Optimal. Leaf size=94 \[ \frac {357 \sqrt {1-2 x}}{1375}+\frac {119 (1-2 x)^{3/2}}{3025}-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac {131 (1-2 x)^{5/2}}{6050 (3+5 x)}-\frac {357 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}} \]
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Rubi [A]
time = 0.02, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {131 (1-2 x)^{5/2}}{6050 (5 x+3)}-\frac {(1-2 x)^{5/2}}{550 (5 x+3)^2}+\frac {119 (1-2 x)^{3/2}}{3025}+\frac {357 \sqrt {1-2 x}}{1375}-\frac {357 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}} \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 {(1-2 x)^{3/2} (2+3 x)^2}{(3+5 x)^3} \, dx &=-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}+\frac {1}{550} \int \frac {(1-2 x)^{3/2} (725+990 x)}{(3+5 x)^2} \, dx\\ &=-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac {131 (1-2 x)^{5/2}}{6050 (3+5 x)}+\frac {357 \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx}{1210}\\ &=\frac {119 (1-2 x)^{3/2}}{3025}-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac {131 (1-2 x)^{5/2}}{6050 (3+5 x)}+\frac {357}{550} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {357 \sqrt {1-2 x}}{1375}+\frac {119 (1-2 x)^{3/2}}{3025}-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac {131 (1-2 x)^{5/2}}{6050 (3+5 x)}+\frac {357}{250} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {357 \sqrt {1-2 x}}{1375}+\frac {119 (1-2 x)^{3/2}}{3025}-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac {131 (1-2 x)^{5/2}}{6050 (3+5 x)}-\frac {357}{250} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {357 \sqrt {1-2 x}}{1375}+\frac {119 (1-2 x)^{3/2}}{3025}-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac {131 (1-2 x)^{5/2}}{6050 (3+5 x)}-\frac {357 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 63, normalized size = 0.67 \begin {gather*} \frac {\sqrt {1-2 x} \left (656+2105 x+1320 x^2-600 x^3\right )}{250 (3+5 x)^2}-\frac {357 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 66, normalized size = 0.70
method | result | size |
risch | \(\frac {1200 x^{4}-3240 x^{3}-2890 x^{2}+793 x +656}{250 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {357 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6875}\) | \(56\) |
derivativedivides | \(\frac {6 \left (1-2 x \right )^{\frac {3}{2}}}{125}+\frac {174 \sqrt {1-2 x}}{625}+\frac {\frac {127 \left (1-2 x \right )^{\frac {3}{2}}}{125}-\frac {1419 \sqrt {1-2 x}}{625}}{\left (-6-10 x \right )^{2}}-\frac {357 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6875}\) | \(66\) |
default | \(\frac {6 \left (1-2 x \right )^{\frac {3}{2}}}{125}+\frac {174 \sqrt {1-2 x}}{625}+\frac {\frac {127 \left (1-2 x \right )^{\frac {3}{2}}}{125}-\frac {1419 \sqrt {1-2 x}}{625}}{\left (-6-10 x \right )^{2}}-\frac {357 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6875}\) | \(66\) |
trager | \(-\frac {\left (600 x^{3}-1320 x^{2}-2105 x -656\right ) \sqrt {1-2 x}}{250 \left (3+5 x \right )^{2}}+\frac {357 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{13750}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 92, normalized size = 0.98 \begin {gather*} \frac {6}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {357}{13750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {174}{625} \, \sqrt {-2 \, x + 1} + \frac {635 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1419 \, \sqrt {-2 \, x + 1}}{625 \, {\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 = 0.60, size = 79, normalized size = 0.84 \begin {gather*} \frac {357 \, \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 (600 \, x^{3} - 1320 \, x^{2} - 2105 \, x - 656\right )} \sqrt {-2 \, x + 1}}{13750 \, {\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 = 0.52, size = 86, normalized size = 0.91 \begin {gather*} \frac {6}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {357}{13750} \, \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 {174}{625} \, \sqrt {-2 \, x + 1} + \frac {635 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1419 \, \sqrt {-2 \, x + 1}}{2500 \, {\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 = 1.20, size = 74, normalized size = 0.79 \begin {gather*} \frac {174\,\sqrt {1-2\,x}}{625}+\frac {6\,{\left (1-2\,x\right )}^{3/2}}{125}-\frac {\frac {1419\,\sqrt {1-2\,x}}{15625}-\frac {127\,{\left (1-2\,x\right )}^{3/2}}{3125}}{\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 )\,357{}\mathrm {i}}{6875} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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