Optimal. Leaf size=59 \[ \frac {x \sqrt {1+\frac {d x^8}{c}} F_1\left (\frac {1}{8};1,\frac {1}{2};\frac {9}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{a \sqrt {c+d x^8}} \]
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Rubi [A]
time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {441, 440}
\begin {gather*} \frac {x \sqrt {\frac {d x^8}{c}+1} F_1\left (\frac {1}{8};1,\frac {1}{2};\frac {9}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{a \sqrt {c+d x^8}} \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 441
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx &=\frac {\sqrt {1+\frac {d x^8}{c}} \int \frac {1}{\left (a+b x^8\right ) \sqrt {1+\frac {d x^8}{c}}} \, dx}{\sqrt {c+d x^8}}\\ &=\frac {x \sqrt {1+\frac {d x^8}{c}} F_1\left (\frac {1}{8};1,\frac {1}{2};\frac {9}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{a \sqrt {c+d x^8}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(161\) vs. \(2(59)=118\).
time = 10.14, size = 161, normalized size = 2.73 \begin {gather*} -\frac {9 a c x F_1\left (\frac {1}{8};\frac {1}{2},1;\frac {9}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )}{\left (a+b x^8\right ) \sqrt {c+d x^8} \left (-9 a c F_1\left (\frac {1}{8};\frac {1}{2},1;\frac {9}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+4 x^8 \left (2 b c F_1\left (\frac {9}{8};\frac {1}{2},2;\frac {17}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+a d F_1\left (\frac {9}{8};\frac {3}{2},1;\frac {17}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \,x^{8}+a \right ) \sqrt {d \,x^{8}+c}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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}{\left (a + b x^{8}\right ) \sqrt {c + d x^{8}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (b\,x^8+a\right )\,\sqrt {d\,x^8+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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