed out. \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {8}{3}}}{{\left (d x^{3} + c\right )}^{3}} \,d x } \]
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\[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {8}{3}}}{{\left (d x^{3} + c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^3} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{8/3}}{{\left (d\,x^3+c\right )}^3} \,d x \]
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