Optimal. Leaf size=160 \[ -\frac{2252 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{23625}+\frac{2}{35} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{31}{525} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{2252 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{4725}-\frac{148831 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47250} \]
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Rubi [A] time = 0.0517179, 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} \[ \frac{2}{35} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{31}{525} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{2252 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{4725}-\frac{2252 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{23625}-\frac{148831 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47250} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2} \, dx &=\frac{2}{35} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{2}{35} \int \frac{\left (-\frac{23}{2}-\frac{31 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{31}{525} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{2}{35} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{525} \int \frac{\sqrt{3+5 x} \left (\frac{2907}{4}+1126 x\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{2252 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{4725}-\frac{31}{525} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{2}{35} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{2 \int \frac{-\frac{94253}{4}-\frac{148831 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{4725}\\ &=-\frac{2252 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{4725}-\frac{31}{525} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{2}{35} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{12386 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{23625}+\frac{148831 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{47250}\\ &=-\frac{2252 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{4725}-\frac{31}{525} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{2}{35} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{148831 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47250}-\frac{2252 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{23625}\\ \end{align*}
Mathematica [A] time = 0.217705, size = 97, normalized size = 0.61 \[ \frac{-74515 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (6750 x^2+6705 x-659\right )+148831 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{70875 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 150, normalized size = 0.9 \begin{align*}{\frac{1}{4252500\,{x}^{3}+3260250\,{x}^{2}-992250\,x-850500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 74515\,\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 ) -148831\,\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 ) +6075000\,{x}^{5}+10692000\,{x}^{4}+2615850\,{x}^{3}-3077760\,{x}^{2}-1068510\,x+118620 \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{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}\,{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 (5 \, x + 3\right )}^{\frac{3}{2}} \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{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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