Optimal. Leaf size=142 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^4}{11 \sqrt{1-2 x}}+\frac{939}{880} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^3+\frac{76587 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{17600}+\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (7645620 x+18424549)}{2816000}-\frac{291096141 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{256000 \sqrt{10}} \]
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Rubi [A] time = 0.0440307, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {98, 153, 147, 54, 216} \[ \frac{7 \sqrt{5 x+3} (3 x+2)^4}{11 \sqrt{1-2 x}}+\frac{939}{880} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^3+\frac{76587 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{17600}+\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (7645620 x+18424549)}{2816000}-\frac{291096141 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{256000 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 153
Rule 147
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(2+3 x)^5}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx &=\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}-\frac{1}{11} \int \frac{(2+3 x)^3 \left (285+\frac{939 x}{2}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{939}{880} \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}+\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{1}{440} \int \frac{\left (-35007-\frac{229761 x}{4}\right ) (2+3 x)^2}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{76587 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{17600}+\frac{939}{880} \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}+\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}-\frac{\int \frac{(2+3 x) \left (\frac{12307617}{4}+\frac{40139505 x}{8}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{13200}\\ &=\frac{76587 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{17600}+\frac{939}{880} \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}+\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{21 \sqrt{1-2 x} \sqrt{3+5 x} (18424549+7645620 x)}{2816000}-\frac{291096141 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{512000}\\ &=\frac{76587 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{17600}+\frac{939}{880} \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}+\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{21 \sqrt{1-2 x} \sqrt{3+5 x} (18424549+7645620 x)}{2816000}-\frac{291096141 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{256000 \sqrt{5}}\\ &=\frac{76587 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{17600}+\frac{939}{880} \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}+\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{21 \sqrt{1-2 x} \sqrt{3+5 x} (18424549+7645620 x)}{2816000}-\frac{291096141 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{256000 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.104969, size = 74, normalized size = 0.52 \[ \frac{3202057551 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (17107200 x^4+76887360 x^3+171939240 x^2+332129358 x-488641609\right )}{28160000 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 140, normalized size = 1. \begin{align*} -{\frac{1}{112640000\,x-56320000} \left ( -342144000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-1537747200\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+6404115102\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-3438784800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-3202057551\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -6642587160\,x\sqrt{-10\,{x}^{2}-x+3}+9772832180\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.80041, size = 134, normalized size = 0.94 \begin{align*} \frac{243}{80} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + \frac{24273}{1600} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{291096141}{5120000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{487863}{12800} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{19975419}{256000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{16807 \, \sqrt{-10 \, x^{2} - x + 3}}{176 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75298, size = 323, normalized size = 2.27 \begin{align*} \frac{3202057551 \, \sqrt{10}{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \,{\left (17107200 \, x^{4} + 76887360 \, x^{3} + 171939240 \, x^{2} + 332129358 \, x - 488641609\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{56320000 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.07281, size = 131, normalized size = 0.92 \begin{align*} -\frac{291096141}{2560000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (198 \,{\left (12 \,{\left (8 \,{\left (36 \, \sqrt{5}{\left (5 \, x + 3\right )} + 377 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 29669 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4900505 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 16010291851 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{352000000 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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