Optimal. Leaf size=202 \[ \frac {2 \tanh ^{-1}\left (\frac {(c-2 d x)^2}{3 \sqrt {c} \sqrt {c^3-8 d^3 x^3}}\right )}{9 \sqrt {c} d^2}-\frac {\sqrt {2+\sqrt {3}} (c-2 d x) \sqrt {\frac {c^2+2 c d x+4 d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) c-2 d x}{\left (1+\sqrt {3}\right ) c-2 d x}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d^2 \sqrt {\frac {c (c-2 d x)}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} \sqrt {c^3-8 d^3 x^3}} \]
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Rubi [A] time = 0.26, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2139, 218, 2138, 206} \[ \frac {2 \tanh ^{-1}\left (\frac {(c-2 d x)^2}{3 \sqrt {c} \sqrt {c^3-8 d^3 x^3}}\right )}{9 \sqrt {c} d^2}-\frac {\sqrt {2+\sqrt {3}} (c-2 d x) \sqrt {\frac {c^2+2 c d x+4 d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) c-2 d x}{\left (1+\sqrt {3}\right ) c-2 d x}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d^2 \sqrt {\frac {c (c-2 d x)}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} \sqrt {c^3-8 d^3 x^3}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 218
Rule 2138
Rule 2139
Rubi steps
\begin {align*} \int \frac {x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx &=\frac {\int \frac {1}{\sqrt {c^3-8 d^3 x^3}} \, dx}{3 d}-\frac {\int \frac {c-2 d x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx}{3 d}\\ &=-\frac {\sqrt {2+\sqrt {3}} (c-2 d x) \sqrt {\frac {c^2+2 c d x+4 d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) c-2 d x}{\left (1+\sqrt {3}\right ) c-2 d x}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d^2 \sqrt {\frac {c (c-2 d x)}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} \sqrt {c^3-8 d^3 x^3}}+\frac {(2 c) \operatorname {Subst}\left (\int \frac {1}{9-c^3 x^2} \, dx,x,\frac {\left (1-\frac {2 d x}{c}\right )^2}{\sqrt {c^3-8 d^3 x^3}}\right )}{3 d^2}\\ &=\frac {2 \tanh ^{-1}\left (\frac {(c-2 d x)^2}{3 \sqrt {c} \sqrt {c^3-8 d^3 x^3}}\right )}{9 \sqrt {c} d^2}-\frac {\sqrt {2+\sqrt {3}} (c-2 d x) \sqrt {\frac {c^2+2 c d x+4 d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) c-2 d x}{\left (1+\sqrt {3}\right ) c-2 d x}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d^2 \sqrt {\frac {c (c-2 d x)}{\left (\left (1+\sqrt {3}\right ) c-2 d x\right )^2}} \sqrt {c^3-8 d^3 x^3}}\\ \end {align*}
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Mathematica [C] time = 0.67, size = 295, normalized size = 1.46 \[ \frac {\sqrt {\frac {c-2 d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (\left (\sqrt [3]{-1}-2\right ) \left (\sqrt [3]{-1} c+2 d x\right ) \sqrt {\frac {\sqrt [3]{-1} \left (c+2 \sqrt [3]{-1} d x\right )}{\left (1+\sqrt [3]{-1}\right ) c}} F\left (\sin ^{-1}\left (\sqrt {\frac {c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}\right )|\sqrt [3]{-1}\right )+\frac {2 \sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) c \sqrt {\frac {c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt {\frac {c^2+2 c d x+4 d^2 x^2}{c^2}} \Pi \left (\frac {2 \sqrt {3}}{3 i+\sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}\right )|\sqrt [3]{-1}\right )}{\sqrt {3}}\right )}{\left (\sqrt [3]{-1}-2\right ) d^2 \sqrt {\frac {c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt {c^3-8 d^3 x^3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-8 \, d^{3} x^{3} + c^{3}} x}{8 \, d^{4} x^{4} + 8 \, c d^{3} x^{3} - c^{3} d x - c^{4}}, x\right ) \]
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 \[ \int \frac {x}{\sqrt {-8 \, d^{3} x^{3} + c^{3}} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 653, normalized size = 3.23 \[ -\frac {2 \left (\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}\right ) \sqrt {\frac {x -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, c \EllipticPi \left (\sqrt {\frac {x -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}, \frac {\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}+\frac {c}{d}}, \sqrt {\frac {\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}\right )}{\sqrt {-8 d^{3} x^{3}+c^{3}}\, \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}+\frac {c}{d}\right ) d^{2}}+\frac {2 \left (\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}\right ) \sqrt {\frac {x -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \EllipticF \left (\sqrt {\frac {x -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}, \sqrt {\frac {\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}\right )}{\sqrt {-8 d^{3} x^{3}+c^{3}}\, d} \]
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 \[ \int \frac {x}{\sqrt {-8 \, d^{3} x^{3} + c^{3}} {\left (d x + c\right )}}\,{d x} \]
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 \[ \int \frac {x}{\sqrt {c^3-8\,d^3\,x^3}\,\left (c+d\,x\right )} \,d x \]
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 \[ \int \frac {x}{\sqrt {- \left (- c + 2 d x\right ) \left (c^{2} + 2 c d x + 4 d^{2} x^{2}\right )} \left (c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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