Optimal. Leaf size=62 \[ -\frac {\sqrt {\frac {d x^8}{c}+1} F_1\left (-\frac {1}{8};2,\frac {1}{2};\frac {7}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{a^2 x \sqrt {c+d x^8}} \]
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Rubi [A] time = 0.05, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {511, 510} \[ -\frac {\sqrt {\frac {d x^8}{c}+1} F_1\left (-\frac {1}{8};2,\frac {1}{2};\frac {7}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{a^2 x \sqrt {c+d x^8}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx &=\frac {\sqrt {1+\frac {d x^8}{c}} \int \frac {1}{x^2 \left (a+b x^8\right )^2 \sqrt {1+\frac {d x^8}{c}}} \, dx}{\sqrt {c+d x^8}}\\ &=-\frac {\sqrt {1+\frac {d x^8}{c}} F_1\left (-\frac {1}{8};2,\frac {1}{2};\frac {7}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{a^2 x \sqrt {c+d x^8}}\\ \end {align*}
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Mathematica [B] time = 0.32, size = 226, normalized size = 3.65 \[ \frac {-5 x^8 \left (a+b x^8\right ) \sqrt {\frac {d x^8}{c}+1} \left (24 a^2 d^2-40 a b c d+9 b^2 c^2\right ) F_1\left (\frac {7}{8};\frac {1}{2},1;\frac {15}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+35 a \left (c+d x^8\right ) \left (8 a^2 d-8 a b \left (c-d x^8\right )-9 b^2 c x^8\right )+7 b d x^{16} \left (a+b x^8\right ) \sqrt {\frac {d x^8}{c}+1} (9 b c-8 a d) F_1\left (\frac {15}{8};\frac {1}{2},1;\frac {23}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )}{280 a^3 c x \left (a+b x^8\right ) \sqrt {c+d x^8} (b c-a d)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.23, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x^{8} + c}}{b^{2} d x^{26} + {\left (b^{2} c + 2 \, a b d\right )} x^{18} + {\left (2 \, a b c + a^{2} d\right )} x^{10} + a^{2} c x^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.45, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}\, x^{2}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c} x^{2}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,{\left (b\,x^8+a\right )}^2\,\sqrt {d\,x^8+c}} \,d x \]
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 \[ \int \frac {1}{x^{2} \left (a + b x^{8}\right )^{2} \sqrt {c + d x^{8}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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