Optimal. Leaf size=82 \[ -\frac {2}{81} \sqrt {1-2 x}-\frac {5135}{324} (1-2 x)^{3/2}+\frac {80}{9} (1-2 x)^{5/2}-\frac {125}{84} (1-2 x)^{7/2}+\frac {2}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {90, 52, 65, 212}
\begin {gather*} -\frac {125}{84} (1-2 x)^{7/2}+\frac {80}{9} (1-2 x)^{5/2}-\frac {5135}{324} (1-2 x)^{3/2}-\frac {2}{81} \sqrt {1-2 x}+\frac {2}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 90
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^3}{2+3 x} \, dx &=\int \left (\frac {5135}{108} \sqrt {1-2 x}-\frac {400}{9} (1-2 x)^{3/2}+\frac {125}{12} (1-2 x)^{5/2}-\frac {\sqrt {1-2 x}}{27 (2+3 x)}\right ) \, dx\\ &=-\frac {5135}{324} (1-2 x)^{3/2}+\frac {80}{9} (1-2 x)^{5/2}-\frac {125}{84} (1-2 x)^{7/2}-\frac {1}{27} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=-\frac {2}{81} \sqrt {1-2 x}-\frac {5135}{324} (1-2 x)^{3/2}+\frac {80}{9} (1-2 x)^{5/2}-\frac {125}{84} (1-2 x)^{7/2}-\frac {7}{81} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {2}{81} \sqrt {1-2 x}-\frac {5135}{324} (1-2 x)^{3/2}+\frac {80}{9} (1-2 x)^{5/2}-\frac {125}{84} (1-2 x)^{7/2}+\frac {7}{81} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {2}{81} \sqrt {1-2 x}-\frac {5135}{324} (1-2 x)^{3/2}+\frac {80}{9} (1-2 x)^{5/2}-\frac {125}{84} (1-2 x)^{7/2}+\frac {2}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 56, normalized size = 0.68 \begin {gather*} \frac {3 \sqrt {1-2 x} \left (-4804+2875 x+10035 x^2+6750 x^3\right )+14 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1701} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 56, normalized size = 0.68
method | result | size |
risch | \(-\frac {\left (6750 x^{3}+10035 x^{2}+2875 x -4804\right ) \left (-1+2 x \right )}{567 \sqrt {1-2 x}}+\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(49\) |
derivativedivides | \(-\frac {5135 \left (1-2 x \right )^{\frac {3}{2}}}{324}+\frac {80 \left (1-2 x \right )^{\frac {5}{2}}}{9}-\frac {125 \left (1-2 x \right )^{\frac {7}{2}}}{84}+\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}-\frac {2 \sqrt {1-2 x}}{81}\) | \(56\) |
default | \(-\frac {5135 \left (1-2 x \right )^{\frac {3}{2}}}{324}+\frac {80 \left (1-2 x \right )^{\frac {5}{2}}}{9}-\frac {125 \left (1-2 x \right )^{\frac {7}{2}}}{84}+\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}-\frac {2 \sqrt {1-2 x}}{81}\) | \(56\) |
trager | \(\left (\frac {250}{21} x^{3}+\frac {1115}{63} x^{2}+\frac {2875}{567} x -\frac {4804}{567}\right ) \sqrt {1-2 x}-\frac {\RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{243}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 73, normalized size = 0.89 \begin {gather*} -\frac {125}{84} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {80}{9} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {5135}{324} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {1}{243} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2}{81} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.83, size = 62, normalized size = 0.76 \begin {gather*} \frac {1}{243} \, \sqrt {7} \sqrt {3} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + \frac {1}{567} \, {\left (6750 \, x^{3} + 10035 \, x^{2} + 2875 \, x - 4804\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.25, size = 107, normalized size = 1.30 \begin {gather*} - \frac {125 \left (1 - 2 x\right )^{\frac {7}{2}}}{84} + \frac {80 \left (1 - 2 x\right )^{\frac {5}{2}}}{9} - \frac {5135 \left (1 - 2 x\right )^{\frac {3}{2}}}{324} - \frac {2 \sqrt {1 - 2 x}}{81} - \frac {14 \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 )}{81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.60, size = 90, normalized size = 1.10 \begin {gather*} \frac {125}{84} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {80}{9} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {5135}{324} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {1}{243} \, \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 {2}{81} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 57, normalized size = 0.70 \begin {gather*} \frac {80\,{\left (1-2\,x\right )}^{5/2}}{9}-\frac {5135\,{\left (1-2\,x\right )}^{3/2}}{324}-\frac {2\,\sqrt {1-2\,x}}{81}-\frac {125\,{\left (1-2\,x\right )}^{7/2}}{84}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,2{}\mathrm {i}}{243} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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