Optimal. Leaf size=100 \[ \frac {7 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)}-\frac {38 (3 x+2)^2}{1815 \sqrt {1-2 x} (5 x+3)}-\frac {3 (40912-24739 x)}{33275 \sqrt {1-2 x}}-\frac {274 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{33275 \sqrt {55}} \]
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Rubi [A] time = 0.03, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {98, 149, 146, 63, 206} \[ \frac {7 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)}-\frac {38 (3 x+2)^2}{1815 \sqrt {1-2 x} (5 x+3)}-\frac {3 (40912-24739 x)}{33275 \sqrt {1-2 x}}-\frac {274 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{33275 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 146
Rule 149
Rule 206
Rubi steps
\begin {align*} \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^2} \, dx &=\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {1}{33} \int \frac {(2+3 x)^2 (113+201 x)}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\\ &=-\frac {38 (2+3 x)^2}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {\int \frac {(2+3 x) (3966+6747 x)}{(1-2 x)^{3/2} (3+5 x)} \, dx}{1815}\\ &=-\frac {3 (40912-24739 x)}{33275 \sqrt {1-2 x}}-\frac {38 (2+3 x)^2}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)}+\frac {137 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{33275}\\ &=-\frac {3 (40912-24739 x)}{33275 \sqrt {1-2 x}}-\frac {38 (2+3 x)^2}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {137 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{33275}\\ &=-\frac {3 (40912-24739 x)}{33275 \sqrt {1-2 x}}-\frac {38 (2+3 x)^2}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {274 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{33275 \sqrt {55}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 89, normalized size = 0.89 \[ -\frac {-288 \left (10 x^2+x-3\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {5}{11} (1-2 x)\right )-266 (5 x+3) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {5}{11} (1-2 x)\right )+33 \left (22275 x^3-63855 x^2-24619 x+13028\right )}{45375 (1-2 x)^{3/2} (5 x+3)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 89, normalized size = 0.89 \[ \frac {411 \, \sqrt {55} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (1617165 \, x^{3} - 4634229 \, x^{2} - 1790101 \, x + 943584\right )} \sqrt {-2 \, x + 1}}{5490375 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.26, size = 86, normalized size = 0.86 \[ \frac {137}{1830125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {81}{100} \, \sqrt {-2 \, x + 1} - \frac {343 \, {\left (372 \, x - 109\right )}}{15972 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {\sqrt {-2 \, x + 1}}{33275 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 63, normalized size = 0.63 \[ -\frac {274 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1830125}-\frac {81 \sqrt {-2 x +1}}{100}+\frac {2401}{1452 \left (-2 x +1\right )^{\frac {3}{2}}}-\frac {10633}{2662 \sqrt {-2 x +1}}+\frac {2 \sqrt {-2 x +1}}{166375 \left (-2 x -\frac {6}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 83, normalized size = 0.83 \[ \frac {137}{1830125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {81}{100} \, \sqrt {-2 \, x + 1} - \frac {3987363 \, {\left (2 \, x - 1\right )}^{2} + 20845825 \, x - 6791400}{199650 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 11 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 64, normalized size = 0.64 \[ \frac {\frac {75803\,x}{3630}+\frac {1329121\,{\left (2\,x-1\right )}^2}{332750}-\frac {4116}{605}}{\frac {11\,{\left (1-2\,x\right )}^{3/2}}{5}-{\left (1-2\,x\right )}^{5/2}}-\frac {274\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1830125}-\frac {81\,\sqrt {1-2\,x}}{100} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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