Optimal. Leaf size=160 \[ \frac{988}{945} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{74 \sqrt{1-2 x} (5 x+3)^{3/2}}{45 (3 x+2)^{3/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}+\frac{988 \sqrt{1-2 x} \sqrt{5 x+3}}{945 \sqrt{3 x+2}}-\frac{4418}{945} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0495026, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {97, 150, 158, 113, 119} \[ \frac{74 \sqrt{1-2 x} (5 x+3)^{3/2}}{45 (3 x+2)^{3/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}+\frac{988 \sqrt{1-2 x} \sqrt{5 x+3}}{945 \sqrt{3 x+2}}+\frac{988}{945} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{4418}{945} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac{2}{15} \int \frac{\left (-\frac{3}{2}-30 x\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^{5/2}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac{74 \sqrt{1-2 x} (3+5 x)^{3/2}}{45 (2+3 x)^{3/2}}-\frac{4}{135} \int \frac{\left (-\frac{243}{2}-\frac{735 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{3/2}} \, dx\\ &=\frac{988 \sqrt{1-2 x} \sqrt{3+5 x}}{945 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac{74 \sqrt{1-2 x} (3+5 x)^{3/2}}{45 (2+3 x)^{3/2}}-\frac{8 \int \frac{-\frac{5865}{2}-\frac{33135 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2835}\\ &=\frac{988 \sqrt{1-2 x} \sqrt{3+5 x}}{945 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac{74 \sqrt{1-2 x} (3+5 x)^{3/2}}{45 (2+3 x)^{3/2}}+\frac{4418}{945} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx-\frac{5434}{945} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{988 \sqrt{1-2 x} \sqrt{3+5 x}}{945 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac{74 \sqrt{1-2 x} (3+5 x)^{3/2}}{45 (2+3 x)^{3/2}}-\frac{4418}{945} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{988}{945} \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 = 0.135946, size = 99, normalized size = 0.62 \[ \frac{2 \left (\sqrt{2} \left (2209 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-10360 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (16731 x^2+20754 x+6449\right )}{(3 x+2)^{5/2}}\right )}{2835} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.019, size = 314, normalized size = 2. \begin{align*}{\frac{2}{28350\,{x}^{2}+2835\,x-8505} \left ( 93240\,\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}-19881\,\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}+124320\,\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}-26508\,\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}+41440\,\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 ) -8836\,\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 ) +501930\,{x}^{4}+672813\,{x}^{3}+105153\,{x}^{2}-167439\,x-58041 \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 (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\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 (10 \, x^{2} + x - 3\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16}, 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{3}{2}}}{{\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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