Optimal. Leaf size=186 \[ \frac{673523 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{8750 \sqrt{33}}+\frac{\sqrt{5 x+3} (3 x+2)^{7/2}}{\sqrt{1-2 x}}+\frac{12}{7} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{2517}{350} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{29293}{875} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{4071079 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17500} \]
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Rubi [A] time = 0.059548, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {97, 154, 158, 113, 119} \[ \frac{\sqrt{5 x+3} (3 x+2)^{7/2}}{\sqrt{1-2 x}}+\frac{12}{7} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{2517}{350} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{29293}{875} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{673523 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8750 \sqrt{33}}+\frac{4071079 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17500} \]
Antiderivative was successfully verified.
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Rule 97
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{7/2} \sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac{(2+3 x)^{7/2} \sqrt{3+5 x}}{\sqrt{1-2 x}}-\int \frac{(2+3 x)^{5/2} \left (\frac{73}{2}+60 x\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{12}{7} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{(2+3 x)^{7/2} \sqrt{3+5 x}}{\sqrt{1-2 x}}+\frac{1}{35} \int \frac{\left (-3845-\frac{12585 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{2517}{350} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{12}{7} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{(2+3 x)^{7/2} \sqrt{3+5 x}}{\sqrt{1-2 x}}-\frac{1}{875} \int \frac{\sqrt{2+3 x} \left (\frac{1083625}{4}+439395 x\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{29293}{875} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2517}{350} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{12}{7} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{(2+3 x)^{7/2} \sqrt{3+5 x}}{\sqrt{1-2 x}}+\frac{\int \frac{-9665070-\frac{61066185 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{13125}\\ &=\frac{29293}{875} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2517}{350} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{12}{7} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{(2+3 x)^{7/2} \sqrt{3+5 x}}{\sqrt{1-2 x}}-\frac{673523 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{17500}-\frac{4071079 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{17500}\\ &=\frac{29293}{875} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2517}{350} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{12}{7} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{(2+3 x)^{7/2} \sqrt{3+5 x}}{\sqrt{1-2 x}}+\frac{4071079 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17500}+\frac{673523 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8750 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.215923, size = 115, normalized size = 0.62 \[ \frac{2050510 \sqrt{2-4 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-30 \sqrt{3 x+2} \sqrt{5 x+3} \left (6750 x^3+26010 x^2+54757 x-109756\right )-4071079 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{52500 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.034, size = 150, normalized size = 0.8 \begin{align*} -{\frac{1}{1575000\,{x}^{3}+1207500\,{x}^{2}-367500\,x-315000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 2050510\,\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 ) -4071079\,\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 ) -3037500\,{x}^{5}-15552000\,{x}^{4}-40681350\,{x}^{3}+13496910\,{x}^{2}+52704660\,x+19756080 \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{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{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}}{4 \, x^{2} - 4 \, x + 1}, 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{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{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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