Optimal. Leaf size=108 \[ \frac {192 \sqrt {1-2 x}}{3125}+\frac {27}{175} (1-2 x)^{3/2} (2+3 x)^2-\frac {(1-2 x)^{3/2} (2+3 x)^3}{5 (3+5 x)}-\frac {6}{625} (1-2 x)^{3/2} (29+9 x)-\frac {192 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \]
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Rubi [A]
time = 0.02, antiderivative size = 108, 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 = {99, 158, 152,
52, 65, 212} \begin {gather*} -\frac {(1-2 x)^{3/2} (3 x+2)^3}{5 (5 x+3)}+\frac {27}{175} (1-2 x)^{3/2} (3 x+2)^2-\frac {6}{625} (1-2 x)^{3/2} (9 x+29)+\frac {192 \sqrt {1-2 x}}{3125}-\frac {192 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 99
Rule 152
Rule 158
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{3/2} (2+3 x)^3}{5 (3+5 x)}+\frac {1}{5} \int \frac {(3-27 x) \sqrt {1-2 x} (2+3 x)^2}{3+5 x} \, dx\\ &=\frac {27}{175} (1-2 x)^{3/2} (2+3 x)^2-\frac {(1-2 x)^{3/2} (2+3 x)^3}{5 (3+5 x)}-\frac {1}{175} \int \frac {(-210-126 x) \sqrt {1-2 x} (2+3 x)}{3+5 x} \, dx\\ &=\frac {27}{175} (1-2 x)^{3/2} (2+3 x)^2-\frac {(1-2 x)^{3/2} (2+3 x)^3}{5 (3+5 x)}-\frac {6}{625} (1-2 x)^{3/2} (29+9 x)+\frac {96}{625} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {192 \sqrt {1-2 x}}{3125}+\frac {27}{175} (1-2 x)^{3/2} (2+3 x)^2-\frac {(1-2 x)^{3/2} (2+3 x)^3}{5 (3+5 x)}-\frac {6}{625} (1-2 x)^{3/2} (29+9 x)+\frac {1056 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{3125}\\ &=\frac {192 \sqrt {1-2 x}}{3125}+\frac {27}{175} (1-2 x)^{3/2} (2+3 x)^2-\frac {(1-2 x)^{3/2} (2+3 x)^3}{5 (3+5 x)}-\frac {6}{625} (1-2 x)^{3/2} (29+9 x)-\frac {1056 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3125}\\ &=\frac {192 \sqrt {1-2 x}}{3125}+\frac {27}{175} (1-2 x)^{3/2} (2+3 x)^2-\frac {(1-2 x)^{3/2} (2+3 x)^3}{5 (3+5 x)}-\frac {6}{625} (1-2 x)^{3/2} (29+9 x)-\frac {192 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 68, normalized size = 0.63 \begin {gather*} \frac {-\frac {5 \sqrt {1-2 x} \left (8738-27640 x-57165 x^2+62100 x^3+67500 x^4\right )}{3+5 x}-1344 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{109375} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 72, normalized size = 0.67
method | result | size |
risch | \(\frac {135000 x^{5}+56700 x^{4}-176430 x^{3}+1885 x^{2}+45116 x -8738}{21875 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {192 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}\) | \(61\) |
derivativedivides | \(\frac {27 \left (1-2 x \right )^{\frac {7}{2}}}{350}-\frac {351 \left (1-2 x \right )^{\frac {5}{2}}}{1250}+\frac {6 \left (1-2 x \right )^{\frac {3}{2}}}{625}+\frac {194 \sqrt {1-2 x}}{3125}+\frac {22 \sqrt {1-2 x}}{15625 \left (-\frac {6}{5}-2 x \right )}-\frac {192 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}\) | \(72\) |
default | \(\frac {27 \left (1-2 x \right )^{\frac {7}{2}}}{350}-\frac {351 \left (1-2 x \right )^{\frac {5}{2}}}{1250}+\frac {6 \left (1-2 x \right )^{\frac {3}{2}}}{625}+\frac {194 \sqrt {1-2 x}}{3125}+\frac {22 \sqrt {1-2 x}}{15625 \left (-\frac {6}{5}-2 x \right )}-\frac {192 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}\) | \(72\) |
trager | \(-\frac {\left (67500 x^{4}+62100 x^{3}-57165 x^{2}-27640 x +8738\right ) \sqrt {1-2 x}}{21875 \left (3+5 x \right )}-\frac {96 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{15625}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 89, normalized size = 0.82 \begin {gather*} \frac {27}{350} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {351}{1250} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {6}{625} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {96}{15625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {194}{3125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{3125 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.90, size = 80, normalized size = 0.74 \begin {gather*} \frac {672 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 5 \, {\left (67500 \, x^{4} + 62100 \, x^{3} - 57165 \, x^{2} - 27640 \, x + 8738\right )} \sqrt {-2 \, x + 1}}{109375 \, {\left (5 \, 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 = 0.54, size = 106, normalized size = 0.98 \begin {gather*} -\frac {27}{350} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {351}{1250} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {6}{625} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {96}{15625} \, \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 {194}{3125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{3125 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 73, normalized size = 0.68 \begin {gather*} \frac {194\,\sqrt {1-2\,x}}{3125}-\frac {22\,\sqrt {1-2\,x}}{15625\,\left (2\,x+\frac {6}{5}\right )}+\frac {6\,{\left (1-2\,x\right )}^{3/2}}{625}-\frac {351\,{\left (1-2\,x\right )}^{5/2}}{1250}+\frac {27\,{\left (1-2\,x\right )}^{7/2}}{350}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,192{}\mathrm {i}}{15625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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