Optimal. Leaf size=121 \[ \frac {655}{28812 \sqrt {1-2 x}}-\frac {655}{24696 \sqrt {1-2 x} (3 x+2)}-\frac {131}{3528 \sqrt {1-2 x} (3 x+2)^2}-\frac {131}{1764 \sqrt {1-2 x} (3 x+2)^3}+\frac {1}{84 \sqrt {1-2 x} (3 x+2)^4}-\frac {655 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9604 \sqrt {21}} \]
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Rubi [A] time = 0.04, antiderivative size = 128, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ -\frac {655 \sqrt {1-2 x}}{19208 (3 x+2)}-\frac {655 \sqrt {1-2 x}}{8232 (3 x+2)^2}-\frac {131 \sqrt {1-2 x}}{588 (3 x+2)^3}+\frac {131}{294 \sqrt {1-2 x} (3 x+2)^3}+\frac {1}{84 \sqrt {1-2 x} (3 x+2)^4}-\frac {655 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9604 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 206
Rubi steps
\begin {align*} \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^5} \, dx &=\frac {1}{84 \sqrt {1-2 x} (2+3 x)^4}+\frac {131}{84} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\\ &=\frac {1}{84 \sqrt {1-2 x} (2+3 x)^4}+\frac {131}{294 \sqrt {1-2 x} (2+3 x)^3}+\frac {131}{28} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=\frac {1}{84 \sqrt {1-2 x} (2+3 x)^4}+\frac {131}{294 \sqrt {1-2 x} (2+3 x)^3}-\frac {131 \sqrt {1-2 x}}{588 (2+3 x)^3}+\frac {655}{588} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {1}{84 \sqrt {1-2 x} (2+3 x)^4}+\frac {131}{294 \sqrt {1-2 x} (2+3 x)^3}-\frac {131 \sqrt {1-2 x}}{588 (2+3 x)^3}-\frac {655 \sqrt {1-2 x}}{8232 (2+3 x)^2}+\frac {655 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{2744}\\ &=\frac {1}{84 \sqrt {1-2 x} (2+3 x)^4}+\frac {131}{294 \sqrt {1-2 x} (2+3 x)^3}-\frac {131 \sqrt {1-2 x}}{588 (2+3 x)^3}-\frac {655 \sqrt {1-2 x}}{8232 (2+3 x)^2}-\frac {655 \sqrt {1-2 x}}{19208 (2+3 x)}+\frac {655 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{19208}\\ &=\frac {1}{84 \sqrt {1-2 x} (2+3 x)^4}+\frac {131}{294 \sqrt {1-2 x} (2+3 x)^3}-\frac {131 \sqrt {1-2 x}}{588 (2+3 x)^3}-\frac {655 \sqrt {1-2 x}}{8232 (2+3 x)^2}-\frac {655 \sqrt {1-2 x}}{19208 (2+3 x)}-\frac {655 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{19208}\\ &=\frac {1}{84 \sqrt {1-2 x} (2+3 x)^4}+\frac {131}{294 \sqrt {1-2 x} (2+3 x)^3}-\frac {131 \sqrt {1-2 x}}{588 (2+3 x)^3}-\frac {655 \sqrt {1-2 x}}{8232 (2+3 x)^2}-\frac {655 \sqrt {1-2 x}}{19208 (2+3 x)}-\frac {655 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9604 \sqrt {21}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 42, normalized size = 0.35 \[ \frac {2096 \, _2F_1\left (-\frac {1}{2},4;\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )+\frac {2401}{(3 x+2)^4}}{201684 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 114, normalized size = 0.94 \[ \frac {655 \, \sqrt {21} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (35370 \, x^{4} + 80565 \, x^{3} + 60391 \, x^{2} + 10742 \, x - 2566\right )} \sqrt {-2 \, x + 1}}{403368 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.27, size = 109, normalized size = 0.90 \[ \frac {655}{403368} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {176}{16807 \, \sqrt {-2 \, x + 1}} - \frac {66771 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 526911 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 1417325 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 1281105 \, \sqrt {-2 \, x + 1}}{1075648 \, {\left (3 \, x + 2\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 75, normalized size = 0.62 \[ -\frac {655 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{201684}+\frac {176}{16807 \sqrt {-2 x +1}}+\frac {\frac {66771 \left (-2 x +1\right )^{\frac {7}{2}}}{67228}-\frac {75273 \left (-2 x +1\right )^{\frac {5}{2}}}{9604}+\frac {28925 \left (-2 x +1\right )^{\frac {3}{2}}}{1372}-\frac {3735 \sqrt {-2 x +1}}{196}}{\left (-6 x -4\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.50, size = 119, normalized size = 0.98 \[ \frac {655}{403368} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {17685 \, {\left (2 \, x - 1\right )}^{4} + 151305 \, {\left (2 \, x - 1\right )}^{3} + 468587 \, {\left (2 \, x - 1\right )}^{2} + 1193934 \, x - 355495}{9604 \, {\left (81 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 756 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 2646 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 4116 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2401 \, \sqrt {-2 \, x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 98, normalized size = 0.81 \[ \frac {\frac {4061\,x}{2646}+\frac {9563\,{\left (2\,x-1\right )}^2}{15876}+\frac {7205\,{\left (2\,x-1\right )}^3}{37044}+\frac {655\,{\left (2\,x-1\right )}^4}{28812}-\frac {7255}{15876}}{\frac {2401\,\sqrt {1-2\,x}}{81}-\frac {1372\,{\left (1-2\,x\right )}^{3/2}}{27}+\frac {98\,{\left (1-2\,x\right )}^{5/2}}{3}-\frac {28\,{\left (1-2\,x\right )}^{7/2}}{3}+{\left (1-2\,x\right )}^{9/2}}-\frac {655\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{201684} \]
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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