Integrand size = 21, antiderivative size = 377 \[ \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^3} \, dx=-\frac {d x}{6 c (b c-a d) \sqrt [3]{a+b x^3} \left (c+d x^3\right )^2}+\frac {b (6 b c+a d) x}{6 a c (b c-a d)^2 \sqrt [3]{a+b x^3} \left (c+d x^3\right )}+\frac {d \left (18 b^2 c^2+15 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{2/3}}{18 a c^2 (b c-a d)^3 \left (c+d x^3\right )}-\frac {d \left (27 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} (b c-a d)^{10/3}}-\frac {d \left (27 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} (b c-a d)^{10/3}}+\frac {d \left (27 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} (b c-a d)^{10/3}} \]
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Time = 0.30 (sec) , antiderivative size = 377, 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} \[ \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^3} \, dx=-\frac {d \left (5 a^2 d^2-18 a b c d+27 b^2 c^2\right ) \arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} (b c-a d)^{10/3}}+\frac {d x \left (a+b x^3\right )^{2/3} \left (-5 a^2 d^2+15 a b c d+18 b^2 c^2\right )}{18 a c^2 \left (c+d x^3\right ) (b c-a d)^3}-\frac {d \left (5 a^2 d^2-18 a b c d+27 b^2 c^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} (b c-a d)^{10/3}}+\frac {d \left (5 a^2 d^2-18 a b c d+27 b^2 c^2\right ) \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} (b c-a d)^{10/3}}+\frac {b x (a d+6 b c)}{6 a c \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)^2}-\frac {d x}{6 c \sqrt [3]{a+b x^3} \left (c+d x^3\right )^2 (b c-a d)} \]
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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) \sqrt [3]{a+b x^3} \left (c+d x^3\right )^2}+\frac {\int \frac {6 b c-5 a d-6 b d x^3}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2} \, dx}{6 c (b c-a d)} \\ & = -\frac {d x}{6 c (b c-a d) \sqrt [3]{a+b x^3} \left (c+d x^3\right )^2}+\frac {b (6 b c+a d) x}{6 a c (b c-a d)^2 \sqrt [3]{a+b x^3} \left (c+d x^3\right )}-\frac {\int \frac {a d (12 b c-5 a d)-3 b d (6 b c+a d) x^3}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )^2} \, dx}{6 a c (b c-a d)^2} \\ & = -\frac {d x}{6 c (b c-a d) \sqrt [3]{a+b x^3} \left (c+d x^3\right )^2}+\frac {b (6 b c+a d) x}{6 a c (b c-a d)^2 \sqrt [3]{a+b x^3} \left (c+d x^3\right )}+\frac {d \left (18 b^2 c^2+15 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{2/3}}{18 a c^2 (b c-a d)^3 \left (c+d x^3\right )}-\frac {\int \frac {2 a d \left (27 b^2 c^2-18 a b c d+5 a^2 d^2\right )}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{18 a c^2 (b c-a d)^3} \\ & = -\frac {d x}{6 c (b c-a d) \sqrt [3]{a+b x^3} \left (c+d x^3\right )^2}+\frac {b (6 b c+a d) x}{6 a c (b c-a d)^2 \sqrt [3]{a+b x^3} \left (c+d x^3\right )}+\frac {d \left (18 b^2 c^2+15 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{2/3}}{18 a c^2 (b c-a d)^3 \left (c+d x^3\right )}-\frac {\left (d \left (27 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 (b c-a d)^3} \\ & = -\frac {d x}{6 c (b c-a d) \sqrt [3]{a+b x^3} \left (c+d x^3\right )^2}+\frac {b (6 b c+a d) x}{6 a c (b c-a d)^2 \sqrt [3]{a+b x^3} \left (c+d x^3\right )}+\frac {d \left (18 b^2 c^2+15 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{2/3}}{18 a c^2 (b c-a d)^3 \left (c+d x^3\right )}-\frac {d \left (27 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} (b c-a d)^{10/3}}-\frac {d \left (27 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} (b c-a d)^{10/3}}+\frac {d \left (27 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} (b c-a d)^{10/3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 13.68 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^3} \, dx=-\frac {65 c^2 \left (a+b x^3\right )^2 \left (-14000 a^2 c^5-21896 a b c^5 x^3-48104 a^2 c^4 d x^3-8391 b^2 c^5 x^6-70802 a b c^4 d x^6-60807 a^2 c^3 d^2 x^6-24417 b^2 c^4 d x^9-81534 a b c^3 d^2 x^9-33657 a^2 c^2 d^3 x^9-23409 b^2 c^3 d^2 x^{12}-38652 a b c^2 d^3 x^{12}-7155 a^2 c d^4 x^{12}-7425 b^2 c^2 d^3 x^{15}-5940 a b c d^4 x^{15}-243 a^2 d^5 x^{15}+28 \left (c+d x^3\right )^2 \left (27 b^2 c^2 x^6 \left (7 c+6 d x^3\right )+9 a b c x^3 \left (73 c^2+104 c d x^3+33 d^2 x^6\right )+a^2 \left (500 c^3+843 c^2 d x^3+375 c d^2 x^6+27 d^3 x^9\right )\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )\right )+486 (b c-a d)^4 x^{12} \left (c+d x^3\right )^3 \, _4F_3\left (2,2,2,\frac {7}{3};1,1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{16380 c^5 (-b c+a d)^3 x^8 \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2} \]
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Time = 4.46 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.13
method | result | size |
pseudoelliptic | \(-\frac {-2 \left (5 a^{2} d^{2}-18 a b c d +27 b^{2} c^{2}\right ) d a \left (d \,x^{3}+c \right )^{2} \left (b \,x^{3}+a \right )^{\frac {1}{3}} \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-3 x c \left (5 a^{2} b \,d^{4} x^{6}-15 a \,b^{2} c \,d^{3} x^{6}-18 b^{3} c^{2} d^{2} x^{6}+5 a^{3} d^{4} x^{3}-7 a^{2} b c \,d^{3} x^{3}-18 a \,b^{2} c^{2} d^{2} x^{3}-36 b^{3} c^{3} d \,x^{3}+8 a^{3} c \,d^{3}-18 a^{2} b \,c^{2} d^{2}-18 c^{4} b^{3}\right ) \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}+\left (5 a^{2} d^{2}-18 a b c d +27 b^{2} c^{2}\right ) d a \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right ) \sqrt {3}+\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )\right ) \left (d \,x^{3}+c \right )^{2} \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{54 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (d \,x^{3}+c \right )^{2} c^{3} \left (a d -b c \right )^{3} a}\) | \(426\) |
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Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^3} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}^{3}} \,d x } \]
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\[ \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^3} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^3} \, dx=\int \frac {1}{{\left (b\,x^3+a\right )}^{4/3}\,{\left (d\,x^3+c\right )}^3} \,d x \]
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