Optimal. Leaf size=185 \[ \frac {13 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{497664 c^{11/2}}-\frac {33 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{2048 c^{11/2}}+\frac {665 d^2}{41472 c^5 \sqrt {c+d x^3}}-\frac {71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \]
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Rubi [A] time = 0.16, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \begin {gather*} -\frac {71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {665 d^2}{41472 c^5 \sqrt {c+d x^3}}+\frac {13 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{497664 c^{11/2}}-\frac {33 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{2048 c^{11/2}}+\frac {17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \end {gather*}
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 )^2 \left (c+d x^3\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^3 (8 c-d x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {\operatorname {Subst}\left (\int \frac {17 c d-\frac {7 d^2 x}{2}}{x^2 (8 c-d x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )}{48 c^2}\\ &=-\frac {1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {\operatorname {Subst}\left (\int \frac {198 c^2 d^2-\frac {85}{2} c d^3 x}{x (8 c-d x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )}{384 c^4}\\ &=-\frac {71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {\operatorname {Subst}\left (\int \frac {-1782 c^3 d^3+213 c^2 d^4 x}{x (8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{27648 c^6 d}\\ &=\frac {665 d^2}{41472 c^5 \sqrt {c+d x^3}}-\frac {71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {\operatorname {Subst}\left (\int \frac {-8019 c^4 d^4+\frac {1995}{2} c^3 d^5 x}{x (8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{124416 c^8 d^2}\\ &=\frac {665 d^2}{41472 c^5 \sqrt {c+d x^3}}-\frac {71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {\left (33 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{4096 c^5}+\frac {\left (13 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{331776 c^5}\\ &=\frac {665 d^2}{41472 c^5 \sqrt {c+d x^3}}-\frac {71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {(33 d) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{2048 c^5}+\frac {\left (13 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{165888 c^5}\\ &=\frac {665 d^2}{41472 c^5 \sqrt {c+d x^3}}-\frac {71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {13 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{497664 c^{11/2}}-\frac {33 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{2048 c^{11/2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 135, normalized size = 0.73 \begin {gather*} \frac {13 d^2 x^6 \left (d x^3-8 c\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d x^3+c}{9 c}\right )-3 \left (4 c \left (288 c^2-612 c d x^3+71 d^2 x^6\right )+891 d^2 x^6 \left (d x^3-8 c\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d x^3}{c}+1\right )\right )}{165888 c^5 x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 208, normalized size = 1.12 \begin {gather*} \frac {x^6 \sqrt {c+d x^3} \left (\frac {13 d^2}{62208 c^{9/2}}-\frac {13 d^3 x^3}{497664 c^{11/2}}\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )+x^6 \sqrt {c+d x^3} \left (\frac {33 d^3 x^3}{2048 c^{11/2}}-\frac {33 d^2}{256 c^{9/2}}\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )-\frac {665 d^3 x^9}{41472 c^5}+\frac {5107 d^2 x^6}{41472 c^4}+\frac {17 d x^3}{384 c^3}-\frac {1}{48 c^2}}{x^6 \left (8 c \sqrt {c+d x^3}-d x^3 \sqrt {c+d x^3}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 398, normalized size = 2.15 \begin {gather*} \left [\frac {13 \, {\left (d^{4} x^{12} - 7 \, c d^{3} x^{9} - 8 \, c^{2} 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 ) + 8019 \, {\left (d^{4} x^{12} - 7 \, c d^{3} x^{9} - 8 \, c^{2} 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 (665 \, c d^{3} x^{9} - 5107 \, c^{2} d^{2} x^{6} - 1836 \, c^{3} d x^{3} + 864 \, c^{4}\right )} \sqrt {d x^{3} + c}}{995328 \, {\left (c^{6} d^{2} x^{12} - 7 \, c^{7} d x^{9} - 8 \, c^{8} x^{6}\right )}}, \frac {8019 \, {\left (d^{4} x^{12} - 7 \, c d^{3} x^{9} - 8 \, c^{2} d^{2} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) - 13 \, {\left (d^{4} x^{12} - 7 \, c d^{3} x^{9} - 8 \, c^{2} d^{2} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 12 \, {\left (665 \, c d^{3} x^{9} - 5107 \, c^{2} d^{2} x^{6} - 1836 \, c^{3} d x^{3} + 864 \, c^{4}\right )} \sqrt {d x^{3} + c}}{497664 \, {\left (c^{6} d^{2} x^{12} - 7 \, c^{7} d x^{9} - 8 \, c^{8} x^{6}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 149, normalized size = 0.81 \begin {gather*} \frac {33 \, d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{2048 \, \sqrt {-c} c^{5}} - \frac {13 \, d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{497664 \, \sqrt {-c} c^{5}} + \frac {341 \, {\left (d x^{3} + c\right )} d^{2} - 3072 \, c d^{2}}{41472 \, {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} - 9 \, \sqrt {d x^{3} + c} c\right )} c^{5}} + \frac {3 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} d^{2} - 4 \, \sqrt {d x^{3} + c} c d^{2}}{384 \, c^{5} d^{2} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 1106, normalized size = 5.98
result too large to display
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 )}^{2} x^{7}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.76, size = 171, normalized size = 0.92 \begin {gather*} \frac {\frac {2\,d^2}{9\,c^2}-\frac {10373\,d^2\,\left (d\,x^3+c\right )}{13824\,c^3}+\frac {3551\,d^2\,{\left (d\,x^3+c\right )}^2}{6912\,c^4}-\frac {665\,d^2\,{\left (d\,x^3+c\right )}^3}{13824\,c^5}}{33\,c\,{\left (d\,x^3+c\right )}^{5/2}-3\,{\left (d\,x^3+c\right )}^{7/2}+27\,c^3\,\sqrt {d\,x^3+c}-57\,c^2\,{\left (d\,x^3+c\right )}^{3/2}}+\frac {d^2\,\left (\mathrm {atanh}\left (\frac {c^5\,\sqrt {d\,x^3+c}}{\sqrt {c^{11}}}\right )\,1{}\mathrm {i}-\frac {\mathrm {atanh}\left (\frac {c^5\,\sqrt {d\,x^3+c}}{3\,\sqrt {c^{11}}}\right )\,13{}\mathrm {i}}{8019}\right )\,33{}\mathrm {i}}{2048\,\sqrt {c^{11}}} \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}{x^{7} \left (- 8 c + d x^{3}\right )^{2} \left (c + d x^{3}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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