Optimal. Leaf size=106 \[ \frac {81}{160} (1-2 x)^{9/2}-\frac {43011 (1-2 x)^{7/2}}{5600}+\frac {507627 (1-2 x)^{5/2}}{10000}-\frac {1997451 (1-2 x)^{3/2}}{10000}+\frac {70752609 \sqrt {1-2 x}}{100000}+\frac {117649}{352 \sqrt {1-2 x}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{34375 \sqrt {55}} \]
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Rubi [A] time = 0.08, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {87, 43, 63, 206} \[ \frac {81}{160} (1-2 x)^{9/2}-\frac {43011 (1-2 x)^{7/2}}{5600}+\frac {507627 (1-2 x)^{5/2}}{10000}-\frac {1997451 (1-2 x)^{3/2}}{10000}+\frac {70752609 \sqrt {1-2 x}}{100000}+\frac {117649}{352 \sqrt {1-2 x}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{34375 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 63
Rule 87
Rule 206
Rubi steps
\begin {align*} \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)} \, dx &=\int \left (\frac {117649}{352 (1-2 x)^{3/2}}-\frac {31289679}{100000 \sqrt {1-2 x}}-\frac {4693491 x}{10000 \sqrt {1-2 x}}-\frac {479439 x^2}{1000 \sqrt {1-2 x}}-\frac {28431 x^3}{100 \sqrt {1-2 x}}-\frac {729 x^4}{10 \sqrt {1-2 x}}+\frac {1}{34375 \sqrt {1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac {117649}{352 \sqrt {1-2 x}}+\frac {31289679 \sqrt {1-2 x}}{100000}+\frac {\int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{34375}-\frac {729}{10} \int \frac {x^4}{\sqrt {1-2 x}} \, dx-\frac {28431}{100} \int \frac {x^3}{\sqrt {1-2 x}} \, dx-\frac {4693491 \int \frac {x}{\sqrt {1-2 x}} \, dx}{10000}-\frac {479439 \int \frac {x^2}{\sqrt {1-2 x}} \, dx}{1000}\\ &=\frac {117649}{352 \sqrt {1-2 x}}+\frac {31289679 \sqrt {1-2 x}}{100000}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{34375}-\frac {729}{10} \int \left (\frac {1}{16 \sqrt {1-2 x}}-\frac {1}{4} \sqrt {1-2 x}+\frac {3}{8} (1-2 x)^{3/2}-\frac {1}{4} (1-2 x)^{5/2}+\frac {1}{16} (1-2 x)^{7/2}\right ) \, dx-\frac {28431}{100} \int \left (\frac {1}{8 \sqrt {1-2 x}}-\frac {3}{8} \sqrt {1-2 x}+\frac {3}{8} (1-2 x)^{3/2}-\frac {1}{8} (1-2 x)^{5/2}\right ) \, dx-\frac {4693491 \int \left (\frac {1}{2 \sqrt {1-2 x}}-\frac {1}{2} \sqrt {1-2 x}\right ) \, dx}{10000}-\frac {479439 \int \left (\frac {1}{4 \sqrt {1-2 x}}-\frac {1}{2} \sqrt {1-2 x}+\frac {1}{4} (1-2 x)^{3/2}\right ) \, dx}{1000}\\ &=\frac {117649}{352 \sqrt {1-2 x}}+\frac {70752609 \sqrt {1-2 x}}{100000}-\frac {1997451 (1-2 x)^{3/2}}{10000}+\frac {507627 (1-2 x)^{5/2}}{10000}-\frac {43011 (1-2 x)^{7/2}}{5600}+\frac {81}{160} (1-2 x)^{9/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{34375 \sqrt {55}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 60, normalized size = 0.57 \[ \frac {14 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {5}{11} (1-2 x)\right )-99 \left (196875 x^5+1001250 x^4+2440575 x^3+4301000 x^2+10503235 x-10762494\right )}{1203125 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 79, normalized size = 0.75 \[ \frac {7 \, \sqrt {55} {\left (2 \, x - 1\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (3898125 \, x^{5} + 19824750 \, x^{4} + 48323385 \, x^{3} + 85159800 \, x^{2} + 207964053 \, x - 213097384\right )} \sqrt {-2 \, x + 1}}{13234375 \, {\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.31, size = 115, normalized size = 1.08 \[ \frac {81}{160} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {43011}{5600} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {507627}{10000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {1997451}{10000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{1890625} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {70752609}{100000} \, \sqrt {-2 \, x + 1} + \frac {117649}{352 \, \sqrt {-2 \, x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 74, normalized size = 0.70 \[ -\frac {2 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1890625}-\frac {1997451 \left (-2 x +1\right )^{\frac {3}{2}}}{10000}+\frac {507627 \left (-2 x +1\right )^{\frac {5}{2}}}{10000}-\frac {43011 \left (-2 x +1\right )^{\frac {7}{2}}}{5600}+\frac {81 \left (-2 x +1\right )^{\frac {9}{2}}}{160}+\frac {117649}{352 \sqrt {-2 x +1}}+\frac {70752609 \sqrt {-2 x +1}}{100000} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 91, normalized size = 0.86 \[ \frac {81}{160} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {43011}{5600} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {507627}{10000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {1997451}{10000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{1890625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {70752609}{100000} \, \sqrt {-2 \, x + 1} + \frac {117649}{352 \, \sqrt {-2 \, x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 75, normalized size = 0.71 \[ \frac {117649}{352\,\sqrt {1-2\,x}}+\frac {70752609\,\sqrt {1-2\,x}}{100000}-\frac {1997451\,{\left (1-2\,x\right )}^{3/2}}{10000}+\frac {507627\,{\left (1-2\,x\right )}^{5/2}}{10000}-\frac {43011\,{\left (1-2\,x\right )}^{7/2}}{5600}+\frac {81\,{\left (1-2\,x\right )}^{9/2}}{160}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{1890625} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 171.93, size = 138, normalized size = 1.30 \[ \frac {81 \left (1 - 2 x\right )^{\frac {9}{2}}}{160} - \frac {43011 \left (1 - 2 x\right )^{\frac {7}{2}}}{5600} + \frac {507627 \left (1 - 2 x\right )^{\frac {5}{2}}}{10000} - \frac {1997451 \left (1 - 2 x\right )^{\frac {3}{2}}}{10000} + \frac {70752609 \sqrt {1 - 2 x}}{100000} + \frac {2 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 < - \frac {11}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 > - \frac {11}{5} \end {cases}\right )}{34375} + \frac {117649}{352 \sqrt {1 - 2 x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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