Optimal. Leaf size=100 \[ \frac {d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{2592 c^{7/2}}+\frac {11 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{96 c^{7/2}}-\frac {25 d}{216 c^3 \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \sqrt {c+d x^3}} \]
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Rubi [A] time = 0.10, 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 = {446, 103, 152, 156, 63, 208, 206} \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 103
Rule 152
Rule 156
Rule 206
Rule 208
Rule 446
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} \operatorname {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 {\operatorname {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 {\operatorname {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) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{192 c^3}+\frac {d^2 \operatorname {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 \operatorname {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 \operatorname {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 [C] time = 0.04, size = 77, normalized size = 0.77 \begin {gather*} \frac {-d x^3 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d x^3+c}{9 c}\right )-99 d x^3 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d x^3}{c}+1\right )-36 c}{864 c^3 x^3 \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 91, normalized size = 0.91 \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 {-9 c-25 d x^3}{216 c^3 x^3 \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, 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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giac [A] time = 0.18, 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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maple [C] time = 0.19, size = 549, normalized size = 5.49 \begin {gather*} -\frac {\left (\frac {2}{27 \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}\, c d}+\frac {i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \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 ) d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {\left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right ) d}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i \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 ) d}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (2 \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )^{2} d^{2}+i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right ) d -\left (-c \,d^{2}\right )^{\frac {1}{3}} \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right ) d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}-\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}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )^{2} d +i \sqrt {3}\, c d -3 c d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )-3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )}{18 c d}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{\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 ) d}}\right )}{243 c^{2} d^{3} \sqrt {d \,x^{3}+c}}\right ) d^{2}}{64 c^{2}}+\frac {\frac {d \arctanh \left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{c^{\frac {5}{2}}}-\frac {2 d}{3 \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}\, c^{2}}-\frac {\sqrt {d \,x^{3}+c}}{3 c^{2} x^{3}}}{8 c}+\frac {\left (-\frac {2 \arctanh \left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3 c^{\frac {3}{2}}}+\frac {2}{3 \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}\, c}\right ) d}{64 c^{2}} \end {gather*}
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 {1}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} {\left (d x^{3} - 8 \, c\right )} x^{4}}\,{d x} \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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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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