Optimal. Leaf size=82 \[ \frac {8 \left (c+d x^3\right )^{3/2}}{27 d^2 \left (8 c-d x^3\right )}+\frac {26 \sqrt {c+d x^3}}{27 d^2}-\frac {26 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^2} \]
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Rubi [A] time = 0.06, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {446, 78, 50, 63, 206} \begin {gather*} \frac {8 \left (c+d x^3\right )^{3/2}}{27 d^2 \left (8 c-d x^3\right )}+\frac {26 \sqrt {c+d x^3}}{27 d^2}-\frac {26 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 206
Rule 446
Rubi steps
\begin {align*} \int \frac {x^5 \sqrt {c+d x^3}}{\left (8 c-d x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x \sqrt {c+d x}}{(8 c-d x)^2} \, dx,x,x^3\right )\\ &=\frac {8 \left (c+d x^3\right )^{3/2}}{27 d^2 \left (8 c-d x^3\right )}-\frac {13 \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{8 c-d x} \, dx,x,x^3\right )}{27 d}\\ &=\frac {26 \sqrt {c+d x^3}}{27 d^2}+\frac {8 \left (c+d x^3\right )^{3/2}}{27 d^2 \left (8 c-d x^3\right )}-\frac {(13 c) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{3 d}\\ &=\frac {26 \sqrt {c+d x^3}}{27 d^2}+\frac {8 \left (c+d x^3\right )^{3/2}}{27 d^2 \left (8 c-d x^3\right )}-\frac {(26 c) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{3 d^2}\\ &=\frac {26 \sqrt {c+d x^3}}{27 d^2}+\frac {8 \left (c+d x^3\right )^{3/2}}{27 d^2 \left (8 c-d x^3\right )}-\frac {26 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 79, normalized size = 0.96 \begin {gather*} \frac {6 \sqrt {c+d x^3} \left (d x^3-12 c\right )+26 \sqrt {c} \left (8 c-d x^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^2 \left (d x^3-8 c\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 73, normalized size = 0.89 \begin {gather*} -\frac {2 \sqrt {c+d x^3} \left (12 c-d x^3\right )}{3 d^2 \left (d x^3-8 c\right )}-\frac {26 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 165, normalized size = 2.01 \begin {gather*} \left [\frac {13 \, {\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 ) + 6 \, \sqrt {d x^{3} + c} {\left (d x^{3} - 12 \, c\right )}}{9 \, {\left (d^{3} x^{3} - 8 \, c d^{2}\right )}}, \frac {2 \, {\left (13 \, {\left (d x^{3} - 8 \, c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 3 \, \sqrt {d x^{3} + c} {\left (d x^{3} - 12 \, c\right )}\right )}}{9 \, {\left (d^{3} x^{3} - 8 \, c d^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 69, normalized size = 0.84 \begin {gather*} \frac {26 \, c \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{9 \, \sqrt {-c} d^{2}} + \frac {2 \, \sqrt {d x^{3} + c}}{3 \, d^{2}} - \frac {8 \, \sqrt {d x^{3} + c} c}{3 \, {\left (d x^{3} - 8 \, c\right )} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.17, size = 874, normalized size = 10.66
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, size = 79, normalized size = 0.96 \begin {gather*} \frac {13 \, \sqrt {c} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 6 \, \sqrt {d x^{3} + c} - \frac {24 \, \sqrt {d x^{3} + c} c}{d x^{3} - 8 \, c}}{9 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.99, size = 87, normalized size = 1.06 \begin {gather*} \frac {2\,\sqrt {d\,x^3+c}}{3\,d^2}+\frac {13\,\sqrt {c}\,\ln \left (\frac {10\,c+d\,x^3-6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{9\,d^2}+\frac {8\,c\,\sqrt {d\,x^3+c}}{3\,d^2\,\left (8\,c-d\,x^3\right )} \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 {x^{5} \sqrt {c + d x^{3}}}{\left (- 8 c + d x^{3}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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