Optimal. Leaf size=156 \[ -\frac{164 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{847 \sqrt{33}}+\frac{19480 \sqrt{1-2 x} \sqrt{3 x+2}}{27951 \sqrt{5 x+3}}-\frac{410 \sqrt{1-2 x} \sqrt{3 x+2}}{2541 (5 x+3)^{3/2}}+\frac{4 \sqrt{3 x+2}}{77 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{3896 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{847 \sqrt{33}} \]
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Rubi [A] time = 0.0520021, 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 = {104, 152, 158, 113, 119} \[ \frac{19480 \sqrt{1-2 x} \sqrt{3 x+2}}{27951 \sqrt{5 x+3}}-\frac{410 \sqrt{1-2 x} \sqrt{3 x+2}}{2541 (5 x+3)^{3/2}}+\frac{4 \sqrt{3 x+2}}{77 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{164 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{847 \sqrt{33}}-\frac{3896 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{847 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 104
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx &=\frac{4 \sqrt{2+3 x}}{77 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{2}{77} \int \frac{-\frac{95}{2}-45 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{4 \sqrt{2+3 x}}{77 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{410 \sqrt{1-2 x} \sqrt{2+3 x}}{2541 (3+5 x)^{3/2}}+\frac{4 \int \frac{-\frac{605}{2}+\frac{615 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{2541}\\ &=\frac{4 \sqrt{2+3 x}}{77 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{410 \sqrt{1-2 x} \sqrt{2+3 x}}{2541 (3+5 x)^{3/2}}+\frac{19480 \sqrt{1-2 x} \sqrt{2+3 x}}{27951 \sqrt{3+5 x}}-\frac{8 \int \frac{-\frac{18885}{4}-7305 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{27951}\\ &=\frac{4 \sqrt{2+3 x}}{77 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{410 \sqrt{1-2 x} \sqrt{2+3 x}}{2541 (3+5 x)^{3/2}}+\frac{19480 \sqrt{1-2 x} \sqrt{2+3 x}}{27951 \sqrt{3+5 x}}+\frac{82}{847} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{3896 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{9317}\\ &=\frac{4 \sqrt{2+3 x}}{77 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{410 \sqrt{1-2 x} \sqrt{2+3 x}}{2541 (3+5 x)^{3/2}}+\frac{19480 \sqrt{1-2 x} \sqrt{2+3 x}}{27951 \sqrt{3+5 x}}-\frac{3896 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{847 \sqrt{33}}-\frac{164 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{847 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.141358, size = 98, normalized size = 0.63 \[ \frac{2 \left (\sqrt{2} \left (1948 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-595 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{\sqrt{3 x+2} \left (-97400 x^2-5230 x+27691\right )}{\sqrt{1-2 x} (5 x+3)^{3/2}}\right )}{27951} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.022, size = 219, normalized size = 1.4 \begin{align*}{\frac{2}{167706\,{x}^{2}+27951\,x-55902}\sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 2975\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-9740\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1785\,\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 ) -5844\,\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 ) +292200\,{x}^{3}+210490\,{x}^{2}-72613\,x-55382 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \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{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{1500 \, x^{6} + 2200 \, x^{5} + 95 \, x^{4} - 1091 \, x^{3} - 333 \, x^{2} + 135 \, x + 54}, 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{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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