Optimal. Leaf size=64 \[ \frac {8 \sqrt {c+d x^3}}{27 d^2 \left (8 c-d x^3\right )}-\frac {10 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 \sqrt {c} d^2} \]
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Rubi [A] time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {446, 78, 63, 206} \[ \frac {8 \sqrt {c+d x^3}}{27 d^2 \left (8 c-d x^3\right )}-\frac {10 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 \sqrt {c} d^2} \]
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 206
Rule 446
Rubi steps
\begin {align*} \int \frac {x^5}{\left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{(8 c-d x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )\\ &=\frac {8 \sqrt {c+d x^3}}{27 d^2 \left (8 c-d x^3\right )}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{27 d}\\ &=\frac {8 \sqrt {c+d x^3}}{27 d^2 \left (8 c-d x^3\right )}-\frac {10 \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{27 d^2}\\ &=\frac {8 \sqrt {c+d x^3}}{27 d^2 \left (8 c-d x^3\right )}-\frac {10 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 \sqrt {c} d^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 63, normalized size = 0.98 \[ -\frac {8 \sqrt {c+d x^3}}{27 d^2 \left (d x^3-8 c\right )}-\frac {10 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 \sqrt {c} d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 155, normalized size = 2.42 \[ \left [\frac {5 \, {\left (d x^{3} - 8 \, c\right )} \sqrt {c} \log \left (\frac {d x^{3} - 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) - 24 \, \sqrt {d x^{3} + c} c}{81 \, {\left (c d^{3} x^{3} - 8 \, c^{2} d^{2}\right )}}, \frac {2 \, {\left (5 \, {\left (d x^{3} - 8 \, c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) - 12 \, \sqrt {d x^{3} + c} c\right )}}{81 \, {\left (c d^{3} x^{3} - 8 \, c^{2} d^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 58, normalized size = 0.91 \[ \frac {2 \, {\left (\frac {5 \, \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{\sqrt {-c} d} - \frac {12 \, \sqrt {d x^{3} + c}}{{\left (d x^{3} - 8 \, c\right )} d}\right )}}{81 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.17, size = 861, normalized size = 13.45 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 67, normalized size = 1.05 \[ \frac {\frac {5 \, \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right )}{\sqrt {c}} - \frac {24 \, \sqrt {d x^{3} + c}}{d x^{3} - 8 \, c}}{81 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.01, size = 72, normalized size = 1.12 \[ \frac {5\,\ln \left (\frac {10\,c+d\,x^3-6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{81\,\sqrt {c}\,d^2}+\frac {8\,\sqrt {d\,x^3+c}}{27\,d^2\,\left (8\,c-d\,x^3\right )} \]
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 \[ \int \frac {x^{5}}{\left (- 8 c + d x^{3}\right )^{2} \sqrt {c + d x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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