Optimal. Leaf size=160 \[ -\frac{2657 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{118125}+\frac{2}{35} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{194 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{2625}-\frac{2657 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{23625}-\frac{118898 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{118125} \]
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Rubi [A] time = 0.0508421, 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 = {101, 154, 158, 113, 119} \[ \frac{2}{35} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{194 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{2625}-\frac{2657 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{23625}-\frac{2657 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{118125}-\frac{118898 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{118125} \]
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} \sqrt{2+3 x} \sqrt{3+5 x} \, dx &=\frac{2}{35} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2}{35} \int \frac{\left (-\frac{67}{2}-\frac{97 x}{2}\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{\sqrt{2+3 x}} \, dx\\ &=\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{2625}+\frac{2}{35} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{4 \int \frac{\left (-\frac{1203}{2}-\frac{2657 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{2625}\\ &=-\frac{2657 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{23625}+\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{2625}+\frac{2}{35} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{4 \int \frac{\frac{148523}{8}+\frac{59449 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{23625}\\ &=-\frac{2657 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{23625}+\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{2625}+\frac{2}{35} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{29227 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{236250}+\frac{118898 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{118125}\\ &=-\frac{2657 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{23625}+\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{2625}+\frac{2}{35} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{118898 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{118125}-\frac{2657 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{118125}\\ \end{align*}
Mathematica [A] time = 0.220393, size = 97, normalized size = 0.61 \[ \frac{-150115 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (-13500 x^2+7380 x+6631\right )+237796 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{354375 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 150, normalized size = 0.9 \begin{align*}{\frac{1}{21262500\,{x}^{3}+16301250\,{x}^{2}-4961250\,x-4252500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 150115\,\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 ) -237796\,\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 ) -12150000\,{x}^{5}-2673000\,{x}^{4}+13895100\,{x}^{3}+5455590\,{x}^{2}-2720910\,x-1193580 \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} \sqrt{3 \, x + 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 (\sqrt{5 \, x + 3} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}, 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} \sqrt{3 \, x + 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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