Optimal. Leaf size=80 \[ \frac {7 (2+3 x)^2}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {2 (10309+17112 x)}{19965 \sqrt {1-2 x} (3+5 x)}-\frac {208 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6655 \sqrt {55}} \]
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Rubi [A]
time = 0.01, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {100, 149, 65,
212} \begin {gather*} \frac {7 (3 x+2)^2}{33 (1-2 x)^{3/2} (5 x+3)}-\frac {2 (17112 x+10309)}{19965 \sqrt {1-2 x} (5 x+3)}-\frac {208 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6655 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 149
Rule 212
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)^2} \, dx &=\frac {7 (2+3 x)^2}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {1}{33} \int \frac {(2+3 x) (50+96 x)}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\\ &=\frac {7 (2+3 x)^2}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {2 (10309+17112 x)}{19965 \sqrt {1-2 x} (3+5 x)}+\frac {104 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{6655}\\ &=\frac {7 (2+3 x)^2}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {2 (10309+17112 x)}{19965 \sqrt {1-2 x} (3+5 x)}-\frac {104 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{6655}\\ &=\frac {7 (2+3 x)^2}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {2 (10309+17112 x)}{19965 \sqrt {1-2 x} (3+5 x)}-\frac {208 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6655 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 58, normalized size = 0.72 \begin {gather*} \frac {\frac {55 \left (-3678+57832 x+106563 x^2\right )}{(1-2 x)^{3/2} (3+5 x)}-624 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1098075} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 54, normalized size = 0.68
method | result | size |
risch | \(-\frac {106563 x^{2}+57832 x -3678}{19965 \left (3+5 x \right ) \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {208 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{366025}\) | \(53\) |
derivativedivides | \(\frac {2 \sqrt {1-2 x}}{33275 \left (-\frac {6}{5}-2 x \right )}-\frac {208 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{366025}+\frac {343}{726 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {1421}{2662 \sqrt {1-2 x}}\) | \(54\) |
default | \(\frac {2 \sqrt {1-2 x}}{33275 \left (-\frac {6}{5}-2 x \right )}-\frac {208 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{366025}+\frac {343}{726 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {1421}{2662 \sqrt {1-2 x}}\) | \(54\) |
trager | \(\frac {\left (106563 x^{2}+57832 x -3678\right ) \sqrt {1-2 x}}{19965 \left (-1+2 x \right )^{2} \left (3+5 x \right )}+\frac {104 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{366025}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 74, normalized size = 0.92 \begin {gather*} \frac {104}{366025} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {106563 \, {\left (2 \, x - 1\right )}^{2} + 657580 \, x - 121275}{39930 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 11 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.29, size = 84, normalized size = 1.05 \begin {gather*} \frac {312 \, \sqrt {55} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (106563 \, x^{2} + 57832 \, x - 3678\right )} \sqrt {-2 \, x + 1}}{1098075 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.03, size = 77, normalized size = 0.96 \begin {gather*} \frac {104}{366025} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {49 \, {\left (87 \, x - 5\right )}}{3993 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {\sqrt {-2 \, x + 1}}{6655 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 55, normalized size = 0.69 \begin {gather*} \frac {\frac {5978\,x}{1815}+\frac {35521\,{\left (2\,x-1\right )}^2}{66550}-\frac {147}{242}}{\frac {11\,{\left (1-2\,x\right )}^{3/2}}{5}-{\left (1-2\,x\right )}^{5/2}}-\frac {208\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{366025} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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