Optimal. Leaf size=143 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt {c+d x}}\right )}{2 \sqrt {3} c^{5/6} d^{2/3}}+\frac {\tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x}}\right )}{6 c^{5/6} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x}}{3 \sqrt {c}}\right )}{6 c^{5/6} d^{2/3}} \]
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Rubi [A] time = 0.46, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {130, 486, 444, 63, 206, 2138, 2145, 205} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt {c+d x}}\right )}{2 \sqrt {3} c^{5/6} d^{2/3}}+\frac {\tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x}}\right )}{6 c^{5/6} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x}}{3 \sqrt {c}}\right )}{6 c^{5/6} d^{2/3}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 130
Rule 205
Rule 206
Rule 444
Rule 486
Rule 2138
Rule 2145
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx &=3 \operatorname {Subst}\left (\int \frac {x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {2 \sqrt [3]{c} d^{2/3}-2 d x-\frac {d^{4/3} x^2}{\sqrt [3]{c}}}{\left (4+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+\frac {d^{2/3} x^2}{c^{2/3}}\right ) \sqrt {c+d x^3}} \, dx,x,\sqrt [3]{x}\right )}{4 c d}+\frac {\operatorname {Subst}\left (\int \frac {1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (2-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt {c+d x^3}} \, dx,x,\sqrt [3]{x}\right )}{4 c^{2/3} \sqrt [3]{d}}-\frac {\left (3 \sqrt [3]{d}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx,x,\sqrt [3]{x}\right )}{4 \sqrt [3]{c}}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{9-c x^2} \, dx,x,\frac {\left (1+\frac {\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}\right )^2}{\sqrt {c+d x}}\right )}{2 \sqrt [3]{c} d^{2/3}}-\frac {\sqrt [3]{d} \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x\right )}{4 \sqrt [3]{c}}+\frac {d^{4/3} \operatorname {Subst}\left (\int \frac {1}{-\frac {2 d^2}{c}-6 d^2 x^2} \, dx,x,\frac {1+\frac {\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}}{\sqrt {c+d x}}\right )}{c^{4/3}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {c} \left (1+\frac {\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}\right )}{\sqrt {c+d x}}\right )}{2 \sqrt {3} c^{5/6} d^{2/3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} \left (1+\frac {\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}\right )^2}{3 \sqrt {c+d x}}\right )}{6 c^{5/6} d^{2/3}}-\frac {\operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt [3]{c} d^{2/3}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {c} \left (1+\frac {\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}\right )}{\sqrt {c+d x}}\right )}{2 \sqrt {3} c^{5/6} d^{2/3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} \left (1+\frac {\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}\right )^2}{3 \sqrt {c+d x}}\right )}{6 c^{5/6} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x}}{3 \sqrt {c}}\right )}{6 c^{5/6} d^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 61, normalized size = 0.43 \[ \frac {3 x^{2/3} \sqrt {\frac {c+d x}{c}} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {d x}{c},\frac {d x}{8 c}\right )}{16 c \sqrt {c+d x}} \]
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 \[ \int -\frac {1}{\sqrt {d x + c} {\left (d x - 8 \, c\right )} x^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-d x +8 c \right ) \sqrt {d x +c}\, x^{\frac {1}{3}}}\, dx \]
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 \[ -\int \frac {1}{\sqrt {d x + c} {\left (d x - 8 \, c\right )} x^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^{1/3}\,\left (8\,c-d\,x\right )\,\sqrt {c+d\,x}} \,d x \]
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 \[ - \int \frac {1}{- 8 c \sqrt [3]{x} \sqrt {c + d x} + d x^{\frac {4}{3}} \sqrt {c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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