Optimal. Leaf size=128 \[ -\frac {(1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}+\frac {11}{75} (1-2 x)^{3/2} (3 x+2)^3-\frac {2}{875} (1-2 x)^{3/2} (3 x+2)^2-\frac {(1-2 x)^{3/2} (3663 x+5678)}{9375}+\frac {258 \sqrt {1-2 x}}{15625}-\frac {258 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \]
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Rubi [A] time = 0.04, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {97, 153, 147, 50, 63, 206} \begin {gather*} -\frac {(1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}+\frac {11}{75} (1-2 x)^{3/2} (3 x+2)^3-\frac {2}{875} (1-2 x)^{3/2} (3 x+2)^2-\frac {(1-2 x)^{3/2} (3663 x+5678)}{9375}+\frac {258 \sqrt {1-2 x}}{15625}-\frac {258 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 97
Rule 147
Rule 153
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}+\frac {1}{5} \int \frac {(6-33 x) \sqrt {1-2 x} (2+3 x)^3}{3+5 x} \, dx\\ &=\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac {1}{225} \int \frac {(-243-18 x) \sqrt {1-2 x} (2+3 x)^2}{3+5 x} \, dx\\ &=-\frac {2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}+\frac {\int \frac {\sqrt {1-2 x} (2+3 x) (17010+25641 x)}{3+5 x} \, dx}{7875}\\ &=-\frac {2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{3/2} (5678+3663 x)}{9375}+\frac {129 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{3125}\\ &=\frac {258 \sqrt {1-2 x}}{15625}-\frac {2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{3/2} (5678+3663 x)}{9375}+\frac {1419 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{15625}\\ &=\frac {258 \sqrt {1-2 x}}{15625}-\frac {2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{3/2} (5678+3663 x)}{9375}-\frac {1419 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{15625}\\ &=\frac {258 \sqrt {1-2 x}}{15625}-\frac {2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{3/2} (5678+3663 x)}{9375}-\frac {258 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 78, normalized size = 0.61 \begin {gather*} -\frac {5 \sqrt {1-2 x} \left (787500 x^5+1395000 x^4+157275 x^3-924335 x^2-143235 x+161312\right )+1806 \sqrt {55} (5 x+3) \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{546875 (5 x+3)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 99, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {1-2 x} \left (196875 (1-2 x)^5-1681875 (1-2 x)^4+4916025 (1-2 x)^3-4776905 (1-2 x)^2-24080 (1-2 x)+79464\right )}{437500 (5 (1-2 x)-11)}-\frac {258 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.43, size = 85, normalized size = 0.66 \begin {gather*} \frac {903 \, \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 (787500 \, x^{5} + 1395000 \, x^{4} + 157275 \, x^{3} - 924335 \, x^{2} - 143235 \, x + 161312\right )} \sqrt {-2 \, x + 1}}{546875 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 122, normalized size = 0.95 \begin {gather*} -\frac {9}{100} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {999}{1750} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {12393}{12500} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {8}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {129}{78125} \, \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 {52}{3125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{15625 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 81, normalized size = 0.63 \begin {gather*} -\frac {258 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{78125}-\frac {9 \left (-2 x +1\right )^{\frac {9}{2}}}{100}+\frac {999 \left (-2 x +1\right )^{\frac {7}{2}}}{1750}-\frac {12393 \left (-2 x +1\right )^{\frac {5}{2}}}{12500}+\frac {8 \left (-2 x +1\right )^{\frac {3}{2}}}{3125}+\frac {52 \sqrt {-2 x +1}}{3125}+\frac {22 \sqrt {-2 x +1}}{78125 \left (-2 x -\frac {6}{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, size = 98, normalized size = 0.77 \begin {gather*} -\frac {9}{100} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {999}{1750} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {12393}{12500} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {8}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {129}{78125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {52}{3125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{15625 \, {\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 = 82, normalized size = 0.64 \begin {gather*} \frac {52\,\sqrt {1-2\,x}}{3125}-\frac {22\,\sqrt {1-2\,x}}{78125\,\left (2\,x+\frac {6}{5}\right )}+\frac {8\,{\left (1-2\,x\right )}^{3/2}}{3125}-\frac {12393\,{\left (1-2\,x\right )}^{5/2}}{12500}+\frac {999\,{\left (1-2\,x\right )}^{7/2}}{1750}-\frac {9\,{\left (1-2\,x\right )}^{9/2}}{100}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,258{}\mathrm {i}}{78125} \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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