Optimal. Leaf size=148 \[ \frac {(1-2 x)^{7/2}}{126 (3 x+2)^6}-\frac {41 (1-2 x)^{5/2}}{378 (3 x+2)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (3 x+2)^4}+\frac {205 \sqrt {1-2 x}}{444528 (3 x+2)}+\frac {205 \sqrt {1-2 x}}{190512 (3 x+2)^2}-\frac {205 \sqrt {1-2 x}}{13608 (3 x+2)^3}+\frac {205 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{222264 \sqrt {21}} \]
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Rubi [A] time = 0.05, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {78, 47, 51, 63, 206} \[ \frac {(1-2 x)^{7/2}}{126 (3 x+2)^6}-\frac {41 (1-2 x)^{5/2}}{378 (3 x+2)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (3 x+2)^4}+\frac {205 \sqrt {1-2 x}}{444528 (3 x+2)}+\frac {205 \sqrt {1-2 x}}{190512 (3 x+2)^2}-\frac {205 \sqrt {1-2 x}}{13608 (3 x+2)^3}+\frac {205 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{222264 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 78
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx &=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}+\frac {205}{126} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6} \, dx\\ &=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}-\frac {205}{378} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5} \, dx\\ &=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac {205 \int \frac {\sqrt {1-2 x}}{(2+3 x)^4} \, dx}{1512}\\ &=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}-\frac {205 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{13608}\\ &=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}+\frac {205 \sqrt {1-2 x}}{190512 (2+3 x)^2}-\frac {205 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{63504}\\ &=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}+\frac {205 \sqrt {1-2 x}}{190512 (2+3 x)^2}+\frac {205 \sqrt {1-2 x}}{444528 (2+3 x)}-\frac {205 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{444528}\\ &=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}+\frac {205 \sqrt {1-2 x}}{190512 (2+3 x)^2}+\frac {205 \sqrt {1-2 x}}{444528 (2+3 x)}+\frac {205 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{444528}\\ &=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}+\frac {205 \sqrt {1-2 x}}{190512 (2+3 x)^2}+\frac {205 \sqrt {1-2 x}}{444528 (2+3 x)}+\frac {205 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{222264 \sqrt {21}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 42, normalized size = 0.28 \[ \frac {(1-2 x)^{7/2} \left (\frac {823543}{(3 x+2)^6}-13120 \, _2F_1\left (\frac {7}{2},6;\frac {9}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{103766418} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 130, normalized size = 0.88 \[ \frac {205 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (49815 \, x^{5} + 204795 \, x^{4} - 824526 \, x^{3} - 176850 \, x^{2} + 154312 \, x - 51904\right )} \sqrt {-2 \, x + 1}}{9335088 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 132, normalized size = 0.89 \[ -\frac {205}{9335088} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {49815 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 658665 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 1161594 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 8353422 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 8367485 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 3445435 \, \sqrt {-2 \, x + 1}}{14224896 \, {\left (3 \, x + 2\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 0.57 \[ \frac {205 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{4667544}-\frac {46656 \left (\frac {205 \left (-2 x +1\right )^{\frac {11}{2}}}{42674688}-\frac {3485 \left (-2 x +1\right )^{\frac {9}{2}}}{54867456}-\frac {439 \left (-2 x +1\right )^{\frac {7}{2}}}{3919104}+\frac {451 \left (-2 x +1\right )^{\frac {5}{2}}}{559872}-\frac {24395 \left (-2 x +1\right )^{\frac {3}{2}}}{30233088}+\frac {10045 \sqrt {-2 x +1}}{30233088}\right )}{\left (-6 x -4\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 146, normalized size = 0.99 \[ -\frac {205}{9335088} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {49815 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 658665 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 1161594 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 8353422 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 8367485 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 3445435 \, \sqrt {-2 \, x + 1}}{222264 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 126, normalized size = 0.85 \[ \frac {205\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{4667544}-\frac {\frac {10045\,\sqrt {1-2\,x}}{472392}-\frac {24395\,{\left (1-2\,x\right )}^{3/2}}{472392}+\frac {451\,{\left (1-2\,x\right )}^{5/2}}{8748}-\frac {439\,{\left (1-2\,x\right )}^{7/2}}{61236}-\frac {3485\,{\left (1-2\,x\right )}^{9/2}}{857304}+\frac {205\,{\left (1-2\,x\right )}^{11/2}}{666792}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}} \]
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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