Optimal. Leaf size=172 \[ -\frac{3}{70} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{7/2}-\frac{3 (1-2 x)^{3/2} (1140 x+1963) (5 x+3)^{7/2}}{8000}-\frac{296633 (1-2 x)^{3/2} (5 x+3)^{5/2}}{128000}-\frac{3262963 (1-2 x)^{3/2} (5 x+3)^{3/2}}{307200}-\frac{35892593 (1-2 x)^{3/2} \sqrt{5 x+3}}{819200}+\frac{394818523 \sqrt{1-2 x} \sqrt{5 x+3}}{8192000}+\frac{4343003753 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8192000 \sqrt{10}} \]
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Rubi [A] time = 0.0526206, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {100, 147, 50, 54, 216} \[ -\frac{3}{70} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{7/2}-\frac{3 (1-2 x)^{3/2} (1140 x+1963) (5 x+3)^{7/2}}{8000}-\frac{296633 (1-2 x)^{3/2} (5 x+3)^{5/2}}{128000}-\frac{3262963 (1-2 x)^{3/2} (5 x+3)^{3/2}}{307200}-\frac{35892593 (1-2 x)^{3/2} \sqrt{5 x+3}}{819200}+\frac{394818523 \sqrt{1-2 x} \sqrt{5 x+3}}{8192000}+\frac{4343003753 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8192000 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 100
Rule 147
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2} \, dx &=-\frac{3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{1}{70} \int \left (-385-\frac{1197 x}{2}\right ) \sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx\\ &=-\frac{3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac{296633 \int \sqrt{1-2 x} (3+5 x)^{5/2} \, dx}{16000}\\ &=-\frac{296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac{3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac{3262963 \int \sqrt{1-2 x} (3+5 x)^{3/2} \, dx}{51200}\\ &=-\frac{3262963 (1-2 x)^{3/2} (3+5 x)^{3/2}}{307200}-\frac{296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac{3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac{35892593 \int \sqrt{1-2 x} \sqrt{3+5 x} \, dx}{204800}\\ &=-\frac{35892593 (1-2 x)^{3/2} \sqrt{3+5 x}}{819200}-\frac{3262963 (1-2 x)^{3/2} (3+5 x)^{3/2}}{307200}-\frac{296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac{3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac{394818523 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{1638400}\\ &=\frac{394818523 \sqrt{1-2 x} \sqrt{3+5 x}}{8192000}-\frac{35892593 (1-2 x)^{3/2} \sqrt{3+5 x}}{819200}-\frac{3262963 (1-2 x)^{3/2} (3+5 x)^{3/2}}{307200}-\frac{296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac{3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac{4343003753 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{16384000}\\ &=\frac{394818523 \sqrt{1-2 x} \sqrt{3+5 x}}{8192000}-\frac{35892593 (1-2 x)^{3/2} \sqrt{3+5 x}}{819200}-\frac{3262963 (1-2 x)^{3/2} (3+5 x)^{3/2}}{307200}-\frac{296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac{3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac{4343003753 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{8192000 \sqrt{5}}\\ &=\frac{394818523 \sqrt{1-2 x} \sqrt{3+5 x}}{8192000}-\frac{35892593 (1-2 x)^{3/2} \sqrt{3+5 x}}{819200}-\frac{3262963 (1-2 x)^{3/2} (3+5 x)^{3/2}}{307200}-\frac{296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac{3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac{4343003753 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{8192000 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.192634, size = 89, normalized size = 0.52 \[ -\frac{10 \sqrt{5 x+3} \left (33177600000 x^7+107550720000 x^6+127277568000 x^5+50509190400 x^4-23917446080 x^3-34142598520 x^2-20160334154 x+12531569067\right )+91203078813 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1720320000 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 155, normalized size = 0.9 \begin{align*}{\frac{1}{3440640000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 331776000000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}+1241395200000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}+1893473280000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+1451828544000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+486739811200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+91203078813\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -98056079600\,x\sqrt{-10\,{x}^{2}-x+3}-250631381340\,\sqrt{-10\,{x}^{2}-x+3} \right ){\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 = 3.92328, size = 163, normalized size = 0.95 \begin{align*} -\frac{135}{14} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} - \frac{3933}{112} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} - \frac{121887}{2240} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{8474351}{179200} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{55355473}{2150400} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{35892593}{409600} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{4343003753}{163840000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{35892593}{8192000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88407, size = 367, normalized size = 2.13 \begin{align*} \frac{1}{172032000} \,{\left (16588800000 \, x^{6} + 62069760000 \, x^{5} + 94673664000 \, x^{4} + 72591427200 \, x^{3} + 24336990560 \, x^{2} - 4902803980 \, x - 12531569067\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{4343003753}{163840000} \, \sqrt{10} \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 ) \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 [B] time = 2.54868, size = 548, normalized size = 3.19 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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