Optimal. Leaf size=120 \[ \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)}-\frac {38 (3 x+2)^3}{1815 \sqrt {1-2 x} (5 x+3)}-\frac {7588 (3 x+2)^2}{6655 \sqrt {1-2 x}}-\frac {6 \sqrt {1-2 x} (38025 x+114092)}{33275}-\frac {68 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{33275 \sqrt {55}} \]
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Rubi [A] time = 0.04, antiderivative size = 120, 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 = {98, 149, 150, 147, 63, 206} \[ \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)}-\frac {38 (3 x+2)^3}{1815 \sqrt {1-2 x} (5 x+3)}-\frac {7588 (3 x+2)^2}{6655 \sqrt {1-2 x}}-\frac {6 \sqrt {1-2 x} (38025 x+114092)}{33275}-\frac {68 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{33275 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 147
Rule 149
Rule 150
Rule 206
Rubi steps
\begin {align*} \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^2} \, dx &=\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {1}{33} \int \frac {(2+3 x)^3 (176+306 x)}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\\ &=-\frac {38 (2+3 x)^3}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {\int \frac {(2+3 x)^2 (6162+10440 x)}{(1-2 x)^{3/2} (3+5 x)} \, dx}{1815}\\ &=-\frac {7588 (2+3 x)^2}{6655 \sqrt {1-2 x}}-\frac {38 (2+3 x)^3}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {\int \frac {(-410772-684450 x) (2+3 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{19965}\\ &=-\frac {7588 (2+3 x)^2}{6655 \sqrt {1-2 x}}-\frac {38 (2+3 x)^3}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {6 \sqrt {1-2 x} (114092+38025 x)}{33275}+\frac {34 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{33275}\\ &=-\frac {7588 (2+3 x)^2}{6655 \sqrt {1-2 x}}-\frac {38 (2+3 x)^3}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {6 \sqrt {1-2 x} (114092+38025 x)}{33275}-\frac {34 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{33275}\\ &=-\frac {7588 (2+3 x)^2}{6655 \sqrt {1-2 x}}-\frac {38 (2+3 x)^3}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {6 \sqrt {1-2 x} (114092+38025 x)}{33275}-\frac {68 \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 = 94, normalized size = 0.78 \[ -\frac {-270 \left (10 x^2+x-3\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {5}{11} (1-2 x)\right )-266 (5 x+3) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {5}{11} (1-2 x)\right )+33 \left (111375 x^4+1113750 x^3-1975050 x^2-734880 x+496226\right )}{226875 (1-2 x)^{3/2} (5 x+3)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 94, normalized size = 0.78 \[ \frac {102 \, \sqrt {55} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (1617165 \, x^{4} + 16171650 \, x^{3} - 28677318 \, x^{2} - 10671002 \, x + 7204728\right )} \sqrt {-2 \, x + 1}}{5490375 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.29, size = 95, normalized size = 0.79 \[ \frac {81}{200} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {34}{1830125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {8829}{1000} \, \sqrt {-2 \, x + 1} - \frac {2401 \, {\left (285 \, x - 104\right )}}{15972 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {\sqrt {-2 \, x + 1}}{166375 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 72, normalized size = 0.60 \[ -\frac {68 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1830125}+\frac {81 \left (-2 x +1\right )^{\frac {3}{2}}}{200}-\frac {8829 \sqrt {-2 x +1}}{1000}+\frac {16807}{2904 \left (-2 x +1\right )^{\frac {3}{2}}}-\frac {228095}{10648 \sqrt {-2 x +1}}+\frac {2 \sqrt {-2 x +1}}{831875 \left (-2 x -\frac {6}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 92, normalized size = 0.77 \[ \frac {81}{200} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {34}{1830125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {8829}{1000} \, \sqrt {-2 \, x + 1} - \frac {427678077 \, {\left (2 \, x - 1\right )}^{2} + 2112880000 \, x - 802234125}{3993000 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 11 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 75, normalized size = 0.62 \[ \frac {\frac {38416\,x}{363}+\frac {142559359\,{\left (2\,x-1\right )}^2}{6655000}-\frac {194481}{4840}}{\frac {11\,{\left (1-2\,x\right )}^{3/2}}{5}-{\left (1-2\,x\right )}^{5/2}}-\frac {8829\,\sqrt {1-2\,x}}{1000}+\frac {81\,{\left (1-2\,x\right )}^{3/2}}{200}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,68{}\mathrm {i}}{1830125} \]
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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