Optimal. Leaf size=71 \[ -\frac {2 c}{27 d^3 \sqrt {c+d x^3}}-\frac {2 \sqrt {c+d x^3}}{3 d^3}+\frac {128 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 d^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {457, 89, 65,
212} \begin {gather*} -\frac {2 c}{27 d^3 \sqrt {c+d x^3}}-\frac {2 \sqrt {c+d x^3}}{3 d^3}+\frac {128 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 89
Rule 212
Rule 457
Rubi steps
\begin {align*} \int \frac {x^8}{\left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{(8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {c}{9 d^2 (c+d x)^{3/2}}-\frac {1}{d^2 \sqrt {c+d x}}+\frac {64 c}{9 d^2 (8 c-d x) \sqrt {c+d x}}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 c}{27 d^3 \sqrt {c+d x^3}}-\frac {2 \sqrt {c+d x^3}}{3 d^3}+\frac {(64 c) \text {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{27 d^2}\\ &=-\frac {2 c}{27 d^3 \sqrt {c+d x^3}}-\frac {2 \sqrt {c+d x^3}}{3 d^3}+\frac {(128 c) \text {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{27 d^3}\\ &=-\frac {2 c}{27 d^3 \sqrt {c+d x^3}}-\frac {2 \sqrt {c+d x^3}}{3 d^3}+\frac {128 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 59, normalized size = 0.83 \begin {gather*} \frac {2 \left (-\frac {3 \left (10 c+9 d x^3\right )}{\sqrt {c+d x^3}}+64 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )\right )}{81 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.42, size = 498, normalized size = 7.01 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 68, normalized size = 0.96 \begin {gather*} -\frac {2 \, {\left (32 \, \sqrt {c} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 27 \, \sqrt {d x^{3} + c} + \frac {3 \, c}{\sqrt {d x^{3} + c}}\right )}}{81 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.48, size = 161, normalized size = 2.27 \begin {gather*} \left [\frac {2 \, {\left (32 \, {\left (d x^{3} + c\right )} \sqrt {c} \log \left (\frac {d x^{3} + 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) - 3 \, {\left (9 \, d x^{3} + 10 \, c\right )} \sqrt {d x^{3} + c}\right )}}{81 \, {\left (d^{4} x^{3} + c d^{3}\right )}}, -\frac {2 \, {\left (64 \, {\left (d x^{3} + c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 3 \, {\left (9 \, d x^{3} + 10 \, c\right )} \sqrt {d x^{3} + c}\right )}}{81 \, {\left (d^{4} x^{3} + c d^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [A]
time = 0.75, size = 58, normalized size = 0.82 \begin {gather*} -\frac {128 \, c \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{81 \, \sqrt {-c} d^{3}} - \frac {2 \, \sqrt {d x^{3} + c}}{3 \, d^{3}} - \frac {2 \, c}{27 \, \sqrt {d x^{3} + c} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.71, size = 75, normalized size = 1.06 \begin {gather*} \frac {64\,\sqrt {c}\,\ln \left (\frac {10\,c+d\,x^3+6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{81\,d^3}-\frac {2\,c}{27\,d^3\,\sqrt {d\,x^3+c}}-\frac {2\,\sqrt {d\,x^3+c}}{3\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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