Optimal. Leaf size=128 \[ \frac {245 d^2}{1728 c^4 \sqrt {c+d x^3}}-\frac {1}{48 c^2 x^6 \sqrt {c+d x^3}}+\frac {3 d}{64 c^3 x^3 \sqrt {c+d x^3}}+\frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{20736 c^{9/2}}-\frac {109 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{768 c^{9/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {457, 105, 156,
157, 162, 65, 214, 212} \begin {gather*} \frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{20736 c^{9/2}}-\frac {109 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{768 c^{9/2}}+\frac {245 d^2}{1728 c^4 \sqrt {c+d x^3}}+\frac {3 d}{64 c^3 x^3 \sqrt {c+d x^3}}-\frac {1}{48 c^2 x^6 \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 105
Rule 156
Rule 157
Rule 162
Rule 212
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {1}{x^7 \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^3 (8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {1}{48 c^2 x^6 \sqrt {c+d x^3}}-\frac {\text {Subst}\left (\int \frac {18 c d-\frac {5 d^2 x}{2}}{x^2 (8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{48 c^2}\\ &=-\frac {1}{48 c^2 x^6 \sqrt {c+d x^3}}+\frac {3 d}{64 c^3 x^3 \sqrt {c+d x^3}}+\frac {\text {Subst}\left (\int \frac {218 c^2 d^2-27 c d^3 x}{x (8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{384 c^4}\\ &=\frac {245 d^2}{1728 c^4 \sqrt {c+d x^3}}-\frac {1}{48 c^2 x^6 \sqrt {c+d x^3}}+\frac {3 d}{64 c^3 x^3 \sqrt {c+d x^3}}+\frac {\text {Subst}\left (\int \frac {981 c^3 d^3-\frac {245}{2} c^2 d^4 x}{x (8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{1728 c^6 d}\\ &=\frac {245 d^2}{1728 c^4 \sqrt {c+d x^3}}-\frac {1}{48 c^2 x^6 \sqrt {c+d x^3}}+\frac {3 d}{64 c^3 x^3 \sqrt {c+d x^3}}+\frac {\left (109 d^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{1536 c^4}+\frac {d^3 \text {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{13824 c^4}\\ &=\frac {245 d^2}{1728 c^4 \sqrt {c+d x^3}}-\frac {1}{48 c^2 x^6 \sqrt {c+d x^3}}+\frac {3 d}{64 c^3 x^3 \sqrt {c+d x^3}}+\frac {(109 d) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{768 c^4}+\frac {d^2 \text {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{6912 c^4}\\ &=\frac {245 d^2}{1728 c^4 \sqrt {c+d x^3}}-\frac {1}{48 c^2 x^6 \sqrt {c+d x^3}}+\frac {3 d}{64 c^3 x^3 \sqrt {c+d x^3}}+\frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{20736 c^{9/2}}-\frac {109 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{768 c^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 100, normalized size = 0.78 \begin {gather*} \frac {\frac {12 \sqrt {c} \left (-36 c^2+81 c d x^3+245 d^2 x^6\right )}{x^6 \sqrt {c+d x^3}}+d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )-2943 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{20736 c^{9/2}} \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 = 636, normalized size = 4.97
method | result | size |
risch | \(-\frac {\sqrt {d \,x^{3}+c}\, \left (-13 d \,x^{3}+4 c \right )}{192 c^{4} x^{6}}+\frac {d^{2} \left (-\frac {109 \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 {256}{27 \sqrt {d \,x^{3}+c}}\right )}{128 c^{4}}\) | \(480\) |
default | \(\frac {-\frac {\sqrt {d \,x^{3}+c}}{6 c^{2} x^{6}}+\frac {7 d \sqrt {d \,x^{3}+c}}{12 c^{3} x^{3}}+\frac {2 d^{2}}{3 c^{3} \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}}-\frac {5 d^{2} \arctanh \left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{4 c^{\frac {7}{2}}}}{8 c}-\frac {d^{3} \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 )}{512 c^{3}}+\frac {d \left (-\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}}}\right )}{64 c^{2}}+\frac {d^{2} \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 )}{512 c^{3}}\) | \(636\) |
elliptic | \(\text {Expression too large to display}\) | \(1572\) |
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 = 2.99, size = 303, normalized size = 2.37 \begin {gather*} \left [\frac {{\left (d^{3} x^{9} + c d^{2} x^{6}\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 ) + 2943 \, {\left (d^{3} x^{9} + c d^{2} x^{6}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + 24 \, {\left (245 \, c d^{2} x^{6} + 81 \, c^{2} d x^{3} - 36 \, c^{3}\right )} \sqrt {d x^{3} + c}}{41472 \, {\left (c^{5} d x^{9} + c^{6} x^{6}\right )}}, \frac {2943 \, {\left (d^{3} x^{9} + c d^{2} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) - {\left (d^{3} x^{9} + c d^{2} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 12 \, {\left (245 \, c d^{2} x^{6} + 81 \, c^{2} d x^{3} - 36 \, c^{3}\right )} \sqrt {d x^{3} + c}}{20736 \, {\left (c^{5} d x^{9} + c^{6} x^{6}\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^{7} \sqrt {c + d x^{3}} - 7 c d x^{10} \sqrt {c + d x^{3}} + d^{2} x^{13} \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 = 0.78, size = 118, normalized size = 0.92 \begin {gather*} \frac {109 \, d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{768 \, \sqrt {-c} c^{4}} - \frac {d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{20736 \, \sqrt {-c} c^{4}} + \frac {2 \, d^{2}}{27 \, \sqrt {d x^{3} + c} c^{4}} + \frac {13 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} d^{2} - 17 \, \sqrt {d x^{3} + c} c d^{2}}{192 \, c^{4} d^{2} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.03, size = 112, normalized size = 0.88 \begin {gather*} \frac {245\,d^2}{1728\,c^4\,\sqrt {d\,x^3+c}}-\frac {109\,d^2\,\mathrm {atanh}\left (\frac {c^4\,\sqrt {d\,x^3+c}}{\sqrt {c^9}}\right )}{768\,\sqrt {c^9}}+\frac {d^2\,\mathrm {atanh}\left (\frac {c^4\,\sqrt {d\,x^3+c}}{3\,\sqrt {c^9}}\right )}{20736\,\sqrt {c^9}}-\frac {1}{48\,c^2\,x^6\,\sqrt {d\,x^3+c}}+\frac {3\,d}{64\,c^3\,x^3\,\sqrt {d\,x^3+c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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