Optimal. Leaf size=156 \[ \frac{2657 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{15125 \sqrt{33}}+\frac{7 (3 x+2)^{5/2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^{3/2}}{1815 (5 x+3)^{3/2}}-\frac{4289 \sqrt{1-2 x} \sqrt{3 x+2}}{99825 \sqrt{5 x+3}}+\frac{118898 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15125 \sqrt{33}} \]
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Rubi [A] time = 0.050084, 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} \[ \frac{7 (3 x+2)^{5/2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^{3/2}}{1815 (5 x+3)^{3/2}}-\frac{4289 \sqrt{1-2 x} \sqrt{3 x+2}}{99825 \sqrt{5 x+3}}+\frac{2657 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15125 \sqrt{33}}+\frac{118898 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15125 \sqrt{33}} \]
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)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac{7 (2+3 x)^{5/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{1}{11} \int \frac{(2+3 x)^{3/2} \left (\frac{101}{2}+102 x\right )}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^{3/2}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{5/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{2 \int \frac{\sqrt{2+3 x} \left (\frac{11645}{4}+\frac{10419 x}{2}\right )}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx}{1815}\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^{3/2}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{5/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4289 \sqrt{1-2 x} \sqrt{2+3 x}}{99825 \sqrt{3+5 x}}-\frac{4 \int \frac{\frac{445569}{8}+\frac{178347 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{99825}\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^{3/2}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{5/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4289 \sqrt{1-2 x} \sqrt{2+3 x}}{99825 \sqrt{3+5 x}}-\frac{2657 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{30250}-\frac{118898 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{166375}\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^{3/2}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{5/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4289 \sqrt{1-2 x} \sqrt{2+3 x}}{99825 \sqrt{3+5 x}}+\frac{118898 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15125 \sqrt{33}}+\frac{2657 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15125 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.127775, size = 102, normalized size = 0.65 \[ \frac{150115 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{10 \sqrt{3 x+2} \left (649925 x^2+772474 x+229463\right )}{\sqrt{1-2 x} (5 x+3)^{3/2}}-237796 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{998250} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 219, normalized size = 1.4 \begin{align*} -{\frac{1}{5989500\,{x}^{2}+998250\,x-1996500}\sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 750575\,\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}-1188980\,\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}+450345\,\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 ) -713388\,\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 ) +19497750\,{x}^{3}+36172720\,{x}^{2}+22333370\,x+4589260 \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{7}{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 (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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{7}{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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