Optimal. Leaf size=253 \[ -\frac{1313411 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{3150}+\frac{(5 x+3)^{5/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}-\frac{203 (5 x+3)^{5/2} (3 x+2)^{5/2}}{33 \sqrt{1-2 x}}-\frac{225}{22} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}-\frac{6277}{154} \sqrt{1-2 x} (5 x+3)^{5/2} \sqrt{3 x+2}-\frac{1310203 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{4620}-\frac{1313411}{630} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{174654791 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12600} \]
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Rubi [A] time = 0.0994895, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {97, 150, 154, 158, 113, 119} \[ \frac{(5 x+3)^{5/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}-\frac{203 (5 x+3)^{5/2} (3 x+2)^{5/2}}{33 \sqrt{1-2 x}}-\frac{225}{22} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}-\frac{6277}{154} \sqrt{1-2 x} (5 x+3)^{5/2} \sqrt{3 x+2}-\frac{1310203 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{4620}-\frac{1313411}{630} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{1313411 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3150}-\frac{174654791 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12600} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
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)^{5/2}} \, dx &=\frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{1}{3} \int \frac{(2+3 x)^{5/2} (3+5 x)^{3/2} \left (\frac{113}{2}+90 x\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac{203 (2+3 x)^{5/2} (3+5 x)^{5/2}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{1}{33} \int \frac{\left (-\frac{19235}{2}-\frac{30375 x}{2}\right ) (2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{225}{22} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{203 (2+3 x)^{5/2} (3+5 x)^{5/2}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac{\int \frac{\sqrt{2+3 x} (3+5 x)^{3/2} \left (\frac{5436675}{4}+\frac{4236975 x}{2}\right )}{\sqrt{1-2 x}} \, dx}{1485}\\ &=-\frac{6277}{154} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{225}{22} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{203 (2+3 x)^{5/2} (3+5 x)^{5/2}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{\int \frac{\left (-\frac{579705075}{4}-\frac{884387025 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{51975}\\ &=-\frac{1310203 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{4620}-\frac{6277}{154} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{225}{22} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{203 (2+3 x)^{5/2} (3+5 x)^{5/2}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac{\int \frac{\sqrt{3+5 x} \left (\frac{76051906425}{8}+\frac{29256230025 x}{2}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{779625}\\ &=-\frac{1313411}{630} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{1310203 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{4620}-\frac{6277}{154} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{225}{22} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{203 (2+3 x)^{5/2} (3+5 x)^{5/2}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{\int \frac{-\frac{2462988693825}{8}-\frac{3890435469525 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{7016625}\\ &=-\frac{1313411}{630} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{1310203 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{4620}-\frac{6277}{154} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{225}{22} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{203 (2+3 x)^{5/2} (3+5 x)^{5/2}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac{14447521 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{6300}+\frac{174654791 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{12600}\\ &=-\frac{1313411}{630} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{1310203 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{4620}-\frac{6277}{154} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{225}{22} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{203 (2+3 x)^{5/2} (3+5 x)^{5/2}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{174654791 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12600}-\frac{1313411 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3150}\\ \end{align*}
Mathematica [A] time = 0.240332, size = 135, normalized size = 0.53 \[ -\frac{-87969665 \sqrt{2-4 x} (2 x-1) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+30 \sqrt{3 x+2} \sqrt{5 x+3} \left (94500 x^5+486900 x^4+1279350 x^3+2783146 x^2-12151171 x+4641769\right )+174654791 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{37800 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 253, normalized size = 1. \begin{align*}{\frac{1}{ \left ( 1134000\,{x}^{3}+869400\,{x}^{2}-264600\,x-226800 \right ) \left ( 2\,x-1 \right ) } \left ( -42525000\,{x}^{7}+175939330\,\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}-349309582\,\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}-272970000\,{x}^{6}-87969665\,\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 ) +174654791\,\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 ) -870250500\,{x}^{5}-2069287200\,{x}^{4}+3651350730\,{x}^{3}+4336405140\,{x}^{2}-458597550\,x-835518420 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}} \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{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 (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}}{8 \, x^{3} - 12 \, x^{2} + 6 \, 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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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