Optimal. Leaf size=108 \[ \frac {277 (1-2 x)^{5/2}}{5292 (3 x+2)^3}-\frac {(1-2 x)^{5/2}}{252 (3 x+2)^4}-\frac {14423 (1-2 x)^{3/2}}{31752 (3 x+2)^2}+\frac {14423 \sqrt {1-2 x}}{31752 (3 x+2)}-\frac {14423 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15876 \sqrt {21}} \]
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Rubi [A] time = 0.03, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {89, 78, 47, 63, 206} \[ \frac {277 (1-2 x)^{5/2}}{5292 (3 x+2)^3}-\frac {(1-2 x)^{5/2}}{252 (3 x+2)^4}-\frac {14423 (1-2 x)^{3/2}}{31752 (3 x+2)^2}+\frac {14423 \sqrt {1-2 x}}{31752 (3 x+2)}-\frac {14423 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15876 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 78
Rule 89
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^5} \, dx &=-\frac {(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac {1}{252} \int \frac {(1-2 x)^{3/2} (1123+2100 x)}{(2+3 x)^4} \, dx\\ &=-\frac {(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac {277 (1-2 x)^{5/2}}{5292 (2+3 x)^3}+\frac {14423 \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3} \, dx}{5292}\\ &=-\frac {(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac {277 (1-2 x)^{5/2}}{5292 (2+3 x)^3}-\frac {14423 (1-2 x)^{3/2}}{31752 (2+3 x)^2}-\frac {14423 \int \frac {\sqrt {1-2 x}}{(2+3 x)^2} \, dx}{10584}\\ &=-\frac {(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac {277 (1-2 x)^{5/2}}{5292 (2+3 x)^3}-\frac {14423 (1-2 x)^{3/2}}{31752 (2+3 x)^2}+\frac {14423 \sqrt {1-2 x}}{31752 (2+3 x)}+\frac {14423 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{31752}\\ &=-\frac {(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac {277 (1-2 x)^{5/2}}{5292 (2+3 x)^3}-\frac {14423 (1-2 x)^{3/2}}{31752 (2+3 x)^2}+\frac {14423 \sqrt {1-2 x}}{31752 (2+3 x)}-\frac {14423 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{31752}\\ &=-\frac {(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac {277 (1-2 x)^{5/2}}{5292 (2+3 x)^3}-\frac {14423 (1-2 x)^{3/2}}{31752 (2+3 x)^2}+\frac {14423 \sqrt {1-2 x}}{31752 (2+3 x)}-\frac {14423 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15876 \sqrt {21}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 79, normalized size = 0.73 \[ -\frac {21 \left (1337958 x^4+1307091 x^3-80575 x^2-331950 x-60890\right )-28846 (3 x+2)^4 \sqrt {42 x-21} \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {2 x-1}\right )}{666792 \sqrt {1-2 x} (3 x+2)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 99, normalized size = 0.92 \[ \frac {14423 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (668979 \, x^{3} + 988035 \, x^{2} + 453730 \, x + 60890\right )} \sqrt {-2 \, x + 1}}{666792 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 100, normalized size = 0.93 \[ \frac {14423}{666792} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {668979 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 3983007 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 7773997 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 4947089 \, \sqrt {-2 \, x + 1}}{254016 \, {\left (3 \, x + 2\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 66, normalized size = 0.61 \[ -\frac {14423 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{333396}+\frac {-\frac {8259 \left (-2 x +1\right )^{\frac {7}{2}}}{196}+\frac {189667 \left (-2 x +1\right )^{\frac {5}{2}}}{756}-\frac {158653 \left (-2 x +1\right )^{\frac {3}{2}}}{324}+\frac {100961 \sqrt {-2 x +1}}{324}}{\left (-6 x -4\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 110, normalized size = 1.02 \[ \frac {14423}{666792} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {668979 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 3983007 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 7773997 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 4947089 \, \sqrt {-2 \, x + 1}}{15876 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 89, normalized size = 0.82 \[ \frac {\frac {100961\,\sqrt {1-2\,x}}{26244}-\frac {158653\,{\left (1-2\,x\right )}^{3/2}}{26244}+\frac {189667\,{\left (1-2\,x\right )}^{5/2}}{61236}-\frac {2753\,{\left (1-2\,x\right )}^{7/2}}{5292}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}}-\frac {14423\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{333396} \]
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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