Optimal. Leaf size=129 \[ \frac{412 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{3375}+\frac{2}{15} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{214}{675} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{4157 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3375} \]
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Rubi [A] time = 0.0391389, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ \frac{2}{15} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{214}{675} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{412 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3375}-\frac{4157 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3375} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{\sqrt{2+3 x}} \, dx &=\frac{2}{15} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{2}{15} \int \frac{\left (-31-\frac{107 x}{2}\right ) \sqrt{1-2 x}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{214}{675} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2}{15} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{4}{675} \int \frac{-\frac{2041}{4}-\frac{4157 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{214}{675} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2}{15} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{2266 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3375}+\frac{4157 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3375}\\ &=\frac{214}{675} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2}{15} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{4157 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3375}+\frac{412 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3375}\\ \end{align*}
Mathematica [A] time = 0.0880568, size = 97, normalized size = 0.75 \[ \frac{-10955 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-60 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} (45 x-76)+4157 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{10125} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 145, normalized size = 1.1 \begin{align*}{\frac{1}{303750\,{x}^{3}+232875\,{x}^{2}-70875\,x-60750}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 10955\,\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 ) -4157\,\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 ) -81000\,{x}^{4}+74700\,{x}^{3}+123780\,{x}^{2}-15720\,x-27360 \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 \frac{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{\sqrt{3 \, x + 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{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{\sqrt{3 \, x + 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 \frac{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{\sqrt{3 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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