Optimal. Leaf size=261 \[ \frac {b (3 b c+a d) x}{3 a c (b c-a d)^2 \sqrt [3]{a+b x^3}}-\frac {d x}{3 c (b c-a d) \sqrt [3]{a+b x^3} \left (c+d x^3\right )}-\frac {2 d (3 b c-a d) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} c^{5/3} (b c-a d)^{7/3}}-\frac {d (3 b c-a d) \log \left (c+d x^3\right )}{9 c^{5/3} (b c-a d)^{7/3}}+\frac {d (3 b c-a d) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{3 c^{5/3} (b c-a d)^{7/3}} \]
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Rubi [A]
time = 0.15, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {425, 541, 12,
384} \begin {gather*} -\frac {2 d (3 b c-a d) \text {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)^{7/3}}-\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 {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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 384
Rule 425
Rule 541
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] Result contains complex when optimal does not.
time = 3.44, size = 370, normalized size = 1.42 \begin {gather*} \frac {\frac {6 c^{2/3} x \left (a^2 d^2+a b d^2 x^3+3 b^2 c \left (c+d x^3\right )\right )}{a (b c-a d)^2 \sqrt [3]{a+b x^3} \left (c+d x^3\right )}+\frac {2 i \left (3 i+\sqrt {3}\right ) d (3 b c-a d) \tanh ^{-1}\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)^{7/3}}+\frac {2 \left (1+i \sqrt {3}\right ) d (-3 b c+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)^{7/3}}+\frac {\left (1+i \sqrt {3}\right ) d (3 b c-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)^{7/3}}}{18 c^{5/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (a + b x^{3}\right )^{\frac {4}{3}} \left (c + d x^{3}\right )^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (b\,x^3+a\right )}^{4/3}\,{\left (d\,x^3+c\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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