Optimal. Leaf size=156 \[ -\frac{1}{5} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )-\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{3 \sqrt{3 x+2}}+\frac{4}{3} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{3}{5} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0494091, antiderivative size = 156, 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 = {97, 154, 158, 113, 119} \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{3 \sqrt{3 x+2}}+\frac{4}{3} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{1}{5} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{3}{5} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 97
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}+\frac{2}{3} \int \frac{\left (\frac{19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{4}{3} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}-\frac{2}{45} \int \frac{\left (-\frac{45}{2}-\frac{405 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{4}{3} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}+\frac{2}{405} \int \frac{\frac{5265}{4}+\frac{3645 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{4}{3} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}+\frac{11}{10} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{9}{5} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{4}{3} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}-\frac{3}{5} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{1}{5} \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 = 0.194825, size = 112, normalized size = 0.72 \[ \frac{15 \sqrt{2} (3 x+2) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+10 \sqrt{1-2 x} x \sqrt{3 x+2} \sqrt{5 x+3} (10 x+7)+18 \sqrt{2} (3 x+2) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{90 x+60} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.015, size = 144, normalized size = 0.9 \begin{align*} -{\frac{1}{900\,{x}^{3}+690\,{x}^{2}-210\,x-180}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 15\,\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 ) +18\,\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 ) -1000\,{x}^{4}-800\,{x}^{3}+230\,{x}^{2}+210\,x \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}}{{\left (3 \, x + 2\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 (\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{9 \, x^{2} + 12 \, x + 4}, 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}}{{\left (3 \, x + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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