Optimal. Leaf size=130 \[ -\frac {\sqrt {\frac {613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4}{\left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right )^2}} \left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right ) x^2 F\left (2 \tan ^{-1}\left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{12 \sqrt [4]{613} \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \]
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Rubi [A]
time = 0.18, antiderivative size = 130, 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,
1110} \begin {gather*} -\frac {\sqrt {\frac {\left (\frac {6}{x}-1\right )^4-182 \left (1-\frac {6}{x}\right )^2+613}{\left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right )^2}} \left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right ) x^2 F\left (2 \text {ArcTan}\left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{12 \sqrt [4]{613} \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1110
Rule 2094
Rule 6851
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx &=-\left (1296 \text {Subst}\left (\int \frac {1}{3 (-6-36 x)^2 \sqrt {\frac {794448-8491392 x^2+1679616 x^4}{(-6-36 x)^4}}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )\right )\\ &=-\left (432 \text {Subst}\left (\int \frac {1}{(-6-36 x)^2 \sqrt {\frac {794448-8491392 x^2+1679616 x^4}{(-6-36 x)^4}}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )\right )\\ &=-\frac {\left (\sqrt {794448-8491392 \left (-\frac {1}{6}+\frac {1}{x}\right )^2+1679616 \left (-\frac {1}{6}+\frac {1}{x}\right )^4} x^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {794448-8491392 x^2+1679616 x^4}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}\\ &=-\frac {\sqrt {\frac {613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4}{\left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right )^2}} \left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right ) x^2 F\left (2 \tan ^{-1}\left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{12 \sqrt [4]{613} \sqrt {9-6 x-44 x^2+15 x^3+3 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 = 10.10, size = 826, normalized size = 6.35 \begin {gather*} -\frac {2 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,1\right ]\right ) \left (\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,4\right ]\right )}{\left (x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]\right ) \left (\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,1\right ]-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,4\right ]\right )}}\right )|\frac {\left (\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,3\right ]\right ) \left (\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,1\right ]-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,4\right ]\right )}{\left (\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,1\right ]-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,3\right ]\right ) \left (\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,4\right ]\right )}\right ) \sqrt {\frac {x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,1\right ]}{x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]}} \left (x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]\right )^2 \sqrt {\frac {x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,3\right ]}{x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]}} \sqrt {\frac {x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,4\right ]}{x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]}}}{\sqrt {\left (9-6 x-44 x^2+15 x^3+3 x^4\right ) \left (\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,1\right ]-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,3\right ]\right ) \left (\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,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 = 0.41, size = 1182, normalized size = 9.09
method | result | size |
default | \(\text {Expression too large to display}\) | \(1182\) |
elliptic | \(\text {Expression too large to display}\) | \(1182\) |
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 {3 x^{4} + 15 x^{3} - 44 x^{2} - 6 x + 9}}\, 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 {3\,x^4+15\,x^3-44\,x^2-6\,x+9}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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