Optimal. Leaf size=156 \[ -\frac{2252 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{3025 \sqrt{33}}+\frac{7 (3 x+2)^{5/2}}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{434 (3 x+2)^{3/2}}{363 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{2129 \sqrt{1-2 x} \sqrt{3 x+2}}{19965 \sqrt{5 x+3}}-\frac{148831 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6050 \sqrt{33}} \]
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Rubi [A] time = 0.0508517, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {98, 150, 158, 113, 119} \[ \frac{7 (3 x+2)^{5/2}}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{434 (3 x+2)^{3/2}}{363 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{2129 \sqrt{1-2 x} \sqrt{3 x+2}}{19965 \sqrt{5 x+3}}-\frac{2252 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3025 \sqrt{33}}-\frac{148831 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6050 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac{7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{1}{33} \int \frac{(2+3 x)^{3/2} \left (\frac{233}{2}+201 x\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac{434 (2+3 x)^{3/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{1}{363} \int \frac{\left (-3730-\frac{13143 x}{2}\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{2129 \sqrt{1-2 x} \sqrt{2+3 x}}{19965 \sqrt{3+5 x}}-\frac{434 (2+3 x)^{3/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{2 \int \frac{-\frac{282759}{4}-\frac{446493 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{19965}\\ &=\frac{2129 \sqrt{1-2 x} \sqrt{2+3 x}}{19965 \sqrt{3+5 x}}-\frac{434 (2+3 x)^{3/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}+\frac{1126 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3025}+\frac{148831 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{66550}\\ &=\frac{2129 \sqrt{1-2 x} \sqrt{2+3 x}}{19965 \sqrt{3+5 x}}-\frac{434 (2+3 x)^{3/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{148831 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6050 \sqrt{33}}-\frac{2252 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3025 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.239255, size = 97, normalized size = 0.62 \[ \frac{-74515 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{5 \sqrt{6 x+4} \left (189851 x^2+66174 x-28671\right )}{(1-2 x)^{3/2} \sqrt{5 x+3}}+148831 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{99825 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.022, size = 228, normalized size = 1.5 \begin{align*}{\frac{1}{199650\, \left ( 2\,x-1 \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) }\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 149030\,\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}-297662\,\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}-74515\,\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 ) +148831\,\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 ) +5695530\,{x}^{3}+5782240\,{x}^{2}+463350\,x-573420 \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{7}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\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{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{200 \, x^{5} - 60 \, x^{4} - 138 \, x^{3} + 47 \, x^{2} + 24 \, 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{7}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\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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