Optimal. Leaf size=101 \[ \frac {2873}{73205 \sqrt {1-2 x}}+\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac {614}{1815 \sqrt {1-2 x} (3+5 x)^2}-\frac {2873}{39930 \sqrt {1-2 x} (3+5 x)}-\frac {2873 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641 \sqrt {55}} \]
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Rubi [A]
time = 0.02, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {91, 79, 44, 53,
65, 212} \begin {gather*} \frac {2873}{73205 \sqrt {1-2 x}}-\frac {2873}{39930 \sqrt {1-2 x} (5 x+3)}-\frac {614}{1815 \sqrt {1-2 x} (5 x+3)^2}+\frac {49}{66 (1-2 x)^{3/2} (5 x+3)^2}-\frac {2873 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 79
Rule 91
Rule 212
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^3} \, dx &=\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac {1}{66} \int \frac {-313+297 x}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\\ &=\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac {614}{1815 \sqrt {1-2 x} (3+5 x)^2}+\frac {2873 \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx}{3630}\\ &=\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac {614}{1815 \sqrt {1-2 x} (3+5 x)^2}+\frac {2873}{19965 \sqrt {1-2 x} (3+5 x)}+\frac {2873 \int \frac {1}{\sqrt {1-2 x} (3+5 x)^2} \, dx}{2662}\\ &=\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac {614}{1815 \sqrt {1-2 x} (3+5 x)^2}+\frac {2873}{19965 \sqrt {1-2 x} (3+5 x)}-\frac {2873 \sqrt {1-2 x}}{29282 (3+5 x)}+\frac {2873 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{29282}\\ &=\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac {614}{1815 \sqrt {1-2 x} (3+5 x)^2}+\frac {2873}{19965 \sqrt {1-2 x} (3+5 x)}-\frac {2873 \sqrt {1-2 x}}{29282 (3+5 x)}-\frac {2873 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{29282}\\ &=\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac {614}{1815 \sqrt {1-2 x} (3+5 x)^2}+\frac {2873}{19965 \sqrt {1-2 x} (3+5 x)}-\frac {2873 \sqrt {1-2 x}}{29282 (3+5 x)}-\frac {2873 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 65, normalized size = 0.64 \begin {gather*} \frac {-\frac {55 \left (-47568-107127 x+57460 x^2+172380 x^3\right )}{2 (1-2 x)^{3/2} (3+5 x)^2}-8619 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{2415765} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 66, normalized size = 0.65
method | result | size |
risch | \(\frac {172380 x^{3}+57460 x^{2}-107127 x -47568}{87846 \left (3+5 x \right )^{2} \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {2873 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{805255}\) | \(58\) |
derivativedivides | \(\frac {\frac {65 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {145 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {2873 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{805255}+\frac {98}{3993 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {546}{14641 \sqrt {1-2 x}}\) | \(66\) |
default | \(\frac {\frac {65 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {145 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {2873 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{805255}+\frac {98}{3993 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {546}{14641 \sqrt {1-2 x}}\) | \(66\) |
trager | \(-\frac {\left (172380 x^{3}+57460 x^{2}-107127 x -47568\right ) \sqrt {1-2 x}}{87846 \left (10 x^{2}+x -3\right )^{2}}+\frac {2873 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1610510}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 92, normalized size = 0.91 \begin {gather*} \frac {2873}{1610510} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {43095 \, {\left (2 \, x - 1\right )}^{3} + 158015 \, {\left (2 \, x - 1\right )}^{2} + 159236 \, x - 210056}{43923 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 121 \, {\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 = 0.71, size = 99, normalized size = 0.98 \begin {gather*} \frac {8619 \, \sqrt {55} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (172380 \, x^{3} + 57460 \, x^{2} - 107127 \, x - 47568\right )} \sqrt {-2 \, x + 1}}{4831530 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, 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 = 0.85, size = 89, normalized size = 0.88 \begin {gather*} \frac {2873}{1610510} \, \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 {28 \, {\left (117 \, x - 97\right )}}{43923 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {5 \, {\left (13 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 29 \, \sqrt {-2 \, x + 1}\right )}}{5324 \, {\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 = 1.22, size = 72, normalized size = 0.71 \begin {gather*} -\frac {2873\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{805255}-\frac {\frac {1316\,x}{9075}+\frac {2873\,{\left (2\,x-1\right )}^2}{19965}+\frac {2873\,{\left (2\,x-1\right )}^3}{73205}-\frac {1736}{9075}}{\frac {121\,{\left (1-2\,x\right )}^{3/2}}{25}-\frac {22\,{\left (1-2\,x\right )}^{5/2}}{5}+{\left (1-2\,x\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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