Optimal. Leaf size=189 \[ -\frac{7738 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{15625}-\frac{2 (3 x+2)^{3/2} (1-2 x)^{5/2}}{15 (5 x+3)^{3/2}}-\frac{178 (3 x+2)^{3/2} (1-2 x)^{3/2}}{75 \sqrt{5 x+3}}-\frac{572}{625} (3 x+2)^{3/2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{8874 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{3125}+\frac{9206 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625} \]
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Rubi [A] time = 0.0667983, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{2 (3 x+2)^{3/2} (1-2 x)^{5/2}}{15 (5 x+3)^{3/2}}-\frac{178 (3 x+2)^{3/2} (1-2 x)^{3/2}}{75 \sqrt{5 x+3}}-\frac{572}{625} (3 x+2)^{3/2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{8874 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{3125}-\frac{7738 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625}+\frac{9206 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625} \]
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} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}+\frac{2}{15} \int \frac{\left (-\frac{11}{2}-24 x\right ) (1-2 x)^{3/2} \sqrt{2+3 x}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac{178 (1-2 x)^{3/2} (2+3 x)^{3/2}}{75 \sqrt{3+5 x}}+\frac{4}{75} \int \frac{\left (-\frac{291}{2}-\frac{1287 x}{2}\right ) \sqrt{1-2 x} \sqrt{2+3 x}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac{178 (1-2 x)^{3/2} (2+3 x)^{3/2}}{75 \sqrt{3+5 x}}-\frac{572}{625} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{8 \int \frac{\left (\frac{5175}{2}-\frac{119799 x}{4}\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{5625}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac{178 (1-2 x)^{3/2} (2+3 x)^{3/2}}{75 \sqrt{3+5 x}}+\frac{8874 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3125}-\frac{572}{625} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{8 \int \frac{\frac{217593}{8}+\frac{372843 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{84375}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac{178 (1-2 x)^{3/2} (2+3 x)^{3/2}}{75 \sqrt{3+5 x}}+\frac{8874 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3125}-\frac{572}{625} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{27618 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{15625}+\frac{42559 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{15625}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac{178 (1-2 x)^{3/2} (2+3 x)^{3/2}}{75 \sqrt{3+5 x}}+\frac{8874 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3125}-\frac{572}{625} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{9206 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625}-\frac{7738 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625}\\ \end{align*}
Mathematica [A] time = 0.282583, size = 107, normalized size = 0.57 \[ \frac{155295 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (4500 x^3-9450 x^2-48650 x-25421\right )}{(5 x+3)^{3/2}}-27618 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{46875} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 229, normalized size = 1.2 \begin{align*}{\frac{1}{281250\,{x}^{2}+46875\,x-93750} \left ( 138090\,\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}-776475\,\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}+82854\,\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 ) -465885\,\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 ) +270000\,{x}^{5}-522000\,{x}^{4}-3103500\,{x}^{3}-1822760\,{x}^{2}+718790\,x+508420 \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{{\left (3 \, x + 2\right )}^{\frac{3}{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 (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\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{{\left (3 \, x + 2\right )}^{\frac{3}{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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