Integrand size = 21, antiderivative size = 541 \[ \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx=-\frac {b (2 b c-a d) \left (18 b^2 c^2-18 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{2/3}}{18 c^2 d^4}+\frac {b \left (18 b^2 c^2-10 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{5/3}}{18 c^2 d^3}-\frac {(b c-a d) x \left (a+b x^3\right )^{11/3}}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (12 b c+5 a d) x \left (a+b x^3\right )^{8/3}}{18 c^2 d^2 \left (c+d x^3\right )}+\frac {b^{8/3} \left (54 b^2 c^2-126 a b c d+77 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} d^5}-\frac {(b c-a d)^{8/3} \left (54 b^2 c^2+18 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} d^5}-\frac {(b c-a d)^{8/3} \left (54 b^2 c^2+18 a b c d+5 a^2 d^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} d^5}+\frac {(b c-a d)^{8/3} \left (54 b^2 c^2+18 a b c d+5 a^2 d^2\right ) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} d^5}-\frac {b^{8/3} \left (54 b^2 c^2-126 a b c d+77 a^2 d^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 d^5} \]
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Time = 0.53 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {424, 540, 542, 544, 245, 384} \[ \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx=-\frac {(b c-a d)^{8/3} \left (5 a^2 d^2+18 a b c d+54 b^2 c^2\right ) \arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} d^5}+\frac {b^{8/3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right ) \left (77 a^2 d^2-126 a b c d+54 b^2 c^2\right )}{9 \sqrt {3} d^5}-\frac {b x \left (a+b x^3\right )^{2/3} (2 b c-a d) \left (-5 a^2 d^2-18 a b c d+18 b^2 c^2\right )}{18 c^2 d^4}+\frac {b x \left (a+b x^3\right )^{5/3} \left (-5 a^2 d^2-10 a b c d+18 b^2 c^2\right )}{18 c^2 d^3}-\frac {(b c-a d)^{8/3} \left (5 a^2 d^2+18 a b c d+54 b^2 c^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} d^5}+\frac {(b c-a d)^{8/3} \left (5 a^2 d^2+18 a b c d+54 b^2 c^2\right ) \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} d^5}-\frac {b^{8/3} \left (77 a^2 d^2-126 a b c d+54 b^2 c^2\right ) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 d^5}-\frac {x \left (a+b x^3\right )^{8/3} (b c-a d) (5 a d+12 b c)}{18 c^2 d^2 \left (c+d x^3\right )}-\frac {x \left (a+b x^3\right )^{11/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2} \]
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Rule 245
Rule 384
Rule 424
Rule 540
Rule 542
Rule 544
Rubi steps \begin{align*} \text {integral}& = -\frac {(b c-a d) x \left (a+b x^3\right )^{11/3}}{6 c d \left (c+d x^3\right )^2}+\frac {\int \frac {\left (a+b x^3\right )^{8/3} \left (a (b c+5 a d)+6 b (2 b c-a d) 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 )^{11/3}}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (12 b c+5 a d) x \left (a+b x^3\right )^{8/3}}{18 c^2 d^2 \left (c+d x^3\right )}-\frac {\int \frac {\left (a+b x^3\right )^{5/3} \left (-2 a \left (6 b^2 c^2-2 a b c d+5 a^2 d^2\right )-6 b \left (18 b^2 c^2-10 a b c d-5 a^2 d^2\right ) x^3\right )}{c+d x^3} \, dx}{18 c^2 d^2} \\ & = \frac {b \left (18 b^2 c^2-10 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{5/3}}{18 c^2 d^3}-\frac {(b c-a d) x \left (a+b x^3\right )^{11/3}}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (12 b c+5 a d) x \left (a+b x^3\right )^{8/3}}{18 c^2 d^2 \left (c+d x^3\right )}-\frac {\int \frac {\left (a+b x^3\right )^{2/3} \left (6 a \left (18 b^3 c^3-22 a b^2 c^2 d-a^2 b c d^2-10 a^3 d^3\right )+18 b (2 b c-a d) \left (18 b^2 c^2-18 a b c d-5 a^2 d^2\right ) x^3\right )}{c+d x^3} \, dx}{108 c^2 d^3} \\ & = -\frac {b (2 b c-a d) \left (18 b^2 c^2-18 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{2/3}}{18 c^2 d^4}+\frac {b \left (18 b^2 c^2-10 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{5/3}}{18 c^2 d^3}-\frac {(b c-a d) x \left (a+b x^3\right )^{11/3}}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (12 b c+5 a d) x \left (a+b x^3\right )^{8/3}}{18 c^2 d^2 \left (c+d x^3\right )}-\frac {\int \frac {-36 a \left (18 b^4 c^4-36 a b^3 c^3 d+15 a^2 b^2 c^2 d^2+3 a^3 b c d^3+5 a^4 d^4\right )-36 b^3 c^2 \left (54 b^2 c^2-126 a b c d+77 a^2 d^2\right ) x^3}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{324 c^2 d^4} \\ & = -\frac {b (2 b c-a d) \left (18 b^2 c^2-18 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{2/3}}{18 c^2 d^4}+\frac {b \left (18 b^2 c^2-10 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{5/3}}{18 c^2 d^3}-\frac {(b c-a d) x \left (a+b x^3\right )^{11/3}}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (12 b c+5 a d) x \left (a+b x^3\right )^{8/3}}{18 c^2 d^2 \left (c+d x^3\right )}-\frac {\left ((b c-a d)^3 \left (54 b^2 c^2+18 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^5}+\frac {\left (b^3 \left (54 b^2 c^2-126 a b c d+77 a^2 d^2\right )\right ) \int \frac {1}{\sqrt [3]{a+b x^3}} \, dx}{9 d^5} \\ & = -\frac {b (2 b c-a d) \left (18 b^2 c^2-18 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{2/3}}{18 c^2 d^4}+\frac {b \left (18 b^2 c^2-10 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{5/3}}{18 c^2 d^3}-\frac {(b c-a d) x \left (a+b x^3\right )^{11/3}}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (12 b c+5 a d) x \left (a+b x^3\right )^{8/3}}{18 c^2 d^2 \left (c+d x^3\right )}+\frac {b^{8/3} \left (54 b^2 c^2-126 a b c d+77 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^5}-\frac {(b c-a d)^{8/3} \left (54 b^2 c^2+18 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^5}-\frac {(b c-a d)^{8/3} \left (54 b^2 c^2+18 a b c d+5 a^2 d^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} d^5}+\frac {(b c-a d)^{8/3} \left (54 b^2 c^2+18 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^5}-\frac {b^{8/3} \left (54 b^2 c^2-126 a b c d+77 a^2 d^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 d^5} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 12.28 (sec) , antiderivative size = 1171, normalized size of antiderivative = 2.16 \[ \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx=\frac {1}{108} \left (\frac {6 x \left (a+b x^3\right )^{2/3} \left (-2 b^3 (9 b c-13 a d)+3 b^4 d x^3+\frac {3 (b c-a d)^4}{c \left (c+d x^3\right )^2}-\frac {(b c-a d)^3 (21 b c+5 a d)}{c^2 \left (c+d x^3\right )}\right )}{d^4}+\frac {162 b^5 c 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^4 \sqrt [3]{a+b x^3}}-\frac {378 a b^4 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^3 \sqrt [3]{a+b x^3}}+\frac {231 a^2 b^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 )}{c d^2 \sqrt [3]{a+b x^3}}+\frac {10 a^5 \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 )}{c^{8/3} \sqrt [3]{b c-a d}}+\frac {36 a b^4 c^{4/3} \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 )}{d^4 \sqrt [3]{b c-a d}}-\frac {72 a^2 b^3 \sqrt [3]{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^3 \sqrt [3]{b c-a d}}+\frac {30 a^3 b^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 )}{c^{2/3} d^2 \sqrt [3]{b c-a d}}+\frac {6 a^4 b \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 )}{c^{5/3} d \sqrt [3]{b c-a d}}\right ) \]
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Time = 5.12 (sec) , antiderivative size = 711, normalized size of antiderivative = 1.31
method | result | size |
pseudoelliptic | \(\frac {\frac {b^{\frac {8}{3}} \left (77 a^{2} d^{2}-126 a b c d +54 b^{2} c^{2}\right ) \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} c^{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}+18 a b c d +54 b^{2} c^{2}\right ) \left (d \,x^{3}+c \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 )-\sqrt {3}\, b^{\frac {8}{3}} \left (77 a^{2} d^{2}-126 a b c d +54 b^{2} c^{2}\right ) \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} c^{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 )-b^{\frac {8}{3}} \left (77 a^{2} d^{2}-126 a b c d +54 b^{2} c^{2}\right ) \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} c^{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 {3 x \left (3 b^{4} c^{2} d^{3} x^{9}+26 a \,b^{3} c^{2} d^{3} x^{6}-12 b^{4} c^{3} d^{2} x^{6}+5 a^{4} d^{5} x^{3}+6 a^{3} b c \,d^{4} x^{3}-48 a^{2} b^{2} c^{2} d^{3} x^{3}+110 a \,b^{3} c^{3} d^{2} x^{3}-54 b^{4} c^{4} d \,x^{3}+8 a^{4} d^{4} c -6 a^{3} b \,d^{3} c^{2}-30 a^{2} b^{2} d^{2} c^{3}+72 a \,b^{3} d \,c^{4}-36 b^{4} c^{5}\right ) d \left (b \,x^{3}+a \right )^{\frac {2}{3}} c \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}{2}-\frac {\left (5 a^{2} d^{2}+18 a b c d +54 b^{2} c^{2}\right ) \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 (d \,x^{3}+c \right )^{2} \left (a d -b c \right )^{3}}{2}}{27 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} d^{5} c^{3} \left (d \,x^{3}+c \right )^{2}}\) | \(711\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1555 vs. \(2 (473) = 946\).
Time = 94.18 (sec) , antiderivative size = 1555, normalized size of antiderivative = 2.87 \[ \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {14}{3}}}{{\left (d x^{3} + c\right )}^{3}} \,d x } \]
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\[ \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {14}{3}}}{{\left (d x^{3} + c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{14/3}}{{\left (d\,x^3+c\right )}^3} \,d x \]
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