Optimal. Leaf size=199 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt {c+d x}}\right )}{2^{2/3} \sqrt {3} c^{5/6} d^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {3} \sqrt {c}}\right )}{2^{2/3} \sqrt {3} c^{5/6} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt {c+d x}}\right )}{2^{2/3} c^{5/6} d^{2/3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{3\ 2^{2/3} c^{5/6} d^{2/3}} \]
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Rubi [A] time = 0.10, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {130, 484} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt {c+d x}}\right )}{2^{2/3} \sqrt {3} c^{5/6} d^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {3} \sqrt {c}}\right )}{2^{2/3} \sqrt {3} c^{5/6} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt {c+d x}}\right )}{2^{2/3} c^{5/6} d^{2/3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{3\ 2^{2/3} c^{5/6} d^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 130
Rule 484
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{x} \sqrt {c+d x} (4 c+d x)} \, dx &=3 \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt {c+d x}}\right )}{2^{2/3} \sqrt {3} c^{5/6} d^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {3} \sqrt {c}}\right )}{2^{2/3} \sqrt {3} c^{5/6} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt {c+d x}}\right )}{2^{2/3} c^{5/6} d^{2/3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{3\ 2^{2/3} c^{5/6} d^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 61, normalized size = 0.31 \begin {gather*} \frac {3 x^{2/3} \sqrt {\frac {c+d x}{c}} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {d x}{c},-\frac {d x}{4 c}\right )}{8 c \sqrt {c+d x}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 30.80, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{x} \sqrt {c+d x} (4 c+d x)} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [F(-1)] 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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (d x + 4 \, c\right )} \sqrt {d x + c} x^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d x +4 c \right ) \sqrt {d x +c}\, x^{\frac {1}{3}}}\, dx \end {gather*}
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*} \int \frac {1}{{\left (d x + 4 \, c\right )} \sqrt {d x + c} x^{\frac {1}{3}}}\,{d x} \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 = -0.01 \begin {gather*} \int \frac {1}{x^{1/3}\,\left (4\,c+d\,x\right )\,\sqrt {c+d\,x}} \,d x \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}{\sqrt [3]{x} \sqrt {c + d x} \left (4 c + d x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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