Optimal. Leaf size=64 \[ -\frac {423 x+1367}{7000 \left (5 x^2+2 x+3\right )^2}+\frac {11015 x+34347}{196000 \left (5 x^2+2 x+3\right )}+\frac {339 \tan ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{1568 \sqrt {14}} \]
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Rubi [A] time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1660, 12, 618, 204} \[ -\frac {423 x+1367}{7000 \left (5 x^2+2 x+3\right )^2}+\frac {11015 x+34347}{196000 \left (5 x^2+2 x+3\right )}+\frac {339 \tan ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{1568 \sqrt {14}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 618
Rule 1660
Rubi steps
\begin {align*} \int \frac {2+x+3 x^2-5 x^3+4 x^4}{\left (3+2 x+5 x^2\right )^3} \, dx &=-\frac {1367+423 x}{7000 \left (3+2 x+5 x^2\right )^2}+\frac {1}{112} \int \frac {\frac {6534}{125}-\frac {3696 x}{25}+\frac {448 x^2}{5}}{\left (3+2 x+5 x^2\right )^2} \, dx\\ &=-\frac {1367+423 x}{7000 \left (3+2 x+5 x^2\right )^2}+\frac {34347+11015 x}{196000 \left (3+2 x+5 x^2\right )}+\frac {\int \frac {1356}{3+2 x+5 x^2} \, dx}{6272}\\ &=-\frac {1367+423 x}{7000 \left (3+2 x+5 x^2\right )^2}+\frac {34347+11015 x}{196000 \left (3+2 x+5 x^2\right )}+\frac {339 \int \frac {1}{3+2 x+5 x^2} \, dx}{1568}\\ &=-\frac {1367+423 x}{7000 \left (3+2 x+5 x^2\right )^2}+\frac {34347+11015 x}{196000 \left (3+2 x+5 x^2\right )}-\frac {339}{784} \operatorname {Subst}\left (\int \frac {1}{-56-x^2} \, dx,x,2+10 x\right )\\ &=-\frac {1367+423 x}{7000 \left (3+2 x+5 x^2\right )^2}+\frac {34347+11015 x}{196000 \left (3+2 x+5 x^2\right )}+\frac {339 \tan ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{1568 \sqrt {14}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 53, normalized size = 0.83 \[ \frac {\frac {14 \left (11015 x^3+38753 x^2+17979 x+12953\right )}{\left (5 x^2+2 x+3\right )^2}+8475 \sqrt {14} \tan ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{548800} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 75, normalized size = 1.17 \[ \frac {154210 \, x^{3} + 8475 \, \sqrt {14} {\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + 542542 \, x^{2} + 251706 \, x + 181342}{548800 \, {\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 46, normalized size = 0.72 \[ \frac {339}{21952} \, \sqrt {14} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {11015 \, x^{3} + 38753 \, x^{2} + 17979 \, x + 12953}{39200 \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 0.73 \[ \frac {339 \sqrt {14}\, \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{21952}+\frac {\frac {2203}{7840} x^{3}+\frac {38753}{39200} x^{2}+\frac {17979}{39200} x +\frac {12953}{39200}}{\left (5 x^{2}+2 x +3\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 56, normalized size = 0.88 \[ \frac {339}{21952} \, \sqrt {14} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {11015 \, x^{3} + 38753 \, x^{2} + 17979 \, x + 12953}{39200 \, {\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 55, normalized size = 0.86 \[ \frac {339\,\sqrt {14}\,\mathrm {atan}\left (\frac {5\,\sqrt {14}\,x}{14}+\frac {\sqrt {14}}{14}\right )}{21952}+\frac {\frac {2203\,x^3}{196000}+\frac {38753\,x^2}{980000}+\frac {17979\,x}{980000}+\frac {12953}{980000}}{x^4+\frac {4\,x^3}{5}+\frac {34\,x^2}{25}+\frac {12\,x}{25}+\frac {9}{25}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 61, normalized size = 0.95 \[ \frac {11015 x^{3} + 38753 x^{2} + 17979 x + 12953}{980000 x^{4} + 784000 x^{3} + 1332800 x^{2} + 470400 x + 352800} + \frac {339 \sqrt {14} \operatorname {atan}{\left (\frac {5 \sqrt {14} x}{14} + \frac {\sqrt {14}}{14} \right )}}{21952} \]
Verification of antiderivative is not currently implemented for this CAS.
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