Optimal. Leaf size=100 \[ -\frac {\sqrt {1-2 x} (2+3 x)^3}{10 (3+5 x)^2}-\frac {49 \sqrt {1-2 x} (2+3 x)^2}{275 (3+5 x)}+\frac {21 \sqrt {1-2 x} (44+75 x)}{2750}-\frac {1267 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1375 \sqrt {55}} \]
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Rubi [A]
time = 0.02, antiderivative size = 100, 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 = {99, 154, 152,
65, 212} \begin {gather*} -\frac {\sqrt {1-2 x} (3 x+2)^3}{10 (5 x+3)^2}-\frac {49 \sqrt {1-2 x} (3 x+2)^2}{275 (5 x+3)}+\frac {21 \sqrt {1-2 x} (75 x+44)}{2750}-\frac {1267 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1375 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 99
Rule 152
Rule 154
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^3} \, dx &=-\frac {\sqrt {1-2 x} (2+3 x)^3}{10 (3+5 x)^2}+\frac {1}{10} \int \frac {(7-21 x) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=-\frac {\sqrt {1-2 x} (2+3 x)^3}{10 (3+5 x)^2}-\frac {49 \sqrt {1-2 x} (2+3 x)^2}{275 (3+5 x)}+\frac {1}{550} \int \frac {(322-1575 x) (2+3 x)}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {\sqrt {1-2 x} (2+3 x)^3}{10 (3+5 x)^2}-\frac {49 \sqrt {1-2 x} (2+3 x)^2}{275 (3+5 x)}+\frac {21 \sqrt {1-2 x} (44+75 x)}{2750}+\frac {1267 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{2750}\\ &=-\frac {\sqrt {1-2 x} (2+3 x)^3}{10 (3+5 x)^2}-\frac {49 \sqrt {1-2 x} (2+3 x)^2}{275 (3+5 x)}+\frac {21 \sqrt {1-2 x} (44+75 x)}{2750}-\frac {1267 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2750}\\ &=-\frac {\sqrt {1-2 x} (2+3 x)^3}{10 (3+5 x)^2}-\frac {49 \sqrt {1-2 x} (2+3 x)^2}{275 (3+5 x)}+\frac {21 \sqrt {1-2 x} (44+75 x)}{2750}-\frac {1267 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1375 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 63, normalized size = 0.63 \begin {gather*} \frac {\frac {55 \sqrt {1-2 x} \left (236+4555 x+12870 x^2+9900 x^3\right )}{(3+5 x)^2}-2534 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{151250} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 66, normalized size = 0.66
method | result | size |
risch | \(-\frac {19800 x^{4}+15840 x^{3}-3760 x^{2}-4083 x -236}{2750 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {1267 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{75625}\) | \(56\) |
derivativedivides | \(-\frac {9 \left (1-2 x \right )^{\frac {3}{2}}}{125}+\frac {54 \sqrt {1-2 x}}{625}+\frac {\frac {197 \left (1-2 x \right )^{\frac {3}{2}}}{1375}-\frac {199 \sqrt {1-2 x}}{625}}{\left (-6-10 x \right )^{2}}-\frac {1267 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{75625}\) | \(66\) |
default | \(-\frac {9 \left (1-2 x \right )^{\frac {3}{2}}}{125}+\frac {54 \sqrt {1-2 x}}{625}+\frac {\frac {197 \left (1-2 x \right )^{\frac {3}{2}}}{1375}-\frac {199 \sqrt {1-2 x}}{625}}{\left (-6-10 x \right )^{2}}-\frac {1267 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{75625}\) | \(66\) |
trager | \(\frac {\left (9900 x^{3}+12870 x^{2}+4555 x +236\right ) \sqrt {1-2 x}}{2750 \left (3+5 x \right )^{2}}-\frac {1267 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{151250}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 92, normalized size = 0.92 \begin {gather*} -\frac {9}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1267}{151250} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {54}{625} \, \sqrt {-2 \, x + 1} + \frac {985 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2189 \, \sqrt {-2 \, x + 1}}{6875 \, {\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 = 0.75, size = 79, normalized size = 0.79 \begin {gather*} \frac {1267 \, \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 (9900 \, x^{3} + 12870 \, x^{2} + 4555 \, x + 236\right )} \sqrt {-2 \, x + 1}}{151250 \, {\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 = 1.58, size = 86, normalized size = 0.86 \begin {gather*} -\frac {9}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1267}{151250} \, \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 {54}{625} \, \sqrt {-2 \, x + 1} + \frac {985 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2189 \, \sqrt {-2 \, x + 1}}{27500 \, {\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 = 0.07, size = 74, normalized size = 0.74 \begin {gather*} \frac {54\,\sqrt {1-2\,x}}{625}-\frac {9\,{\left (1-2\,x\right )}^{3/2}}{125}-\frac {\frac {199\,\sqrt {1-2\,x}}{15625}-\frac {197\,{\left (1-2\,x\right )}^{3/2}}{34375}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,1267{}\mathrm {i}}{75625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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