Optimal. Leaf size=160 \[ -\frac{131}{945} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{2}{21} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{1}{7} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{131}{189} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{9013 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1890} \]
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Rubi [A] time = 0.0516328, 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} \[ \frac{2}{21} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{1}{7} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{131}{189} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{131}{945} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{9013 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1890} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{\sqrt{2+3 x}} \, dx &=\frac{2}{21} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{2}{21} \int \frac{\left (-8-\frac{45 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{1}{7} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{2}{21} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{315} \int \frac{\sqrt{3+5 x} \left (\frac{2655}{4}+\frac{1965 x}{2}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{131}{189} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{1}{7} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{2}{21} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{2 \int \frac{-21360-\frac{135195 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2835}\\ &=-\frac{131}{189} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{1}{7} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{2}{21} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{1441 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1890}+\frac{9013 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1890}\\ &=-\frac{131}{189} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{1}{7} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{2}{21} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{9013 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1890}-\frac{131}{945} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.213366, size = 97, normalized size = 0.61 \[ \frac{-4690 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+75 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (90 x^2+81 x-10\right )+9013 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2835 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 150, normalized size = 0.9 \begin{align*}{\frac{1}{170100\,{x}^{3}+130410\,{x}^{2}-39690\,x-34020}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 4690\,\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 ) -9013\,\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 ) +405000\,{x}^{5}+675000\,{x}^{4}+139950\,{x}^{3}-200550\,{x}^{2}-62400\,x+9000 \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{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}{\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{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{\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{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}{\sqrt{3 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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