Integrand size = 21, antiderivative size = 324 \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2} \, dx=\frac {b (3 b c+4 a d) x}{12 a c (b c-a d)^2 \left (a+b x^3\right )^{4/3}}+\frac {b \left (9 b^2 c^2-33 a b c d-4 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt [3]{a+b x^3}}-\frac {d x}{3 c (b c-a d) \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}+\frac {d^2 (9 b c-2 a d) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} c^{5/3} (b c-a d)^{10/3}}+\frac {d^2 (9 b c-2 a d) \log \left (c+d x^3\right )}{18 c^{5/3} (b c-a d)^{10/3}}-\frac {d^2 (9 b c-2 a d) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{6 c^{5/3} (b c-a d)^{10/3}} \]
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Time = 0.26 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {425, 541, 12, 384} \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2} \, dx=\frac {b x \left (-4 a^2 d^2-33 a b c d+9 b^2 c^2\right )}{12 a^2 c \sqrt [3]{a+b x^3} (b c-a d)^3}+\frac {d^2 (9 b c-2 a d) \arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c^{5/3} (b c-a d)^{10/3}}+\frac {d^2 (9 b c-2 a d) \log \left (c+d x^3\right )}{18 c^{5/3} (b c-a d)^{10/3}}-\frac {d^2 (9 b c-2 a d) \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{6 c^{5/3} (b c-a d)^{10/3}}-\frac {d x}{3 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}+\frac {b x (4 a d+3 b c)}{12 a c \left (a+b x^3\right )^{4/3} (b c-a d)^2} \]
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Rule 12
Rule 384
Rule 425
Rule 541
Rubi steps \begin{align*} \text {integral}& = -\frac {d x}{3 c (b c-a d) \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}+\frac {\int \frac {3 b c-2 a d-6 b d x^3}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )} \, dx}{3 c (b c-a d)} \\ & = \frac {b (3 b c+4 a d) x}{12 a c (b c-a d)^2 \left (a+b x^3\right )^{4/3}}-\frac {d x}{3 c (b c-a d) \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}-\frac {\int \frac {-9 b^2 c^2+24 a b c d-8 a^2 d^2-3 b d (3 b c+4 a d) x^3}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx}{12 a c (b c-a d)^2} \\ & = \frac {b (3 b c+4 a d) x}{12 a c (b c-a d)^2 \left (a+b x^3\right )^{4/3}}+\frac {b \left (9 b^2 c^2-33 a b c d-4 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt [3]{a+b x^3}}-\frac {d x}{3 c (b c-a d) \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}+\frac {\int \frac {4 a^2 d^2 (9 b c-2 a d)}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{12 a^2 c (b c-a d)^3} \\ & = \frac {b (3 b c+4 a d) x}{12 a c (b c-a d)^2 \left (a+b x^3\right )^{4/3}}+\frac {b \left (9 b^2 c^2-33 a b c d-4 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt [3]{a+b x^3}}-\frac {d x}{3 c (b c-a d) \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}+\frac {\left (d^2 (9 b c-2 a d)\right ) \int \frac {1}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{3 c (b c-a d)^3} \\ & = \frac {b (3 b c+4 a d) x}{12 a c (b c-a d)^2 \left (a+b x^3\right )^{4/3}}+\frac {b \left (9 b^2 c^2-33 a b c d-4 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt [3]{a+b x^3}}-\frac {d x}{3 c (b c-a d) \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}+\frac {d^2 (9 b c-2 a d) \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 )}{3 \sqrt {3} c^{5/3} (b c-a d)^{10/3}}+\frac {d^2 (9 b c-2 a d) \log \left (c+d x^3\right )}{18 c^{5/3} (b c-a d)^{10/3}}-\frac {d^2 (9 b c-2 a d) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{6 c^{5/3} (b c-a d)^{10/3}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.11 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.37 \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2} \, dx=\frac {\frac {3 c^{2/3} x \left (4 a^4 d^3+8 a^3 b d^3 x^3-9 b^4 c^2 x^3 \left (c+d x^3\right )+4 a^2 b^2 d \left (9 c^2+9 c d x^3+d^2 x^6\right )+3 a b^3 c \left (-4 c^2+7 c d x^3+11 d^2 x^6\right )\right )}{a^2 (-b c+a d)^3 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}+\frac {2 i \left (3 i+\sqrt {3}\right ) d^2 (-9 b c+2 a d) \text {arctanh}\left (\frac {i+\frac {\left (-i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d} x}}{\sqrt {3}}\right )}{(b c-a d)^{10/3}}+\frac {2 \left (1+i \sqrt {3}\right ) d^2 (9 b c-2 a d) \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{(b c-a d)^{10/3}}+\frac {\left (1+i \sqrt {3}\right ) d^2 (-9 b c+2 a d) \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{(b c-a d)^{10/3}}}{36 c^{5/3}} \]
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Time = 4.55 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.15
method | result | size |
pseudoelliptic | \(-\frac {-\frac {3 x \left (4 a^{2} b^{2} d^{3} x^{6}+33 a \,b^{3} c \,d^{2} x^{6}-9 b^{4} c^{2} d \,x^{6}+8 a^{3} b \,d^{3} x^{3}+36 a^{2} b^{2} c \,d^{2} x^{3}+21 a \,b^{3} c^{2} d \,x^{3}-9 b^{4} c^{3} x^{3}+4 a^{4} d^{3}+36 a^{2} b^{2} c^{2} d -12 a \,b^{3} c^{3}\right ) c \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}{2}+\left (b \,x^{3}+a \right )^{\frac {4}{3}} a^{2} d^{2} \left (d \,x^{3}+c \right ) \left (2 a d -9 b c \right ) \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right ) \sqrt {3}+\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )\right )}{18 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {4}{3}} c^{2} \left (d \,x^{3}+c \right ) \left (a d -b c \right )^{3} a^{2}}\) | \(371\) |
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Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2} \, dx=\int \frac {1}{\left (a + b x^{3}\right )^{\frac {7}{3}} \left (c + d x^{3}\right )^{2}}\, dx \]
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\[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {7}{3}} {\left (d x^{3} + c\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {7}{3}} {\left (d x^{3} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2} \, dx=\int \frac {1}{{\left (b\,x^3+a\right )}^{7/3}\,{\left (d\,x^3+c\right )}^2} \,d x \]
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