Optimal. Leaf size=130 \[ -\frac {\sqrt {\frac {\left (\frac {6}{x}-1\right )^4-182 \left (1-\frac {6}{x}\right )^2+613}{\left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right )^2}} \left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right ) x^2 F\left (2 \tan ^{-1}\left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{12 \sqrt [4]{613} \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}} \]
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Rubi [A] time = 0.26, 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 = {2069, 12, 6719, 1096} \[ -\frac {\sqrt {\frac {\left (\frac {6}{x}-1\right )^4-182 \left (1-\frac {6}{x}\right )^2+613}{\left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right )^2}} \left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right ) x^2 F\left (2 \tan ^{-1}\left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{12 \sqrt [4]{613} \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 1096
Rule 2069
Rule 6719
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx &=-\left (1296 \operatorname {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 \operatorname {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 ) \operatorname {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] time = 0.12, size = 826, normalized size = 6.35 \[ -\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 (3 x^4+15 x^3-44 x^2-6 x+9\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 )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.50, size = 1182, normalized size = 9.09 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{\sqrt {3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {3\,x^4+15\,x^3-44\,x^2-6\,x+9}} \,d x \]
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 \[ \int \frac {1}{\sqrt {3 x^{4} + 15 x^{3} - 44 x^{2} - 6 x + 9}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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