Optimal. Leaf size=125 \[ -\frac{\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{11 \sqrt{33}}-\frac{62 \sqrt{3 x+2} \sqrt{5 x+3}}{363 \sqrt{1-2 x}}+\frac{7 \sqrt{3 x+2} \sqrt{5 x+3}}{33 (1-2 x)^{3/2}}-\frac{31 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{11 \sqrt{33}} \]
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Rubi [A] time = 0.0381773, antiderivative size = 125, 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} \[ -\frac{62 \sqrt{3 x+2} \sqrt{5 x+3}}{363 \sqrt{1-2 x}}+\frac{7 \sqrt{3 x+2} \sqrt{5 x+3}}{33 (1-2 x)^{3/2}}-\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{11 \sqrt{33}}-\frac{31 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{11 \sqrt{33}} \]
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)^{5/2} \sqrt{3+5 x}} \, dx &=\frac{7 \sqrt{2+3 x} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{1}{33} \int \frac{\frac{121}{2}+96 x}{(1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{7 \sqrt{2+3 x} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{62 \sqrt{2+3 x} \sqrt{3+5 x}}{363 \sqrt{1-2 x}}+\frac{2 \int \frac{\frac{4137}{4}+\frac{3255 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2541}\\ &=\frac{7 \sqrt{2+3 x} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{62 \sqrt{2+3 x} \sqrt{3+5 x}}{363 \sqrt{1-2 x}}+\frac{1}{22} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{31}{121} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{7 \sqrt{2+3 x} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{62 \sqrt{2+3 x} \sqrt{3+5 x}}{363 \sqrt{1-2 x}}-\frac{31 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{11 \sqrt{33}}-\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{11 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.153586, size = 115, normalized size = 0.92 \[ \frac{29 \sqrt{2-4 x} (2 x-1) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+2 \sqrt{3 x+2} \sqrt{5 x+3} (124 x+15)-62 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{726 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 228, normalized size = 1.8 \begin{align*}{\frac{1}{726\, \left ( 2\,x-1 \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 58\,\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}-124\,\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}-29\,\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 ) +62\,\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 ) +3720\,{x}^{3}+5162\,{x}^{2}+2058\,x+180 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}\sqrt{2+3\,x}} \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}}}{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{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}}{40 \, x^{4} - 36 \, x^{3} - 6 \, x^{2} + 13 \, x - 3}, 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}}}{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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