Optimal. Leaf size=261 \[ -\frac {d (3 b c-a d) \log \left (c+d x^3\right )}{9 c^{5/3} (b c-a d)^{7/3}}+\frac {d (3 b c-a d) \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{3 c^{5/3} (b c-a d)^{7/3}}-\frac {2 d (3 b c-a d) \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c^{5/3} (b c-a d)^{7/3}}+\frac {b x (a d+3 b c)}{3 a c \sqrt [3]{a+b x^3} (b c-a d)^2}-\frac {d x}{3 c \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)} \]
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Rubi [C] time = 1.93, antiderivative size = 625, normalized size of antiderivative = 2.39, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {430, 429} \[ -\frac {c \left (a+b x^3\right )^{2/3} \left (-\frac {54 d^2 x^{15} (b c-a d)^3 \, _3F_2\left (2,2,\frac {7}{3};1,\frac {13}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^5 \left (a+b x^3\right )^3}-\frac {108 d x^{12} (b c-a d)^3 \, _3F_2\left (2,2,\frac {7}{3};1,\frac {13}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^4 \left (a+b x^3\right )^3}-\frac {54 x^9 (b c-a d)^3 \, _3F_2\left (2,2,\frac {7}{3};1,\frac {13}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^3 \left (a+b x^3\right )^3}+\frac {2520 d^2 x^9 (b c-a d) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^3 \left (a+b x^3\right )}-\frac {6300 d^2 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^2}-\frac {945 d^2 x^9 (b c-a d)}{c^3 \left (a+b x^3\right )}+\frac {5320 d x^6 (b c-a d) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^2 \left (a+b x^3\right )}-\frac {1890 d x^6 (b c-a d)}{c^2 \left (a+b x^3\right )}-\frac {13720 d x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c}+\frac {2240 x^3 (b c-a d) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c \left (a+b x^3\right )}-6860 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-\frac {525 x^3 (b c-a d)}{c \left (a+b x^3\right )}+\frac {6300 d^2 x^6}{c^2}+\frac {13720 d x^3}{c}+6860\right )}{420 x^5 \left (c+d x^3\right ) (b c-a d)^2} \]
Warning: Unable to verify antiderivative.
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Rule 429
Rule 430
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2} \, dx &=\frac {\sqrt [3]{1+\frac {b x^3}{a}} \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{4/3} \left (c+d x^3\right )^2} \, dx}{a \sqrt [3]{a+b x^3}}\\ &=-\frac {c \left (a+b x^3\right )^{2/3} \left (6860+\frac {13720 d x^3}{c}+\frac {6300 d^2 x^6}{c^2}-\frac {525 (b c-a d) x^3}{c \left (a+b x^3\right )}-\frac {1890 d (b c-a d) x^6}{c^2 \left (a+b x^3\right )}-\frac {945 d^2 (b c-a d) x^9}{c^3 \left (a+b x^3\right )}-6860 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-\frac {13720 d x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c}-\frac {6300 d^2 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^2}+\frac {2240 (b c-a d) x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c \left (a+b x^3\right )}+\frac {5320 d (b c-a d) x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^2 \left (a+b x^3\right )}+\frac {2520 d^2 (b c-a d) x^9 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^3 \left (a+b x^3\right )}-\frac {54 (b c-a d)^3 x^9 \, _3F_2\left (2,2,\frac {7}{3};1,\frac {13}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^3 \left (a+b x^3\right )^3}-\frac {108 d (b c-a d)^3 x^{12} \, _3F_2\left (2,2,\frac {7}{3};1,\frac {13}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^4 \left (a+b x^3\right )^3}-\frac {54 d^2 (b c-a d)^3 x^{15} \, _3F_2\left (2,2,\frac {7}{3};1,\frac {13}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^5 \left (a+b x^3\right )^3}\right )}{420 (b c-a d)^2 x^5 \left (c+d x^3\right )}\\ \end {align*}
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Mathematica [C] time = 2.03, size = 625, normalized size = 2.39 \[ \frac {c \left (a+b x^3\right )^{2/3} \left (\frac {54 d^2 x^{15} (b c-a d)^3 \, _3F_2\left (2,2,\frac {7}{3};1,\frac {13}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^5 \left (a+b x^3\right )^3}+\frac {108 d x^{12} (b c-a d)^3 \, _3F_2\left (2,2,\frac {7}{3};1,\frac {13}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^4 \left (a+b x^3\right )^3}+\frac {54 x^9 (b c-a d)^3 \, _3F_2\left (2,2,\frac {7}{3};1,\frac {13}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^3 \left (a+b x^3\right )^3}+\frac {2520 d^2 x^9 (a d-b c) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^3 \left (a+b x^3\right )}+\frac {945 d^2 x^9 (b c-a d)}{c^3 \left (a+b x^3\right )}+\frac {6300 d^2 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^2}+\frac {5320 d x^6 (a d-b c) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^2 \left (a+b x^3\right )}+\frac {1890 d x^6 (b c-a d)}{c^2 \left (a+b x^3\right )}+\frac {13720 d x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c}-\frac {2240 x^3 (b c-a d) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c \left (a+b x^3\right )}+6860 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+\frac {525 x^3 (b c-a d)}{c \left (a+b x^3\right )}-\frac {6300 d^2 x^6}{c^2}-\frac {13720 d x^3}{c}-6860\right )}{420 x^5 \left (c+d x^3\right ) (b c-a d)^2} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (d \,x^{3}+c \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (b\,x^3+a\right )}^{4/3}\,{\left (d\,x^3+c\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b x^{3}\right )^{\frac {4}{3}} \left (c + d x^{3}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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