Optimal. Leaf size=100 \[ -\frac {25 d}{216 c^3 \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \sqrt {c+d x^3}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{2592 c^{7/2}}+\frac {11 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{96 c^{7/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {457, 105, 157,
162, 65, 214, 212} \begin {gather*} \frac {d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{2592 c^{7/2}}+\frac {11 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{96 c^{7/2}}-\frac {25 d}{216 c^3 \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 105
Rule 157
Rule 162
Rule 212
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x^2 (8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {1}{24 c^2 x^3 \sqrt {c+d x^3}}-\frac {\text {Subst}\left (\int \frac {11 c d-\frac {3 d^2 x}{2}}{x (8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{24 c^2}\\ &=-\frac {25 d}{216 c^3 \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \sqrt {c+d x^3}}-\frac {\text {Subst}\left (\int \frac {\frac {99 c^2 d^2}{2}-\frac {25}{4} c d^3 x}{x (8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{108 c^4 d}\\ &=-\frac {25 d}{216 c^3 \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \sqrt {c+d x^3}}-\frac {(11 d) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{192 c^3}+\frac {d^2 \text {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{1728 c^3}\\ &=-\frac {25 d}{216 c^3 \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \sqrt {c+d x^3}}-\frac {11 \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{96 c^3}+\frac {d \text {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{864 c^3}\\ &=-\frac {25 d}{216 c^3 \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \sqrt {c+d x^3}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{2592 c^{7/2}}+\frac {11 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{96 c^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 85, normalized size = 0.85 \begin {gather*} \frac {-\frac {12 \sqrt {c} \left (9 c+25 d x^3\right )}{x^3 \sqrt {c+d x^3}}+d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )+297 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{2592 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.44, size = 549, normalized size = 5.49
method | result | size |
risch | \(-\frac {\sqrt {d \,x^{3}+c}}{24 c^{3} x^{3}}-\frac {d \left (-\frac {11 \arctanh \left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )}{\sum }\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d}{18 d c}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \sqrt {d \,x^{3}+c}}\right )}{972 d^{2} c}+\frac {32}{27 \sqrt {d \,x^{3}+c}}\right )}{16 c^{3}}\) | \(468\) |
default | \(-\frac {d^{2} \left (\frac {2}{27 d c \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )}{\sum }\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d}{18 d c}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \sqrt {d \,x^{3}+c}}\right )}{243 d^{3} c^{2}}\right )}{64 c^{2}}+\frac {-\frac {2 d}{3 c^{2} \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}}-\frac {\sqrt {d \,x^{3}+c}}{3 c^{2} x^{3}}+\frac {d \arctanh \left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{c^{\frac {5}{2}}}}{8 c}+\frac {d \left (\frac {2}{3 c \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}}-\frac {2 \arctanh \left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3 c^{\frac {3}{2}}}\right )}{64 c^{2}}\) | \(549\) |
elliptic | \(\text {Expression too large to display}\) | \(1542\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.45, size = 272, normalized size = 2.72 \begin {gather*} \left [\frac {{\left (d^{2} x^{6} + c d x^{3}\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 ) + 297 \, {\left (d^{2} x^{6} + c d x^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} + 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) - 24 \, {\left (25 \, c d x^{3} + 9 \, c^{2}\right )} \sqrt {d x^{3} + c}}{5184 \, {\left (c^{4} d x^{6} + c^{5} x^{3}\right )}}, -\frac {297 \, {\left (d^{2} x^{6} + c d x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + {\left (d^{2} x^{6} + c d x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 12 \, {\left (25 \, c d x^{3} + 9 \, c^{2}\right )} \sqrt {d x^{3} + c}}{2592 \, {\left (c^{4} d x^{6} + c^{5} x^{3}\right )}}\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 {1}{- 8 c^{2} x^{4} \sqrt {c + d x^{3}} - 7 c d x^{7} \sqrt {c + d x^{3}} + d^{2} x^{10} \sqrt {c + d x^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.79, size = 100, normalized size = 1.00 \begin {gather*} -\frac {11 \, d \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{96 \, \sqrt {-c} c^{3}} - \frac {d \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{2592 \, \sqrt {-c} c^{3}} - \frac {25 \, {\left (d x^{3} + c\right )} d - 16 \, c d}{216 \, {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} - \sqrt {d x^{3} + c} c\right )} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.80, size = 88, normalized size = 0.88 \begin {gather*} \frac {11\,d\,\mathrm {atanh}\left (\frac {c^3\,\sqrt {d\,x^3+c}}{\sqrt {c^7}}\right )}{96\,\sqrt {c^7}}-\frac {25\,d}{216\,c^3\,\sqrt {d\,x^3+c}}+\frac {d\,\mathrm {atanh}\left (\frac {c^3\,\sqrt {d\,x^3+c}}{3\,\sqrt {c^7}}\right )}{2592\,\sqrt {c^7}}-\frac {1}{24\,c^2\,x^3\,\sqrt {d\,x^3+c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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