Optimal. Leaf size=127 \[ -\frac{52 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{25 \sqrt{33}}+\frac{148 \sqrt{1-2 x} \sqrt{3 x+2}}{15 \sqrt{5 x+3}}-\frac{22 \sqrt{1-2 x} \sqrt{3 x+2}}{15 (5 x+3)^{3/2}}-\frac{148}{25} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0393485, antiderivative size = 127, 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 = {98, 152, 158, 113, 119} \[ \frac{148 \sqrt{1-2 x} \sqrt{3 x+2}}{15 \sqrt{5 x+3}}-\frac{22 \sqrt{1-2 x} \sqrt{3 x+2}}{15 (5 x+3)^{3/2}}-\frac{52 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25 \sqrt{33}}-\frac{148}{25} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2}}{\sqrt{2+3 x} (3+5 x)^{5/2}} \, dx &=-\frac{22 \sqrt{1-2 x} \sqrt{2+3 x}}{15 (3+5 x)^{3/2}}-\frac{2}{15} \int \frac{58-39 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{22 \sqrt{1-2 x} \sqrt{2+3 x}}{15 (3+5 x)^{3/2}}+\frac{148 \sqrt{1-2 x} \sqrt{2+3 x}}{15 \sqrt{3+5 x}}+\frac{4}{165} \int \frac{\frac{1551}{2}+1221 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{22 \sqrt{1-2 x} \sqrt{2+3 x}}{15 (3+5 x)^{3/2}}+\frac{148 \sqrt{1-2 x} \sqrt{2+3 x}}{15 \sqrt{3+5 x}}+\frac{26}{25} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{148}{25} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{22 \sqrt{1-2 x} \sqrt{2+3 x}}{15 (3+5 x)^{3/2}}+\frac{148 \sqrt{1-2 x} \sqrt{2+3 x}}{15 \sqrt{3+5 x}}-\frac{148}{25} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{52 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.256613, size = 97, normalized size = 0.76 \[ \frac{2}{75} \left (-35 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{5 \sqrt{1-2 x} \sqrt{3 x+2} (370 x+211)}{(5 x+3)^{3/2}}+74 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.018, size = 219, normalized size = 1.7 \begin{align*}{\frac{2}{450\,{x}^{2}+75\,x-150} \left ( 175\,\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}-370\,\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}+105\,\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 ) -222\,\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 ) +11100\,{x}^{3}+8180\,{x}^{2}-2645\,x-2110 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x} \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{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{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} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{375 \, x^{4} + 925 \, x^{3} + 855 \, x^{2} + 351 \, 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{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{3 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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