Optimal. Leaf size=246 \[ \frac{269045681 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{94500 \sqrt{33}}+\frac{(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt{1-2 x}}+\frac{18}{11} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{5/2}+\frac{419}{66} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}+\frac{9741}{385} \sqrt{1-2 x} (5 x+3)^{5/2} \sqrt{3 x+2}+\frac{4066493 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{23100}+\frac{269045681 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{207900}+\frac{17888580643 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{189000 \sqrt{33}} \]
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Rubi [A] time = 0.0892238, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt{1-2 x}}+\frac{18}{11} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{5/2}+\frac{419}{66} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}+\frac{9741}{385} \sqrt{1-2 x} (5 x+3)^{5/2} \sqrt{3 x+2}+\frac{4066493 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{23100}+\frac{269045681 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{207900}+\frac{269045681 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{94500 \sqrt{33}}+\frac{17888580643 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{189000 \sqrt{33}} \]
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} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}-\int \frac{(2+3 x)^{5/2} (3+5 x)^{3/2} \left (\frac{113}{2}+90 x\right )}{\sqrt{1-2 x}} \, dx\\ &=\frac{18}{11} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{1}{55} \int \frac{\left (-9950-\frac{31425 x}{2}\right ) (2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=\frac{419}{66} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{18}{11} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}-\frac{\int \frac{\sqrt{2+3 x} (3+5 x)^{3/2} \left (\frac{5624625}{4}+2191725 x\right )}{\sqrt{1-2 x}} \, dx}{2475}\\ &=\frac{9741}{385} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{419}{66} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{18}{11} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{\int \frac{\left (-149936475-\frac{914960925 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{86625}\\ &=\frac{4066493 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{23100}+\frac{9741}{385} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{419}{66} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{18}{11} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}-\frac{\int \frac{\sqrt{3+5 x} \left (\frac{78681075975}{8}+\frac{60535278225 x}{4}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1299375}\\ &=\frac{269045681 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{207900}+\frac{4066493 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{23100}+\frac{9741}{385} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{419}{66} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{18}{11} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{\int \frac{-\frac{637033999725}{2}-\frac{4024930644675 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{11694375}\\ &=\frac{269045681 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{207900}+\frac{4066493 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{23100}+\frac{9741}{385} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{419}{66} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{18}{11} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}-\frac{269045681 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{189000}-\frac{17888580643 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{2079000}\\ &=\frac{269045681 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{207900}+\frac{4066493 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{23100}+\frac{9741}{385} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{419}{66} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{18}{11} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{17888580643 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{189000 \sqrt{33}}+\frac{269045681 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{94500 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.237963, size = 125, normalized size = 0.51 \[ \frac{9010073170 \sqrt{2-4 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-30 \sqrt{3 x+2} \sqrt{5 x+3} \left (12757500 x^5+60196500 x^4+133330950 x^3+198895770 x^2+273928969 x-477155552\right )-17888580643 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{6237000 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.018, size = 160, normalized size = 0.7 \begin{align*} -{\frac{1}{187110000\,{x}^{3}+143451000\,{x}^{2}-43659000\,x-37422000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -5740875000\,{x}^{7}-34360200000\,{x}^{6}+9010073170\,\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 ) -17888580643\,\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 ) -96607282500\,{x}^{5}-176337108000\,{x}^{4}-260638195950\,{x}^{3}+22779247470\,{x}^{2}+222671450220\,x+85887999360 \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{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 (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\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{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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