Optimal. Leaf size=675 \[ -\frac {d^{4/3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{3456 \sqrt {3} c^{23/6}}+\frac {d^{4/3} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{10368 c^{23/6}}-\frac {d^{4/3} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{10368 c^{23/6}}-\frac {113 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{216 \sqrt {2} \sqrt [4]{3} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {113 \sqrt {2-\sqrt {3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{288\ 3^{3/4} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {113 d^{4/3} \sqrt {c+d x^3}}{432 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {113 d \sqrt {c+d x^3}}{432 c^4 x}-\frac {91 \sqrt {c+d x^3}}{864 c^3 x^4}+\frac {2}{27 c^2 x^4 \sqrt {c+d x^3}} \]
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Rubi [A] time = 0.92, antiderivative size = 675, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.482, Rules used = {472, 583, 584, 303, 218, 1877, 486, 444, 63, 206, 2138, 2145, 205} \[ -\frac {113 d^{4/3} \sqrt {c+d x^3}}{432 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {d^{4/3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{3456 \sqrt {3} c^{23/6}}+\frac {d^{4/3} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{10368 c^{23/6}}-\frac {d^{4/3} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{10368 c^{23/6}}-\frac {113 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{216 \sqrt {2} \sqrt [4]{3} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {113 \sqrt {2-\sqrt {3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{288\ 3^{3/4} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {113 d \sqrt {c+d x^3}}{432 c^4 x}-\frac {91 \sqrt {c+d x^3}}{864 c^3 x^4}+\frac {2}{27 c^2 x^4 \sqrt {c+d x^3}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 205
Rule 206
Rule 218
Rule 303
Rule 444
Rule 472
Rule 486
Rule 583
Rule 584
Rule 1877
Rule 2138
Rule 2145
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx &=\frac {2}{27 c^2 x^4 \sqrt {c+d x^3}}-\frac {2 \int \frac {-\frac {91 c d}{2}+\frac {11 d^2 x^3}{2}}{x^5 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{27 c^2 d}\\ &=\frac {2}{27 c^2 x^4 \sqrt {c+d x^3}}-\frac {91 \sqrt {c+d x^3}}{864 c^3 x^4}+\frac {\int \frac {-904 c^2 d^2+\frac {455}{4} c d^3 x^3}{x^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{432 c^4 d}\\ &=\frac {2}{27 c^2 x^4 \sqrt {c+d x^3}}-\frac {91 \sqrt {c+d x^3}}{864 c^3 x^4}+\frac {113 d \sqrt {c+d x^3}}{432 c^4 x}-\frac {\int \frac {x \left (3610 c^3 d^3-452 c^2 d^4 x^3\right )}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{3456 c^6 d}\\ &=\frac {2}{27 c^2 x^4 \sqrt {c+d x^3}}-\frac {91 \sqrt {c+d x^3}}{864 c^3 x^4}+\frac {113 d \sqrt {c+d x^3}}{432 c^4 x}-\frac {\int \left (\frac {452 c^2 d^3 x}{\sqrt {c+d x^3}}-\frac {6 c^3 d^3 x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right ) \, dx}{3456 c^6 d}\\ &=\frac {2}{27 c^2 x^4 \sqrt {c+d x^3}}-\frac {91 \sqrt {c+d x^3}}{864 c^3 x^4}+\frac {113 d \sqrt {c+d x^3}}{432 c^4 x}-\frac {\left (113 d^2\right ) \int \frac {x}{\sqrt {c+d x^3}} \, dx}{864 c^4}+\frac {d^2 \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{576 c^3}\\ &=\frac {2}{27 c^2 x^4 \sqrt {c+d x^3}}-\frac {91 \sqrt {c+d x^3}}{864 c^3 x^4}+\frac {113 d \sqrt {c+d x^3}}{432 c^4 x}-\frac {d \int \frac {2 \sqrt [3]{c} d^{2/3}-2 d x-\frac {d^{4/3} x^2}{\sqrt [3]{c}}}{\left (4+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+\frac {d^{2/3} x^2}{c^{2/3}}\right ) \sqrt {c+d x^3}} \, dx}{6912 c^4}-\frac {\left (113 d^{5/3}\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt {c+d x^3}} \, dx}{864 c^4}+\frac {d^{5/3} \int \frac {1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (2-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt {c+d x^3}} \, dx}{6912 c^{11/3}}-\frac {\left (113 \sqrt {\frac {1}{2} \left (2-\sqrt {3}\right )} d^{5/3}\right ) \int \frac {1}{\sqrt {c+d x^3}} \, dx}{432 c^{11/3}}-\frac {d^{7/3} \int \frac {x^2}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{2304 c^{10/3}}\\ &=\frac {2}{27 c^2 x^4 \sqrt {c+d x^3}}-\frac {91 \sqrt {c+d x^3}}{864 c^3 x^4}+\frac {113 d \sqrt {c+d x^3}}{432 c^4 x}-\frac {113 d^{4/3} \sqrt {c+d x^3}}{432 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {113 \sqrt {2-\sqrt {3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{288\ 3^{3/4} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {113 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{216 \sqrt {2} \sqrt [4]{3} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {d^{4/3} \operatorname {Subst}\left (\int \frac {1}{9-c x^2} \, dx,x,\frac {\left (1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}{\sqrt {c+d x^3}}\right )}{3456 c^{10/3}}-\frac {d^{7/3} \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{6912 c^{10/3}}+\frac {d^{10/3} \operatorname {Subst}\left (\int \frac {1}{-\frac {2 d^2}{c}-6 d^2 x^2} \, dx,x,\frac {1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {c+d x^3}}\right )}{1728 c^{13/3}}\\ &=\frac {2}{27 c^2 x^4 \sqrt {c+d x^3}}-\frac {91 \sqrt {c+d x^3}}{864 c^3 x^4}+\frac {113 d \sqrt {c+d x^3}}{432 c^4 x}-\frac {113 d^{4/3} \sqrt {c+d x^3}}{432 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {d^{4/3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{3456 \sqrt {3} c^{23/6}}+\frac {d^{4/3} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{10368 c^{23/6}}+\frac {113 \sqrt {2-\sqrt {3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{288\ 3^{3/4} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {113 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{216 \sqrt {2} \sqrt [4]{3} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {d^{4/3} \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{3456 c^{10/3}}\\ &=\frac {2}{27 c^2 x^4 \sqrt {c+d x^3}}-\frac {91 \sqrt {c+d x^3}}{864 c^3 x^4}+\frac {113 d \sqrt {c+d x^3}}{432 c^4 x}-\frac {113 d^{4/3} \sqrt {c+d x^3}}{432 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {d^{4/3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{3456 \sqrt {3} c^{23/6}}+\frac {d^{4/3} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{10368 c^{23/6}}-\frac {d^{4/3} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{10368 c^{23/6}}+\frac {113 \sqrt {2-\sqrt {3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{288\ 3^{3/4} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {113 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{216 \sqrt {2} \sqrt [4]{3} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 153, normalized size = 0.23 \[ \frac {452 d^3 x^9 \sqrt {\frac {d x^3}{c}+1} F_1\left (\frac {5}{3};\frac {1}{2},1;\frac {8}{3};-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )-9025 c d^2 x^6 \sqrt {\frac {d x^3}{c}+1} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )+160 c \left (-27 c^2+135 c d x^3+226 d^2 x^6\right )}{138240 c^5 x^4 \sqrt {c+d x^3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 9.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {d x^{3} + c}}{d^{3} x^{14} - 6 \, c d^{2} x^{11} - 15 \, c^{2} d x^{8} - 8 \, c^{3} x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} {\left (d x^{3} - 8 \, c\right )} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 1864, normalized size = 2.76 \[ \text {result too large to display} \]
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 \[ -\int \frac {1}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} {\left (d x^{3} - 8 \, c\right )} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^5\,{\left (d\,x^3+c\right )}^{3/2}\,\left (8\,c-d\,x^3\right )} \,d x \]
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 \[ - \int \frac {1}{- 8 c^{2} x^{5} \sqrt {c + d x^{3}} - 7 c d x^{8} \sqrt {c + d x^{3}} + d^{2} x^{11} \sqrt {c + d x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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