Optimal. Leaf size=187 \[ -\frac{104663 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{171875 \sqrt{33}}-\frac{2 \sqrt{1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}-\frac{668 \sqrt{1-2 x} (3 x+2)^{5/2}}{9075 \sqrt{5 x+3}}+\frac{403 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{75625}-\frac{87476 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{378125}-\frac{6515539 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343750 \sqrt{33}} \]
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Rubi [A] time = 0.0687328, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 154, 158, 113, 119} \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}-\frac{668 \sqrt{1-2 x} (3 x+2)^{5/2}}{9075 \sqrt{5 x+3}}+\frac{403 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{75625}-\frac{87476 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{378125}-\frac{104663 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{171875 \sqrt{33}}-\frac{6515539 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343750 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{9/2}}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac{2}{165} \int \frac{\left (-\frac{227}{2}-\frac{267 x}{2}\right ) (2+3 x)^{5/2}}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac{668 \sqrt{1-2 x} (2+3 x)^{5/2}}{9075 \sqrt{3+5 x}}-\frac{4 \int \frac{(2+3 x)^{3/2} \left (-2721+\frac{1209 x}{4}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{9075}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac{668 \sqrt{1-2 x} (2+3 x)^{5/2}}{9075 \sqrt{3+5 x}}+\frac{403 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{75625}+\frac{4 \int \frac{\sqrt{2+3 x} \left (\frac{1058175}{8}+196821 x\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{226875}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac{668 \sqrt{1-2 x} (2+3 x)^{5/2}}{9075 \sqrt{3+5 x}}-\frac{87476 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{378125}+\frac{403 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{75625}-\frac{4 \int \frac{-\frac{18628119}{4}-\frac{58639851 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3403125}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac{668 \sqrt{1-2 x} (2+3 x)^{5/2}}{9075 \sqrt{3+5 x}}-\frac{87476 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{378125}+\frac{403 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{75625}+\frac{104663 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{343750}+\frac{6515539 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3781250}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac{668 \sqrt{1-2 x} (2+3 x)^{5/2}}{9075 \sqrt{3+5 x}}-\frac{87476 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{378125}+\frac{403 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{75625}-\frac{6515539 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343750 \sqrt{33}}-\frac{104663 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{171875 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.208751, size = 107, normalized size = 0.57 \[ \frac{-3061660 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (3675375 x^3+13721400 x^2+12517925 x+3365042\right )}{(5 x+3)^{3/2}}+6515539 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{11343750} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 229, normalized size = 1.2 \begin{align*}{\frac{1}{68062500\,{x}^{2}+11343750\,x-22687500} \left ( 15308300\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-32577695\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+9184980\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -19546617\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -220522500\,{x}^{5}-860037750\,{x}^{4}-814782000\,{x}^{3}-52653770\,{x}^{2}+216708080\,x+67300840 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{250 \, x^{4} + 325 \, x^{3} + 45 \, x^{2} - 81 \, x - 27}, x\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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