Optimal. Leaf size=43 \[ -\frac {19 x+39}{28 \left (3 x^2+2 x+5\right )}-\frac {19 \tan ^{-1}\left (\frac {3 x+1}{\sqrt {14}}\right )}{28 \sqrt {14}} \]
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Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {638, 618, 204} \[ -\frac {19 x+39}{28 \left (3 x^2+2 x+5\right )}-\frac {19 \tan ^{-1}\left (\frac {3 x+1}{\sqrt {14}}\right )}{28 \sqrt {14}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 638
Rubi steps
\begin {align*} \int \frac {-4+7 x}{\left (5+2 x+3 x^2\right )^2} \, dx &=-\frac {39+19 x}{28 \left (5+2 x+3 x^2\right )}-\frac {19}{28} \int \frac {1}{5+2 x+3 x^2} \, dx\\ &=-\frac {39+19 x}{28 \left (5+2 x+3 x^2\right )}+\frac {19}{14} \operatorname {Subst}\left (\int \frac {1}{-56-x^2} \, dx,x,2+6 x\right )\\ &=-\frac {39+19 x}{28 \left (5+2 x+3 x^2\right )}-\frac {19 \tan ^{-1}\left (\frac {1+3 x}{\sqrt {14}}\right )}{28 \sqrt {14}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 43, normalized size = 1.00 \[ \frac {-19 x-39}{28 \left (3 x^2+2 x+5\right )}-\frac {19 \tan ^{-1}\left (\frac {3 x+1}{\sqrt {14}}\right )}{28 \sqrt {14}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 46, normalized size = 1.07 \[ \frac {-19 x-39}{28 \left (3 x^2+2 x+5\right )}-\frac {19 \tan ^{-1}\left (\frac {3 x}{\sqrt {14}}+\frac {1}{\sqrt {14}}\right )}{28 \sqrt {14}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 45, normalized size = 1.05 \[ -\frac {19 \, \sqrt {14} {\left (3 \, x^{2} + 2 \, x + 5\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (3 \, x + 1\right )}\right ) + 266 \, x + 546}{392 \, {\left (3 \, x^{2} + 2 \, x + 5\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.85, size = 36, normalized size = 0.84 \[ -\frac {19}{392} \, \sqrt {14} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (3 \, x + 1\right )}\right ) - \frac {19 \, x + 39}{28 \, {\left (3 \, x^{2} + 2 \, x + 5\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 34, normalized size = 0.79
method | result | size |
risch | \(\frac {-\frac {19 x}{84}-\frac {13}{28}}{x^{2}+\frac {2}{3} x +\frac {5}{3}}-\frac {19 \arctan \left (\frac {\left (1+3 x \right ) \sqrt {14}}{14}\right ) \sqrt {14}}{392}\) | \(34\) |
default | \(\frac {-38 x -78}{168 x^{2}+112 x +280}-\frac {19 \sqrt {14}\, \arctan \left (\frac {\left (6 x +2\right ) \sqrt {14}}{28}\right )}{392}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 36, normalized size = 0.84 \[ -\frac {19}{392} \, \sqrt {14} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (3 \, x + 1\right )}\right ) - \frac {19 \, x + 39}{28 \, {\left (3 \, x^{2} + 2 \, x + 5\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 36, normalized size = 0.84 \[ -\frac {\frac {19\,x}{84}+\frac {13}{28}}{x^2+\frac {2\,x}{3}+\frac {5}{3}}-\frac {19\,\sqrt {14}\,\mathrm {atan}\left (\frac {3\,\sqrt {14}\,x}{14}+\frac {\sqrt {14}}{14}\right )}{392} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 42, normalized size = 0.98 \[ \frac {- 19 x - 39}{84 x^{2} + 56 x + 140} - \frac {19 \sqrt {14} \operatorname {atan}{\left (\frac {3 \sqrt {14} x}{14} + \frac {\sqrt {14}}{14} \right )}}{392} \]
Verification of antiderivative is not currently implemented for this CAS.
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