Integrand size = 21, antiderivative size = 463 \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=-\frac {d x}{6 c (b c-a d) \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2}+\frac {b (3 b c+2 a d) x}{12 a c (b c-a d)^2 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}+\frac {b \left (9 b^2 c^2-42 a b c d-2 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt [3]{a+b x^3} \left (c+d x^3\right )}+\frac {d \left (27 b^3 c^3-135 a b^2 c^2 d-42 a^2 b c d^2+10 a^3 d^3\right ) x \left (a+b x^3\right )^{2/3}}{36 a^2 c^2 (b c-a d)^4 \left (c+d x^3\right )}+\frac {d^2 \left (54 b^2 c^2-24 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} (b c-a d)^{13/3}}+\frac {d^2 \left (54 b^2 c^2-24 a b c d+5 a^2 d^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} (b c-a d)^{13/3}}-\frac {d^2 \left (54 b^2 c^2-24 a b c d+5 a^2 d^2\right ) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} (b c-a d)^{13/3}} \]
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Time = 0.49 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {425, 541, 12, 384} \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\frac {d^2 \left (5 a^2 d^2-24 a b c d+54 b^2 c^2\right ) \arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} (b c-a d)^{13/3}}+\frac {b x \left (-2 a^2 d^2-42 a b c d+9 b^2 c^2\right )}{12 a^2 c \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)^3}+\frac {d^2 \left (5 a^2 d^2-24 a b c d+54 b^2 c^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} (b c-a d)^{13/3}}-\frac {d^2 \left (5 a^2 d^2-24 a b c d+54 b^2 c^2\right ) \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} (b c-a d)^{13/3}}+\frac {d x \left (a+b x^3\right )^{2/3} \left (10 a^3 d^3-42 a^2 b c d^2-135 a b^2 c^2 d+27 b^3 c^3\right )}{36 a^2 c^2 \left (c+d x^3\right ) (b c-a d)^4}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}+\frac {b x (2 a d+3 b c)}{12 a c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)^2} \]
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Rule 12
Rule 384
Rule 425
Rule 541
Rubi steps \begin{align*} \text {integral}& = -\frac {d x}{6 c (b c-a d) \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2}+\frac {\int \frac {6 b c-5 a d-9 b d x^3}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2} \, dx}{6 c (b c-a d)} \\ & = -\frac {d x}{6 c (b c-a d) \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2}+\frac {b (3 b c+2 a d) x}{12 a c (b c-a d)^2 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}-\frac {\int \frac {-2 \left (9 b^2 c^2-24 a b c d+10 a^2 d^2\right )-12 b d (3 b c+2 a d) x^3}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2} \, dx}{24 a c (b c-a d)^2} \\ & = -\frac {d x}{6 c (b c-a d) \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2}+\frac {b (3 b c+2 a d) x}{12 a c (b c-a d)^2 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}+\frac {b \left (9 b^2 c^2-42 a b c d-2 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt [3]{a+b x^3} \left (c+d x^3\right )}+\frac {\int \frac {2 a d \left (9 b^2 c^2+36 a b c d-10 a^2 d^2\right )+6 b d \left (9 b^2 c^2-42 a b c d-2 a^2 d^2\right ) x^3}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )^2} \, dx}{24 a^2 c (b c-a d)^3} \\ & = -\frac {d x}{6 c (b c-a d) \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2}+\frac {b (3 b c+2 a d) x}{12 a c (b c-a d)^2 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}+\frac {b \left (9 b^2 c^2-42 a b c d-2 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt [3]{a+b x^3} \left (c+d x^3\right )}+\frac {d \left (27 b^3 c^3-135 a b^2 c^2 d-42 a^2 b c d^2+10 a^3 d^3\right ) x \left (a+b x^3\right )^{2/3}}{36 a^2 c^2 (b c-a d)^4 \left (c+d x^3\right )}+\frac {\int \frac {8 a^2 d^2 \left (54 b^2 c^2-24 a b c d+5 a^2 d^2\right )}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{72 a^2 c^2 (b c-a d)^4} \\ & = -\frac {d x}{6 c (b c-a d) \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2}+\frac {b (3 b c+2 a d) x}{12 a c (b c-a d)^2 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}+\frac {b \left (9 b^2 c^2-42 a b c d-2 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt [3]{a+b x^3} \left (c+d x^3\right )}+\frac {d \left (27 b^3 c^3-135 a b^2 c^2 d-42 a^2 b c d^2+10 a^3 d^3\right ) x \left (a+b x^3\right )^{2/3}}{36 a^2 c^2 (b c-a d)^4 \left (c+d x^3\right )}+\frac {\left (d^2 \left (54 b^2 c^2-24 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 (b c-a d)^4} \\ & = -\frac {d x}{6 c (b c-a d) \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2}+\frac {b (3 b c+2 a d) x}{12 a c (b c-a d)^2 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}+\frac {b \left (9 b^2 c^2-42 a b c d-2 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt [3]{a+b x^3} \left (c+d x^3\right )}+\frac {d \left (27 b^3 c^3-135 a b^2 c^2 d-42 a^2 b c d^2+10 a^3 d^3\right ) x \left (a+b x^3\right )^{2/3}}{36 a^2 c^2 (b c-a d)^4 \left (c+d x^3\right )}+\frac {d^2 \left (54 b^2 c^2-24 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} (b c-a d)^{13/3}}+\frac {d^2 \left (54 b^2 c^2-24 a b c d+5 a^2 d^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} (b c-a d)^{13/3}}-\frac {d^2 \left (54 b^2 c^2-24 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} (b c-a d)^{13/3}} \\ \end{align*}
Time = 15.75 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\frac {1}{36} x \left (a+b x^3\right )^{2/3} \left (-\frac {9 b^3}{a (-b c+a d)^3 \left (a+b x^3\right )^2}+\frac {27 b^3 (b c-5 a d)}{a^2 (b c-a d)^4 \left (a+b x^3\right )}-\frac {6 d^3}{c (b c-a d)^3 \left (c+d x^3\right )^2}+\frac {2 d^3 (-21 b c+5 a d)}{c^2 (b c-a d)^4 \left (c+d x^3\right )}\right )+\frac {d^2 \left (54 b^2 c^2-24 a b c d+5 a^2 d^2\right ) \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{54 c^{8/3} (b c-a d)^{13/3}} \]
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Time = 4.69 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.03
method | result | size |
pseudoelliptic | \(\frac {\frac {5 \left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (a^{2} d^{2}-\frac {24}{5} a b c d +\frac {54}{5} b^{2} c^{2}\right ) d^{2} a^{2} \left (d \,x^{3}+c \right )^{2} \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )}{27}+\frac {4 \left (\frac {5 a^{3} x^{3} \left (b \,x^{3}+a \right )^{2} d^{5}}{8}+\left (b \,x^{3}+a \right )^{2} \left (-\frac {21 b \,x^{3}}{8}+a \right ) c \,a^{2} d^{4}-3 b \left (\frac {45}{16} b^{3} x^{9}+4 a \,b^{2} x^{6}+2 a^{2} b \,x^{3}+a^{3}\right ) c^{2} a \,d^{3}-18 x^{3} b^{3} \left (-\frac {3}{32} b^{2} x^{6}+\frac {13}{16} a b \,x^{3}+a^{2}\right ) c^{3} d^{2}-9 b^{3} \left (-\frac {3}{8} b^{2} x^{6}+\frac {7}{16} a b \,x^{3}+a^{2}\right ) c^{4} d +\frac {9 b^{4} \left (\frac {3 b \,x^{3}}{4}+a \right ) c^{5}}{4}\right ) x c \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}{9}+\frac {5 \left (\arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right ) \sqrt {3}-\frac {\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right ) \left (a^{2} d^{2}-\frac {24}{5} a b c d +\frac {54}{5} b^{2} c^{2}\right ) d^{2} a^{2} \left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (d \,x^{3}+c \right )^{2}}{27}}{\left (d \,x^{3}+c \right )^{2} c^{3} \left (a d -b c \right )^{4} \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {4}{3}} a^{2}}\) | \(475\) |
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Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {7}{3}} {\left (d x^{3} + c\right )}^{3}} \,d x } \]
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\[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {7}{3}} {\left (d x^{3} + c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\int \frac {1}{{\left (b\,x^3+a\right )}^{7/3}\,{\left (d\,x^3+c\right )}^3} \,d x \]
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