Optimal. Leaf size=81 \[ \frac {863}{441} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac {139 (1-2 x)^{3/2}}{882 (2+3 x)}-\frac {863 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 \sqrt {21}} \]
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Rubi [A]
time = 0.01, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {91, 79, 52, 65,
212} \begin {gather*} \frac {139 (1-2 x)^{3/2}}{882 (3 x+2)}-\frac {(1-2 x)^{3/2}}{126 (3 x+2)^2}+\frac {863}{441} \sqrt {1-2 x}-\frac {863 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 79
Rule 91
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx &=-\frac {(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac {1}{126} \int \frac {\sqrt {1-2 x} (561+1050 x)}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac {139 (1-2 x)^{3/2}}{882 (2+3 x)}+\frac {863}{294} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=\frac {863}{441} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac {139 (1-2 x)^{3/2}}{882 (2+3 x)}+\frac {863}{126} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {863}{441} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac {139 (1-2 x)^{3/2}}{882 (2+3 x)}-\frac {863}{126} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {863}{441} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac {139 (1-2 x)^{3/2}}{882 (2+3 x)}-\frac {863 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 58, normalized size = 0.72 \begin {gather*} \frac {\sqrt {1-2 x} \left (1025+2941 x+2100 x^2\right )}{126 (2+3 x)^2}-\frac {863 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 57, normalized size = 0.70
method | result | size |
risch | \(-\frac {4200 x^{3}+3782 x^{2}-891 x -1025}{126 \left (2+3 x \right )^{2} \sqrt {1-2 x}}-\frac {863 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1323}\) | \(51\) |
derivativedivides | \(\frac {50 \sqrt {1-2 x}}{27}+\frac {-\frac {47 \left (1-2 x \right )^{\frac {3}{2}}}{21}+\frac {139 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{2}}-\frac {863 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1323}\) | \(57\) |
default | \(\frac {50 \sqrt {1-2 x}}{27}+\frac {-\frac {47 \left (1-2 x \right )^{\frac {3}{2}}}{21}+\frac {139 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{2}}-\frac {863 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1323}\) | \(57\) |
trager | \(\frac {\left (2100 x^{2}+2941 x +1025\right ) \sqrt {1-2 x}}{126 \left (2+3 x \right )^{2}}+\frac {863 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{2646}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.64, size = 83, normalized size = 1.02 \begin {gather*} \frac {863}{2646} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {50}{27} \, \sqrt {-2 \, x + 1} - \frac {423 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 973 \, \sqrt {-2 \, x + 1}}{189 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.45, size = 74, normalized size = 0.91 \begin {gather*} \frac {863 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (2100 \, x^{2} + 2941 \, x + 1025\right )} \sqrt {-2 \, x + 1}}{2646 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 178.23, size = 360, normalized size = 4.44 \begin {gather*} \frac {50 \sqrt {1 - 2 x}}{27} + \frac {32 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{3} + \frac {56 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{27} + \frac {130 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x < - \frac {2}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x > - \frac {2}{3} \end {cases}\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.57, size = 77, normalized size = 0.95 \begin {gather*} \frac {863}{2646} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {50}{27} \, \sqrt {-2 \, x + 1} - \frac {423 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 973 \, \sqrt {-2 \, x + 1}}{756 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 62, normalized size = 0.77 \begin {gather*} \frac {50\,\sqrt {1-2\,x}}{27}-\frac {863\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1323}+\frac {\frac {139\,\sqrt {1-2\,x}}{243}-\frac {47\,{\left (1-2\,x\right )}^{3/2}}{189}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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