Optimal. Leaf size=88 \[ \frac {5 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{288 c^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{96 c^{3/2}}+\frac {\sqrt {c+d x^3}}{24 c \left (8 c-d x^3\right )} \]
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Rubi [A] time = 0.07, antiderivative size = 88, 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, 99, 156, 63, 208, 206} \begin {gather*} \frac {5 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{288 c^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{96 c^{3/2}}+\frac {\sqrt {c+d x^3}}{24 c \left (8 c-d x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 99
Rule 156
Rule 206
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^3}}{x \left (8 c-d x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{x (8 c-d x)^2} \, dx,x,x^3\right )\\ &=\frac {\sqrt {c+d x^3}}{24 c \left (8 c-d x^3\right )}-\frac {\operatorname {Subst}\left (\int \frac {-c-\frac {d x}{2}}{x (8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{24 c}\\ &=\frac {\sqrt {c+d x^3}}{24 c \left (8 c-d x^3\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{192 c}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{192 c}\\ &=\frac {\sqrt {c+d x^3}}{24 c \left (8 c-d x^3\right )}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{96 c}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{96 c d}\\ &=\frac {\sqrt {c+d x^3}}{24 c \left (8 c-d x^3\right )}+\frac {5 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{288 c^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{96 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 102, normalized size = 1.16 \begin {gather*} \frac {12 \sqrt {c} \sqrt {c+d x^3}+5 \left (8 c-d x^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )+3 \left (d x^3-8 c\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{288 c^{3/2} \left (8 c-d x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 88, normalized size = 1.00 \begin {gather*} \frac {5 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{288 c^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{96 c^{3/2}}+\frac {\sqrt {c+d x^3}}{24 c \left (8 c-d x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 226, normalized size = 2.57 \begin {gather*} \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 ) + 3 \, {\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 ) - 24 \, \sqrt {d x^{3} + c} c}{576 \, {\left (c^{2} d x^{3} - 8 \, c^{3}\right )}}, \frac {3 \, {\left (d x^{3} - 8 \, c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) - 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}{288 \, {\left (c^{2} d x^{3} - 8 \, c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 79, normalized size = 0.90 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{96 \, \sqrt {-c} c} - \frac {5 \, \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{288 \, \sqrt {-c} c} - \frac {\sqrt {d x^{3} + c}}{24 \, {\left (d x^{3} - 8 \, c\right )} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 912, normalized size = 10.36
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 {\sqrt {d x^{3} + c}}{{\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 = 3.98, size = 76, normalized size = 0.86 \begin {gather*} \frac {5\,\mathrm {atanh}\left (\frac {c\,\sqrt {d\,x^3+c}}{3\,\sqrt {c^3}}\right )}{288\,\sqrt {c^3}}-\frac {\mathrm {atanh}\left (\frac {c\,\sqrt {d\,x^3+c}}{\sqrt {c^3}}\right )}{96\,\sqrt {c^3}}+\frac {\sqrt {d\,x^3+c}}{8\,c\,\left (24\,c-3\,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 {\sqrt {c + d x^{3}}}{x \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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