Optimal. Leaf size=135 \[ \frac {55 (1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)}-\frac {(1-2 x)^{5/2} (5 x+3)^3}{6 (3 x+2)^2}-\frac {220}{21} (1-2 x)^{3/2} (5 x+3)^2+\frac {55 (1-2 x)^{3/2} (603 x+209)}{1134}-\frac {935}{81} \sqrt {1-2 x}+\frac {935}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {97, 12, 149, 153, 147, 50, 63, 206} \begin {gather*} \frac {55 (1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)}-\frac {(1-2 x)^{5/2} (5 x+3)^3}{6 (3 x+2)^2}-\frac {220}{21} (1-2 x)^{3/2} (5 x+3)^2+\frac {55 (1-2 x)^{3/2} (603 x+209)}{1134}-\frac {935}{81} \sqrt {1-2 x}+\frac {935}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 50
Rule 63
Rule 97
Rule 147
Rule 149
Rule 153
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^3} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac {1}{6} \int -\frac {55 (1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{6 (2+3 x)^2}-\frac {55}{6} \int \frac {(1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)}+\frac {55}{18} \int \frac {\sqrt {1-2 x} (3+5 x)^2 (-3+72 x)}{2+3 x} \, dx\\ &=-\frac {220}{21} (1-2 x)^{3/2} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)}-\frac {55}{378} \int \frac {\sqrt {1-2 x} (3+5 x) (45+603 x)}{2+3 x} \, dx\\ &=-\frac {220}{21} (1-2 x)^{3/2} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)}+\frac {55 (1-2 x)^{3/2} (209+603 x)}{1134}-\frac {935}{54} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=-\frac {935}{81} \sqrt {1-2 x}-\frac {220}{21} (1-2 x)^{3/2} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)}+\frac {55 (1-2 x)^{3/2} (209+603 x)}{1134}-\frac {6545}{162} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {935}{81} \sqrt {1-2 x}-\frac {220}{21} (1-2 x)^{3/2} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)}+\frac {55 (1-2 x)^{3/2} (209+603 x)}{1134}+\frac {6545}{162} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {935}{81} \sqrt {1-2 x}-\frac {220}{21} (1-2 x)^{3/2} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)}+\frac {55 (1-2 x)^{3/2} (209+603 x)}{1134}+\frac {935}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 75, normalized size = 0.56 \begin {gather*} \frac {\sqrt {1-2 x} \left (54000 x^5-24120 x^4-17460 x^3-67962 x^2-152833 x-64943\right )}{1134 (3 x+2)^2}+\frac {935}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 99, normalized size = 0.73 \begin {gather*} \frac {935}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {\left (3375 (1-2 x)^5-13860 (1-2 x)^4+17325 (1-2 x)^3+31416 (1-2 x)^2-229075 (1-2 x)+320705\right ) \sqrt {1-2 x}}{567 (3 (1-2 x)-7)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.26, size = 96, normalized size = 0.71 \begin {gather*} \frac {6545 \, \sqrt {7} \sqrt {3} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 3 \, {\left (54000 \, x^{5} - 24120 \, x^{4} - 17460 \, x^{3} - 67962 \, x^{2} - 152833 \, x - 64943\right )} \sqrt {-2 \, x + 1}}{3402 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 118, normalized size = 0.87 \begin {gather*} \frac {125}{189} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {10}{27} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {370}{243} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {935}{486} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {8198}{729} \, \sqrt {-2 \, x + 1} + \frac {7 \, {\left (657 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1519 \, \sqrt {-2 \, x + 1}\right )}}{2916 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 0.62 \begin {gather*} \frac {935 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{243}-\frac {125 \left (-2 x +1\right )^{\frac {7}{2}}}{189}-\frac {10 \left (-2 x +1\right )^{\frac {5}{2}}}{27}-\frac {370 \left (-2 x +1\right )^{\frac {3}{2}}}{243}-\frac {8198 \sqrt {-2 x +1}}{729}-\frac {14 \left (-\frac {73 \left (-2 x +1\right )^{\frac {3}{2}}}{2}+\frac {1519 \sqrt {-2 x +1}}{18}\right )}{81 \left (-6 x -4\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 110, normalized size = 0.81 \begin {gather*} -\frac {125}{189} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {10}{27} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {370}{243} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {935}{486} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {8198}{729} \, \sqrt {-2 \, x + 1} + \frac {7 \, {\left (657 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1519 \, \sqrt {-2 \, x + 1}\right )}}{729 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 92, normalized size = 0.68 \begin {gather*} -\frac {8198\,\sqrt {1-2\,x}}{729}-\frac {370\,{\left (1-2\,x\right )}^{3/2}}{243}-\frac {10\,{\left (1-2\,x\right )}^{5/2}}{27}-\frac {125\,{\left (1-2\,x\right )}^{7/2}}{189}-\frac {\frac {10633\,\sqrt {1-2\,x}}{6561}-\frac {511\,{\left (1-2\,x\right )}^{3/2}}{729}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,935{}\mathrm {i}}{243} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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