Optimal. Leaf size=126 \[ -\frac {\left (\sqrt {517}+\left (3+\frac {4}{x}\right )^2\right ) \sqrt {\frac {517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4}{\left (\sqrt {517}+\left (3+\frac {4}{x}\right )^2\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac {4+3 x}{\sqrt [4]{517} x}\right )|\frac {517+19 \sqrt {517}}{1034}\right )}{8 \sqrt [4]{517} \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}} \]
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Rubi [A]
time = 0.22, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2094, 12, 6851,
1117} \begin {gather*} -\frac {\left (\left (\frac {4}{x}+3\right )^2+\sqrt {517}\right ) \sqrt {\frac {\left (\frac {4}{x}+3\right )^4-38 \left (\frac {4}{x}+3\right )^2+517}{\left (\left (\frac {4}{x}+3\right )^2+\sqrt {517}\right )^2}} x^2 F\left (2 \text {ArcTan}\left (\frac {3 x+4}{\sqrt [4]{517} x}\right )|\frac {517+19 \sqrt {517}}{1034}\right )}{8 \sqrt [4]{517} \sqrt {8 x^4-15 x^3+8 x^2+24 x+8}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1117
Rule 2094
Rule 6851
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {8+24 x+8 x^2-15 x^3+8 x^4}} \, dx &=-\left (1024 \text {Subst}\left (\int \frac {1}{2 \sqrt {2} (24-32 x)^2 \sqrt {\frac {2117632-2490368 x^2+1048576 x^4}{(24-32 x)^4}}} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )\right )\\ &=-\left (\left (256 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{(24-32 x)^2 \sqrt {\frac {2117632-2490368 x^2+1048576 x^4}{(24-32 x)^4}}} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )\right )\\ &=-\frac {\left (\sqrt {2117632-2490368 \left (\frac {3}{4}+\frac {1}{x}\right )^2+1048576 \left (\frac {3}{4}+\frac {1}{x}\right )^4} x^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2117632-2490368 x^2+1048576 x^4}} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )}{\sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}\\ &=-\frac {\left (\sqrt {517}+\left (3+\frac {4}{x}\right )^2\right ) \sqrt {\frac {517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4}{\left (\sqrt {517}+\left (3+\frac {4}{x}\right )^2\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac {4+3 x}{\sqrt [4]{517} x}\right )|\frac {517+19 \sqrt {517}}{1034}\right )}{8 \sqrt [4]{517} \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in
optimal.
time = 10.67, size = 1148, normalized size = 9.11 \begin {gather*} -\frac {2 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (x-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,1\right ]\right ) \left (\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,2\right ]-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,4\right ]\right )}{\left (x-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,2\right ]\right ) \left (\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,1\right ]-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,4\right ]\right )}}\right )|\frac {\left (\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,2\right ]-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,3\right ]\right ) \left (\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,1\right ]-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,4\right ]\right )}{\left (\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,1\right ]-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,3\right ]\right ) \left (\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,2\right ]-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,4\right ]\right )}\right ) \left (x-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,2\right ]\right )^2 \sqrt {\frac {\left (\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,1\right ]-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,2\right ]\right ) \left (x-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,3\right ]\right )}{\left (x-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,2\right ]\right ) \left (\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,1\right ]-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,3\right ]\right )}} \left (\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,1\right ]-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,4\right ]\right ) \sqrt {\frac {\left (x-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,1\right ]\right ) \left (\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,1\right ]-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,2\right ]\right ) \left (x-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,4\right ]\right ) \left (\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,2\right ]-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,4\right ]\right )}{\left (x-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,2\right ]\right )^2 \left (\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,1\right ]-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,4\right ]\right )^2}}}{\sqrt {8+24 x+8 x^2-15 x^3+8 x^4} \left (-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,1\right ]+\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,2\right ]\right ) \left (\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,2\right ]-\text {Root}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\&,4\right ]\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.92, size = 1180, normalized size = 9.37
method | result | size |
default | \(\text {Expression too large to display}\) | \(1180\) |
elliptic | \(\text {Expression too large to display}\) | \(1180\) |
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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {8 x^{4} - 15 x^{3} + 8 x^{2} + 24 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 = -0.01 \begin {gather*} \int \frac {1}{\sqrt {8\,x^4-15\,x^3+8\,x^2+24\,x+8}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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