Optimal. Leaf size=129 \[ -\frac{2}{55} \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )-\frac{37 \sqrt{1-2 x} \sqrt{3 x+2}}{121 \sqrt{5 x+3}}+\frac{7 \sqrt{3 x+2}}{11 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{37}{55} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0377352, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {98, 152, 158, 113, 119} \[ -\frac{37 \sqrt{1-2 x} \sqrt{3 x+2}}{121 \sqrt{5 x+3}}+\frac{7 \sqrt{3 x+2}}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{2}{55} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{37}{55} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{3/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac{7 \sqrt{2+3 x}}{11 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{1}{11} \int \frac{-\frac{11}{2}-3 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{7 \sqrt{2+3 x}}{11 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{37 \sqrt{1-2 x} \sqrt{2+3 x}}{121 \sqrt{3+5 x}}+\frac{2}{121} \int \frac{-30-\frac{111 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{7 \sqrt{2+3 x}}{11 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{37 \sqrt{1-2 x} \sqrt{2+3 x}}{121 \sqrt{3+5 x}}+\frac{3}{55} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{111}{605} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{7 \sqrt{2+3 x}}{11 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{37 \sqrt{1-2 x} \sqrt{2+3 x}}{121 \sqrt{3+5 x}}+\frac{37}{55} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{2}{55} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.131273, size = 122, normalized size = 0.95 \[ \frac{70 \sqrt{2-4 x} (5 x+3) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+10 \sqrt{3 x+2} \sqrt{5 x+3} (37 x+20)-37 \sqrt{2-4 x} (5 x+3) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{605 \sqrt{1-2 x} (5 x+3)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 135, normalized size = 1.1 \begin{align*} -{\frac{1}{18150\,{x}^{3}+13915\,{x}^{2}-4235\,x-3630}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 70\,\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 ) -37\,\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 ) +1110\,{x}^{2}+1340\,x+400 \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{3}{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{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{3}{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{3}{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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