Optimal. Leaf size=128 \[ \frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{20736 c^{9/2}}-\frac {109 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{768 c^{9/2}}+\frac {245 d^2}{1728 c^4 \sqrt {c+d x^3}}+\frac {3 d}{64 c^3 x^3 \sqrt {c+d x^3}}-\frac {1}{48 c^2 x^6 \sqrt {c+d x^3}} \]
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Rubi [A] time = 0.12, 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 = {446, 103, 151, 152, 156, 63, 208, 206} \[ \frac {245 d^2}{1728 c^4 \sqrt {c+d x^3}}+\frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{20736 c^{9/2}}-\frac {109 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{768 c^{9/2}}+\frac {3 d}{64 c^3 x^3 \sqrt {c+d x^3}}-\frac {1}{48 c^2 x^6 \sqrt {c+d x^3}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 103
Rule 151
Rule 152
Rule 156
Rule 206
Rule 208
Rule 446
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} \operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{1536 c^4}+\frac {d^3 \operatorname {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) \operatorname {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 \operatorname {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 [C] time = 0.08, size = 91, normalized size = 0.71 \[ \frac {-d^2 x^6 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d x^3+c}{9 c}\right )+981 d^2 x^6 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d x^3}{c}+1\right )+36 c \left (9 d x^3-4 c\right )}{6912 c^4 x^6 \sqrt {c+d x^3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 303, normalized size = 2.37 \[ \left [\frac {{\left (d^{3} x^{9} + c d^{2} x^{6}\right )} \sqrt {c} \log \left (\frac {d x^{3} + 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 2943 \, {\left (d^{3} x^{9} + c d^{2} x^{6}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + 24 \, {\left (245 \, c d^{2} x^{6} + 81 \, c^{2} d x^{3} - 36 \, c^{3}\right )} \sqrt {d x^{3} + c}}{41472 \, {\left (c^{5} d x^{9} + c^{6} x^{6}\right )}}, \frac {2943 \, {\left (d^{3} x^{9} + c d^{2} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) - {\left (d^{3} x^{9} + c d^{2} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 12 \, {\left (245 \, c d^{2} x^{6} + 81 \, c^{2} d x^{3} - 36 \, c^{3}\right )} \sqrt {d x^{3} + c}}{20736 \, {\left (c^{5} d x^{9} + c^{6} x^{6}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 118, normalized size = 0.92 \[ \frac {109 \, d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{768 \, \sqrt {-c} c^{4}} - \frac {d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{20736 \, \sqrt {-c} c^{4}} + \frac {2 \, d^{2}}{27 \, \sqrt {d x^{3} + c} c^{4}} + \frac {13 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} d^{2} - 17 \, \sqrt {d x^{3} + c} c d^{2}}{192 \, c^{4} d^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 636, normalized size = 4.97 \[ -\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^{3}}{512 c^{3}}+\frac {-\frac {5 d^{2} \arctanh \left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{4 c^{\frac {7}{2}}}+\frac {2 d^{2}}{3 \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}\, c^{3}}+\frac {7 \sqrt {d \,x^{3}+c}\, d}{12 c^{3} x^{3}}-\frac {\sqrt {d \,x^{3}+c}}{6 c^{2} x^{6}}}{8 c}+\frac {\left (\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}}\right ) d}{64 c^{2}}+\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^{2}}{512 c^{3}} \]
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 \[ -\int \frac {1}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} {\left (d x^{3} - 8 \, c\right )} x^{7}}\,{d x} \]
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 \[ \frac {245\,d^2}{1728\,c^4\,\sqrt {d\,x^3+c}}-\frac {109\,d^2\,\mathrm {atanh}\left (\frac {c^4\,\sqrt {d\,x^3+c}}{\sqrt {c^9}}\right )}{768\,\sqrt {c^9}}+\frac {d^2\,\mathrm {atanh}\left (\frac {c^4\,\sqrt {d\,x^3+c}}{3\,\sqrt {c^9}}\right )}{20736\,\sqrt {c^9}}-\frac {1}{48\,c^2\,x^6\,\sqrt {d\,x^3+c}}+\frac {3\,d}{64\,c^3\,x^3\,\sqrt {d\,x^3+c}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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