Optimal. Leaf size=66 \[ -\frac {\sqrt {\frac {d x^3}{c}+1} F_1\left (-\frac {5}{3};1,\frac {1}{2};-\frac {2}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{40 c x^5 \sqrt {c+d x^3}} \]
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Rubi [A] time = 0.06, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {511, 510} \[ -\frac {\sqrt {\frac {d x^3}{c}+1} F_1\left (-\frac {5}{3};1,\frac {1}{2};-\frac {2}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{40 c x^5 \sqrt {c+d x^3}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx &=\frac {\sqrt {1+\frac {d x^3}{c}} \int \frac {1}{x^6 \left (8 c-d x^3\right ) \sqrt {1+\frac {d x^3}{c}}} \, dx}{\sqrt {c+d x^3}}\\ &=-\frac {\sqrt {1+\frac {d x^3}{c}} F_1\left (-\frac {5}{3};1,\frac {1}{2};-\frac {2}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{40 c x^5 \sqrt {c+d x^3}}\\ \end {align*}
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Mathematica [B] time = 0.25, size = 261, normalized size = 3.95 \[ \frac {64 c \left (\frac {3264 c^2 d^2 x^6 F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (3 d x^3 \left (F_1\left (\frac {4}{3};\frac {1}{2},2;\frac {7}{3};-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )-4 F_1\left (\frac {4}{3};\frac {3}{2},1;\frac {7}{3};-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )\right )+32 c F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )\right )}-16 c^2+7 c d x^3+23 d^2 x^6\right )-23 d^3 x^9 \sqrt {\frac {d x^3}{c}+1} F_1\left (\frac {4}{3};\frac {1}{2},1;\frac {7}{3};-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{40960 c^4 x^5 \sqrt {c+d x^3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 11.25, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {d x^{3} + c}}{d^{2} x^{12} - 7 \, c d x^{9} - 8 \, c^{2} x^{6}}, x\right ) \]
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 \[ \int -\frac {1}{\sqrt {d x^{3} + c} {\left (d x^{3} - 8 \, c\right )} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.20, size = 1047, normalized size = 15.86 \[ \text {result too large to display} \]
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 \[ -\int \frac {1}{\sqrt {d x^{3} + c} {\left (d x^{3} - 8 \, c\right )} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^6\,\sqrt {d\,x^3+c}\,\left (8\,c-d\,x^3\right )} \,d x \]
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 \[ - \int \frac {1}{- 8 c x^{6} \sqrt {c + d x^{3}} + d x^{9} \sqrt {c + d x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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