Optimal. Leaf size=64 \[ -\frac {\sqrt {\frac {d x^8}{c}+1} F_1\left (-\frac {3}{8};1,\frac {1}{2};\frac {5}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{3 a x^3 \sqrt {c+d x^8}} \]
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Rubi [A] time = 0.06, antiderivative size = 64, 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} \[ -\frac {\sqrt {\frac {d x^8}{c}+1} F_1\left (-\frac {3}{8};1,\frac {1}{2};\frac {5}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{3 a x^3 \sqrt {c+d x^8}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx &=\frac {\sqrt {1+\frac {d x^8}{c}} \int \frac {1}{x^4 \left (a+b x^8\right ) \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 {3}{8};1,\frac {1}{2};\frac {5}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{3 a x^3 \sqrt {c+d x^8}}\\ \end {align*}
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Mathematica [B] time = 0.14, size = 141, normalized size = 2.20 \[ \frac {13 x^8 \sqrt {\frac {d x^8}{c}+1} (a d-3 b c) F_1\left (\frac {5}{8};\frac {1}{2},1;\frac {13}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+5 b d x^{16} \sqrt {\frac {d x^8}{c}+1} F_1\left (\frac {13}{8};\frac {1}{2},1;\frac {21}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )-65 a \left (c+d x^8\right )}{195 a^2 c x^3 \sqrt {c+d x^8}} \]
Warning: Unable to verify antiderivative.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.64, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{8}+a \right ) \sqrt {d \,x^{8}+c}\, x^{4}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c} x^{4}}\,{d x} \]
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 \[ \int \frac {1}{x^4\,\left (b\,x^8+a\right )\,\sqrt {d\,x^8+c}} \,d x \]
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 \[ \int \frac {1}{x^{4} \left (a + b x^{8}\right ) \sqrt {c + d x^{8}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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