Optimal. Leaf size=191 \[ -\frac{84134 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1771875}+\frac{2}{45} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{3/2}+\frac{62 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}{1575}-\frac{347 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{39375}-\frac{84134 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{354375}-\frac{5684677 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3543750} \]
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Rubi [A] time = 0.0671685, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {101, 154, 158, 113, 119} \[ \frac{2}{45} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{3/2}+\frac{62 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}{1575}-\frac{347 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{39375}-\frac{84134 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{354375}-\frac{84134 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1771875}-\frac{5684677 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3543750} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x} \, dx &=\frac{2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{2}{45} \int \left (-\frac{69}{2}-\frac{93 x}{2}\right ) \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x} \, dx\\ &=\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac{2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{4 \int \frac{\left (-765-\frac{1041 x}{4}\right ) \sqrt{2+3 x} \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{4725}\\ &=-\frac{347 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{39375}+\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac{2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{4 \int \frac{\sqrt{3+5 x} \left (\frac{334107}{8}+\frac{126201 x}{2}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{118125}\\ &=-\frac{84134 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{354375}-\frac{347 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{39375}+\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac{2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{4 \int \frac{-\frac{10787703}{8}-\frac{17054031 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1063125}\\ &=-\frac{84134 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{354375}-\frac{347 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{39375}+\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac{2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{462737 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1771875}+\frac{5684677 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3543750}\\ &=-\frac{84134 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{354375}-\frac{347 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{39375}+\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac{2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{5684677 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3543750}-\frac{84134 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1771875}\\ \end{align*}
Mathematica [A] time = 0.239729, size = 105, normalized size = 0.55 \[ \frac{5684677 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (581651 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (472500 x^3+153000 x^2-359685 x-84697\right )\right )}{5315625 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.011, size = 155, normalized size = 0.8 \begin{align*}{\frac{1}{318937500\,{x}^{3}+244518750\,{x}^{2}-74418750\,x-63787500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -425250000\,{x}^{6}+2908255\,\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 ) -5684677\,\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 ) -463725000\,{x}^{5}+317371500\,{x}^{4}+441589950\,{x}^{3}+10447080\,{x}^{2}-82529670\,x-15245460 \right ) } \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 \sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{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 (-{\left (6 \, x^{2} + x - 2\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, 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 \sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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