Optimal. Leaf size=219 \[ \frac{1284329 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{18900}+\frac{(3 x+2)^{5/2} (5 x+3)^{5/2}}{\sqrt{1-2 x}}+\frac{5}{3} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac{93}{14} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{4853}{105} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{1284329 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{3780}+\frac{42696881 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18900} \]
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Rubi [A] time = 0.0714172, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{(3 x+2)^{5/2} (5 x+3)^{5/2}}{\sqrt{1-2 x}}+\frac{5}{3} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac{93}{14} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{4853}{105} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{1284329 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{3780}+\frac{1284329 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18900}+\frac{42696881 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18900} \]
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)^{5/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}-\int \frac{(2+3 x)^{3/2} (3+5 x)^{3/2} \left (\frac{95}{2}+75 x\right )}{\sqrt{1-2 x}} \, dx\\ &=\frac{5}{3} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{1}{45} \int \frac{\left (-\frac{13425}{2}-\frac{20925 x}{2}\right ) \sqrt{2+3 x} (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=\frac{93}{14} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{5}{3} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}-\frac{\int \frac{(3+5 x)^{3/2} \left (\frac{2862975}{4}+1091925 x\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1575}\\ &=\frac{4853}{105} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{93}{14} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{5}{3} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{\int \frac{\left (-\frac{187797825}{4}-\frac{288974025 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{23625}\\ &=\frac{1284329 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3780}+\frac{4853}{105} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{93}{14} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{5}{3} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}-\frac{\int \frac{\frac{12163900725}{8}+\frac{9606798225 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{212625}\\ &=\frac{1284329 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3780}+\frac{4853}{105} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{93}{14} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{5}{3} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}-\frac{14127619 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{37800}-\frac{42696881 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{18900}\\ &=\frac{1284329 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3780}+\frac{4853}{105} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{93}{14} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{5}{3} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{42696881 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18900}+\frac{1284329 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18900}\\ \end{align*}
Mathematica [A] time = 0.226137, size = 120, normalized size = 0.55 \[ \frac{43010905 \sqrt{2-4 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-30 \sqrt{3 x+2} \sqrt{5 x+3} \left (94500 x^4+392400 x^3+795150 x^2+1258906 x-2283923\right )-85393762 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{113400 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 155, normalized size = 0.7 \begin{align*} -{\frac{1}{3402000\,{x}^{3}+2608200\,{x}^{2}-793800\,x-680400}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -42525000\,{x}^{6}+43010905\,\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 ) -85393762\,\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 ) -230445000\,{x}^{5}-598495500\,{x}^{4}-1090375200\,{x}^{3}+167061930\,{x}^{2}+1075233030\,x+411106140 \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 (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\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 (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\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{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\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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