Optimal. Leaf size=204 \[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt [6]{c} \left (\sqrt [3]{2} \sqrt [3]{c+3 d x^2}+\sqrt [3]{c}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c+3 d x^2}\right )}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt {c}}\right )}{2\ 2^{2/3} \sqrt {3} c^{5/6} \sqrt {d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c}}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}} \]
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Rubi [A] time = 0.04, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {392} \[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt [6]{c} \left (\sqrt [3]{2} \sqrt [3]{c+3 d x^2}+\sqrt [3]{c}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c+3 d x^2}\right )}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt {c}}\right )}{2\ 2^{2/3} \sqrt {3} c^{5/6} \sqrt {d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c}}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}} \]
Antiderivative was successfully verified.
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Rule 392
Rubi steps
\begin {align*} \int \frac {1}{\left (c-d x^2\right ) \sqrt [3]{c+3 d x^2}} \, dx &=-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt {c}}\right )}{2\ 2^{2/3} \sqrt {3} c^{5/6} \sqrt {d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{c+3 d x^2}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c}}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c+3 d x^2}\right )}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 153, normalized size = 0.75 \[ \frac {3 c x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-\frac {3 d x^2}{c},\frac {d x^2}{c}\right )}{\left (c-d x^2\right ) \sqrt [3]{c+3 d x^2} \left (2 d x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};-\frac {3 d x^2}{c},\frac {d x^2}{c}\right )-F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};-\frac {3 d x^2}{c},\frac {d x^2}{c}\right )\right )+3 c F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-\frac {3 d x^2}{c},\frac {d x^2}{c}\right )\right )} \]
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 \[ \int -\frac {1}{{\left (3 \, d x^{2} + c\right )}^{\frac {1}{3}} {\left (d x^{2} - c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-d \,x^{2}+c \right ) \left (3 d \,x^{2}+c \right )^{\frac {1}{3}}}\, dx \]
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 \[ -\int \frac {1}{{\left (3 \, d x^{2} + c\right )}^{\frac {1}{3}} {\left (d x^{2} - c\right )}}\,{d x} \]
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 \[ \int \frac {1}{\left (c-d\,x^2\right )\,{\left (3\,d\,x^2+c\right )}^{1/3}} \,d x \]
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 \[ - \int \frac {1}{- c \sqrt [3]{c + 3 d x^{2}} + d x^{2} \sqrt [3]{c + 3 d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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