Optimal. Leaf size=85 \[ \frac {3 \sqrt {c+d x^3}}{8 \left (8 c-d x^3\right )}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{32 \sqrt {c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{96 \sqrt {c}} \]
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Rubi [A] time = 0.08, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {446, 98, 156, 63, 208, 206} \begin {gather*} \frac {3 \sqrt {c+d x^3}}{8 \left (8 c-d x^3\right )}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{32 \sqrt {c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{96 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 156
Rule 206
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (c+d x^3\right )^{3/2}}{x \left (8 c-d x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{x (8 c-d x)^2} \, dx,x,x^3\right )\\ &=\frac {3 \sqrt {c+d x^3}}{8 \left (8 c-d x^3\right )}-\frac {\operatorname {Subst}\left (\int \frac {-c^2 d+\frac {7}{2} c d^2 x}{x (8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{24 c d}\\ &=\frac {3 \sqrt {c+d x^3}}{8 \left (8 c-d x^3\right )}+\frac {1}{192} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )-\frac {1}{64} (9 d) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )\\ &=\frac {3 \sqrt {c+d x^3}}{8 \left (8 c-d x^3\right )}-\frac {9}{32} \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{96 d}\\ &=\frac {3 \sqrt {c+d x^3}}{8 \left (8 c-d x^3\right )}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{32 \sqrt {c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{96 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 100, normalized size = 1.18 \begin {gather*} \frac {36 \sqrt {c} \sqrt {c+d x^3}+\left (9 d x^3-72 c\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )+\left (d x^3-8 c\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{96 \sqrt {c} \left (8 c-d x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 85, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt {c+d x^3}}{8 \left (8 c-d x^3\right )}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{32 \sqrt {c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{96 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 220, normalized size = 2.59 \begin {gather*} \left [\frac {9 \, {\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 ) + {\left (d x^{3} - 8 \, c\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) - 72 \, \sqrt {d x^{3} + c} c}{192 \, {\left (c d x^{3} - 8 \, c^{2}\right )}}, \frac {{\left (d x^{3} - 8 \, c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + 9 \, {\left (d x^{3} - 8 \, c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) - 36 \, \sqrt {d x^{3} + c} c}{96 \, {\left (c d x^{3} - 8 \, c^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 70, normalized size = 0.82 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{96 \, \sqrt {-c}} + \frac {3 \, \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{32 \, \sqrt {-c}} - \frac {3 \, \sqrt {d x^{3} + c}}{8 \, {\left (d x^{3} - 8 \, c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.17, size = 956, normalized size = 11.25
result too large to display
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 \begin {gather*} \int \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (d x^{3} - 8 \, c\right )}^{2} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.72, size = 101, normalized size = 1.19 \begin {gather*} \frac {3\,\sqrt {d\,x^3+c}}{8\,\left (8\,c-d\,x^3\right )}+\frac {\ln \left (\frac {{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^3\,\left (\sqrt {d\,x^3+c}+\sqrt {c}\right )\,{\left (10\,c+d\,x^3-6\,\sqrt {c}\,\sqrt {d\,x^3+c}\right )}^9}{x^6\,{\left (8\,c-d\,x^3\right )}^9}\right )}{192\,\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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