Optimal. Leaf size=55 \[ -\frac {25}{128 (1+2 x)^4}+\frac {7}{24 (1+2 x)^3}-\frac {3}{32 (1+2 x)^2}+\frac {1}{8 (1+2 x)}+\frac {1}{32} \log (1+2 x) \]
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Rubi [A]
time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1607, 1634}
\begin {gather*} \frac {1}{8 (2 x+1)}-\frac {3}{32 (2 x+1)^2}+\frac {7}{24 (2 x+1)^3}-\frac {25}{128 (2 x+1)^4}+\frac {1}{32} \log (2 x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 1607
Rule 1634
Rubi steps
\begin {align*} \int \frac {-3 x+x^4}{(1+2 x)^5} \, dx &=\int \frac {x \left (-3+x^3\right )}{(1+2 x)^5} \, dx\\ &=\int \left (\frac {25}{16 (1+2 x)^5}-\frac {7}{4 (1+2 x)^4}+\frac {3}{8 (1+2 x)^3}-\frac {1}{4 (1+2 x)^2}+\frac {1}{16 (1+2 x)}\right ) \, dx\\ &=-\frac {25}{128 (1+2 x)^4}+\frac {7}{24 (1+2 x)^3}-\frac {3}{32 (1+2 x)^2}+\frac {1}{8 (1+2 x)}+\frac {1}{32} \log (1+2 x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 41, normalized size = 0.75 \begin {gather*} \frac {49+368 x+432 x^2+384 x^3+12 (1+2 x)^4 \log (1+2 x)}{384 (1+2 x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 46, normalized size = 0.84
method | result | size |
risch | \(\frac {x^{3}+\frac {9}{8} x^{2}+\frac {23}{24} x +\frac {49}{384}}{\left (1+2 x \right )^{4}}+\frac {\ln \left (1+2 x \right )}{32}\) | \(34\) |
norman | \(\frac {-\frac {37}{12} x^{3}-\frac {31}{16} x^{2}-\frac {1}{16} x -\frac {49}{24} x^{4}}{\left (1+2 x \right )^{4}}+\frac {\ln \left (1+2 x \right )}{32}\) | \(37\) |
default | \(-\frac {25}{128 \left (1+2 x \right )^{4}}+\frac {7}{24 \left (1+2 x \right )^{3}}-\frac {3}{32 \left (1+2 x \right )^{2}}+\frac {1}{8+16 x}+\frac {\ln \left (1+2 x \right )}{32}\) | \(46\) |
meijerg | \(-\frac {x \left (1000 x^{3}+1040 x^{2}+420 x +60\right )}{960 \left (1+2 x \right )^{4}}+\frac {\ln \left (1+2 x \right )}{32}-\frac {x^{2} \left (4 x^{2}+8 x +6\right )}{4 \left (1+2 x \right )^{4}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.14, size = 48, normalized size = 0.87 \begin {gather*} \frac {384 \, x^{3} + 432 \, x^{2} + 368 \, x + 49}{384 \, {\left (16 \, x^{4} + 32 \, x^{3} + 24 \, x^{2} + 8 \, x + 1\right )}} + \frac {1}{32} \, \log \left (2 \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 67, normalized size = 1.22 \begin {gather*} \frac {384 \, x^{3} + 432 \, x^{2} + 12 \, {\left (16 \, x^{4} + 32 \, x^{3} + 24 \, x^{2} + 8 \, x + 1\right )} \log \left (2 \, x + 1\right ) + 368 \, x + 49}{384 \, {\left (16 \, x^{4} + 32 \, x^{3} + 24 \, x^{2} + 8 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 42, normalized size = 0.76 \begin {gather*} \frac {384 x^{3} + 432 x^{2} + 368 x + 49}{6144 x^{4} + 12288 x^{3} + 9216 x^{2} + 3072 x + 384} + \frac {\log {\left (2 x + 1 \right )}}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 55, normalized size = 1.00 \begin {gather*} \frac {1}{8 \, {\left (2 \, x + 1\right )}} - \frac {3}{32 \, {\left (2 \, x + 1\right )}^{2}} + \frac {7}{24 \, {\left (2 \, x + 1\right )}^{3}} - \frac {25}{128 \, {\left (2 \, x + 1\right )}^{4}} - \frac {1}{32} \, \log \left (\frac {{\left | 2 \, x + 1 \right |}}{2 \, {\left (2 \, x + 1\right )}^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 43, normalized size = 0.78 \begin {gather*} \frac {\ln \left (x+\frac {1}{2}\right )}{32}+\frac {\frac {x^3}{16}+\frac {9\,x^2}{128}+\frac {23\,x}{384}+\frac {49}{6144}}{x^4+2\,x^3+\frac {3\,x^2}{2}+\frac {x}{2}+\frac {1}{16}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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