Optimal. Leaf size=97 \[ -\frac {42 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^2}+\frac {8 \left (c+d x^3\right )^{5/2}}{27 d^2 \left (8 c-d x^3\right )}+\frac {14 \left (c+d x^3\right )^{3/2}}{27 d^2}+\frac {14 c \sqrt {c+d x^3}}{d^2} \]
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Rubi [A] time = 0.07, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {42 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^2}+\frac {8 \left (c+d x^3\right )^{5/2}}{27 d^2 \left (8 c-d x^3\right )}+\frac {14 \left (c+d x^3\right )^{3/2}}{27 d^2}+\frac {14 c \sqrt {c+d x^3}}{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 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x (c+d x)^{3/2}}{(8 c-d x)^2} \, dx,x,x^3\right )\\ &=\frac {8 \left (c+d x^3\right )^{5/2}}{27 d^2 \left (8 c-d x^3\right )}-\frac {7 \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{8 c-d x} \, dx,x,x^3\right )}{9 d}\\ &=\frac {14 \left (c+d x^3\right )^{3/2}}{27 d^2}+\frac {8 \left (c+d x^3\right )^{5/2}}{27 d^2 \left (8 c-d x^3\right )}-\frac {(7 c) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{8 c-d x} \, dx,x,x^3\right )}{d}\\ &=\frac {14 c \sqrt {c+d x^3}}{d^2}+\frac {14 \left (c+d x^3\right )^{3/2}}{27 d^2}+\frac {8 \left (c+d x^3\right )^{5/2}}{27 d^2 \left (8 c-d x^3\right )}-\frac {\left (63 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{d}\\ &=\frac {14 c \sqrt {c+d x^3}}{d^2}+\frac {14 \left (c+d x^3\right )^{3/2}}{27 d^2}+\frac {8 \left (c+d x^3\right )^{5/2}}{27 d^2 \left (8 c-d x^3\right )}-\frac {\left (126 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{d^2}\\ &=\frac {14 c \sqrt {c+d x^3}}{d^2}+\frac {14 \left (c+d x^3\right )^{3/2}}{27 d^2}+\frac {8 \left (c+d x^3\right )^{5/2}}{27 d^2 \left (8 c-d x^3\right )}-\frac {42 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 90, normalized size = 0.93 \begin {gather*} \frac {378 c^{3/2} \left (8 c-d x^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )+2 \sqrt {c+d x^3} \left (-524 c^2+44 c d x^3+d^2 x^6\right )}{9 d^2 \left (d x^3-8 c\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 82, normalized size = 0.85 \begin {gather*} -\frac {42 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^2}-\frac {2 \sqrt {c+d x^3} \left (524 c^2-44 c d x^3-d^2 x^6\right )}{9 d^2 \left (d x^3-8 c\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 192, normalized size = 1.98 \begin {gather*} \left [\frac {189 \, {\left (c d x^{3} - 8 \, c^{2}\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 ) + 2 \, {\left (d^{2} x^{6} + 44 \, c d x^{3} - 524 \, c^{2}\right )} \sqrt {d x^{3} + c}}{9 \, {\left (d^{3} x^{3} - 8 \, c d^{2}\right )}}, \frac {2 \, {\left (189 \, {\left (c d x^{3} - 8 \, c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + {\left (d^{2} x^{6} + 44 \, c d x^{3} - 524 \, c^{2}\right )} \sqrt {d x^{3} + c}\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 = 0.17, size = 93, normalized size = 0.96 \begin {gather*} \frac {42 \, c^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{\sqrt {-c} d^{2}} - \frac {24 \, \sqrt {d x^{3} + c} c^{2}}{{\left (d x^{3} - 8 \, c\right )} d^{2}} + \frac {2 \, {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} d^{4} + 51 \, \sqrt {d x^{3} + c} c d^{4}\right )}}{9 \, d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.17, size = 902, normalized size = 9.30
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 93, normalized size = 0.96 \begin {gather*} \frac {189 \, c^{\frac {3}{2}} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 2 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} + 102 \, \sqrt {d x^{3} + c} c - \frac {216 \, \sqrt {d x^{3} + c} c^{2}}{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 = 4.04, size = 107, normalized size = 1.10 \begin {gather*} \frac {104\,c\,\sqrt {d\,x^3+c}}{9\,d^2}+\frac {21\,c^{3/2}\,\ln \left (\frac {10\,c+d\,x^3-6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{d^2}+\frac {2\,x^3\,\sqrt {d\,x^3+c}}{9\,d}+\frac {24\,c^2\,\sqrt {d\,x^3+c}}{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(-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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