Optimal. Leaf size=158 \[ -\frac{1091 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{3125 \sqrt{33}}-\frac{2 \sqrt{1-2 x} (3 x+2)^{5/2}}{5 \sqrt{5 x+3}}+\frac{36}{125} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{13}{625} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{1409 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3125} \]
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Rubi [A] time = 0.0521164, antiderivative size = 158, 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} \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{5/2}}{5 \sqrt{5 x+3}}+\frac{36}{125} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{13}{625} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{1091 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3125 \sqrt{33}}-\frac{1409 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3125} \]
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} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{5/2}}{5 \sqrt{3+5 x}}+\frac{2}{5} \int \frac{\left (\frac{11}{2}-18 x\right ) (2+3 x)^{3/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{5/2}}{5 \sqrt{3+5 x}}+\frac{36}{125} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{2}{125} \int \frac{\sqrt{2+3 x} \left (-50+\frac{39 x}{2}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{5/2}}{5 \sqrt{3+5 x}}+\frac{13}{625} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{36}{125} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{2 \int \frac{\frac{5727}{4}+\frac{4227 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1875}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{5/2}}{5 \sqrt{3+5 x}}+\frac{13}{625} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{36}{125} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{1091 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{6250}+\frac{1409 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3125}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{5/2}}{5 \sqrt{3+5 x}}+\frac{13}{625} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{36}{125} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{1409 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3125}-\frac{1091 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3125 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.263229, size = 102, normalized size = 0.65 \[ \frac{455 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \left (450 x^2+485 x+119\right )}{\sqrt{5 x+3}}+2818 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{18750} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 145, normalized size = 0.9 \begin{align*} -{\frac{1}{562500\,{x}^{3}+431250\,{x}^{2}-131250\,x-112500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 455\,\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 ) +2818\,\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}-100800\,{x}^{3}-8970\,{x}^{2}+25530\,x+7140 \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 (3 \, x + 2\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\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 (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{25 \, x^{2} + 30 \, x + 9}, 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 (3 \, x + 2\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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