Optimal. Leaf size=160 \[ -\frac{496}{135} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{14 \sqrt{5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^{5/2}}+\frac{16564 \sqrt{5 x+3} \sqrt{1-2 x}}{135 \sqrt{3 x+2}}+\frac{1736 \sqrt{5 x+3} \sqrt{1-2 x}}{135 (3 x+2)^{3/2}}-\frac{16564}{135} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0525548, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 152, 158, 113, 119} \[ \frac{14 \sqrt{5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^{5/2}}+\frac{16564 \sqrt{5 x+3} \sqrt{1-2 x}}{135 \sqrt{3 x+2}}+\frac{1736 \sqrt{5 x+3} \sqrt{1-2 x}}{135 (3 x+2)^{3/2}}-\frac{496}{135} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{16564}{135} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^{7/2} \sqrt{3+5 x}} \, dx &=\frac{14 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^{5/2}}+\frac{2}{15} \int \frac{(128-25 x) \sqrt{1-2 x}}{(2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\\ &=\frac{14 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^{5/2}}+\frac{1736 \sqrt{1-2 x} \sqrt{3+5 x}}{135 (2+3 x)^{3/2}}-\frac{4}{135} \int \frac{-\frac{6869}{2}+2095 x}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=\frac{14 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^{5/2}}+\frac{1736 \sqrt{1-2 x} \sqrt{3+5 x}}{135 (2+3 x)^{3/2}}+\frac{16564 \sqrt{1-2 x} \sqrt{3+5 x}}{135 \sqrt{2+3 x}}-\frac{8}{945} \int \frac{-\frac{91735}{2}-\frac{144935 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{14 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^{5/2}}+\frac{1736 \sqrt{1-2 x} \sqrt{3+5 x}}{135 (2+3 x)^{3/2}}+\frac{16564 \sqrt{1-2 x} \sqrt{3+5 x}}{135 \sqrt{2+3 x}}+\frac{2728}{135} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{16564}{135} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{14 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^{5/2}}+\frac{1736 \sqrt{1-2 x} \sqrt{3+5 x}}{135 (2+3 x)^{3/2}}+\frac{16564 \sqrt{1-2 x} \sqrt{3+5 x}}{135 \sqrt{2+3 x}}-\frac{16564}{135} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{496}{135} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.223504, size = 101, normalized size = 0.63 \[ \frac{4}{405} \left (\sqrt{2} \left (4141 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2095 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (74538 x^2+101862 x+34927\right )}{2 (3 x+2)^{5/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 314, normalized size = 2. \begin{align*}{\frac{2}{4050\,{x}^{2}+405\,x-1215} \left ( 37710\,\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}-74538\,\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}+50280\,\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}-99384\,\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}+16760\,\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 ) -33128\,\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 ) +2236140\,{x}^{4}+3279474\,{x}^{3}+682554\,{x}^{2}-811977\,x-314343 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{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 (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{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 (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48}, 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 (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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