Optimal. Leaf size=83 \[ \frac {64 c \sqrt {c+d x^3}}{27 d^3 \left (8 c-d x^3\right )}+\frac {2 \sqrt {c+d x^3}}{3 d^3}-\frac {224 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 d^3} \]
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Rubi [A] time = 0.06, antiderivative size = 83, 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, 89, 80, 63, 206} \begin {gather*} \frac {64 c \sqrt {c+d x^3}}{27 d^3 \left (8 c-d x^3\right )}+\frac {2 \sqrt {c+d x^3}}{3 d^3}-\frac {224 \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 63
Rule 80
Rule 89
Rule 206
Rule 446
Rubi steps
\begin {align*} \int \frac {x^8}{\left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{(8 c-d x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )\\ &=\frac {64 c \sqrt {c+d x^3}}{27 d^3 \left (8 c-d x^3\right )}-\frac {\operatorname {Subst}\left (\int \frac {40 c^2 d+9 c d^2 x}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{27 c d^3}\\ &=\frac {2 \sqrt {c+d x^3}}{3 d^3}+\frac {64 c \sqrt {c+d x^3}}{27 d^3 \left (8 c-d x^3\right )}-\frac {(112 c) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{27 d^2}\\ &=\frac {2 \sqrt {c+d x^3}}{3 d^3}+\frac {64 c \sqrt {c+d x^3}}{27 d^3 \left (8 c-d x^3\right )}-\frac {(224 c) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{27 d^3}\\ &=\frac {2 \sqrt {c+d x^3}}{3 d^3}+\frac {64 c \sqrt {c+d x^3}}{27 d^3 \left (8 c-d x^3\right )}-\frac {224 \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 = 0.05, size = 82, normalized size = 0.99 \begin {gather*} -\frac {64 c \sqrt {c+d x^3}}{27 d^3 \left (d x^3-8 c\right )}+\frac {2 \sqrt {c+d x^3}}{3 d^3}-\frac {224 \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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IntegrateAlgebraic [A] time = 0.07, size = 73, normalized size = 0.88 \begin {gather*} -\frac {2 \sqrt {c+d x^3} \left (104 c-9 d x^3\right )}{27 d^3 \left (d x^3-8 c\right )}-\frac {224 \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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fricas [A] time = 0.43, size = 167, normalized size = 2.01 \begin {gather*} \left [\frac {2 \, {\left (56 \, {\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 ) + 3 \, {\left (9 \, d x^{3} - 104 \, c\right )} \sqrt {d x^{3} + c}\right )}}{81 \, {\left (d^{4} x^{3} - 8 \, c d^{3}\right )}}, \frac {2 \, {\left (112 \, {\left (d x^{3} - 8 \, c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 3 \, {\left (9 \, d x^{3} - 104 \, c\right )} \sqrt {d x^{3} + c}\right )}}{81 \, {\left (d^{4} x^{3} - 8 \, c d^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 69, normalized size = 0.83 \begin {gather*} \frac {224 \, 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 {64 \, \sqrt {d x^{3} + c} c}{27 \, {\left (d x^{3} - 8 \, c\right )} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.20, size = 874, normalized size = 10.53
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.56, size = 79, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (56 \, \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 {96 \, \sqrt {d x^{3} + c} c}{d x^{3} - 8 \, c}\right )}}{81 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.00, size = 87, normalized size = 1.05 \begin {gather*} \frac {2\,\sqrt {d\,x^3+c}}{3\,d^3}+\frac {112\,\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 {64\,c\,\sqrt {d\,x^3+c}}{27\,d^3\,\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^{8}}{\left (- 8 c + d x^{3}\right )^{2} \sqrt {c + d x^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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