Optimal. Leaf size=68 \[ -\frac {\sqrt {1-2 x}}{550 (3+5 x)^2}-\frac {27 \sqrt {1-2 x}}{1210 (3+5 x)}-\frac {2313 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}} \]
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Rubi [A]
time = 0.01, antiderivative size = 68, 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 = {91, 79, 65, 212}
\begin {gather*} -\frac {27 \sqrt {1-2 x}}{1210 (5 x+3)}-\frac {\sqrt {1-2 x}}{550 (5 x+3)^2}-\frac {2313 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 79
Rule 91
Rule 212
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2}{\sqrt {1-2 x} (3+5 x)^3} \, dx &=-\frac {\sqrt {1-2 x}}{550 (3+5 x)^2}+\frac {1}{550} \int \frac {729+990 x}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=-\frac {\sqrt {1-2 x}}{550 (3+5 x)^2}-\frac {27 \sqrt {1-2 x}}{1210 (3+5 x)}+\frac {2313 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{6050}\\ &=-\frac {\sqrt {1-2 x}}{550 (3+5 x)^2}-\frac {27 \sqrt {1-2 x}}{1210 (3+5 x)}-\frac {2313 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{6050}\\ &=-\frac {\sqrt {1-2 x}}{550 (3+5 x)^2}-\frac {27 \sqrt {1-2 x}}{1210 (3+5 x)}-\frac {2313 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 53, normalized size = 0.78 \begin {gather*} \frac {-\frac {55 \sqrt {1-2 x} (416+675 x)}{(3+5 x)^2}-4626 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{332750} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 48, normalized size = 0.71
method | result | size |
risch | \(\frac {1350 x^{2}+157 x -416}{6050 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {2313 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{166375}\) | \(46\) |
derivativedivides | \(\frac {\frac {27 \left (1-2 x \right )^{\frac {3}{2}}}{121}-\frac {137 \sqrt {1-2 x}}{275}}{\left (-6-10 x \right )^{2}}-\frac {2313 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{166375}\) | \(48\) |
default | \(\frac {\frac {27 \left (1-2 x \right )^{\frac {3}{2}}}{121}-\frac {137 \sqrt {1-2 x}}{275}}{\left (-6-10 x \right )^{2}}-\frac {2313 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{166375}\) | \(48\) |
trager | \(-\frac {\left (675 x +416\right ) \sqrt {1-2 x}}{6050 \left (3+5 x \right )^{2}}-\frac {2313 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{332750}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 74, normalized size = 1.09 \begin {gather*} \frac {2313}{332750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {675 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1507 \, \sqrt {-2 \, x + 1}}{3025 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.29, size = 69, normalized size = 1.01 \begin {gather*} \frac {2313 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (675 \, x + 416\right )} \sqrt {-2 \, x + 1}}{332750 \, {\left (25 \, x^{2} + 30 \, x + 9\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 = 1.67, size = 68, normalized size = 1.00 \begin {gather*} \frac {2313}{332750} \, \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 {675 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1507 \, \sqrt {-2 \, x + 1}}{12100 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.23, size = 54, normalized size = 0.79 \begin {gather*} -\frac {2313\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{166375}-\frac {\frac {137\,\sqrt {1-2\,x}}{6875}-\frac {27\,{\left (1-2\,x\right )}^{3/2}}{3025}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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