Optimal. Leaf size=187 \[ \frac{84134 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{75625 \sqrt{33}}+\frac{7 (3 x+2)^{7/2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^{5/2}}{1815 (5 x+3)^{3/2}}-\frac{4421 \sqrt{1-2 x} (3 x+2)^{3/2}}{99825 \sqrt{5 x+3}}+\frac{83093 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{166375}+\frac{5684677 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{151250 \sqrt{33}} \]
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Rubi [A] time = 0.0650503, 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} \[ \frac{7 (3 x+2)^{7/2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^{5/2}}{1815 (5 x+3)^{3/2}}-\frac{4421 \sqrt{1-2 x} (3 x+2)^{3/2}}{99825 \sqrt{5 x+3}}+\frac{83093 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{166375}+\frac{84134 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{75625 \sqrt{33}}+\frac{5684677 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{151250 \sqrt{33}} \]
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}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac{7 (2+3 x)^{7/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{1}{11} \int \frac{(2+3 x)^{5/2} \left (\frac{227}{2}+207 x\right )}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^{5/2}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{7/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{2 \int \frac{(2+3 x)^{3/2} \left (\frac{24863}{4}+10728 x\right )}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx}{1815}\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^{5/2}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{7/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4421 \sqrt{1-2 x} (2+3 x)^{3/2}}{99825 \sqrt{3+5 x}}-\frac{4 \int \frac{\sqrt{2+3 x} \left (\frac{904275}{8}+\frac{747837 x}{4}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{99825}\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^{5/2}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{7/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4421 \sqrt{1-2 x} (2+3 x)^{3/2}}{99825 \sqrt{3+5 x}}+\frac{83093 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{166375}+\frac{4 \int \frac{-\frac{32363109}{8}-\frac{51162093 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1497375}\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^{5/2}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{7/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4421 \sqrt{1-2 x} (2+3 x)^{3/2}}{99825 \sqrt{3+5 x}}+\frac{83093 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{166375}-\frac{42067 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{75625}-\frac{5684677 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1663750}\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^{5/2}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{7/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4421 \sqrt{1-2 x} (2+3 x)^{3/2}}{99825 \sqrt{3+5 x}}+\frac{83093 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{166375}+\frac{5684677 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{151250 \sqrt{33}}+\frac{84134 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{75625 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.158903, size = 136, normalized size = 0.73 \[ \frac{2908255 \sqrt{2-4 x} (5 x+3)^2 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+10 \sqrt{3 x+2} \left (-2695275 x^3+9376775 x^2+14153413 x+4534181\right ) \sqrt{5 x+3}-5684677 \sqrt{2-4 x} (5 x+3)^2 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{4991250 \sqrt{1-2 x} (5 x+3)^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 224, normalized size = 1.2 \begin{align*} -{\frac{1}{29947500\,{x}^{2}+4991250\,x-9982500}\sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 14541275\,\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}-28423385\,\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}+8724765\,\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 ) -17054031\,\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 ) -80858250\,{x}^{4}+227397750\,{x}^{3}+612137890\,{x}^{2}+419093690\,x+90683620 \right ) \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}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{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}}{500 \, x^{5} + 400 \, x^{4} - 235 \, x^{3} - 207 \, x^{2} + 27 \, 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}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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