Optimal. Leaf size=100 \[ \frac {\sqrt {1-2 x} (5 x+3)^2}{84 (3 x+2)^4}+\frac {\sqrt {1-2 x} (4955 x+3168)}{10584 (3 x+2)^3}-\frac {42995 \sqrt {1-2 x}}{74088 (3 x+2)}-\frac {42995 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{37044 \sqrt {21}} \]
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Rubi [A] time = 0.03, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {98, 145, 51, 63, 206} \[ \frac {\sqrt {1-2 x} (5 x+3)^2}{84 (3 x+2)^4}+\frac {\sqrt {1-2 x} (4955 x+3168)}{10584 (3 x+2)^3}-\frac {42995 \sqrt {1-2 x}}{74088 (3 x+2)}-\frac {42995 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{37044 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 98
Rule 145
Rule 206
Rubi steps
\begin {align*} \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^5} \, dx &=\frac {\sqrt {1-2 x} (3+5 x)^2}{84 (2+3 x)^4}-\frac {1}{84} \int \frac {(-389-685 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{84 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3168+4955 x)}{10584 (2+3 x)^3}+\frac {42995 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{10584}\\ &=-\frac {42995 \sqrt {1-2 x}}{74088 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^2}{84 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3168+4955 x)}{10584 (2+3 x)^3}+\frac {42995 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{74088}\\ &=-\frac {42995 \sqrt {1-2 x}}{74088 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^2}{84 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3168+4955 x)}{10584 (2+3 x)^3}-\frac {42995 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{74088}\\ &=-\frac {42995 \sqrt {1-2 x}}{74088 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^2}{84 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3168+4955 x)}{10584 (2+3 x)^3}-\frac {42995 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{37044 \sqrt {21}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 52, normalized size = 0.52 \[ \frac {\sqrt {1-2 x} \left (\frac {343 \left (31500 x^2+41823 x+13885\right )}{(3 x+2)^4}-343960 \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{777924} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 99, normalized size = 0.99 \[ \frac {42995 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (1160865 \, x^{3} + 2195625 \, x^{2} + 1385462 \, x + 291670\right )} \sqrt {-2 \, x + 1}}{1555848 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 100, normalized size = 1.00 \[ \frac {42995}{1555848} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {1160865 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 7873845 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 17806943 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 13427323 \, \sqrt {-2 \, x + 1}}{592704 \, {\left (3 \, x + 2\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 66, normalized size = 0.66 \[ -\frac {42995 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{777924}-\frac {324 \left (-\frac {42995 \left (-2 x +1\right )^{\frac {7}{2}}}{444528}+\frac {374945 \left (-2 x +1\right )^{\frac {5}{2}}}{571536}-\frac {363407 \left (-2 x +1\right )^{\frac {3}{2}}}{244944}+\frac {274027 \sqrt {-2 x +1}}{244944}\right )}{\left (-6 x -4\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 110, normalized size = 1.10 \[ \frac {42995}{1555848} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1160865 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 7873845 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 17806943 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 13427323 \, \sqrt {-2 \, x + 1}}{37044 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 90, normalized size = 0.90 \[ -\frac {42995\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{777924}-\frac {\frac {274027\,\sqrt {1-2\,x}}{61236}-\frac {363407\,{\left (1-2\,x\right )}^{3/2}}{61236}+\frac {374945\,{\left (1-2\,x\right )}^{5/2}}{142884}-\frac {42995\,{\left (1-2\,x\right )}^{7/2}}{111132}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \]
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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