Optimal. Leaf size=42 \[ \frac {(3-x) \sqrt {1+x} \tanh ^{-1}\left (\frac {\sqrt {1+x}}{2}\right )}{\sqrt {9+3 x-5 x^2+x^3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2092, 2089, 65,
212} \begin {gather*} \frac {(3-x) \sqrt {x+1} \tanh ^{-1}\left (\frac {\sqrt {x+1}}{2}\right )}{\sqrt {x^3-5 x^2+3 x+9}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 212
Rule 2089
Rule 2092
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {9+3 x-5 x^2+x^3}} \, dx &=\text {Subst}\left (\int \frac {1}{\sqrt {\frac {128}{27}-\frac {16 x}{3}+x^3}} \, dx,x,-\frac {5}{3}+x\right )\\ &=\frac {\left (128 (3-x) \sqrt {1+x}\right ) \text {Subst}\left (\int \frac {1}{\left (\frac {128}{9}-\frac {32 x}{3}\right ) \sqrt {\frac {128}{9}+\frac {16 x}{3}}} \, dx,x,-\frac {5}{3}+x\right )}{3 \sqrt {3} \sqrt {9+3 x-5 x^2+x^3}}\\ &=\frac {\left (16 (3-x) \sqrt {1+x}\right ) \text {Subst}\left (\int \frac {1}{\frac {128}{3}-2 x^2} \, dx,x,\frac {4 \sqrt {1+x}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {9+3 x-5 x^2+x^3}}\\ &=\frac {(3-x) \sqrt {1+x} \tanh ^{-1}\left (\frac {\sqrt {1+x}}{2}\right )}{\sqrt {9+3 x-5 x^2+x^3}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 37, normalized size = 0.88 \begin {gather*} -\frac {(-3+x) \sqrt {1+x} \tanh ^{-1}\left (\frac {\sqrt {1+x}}{2}\right )}{\sqrt {(-3+x)^2 (1+x)}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.05, size = 45, normalized size = 1.07
method | result | size |
trager | \(-\frac {\ln \left (\frac {x^{2}+4 \sqrt {x^{3}-5 x^{2}+3 x +9}+2 x -15}{\left (-3+x \right )^{2}}\right )}{2}\) | \(35\) |
default | \(\frac {\left (-3+x \right ) \sqrt {1+x}\, \left (\ln \left (\sqrt {1+x}-2\right )-\ln \left (\sqrt {1+x}+2\right )\right )}{2 \sqrt {x^{3}-5 x^{2}+3 x +9}}\) | \(45\) |
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 [A]
time = 0.31, size = 62, normalized size = 1.48 \begin {gather*} -\frac {1}{2} \, \log \left (\frac {2 \, x + \sqrt {x^{3} - 5 \, x^{2} + 3 \, x + 9} - 6}{x - 3}\right ) + \frac {1}{2} \, \log \left (-\frac {2 \, x - \sqrt {x^{3} - 5 \, x^{2} + 3 \, x + 9} - 6}{x - 3}\right ) \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 {x^{3} - 5 x^{2} + 3 x + 9}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 41, normalized size = 0.98 \begin {gather*} 2 \left (-\frac {\ln \left (\sqrt {x+1}+2\right )}{4 \mathrm {sign}\left (x-3\right )}+\frac {\ln \left |\sqrt {x+1}-2\right |}{4 \mathrm {sign}\left (x-3\right )}\right ) \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}{\sqrt {x^3-5\,x^2+3\,x+9}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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