Optimal. Leaf size=121 \[ -\frac {(1-2 x)^{5/2} (3 x+2)^3}{5 (5 x+3)}+\frac {11}{75} (1-2 x)^{5/2} (3 x+2)^2+\frac {188 (1-2 x)^{3/2}}{9375}-\frac {2 (1-2 x)^{5/2} (2850 x+6191)}{65625}+\frac {2068 \sqrt {1-2 x}}{15625}-\frac {2068 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \]
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Rubi [A] time = 0.04, antiderivative size = 121, 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} \[ -\frac {(1-2 x)^{5/2} (3 x+2)^3}{5 (5 x+3)}+\frac {11}{75} (1-2 x)^{5/2} (3 x+2)^2+\frac {188 (1-2 x)^{3/2}}{9375}-\frac {2 (1-2 x)^{5/2} (2850 x+6191)}{65625}+\frac {2068 \sqrt {1-2 x}}{15625}-\frac {2068 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \]
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)^{5/2} (2+3 x)^3}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{5/2} (2+3 x)^3}{5 (3+5 x)}+\frac {1}{5} \int \frac {(-1-33 x) (1-2 x)^{3/2} (2+3 x)^2}{3+5 x} \, dx\\ &=\frac {11}{75} (1-2 x)^{5/2} (2+3 x)^2-\frac {(1-2 x)^{5/2} (2+3 x)^3}{5 (3+5 x)}-\frac {1}{225} \int \frac {(-306-228 x) (1-2 x)^{3/2} (2+3 x)}{3+5 x} \, dx\\ &=\frac {11}{75} (1-2 x)^{5/2} (2+3 x)^2-\frac {(1-2 x)^{5/2} (2+3 x)^3}{5 (3+5 x)}-\frac {2 (1-2 x)^{5/2} (6191+2850 x)}{65625}+\frac {94}{625} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {188 (1-2 x)^{3/2}}{9375}+\frac {11}{75} (1-2 x)^{5/2} (2+3 x)^2-\frac {(1-2 x)^{5/2} (2+3 x)^3}{5 (3+5 x)}-\frac {2 (1-2 x)^{5/2} (6191+2850 x)}{65625}+\frac {1034 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{3125}\\ &=\frac {2068 \sqrt {1-2 x}}{15625}+\frac {188 (1-2 x)^{3/2}}{9375}+\frac {11}{75} (1-2 x)^{5/2} (2+3 x)^2-\frac {(1-2 x)^{5/2} (2+3 x)^3}{5 (3+5 x)}-\frac {2 (1-2 x)^{5/2} (6191+2850 x)}{65625}+\frac {11374 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{15625}\\ &=\frac {2068 \sqrt {1-2 x}}{15625}+\frac {188 (1-2 x)^{3/2}}{9375}+\frac {11}{75} (1-2 x)^{5/2} (2+3 x)^2-\frac {(1-2 x)^{5/2} (2+3 x)^3}{5 (3+5 x)}-\frac {2 (1-2 x)^{5/2} (6191+2850 x)}{65625}-\frac {11374 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{15625}\\ &=\frac {2068 \sqrt {1-2 x}}{15625}+\frac {188 (1-2 x)^{3/2}}{9375}+\frac {11}{75} (1-2 x)^{5/2} (2+3 x)^2-\frac {(1-2 x)^{5/2} (2+3 x)^3}{5 (3+5 x)}-\frac {2 (1-2 x)^{5/2} (6191+2850 x)}{65625}-\frac {2068 \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 = 0.06, size = 73, normalized size = 0.60 \[ \frac {\frac {5 \sqrt {1-2 x} \left (1575000 x^5+427500 x^4-1858950 x^3+152105 x^2+680930 x+16794\right )}{5 x+3}-43428 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1640625} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 85, normalized size = 0.70 \[ \frac {21714 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5 \, {\left (1575000 \, x^{5} + 427500 \, x^{4} - 1858950 \, x^{3} + 152105 \, x^{2} + 680930 \, x + 16794\right )} \sqrt {-2 \, x + 1}}{1640625 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 122, normalized size = 1.01 \[ \frac {3}{50} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {351}{1750} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {18}{3125} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {194}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1034}{78125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {418}{3125} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{15625 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 81, normalized size = 0.67 \[ -\frac {2068 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{78125}+\frac {3 \left (-2 x +1\right )^{\frac {9}{2}}}{50}-\frac {351 \left (-2 x +1\right )^{\frac {7}{2}}}{1750}+\frac {18 \left (-2 x +1\right )^{\frac {5}{2}}}{3125}+\frac {194 \left (-2 x +1\right )^{\frac {3}{2}}}{9375}+\frac {418 \sqrt {-2 x +1}}{3125}+\frac {242 \sqrt {-2 x +1}}{78125 \left (-2 x -\frac {6}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 98, normalized size = 0.81 \[ \frac {3}{50} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {351}{1750} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {18}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {194}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1034}{78125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {418}{3125} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{15625 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 82, normalized size = 0.68 \[ \frac {418\,\sqrt {1-2\,x}}{3125}-\frac {242\,\sqrt {1-2\,x}}{78125\,\left (2\,x+\frac {6}{5}\right )}+\frac {194\,{\left (1-2\,x\right )}^{3/2}}{9375}+\frac {18\,{\left (1-2\,x\right )}^{5/2}}{3125}-\frac {351\,{\left (1-2\,x\right )}^{7/2}}{1750}+\frac {3\,{\left (1-2\,x\right )}^{9/2}}{50}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2068{}\mathrm {i}}{78125} \]
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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