Optimal. Leaf size=191 \[ \frac{175111 \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{68750}+\frac{7 (3 x+2)^{7/2}}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} (3 x+2)^{5/2}}{605 \sqrt{5 x+3}}+\frac{10851 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{15125}+\frac{502941 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{151250}+\frac{2911577 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{34375} \]
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Rubi [A] time = 0.0673624, antiderivative size = 191, 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} \[ \frac{7 (3 x+2)^{7/2}}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} (3 x+2)^{5/2}}{605 \sqrt{5 x+3}}+\frac{10851 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{15125}+\frac{502941 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{151250}+\frac{175111 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{68750}+\frac{2911577 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{34375} \]
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)^{3/2}} \, dx &=\frac{7 (2+3 x)^{7/2}}{11 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{1}{11} \int \frac{(2+3 x)^{5/2} \left (\frac{367}{2}+312 x\right )}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{37 \sqrt{1-2 x} (2+3 x)^{5/2}}{605 \sqrt{3+5 x}}+\frac{7 (2+3 x)^{7/2}}{11 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{2}{605} \int \frac{(2+3 x)^{3/2} \left (\frac{13173}{4}+\frac{10851 x}{2}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{37 \sqrt{1-2 x} (2+3 x)^{5/2}}{605 \sqrt{3+5 x}}+\frac{7 (2+3 x)^{7/2}}{11 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{10851 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{15125}+\frac{2 \int \frac{\left (-\frac{929925}{4}-\frac{1508823 x}{4}\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{15125}\\ &=-\frac{37 \sqrt{1-2 x} (2+3 x)^{5/2}}{605 \sqrt{3+5 x}}+\frac{7 (2+3 x)^{7/2}}{11 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{502941 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{151250}+\frac{10851 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{15125}-\frac{2 \int \frac{\frac{66357261}{8}+\frac{26204193 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{226875}\\ &=-\frac{37 \sqrt{1-2 x} (2+3 x)^{5/2}}{605 \sqrt{3+5 x}}+\frac{7 (2+3 x)^{7/2}}{11 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{502941 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{151250}+\frac{10851 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{15125}-\frac{525333 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{137500}-\frac{8734731 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{378125}\\ &=-\frac{37 \sqrt{1-2 x} (2+3 x)^{5/2}}{605 \sqrt{3+5 x}}+\frac{7 (2+3 x)^{7/2}}{11 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{502941 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{151250}+\frac{10851 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{15125}+\frac{2911577 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{34375}+\frac{175111 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{68750}\\ \end{align*}
Mathematica [A] time = 0.248243, size = 132, normalized size = 0.69 \[ \frac{5867645 \sqrt{2-4 x} (5 x+3) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+10 \sqrt{3 x+2} \sqrt{5 x+3} \left (-490050 x^3-2188890 x^2+3684629 x+2892883\right )-11646308 \sqrt{2-4 x} (5 x+3) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1512500 \sqrt{1-2 x} (5 x+3)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.019, size = 145, normalized size = 0.8 \begin{align*} -{\frac{1}{45375000\,{x}^{3}+34787500\,{x}^{2}-10587500\,x-9075000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 5867645\,\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 ) -11646308\,\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 ) -14701500\,{x}^{4}-75467700\,{x}^{3}+66761070\,{x}^{2}+160479070\,x+57857660 \right ) } \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{3}{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}}{100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9}, 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{3}{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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