Optimal. Leaf size=160 \[ \frac{992 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1875}-\frac{2 \sqrt{3 x+2} (1-2 x)^{5/2}}{15 (5 x+3)^{3/2}}-\frac{46 \sqrt{3 x+2} (1-2 x)^{3/2}}{75 \sqrt{5 x+3}}-\frac{76}{375} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{338 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1875} \]
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Rubi [A] time = 0.0491062, 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 = {97, 150, 154, 158, 113, 119} \[ -\frac{2 \sqrt{3 x+2} (1-2 x)^{5/2}}{15 (5 x+3)^{3/2}}-\frac{46 \sqrt{3 x+2} (1-2 x)^{3/2}}{75 \sqrt{5 x+3}}-\frac{76}{375} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{992 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1875}+\frac{338 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1875} \]
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{(1-2 x)^{5/2} \sqrt{2+3 x}}{(3+5 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} \sqrt{2+3 x}}{15 (3+5 x)^{3/2}}+\frac{2}{15} \int \frac{\left (-\frac{17}{2}-18 x\right ) (1-2 x)^{3/2}}{\sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{2+3 x}}{15 (3+5 x)^{3/2}}-\frac{46 (1-2 x)^{3/2} \sqrt{2+3 x}}{75 \sqrt{3+5 x}}+\frac{4}{75} \int \frac{\left (-78-\frac{171 x}{2}\right ) \sqrt{1-2 x}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{2+3 x}}{15 (3+5 x)^{3/2}}-\frac{46 (1-2 x)^{3/2} \sqrt{2+3 x}}{75 \sqrt{3+5 x}}-\frac{76}{375} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{8 \int \frac{-\frac{5823}{4}-\frac{1521 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3375}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{2+3 x}}{15 (3+5 x)^{3/2}}-\frac{46 (1-2 x)^{3/2} \sqrt{2+3 x}}{75 \sqrt{3+5 x}}-\frac{76}{375} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{338 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1875}-\frac{5456 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1875}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{2+3 x}}{15 (3+5 x)^{3/2}}-\frac{46 (1-2 x)^{3/2} \sqrt{2+3 x}}{75 \sqrt{3+5 x}}-\frac{76}{375} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{338 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1875}+\frac{992 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1875}\\ \end{align*}
Mathematica [A] time = 0.232302, size = 102, normalized size = 0.64 \[ \frac{2 \left (-8015 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{15 \sqrt{1-2 x} \sqrt{3 x+2} \left (100 x^2-925 x-712\right )}{(5 x+3)^{3/2}}-169 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{5625} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 224, normalized size = 1.4 \begin{align*}{\frac{2}{33750\,{x}^{2}+5625\,x-11250} \left ( 40075\,\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}+845\,\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}+24045\,\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 ) +507\,\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 ) +9000\,{x}^{4}-81750\,{x}^{3}-80955\,{x}^{2}+17070\,x+21360 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{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{\sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\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 (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27}, 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{\sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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