Optimal. Leaf size=222 \[ -\frac{95264 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{250047}-\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}+\frac{230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{567 (3 x+2)^{7/2}}+\frac{1532 (5 x+3)^{3/2} \sqrt{1-2 x}}{567 (3 x+2)^{5/2}}+\frac{3545996 \sqrt{5 x+3} \sqrt{1-2 x}}{250047 \sqrt{3 x+2}}-\frac{104036 \sqrt{5 x+3} \sqrt{1-2 x}}{35721 (3 x+2)^{3/2}}-\frac{3545996 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{250047} \]
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Rubi [A] time = 0.0801107, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {97, 150, 152, 158, 113, 119} \[ -\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}+\frac{230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{567 (3 x+2)^{7/2}}+\frac{1532 (5 x+3)^{3/2} \sqrt{1-2 x}}{567 (3 x+2)^{5/2}}+\frac{3545996 \sqrt{5 x+3} \sqrt{1-2 x}}{250047 \sqrt{3 x+2}}-\frac{104036 \sqrt{5 x+3} \sqrt{1-2 x}}{35721 (3 x+2)^{3/2}}-\frac{95264 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{250047}-\frac{3545996 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{250047} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac{2}{27} \int \frac{\left (-\frac{15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^{9/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}-\frac{4}{567} \int \frac{\sqrt{1-2 x} \sqrt{3+5 x} \left (-\frac{1905}{2}+\frac{15 x}{2}\right )}{(2+3 x)^{7/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}+\frac{1532 \sqrt{1-2 x} (3+5 x)^{3/2}}{567 (2+3 x)^{5/2}}+\frac{8 \int \frac{\left (\frac{91845}{4}-14325 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{5/2}} \, dx}{8505}\\ &=-\frac{104036 \sqrt{1-2 x} \sqrt{3+5 x}}{35721 (2+3 x)^{3/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}+\frac{1532 \sqrt{1-2 x} (3+5 x)^{3/2}}{567 (2+3 x)^{5/2}}+\frac{16 \int \frac{\frac{3022395}{8}-\frac{1057575 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{535815}\\ &=-\frac{104036 \sqrt{1-2 x} \sqrt{3+5 x}}{35721 (2+3 x)^{3/2}}+\frac{3545996 \sqrt{1-2 x} \sqrt{3+5 x}}{250047 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}+\frac{1532 \sqrt{1-2 x} (3+5 x)^{3/2}}{567 (2+3 x)^{5/2}}+\frac{32 \int \frac{\frac{41857275}{8}+\frac{66487425 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3750705}\\ &=-\frac{104036 \sqrt{1-2 x} \sqrt{3+5 x}}{35721 (2+3 x)^{3/2}}+\frac{3545996 \sqrt{1-2 x} \sqrt{3+5 x}}{250047 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}+\frac{1532 \sqrt{1-2 x} (3+5 x)^{3/2}}{567 (2+3 x)^{5/2}}+\frac{523952 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{250047}+\frac{3545996 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{250047}\\ &=-\frac{104036 \sqrt{1-2 x} \sqrt{3+5 x}}{35721 (2+3 x)^{3/2}}+\frac{3545996 \sqrt{1-2 x} \sqrt{3+5 x}}{250047 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}+\frac{1532 \sqrt{1-2 x} (3+5 x)^{3/2}}{567 (2+3 x)^{5/2}}-\frac{3545996 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{250047}-\frac{95264 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{250047}\\ \end{align*}
Mathematica [A] time = 0.264694, size = 111, normalized size = 0.5 \[ \frac{4 \left (\sqrt{2} \left (886499 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-493535 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (143612838 x^4+386630766 x^3+391601529 x^2+176436240 x+29785139\right )}{2 (3 x+2)^{9/2}}\right )}{750141} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 504, normalized size = 2.3 \begin{align*}{\frac{2}{7501410\,{x}^{2}+750141\,x-2250423} \left ( 79952670\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}-143612838\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}+213207120\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-382967568\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+213207120\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-382967568\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+94758720\,\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}-170207808\,\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}+4308385140\,{x}^{6}+15793120\,\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 ) -28367968\,\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 ) +12029761494\,{x}^{5}+11615422626\,{x}^{4}+2988214893\,{x}^{3}-2101550871\,{x}^{2}-1498570743\,x-268066251 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{9}{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 (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{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 (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64}, 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{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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