Optimal. Leaf size=160 \[ -\frac{12758 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{3375}-\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{3 \sqrt{3 x+2}}-\frac{8}{15} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{1076}{675} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{31588 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3375} \]
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Rubi [A] time = 0.052174, 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 = {97, 154, 158, 113, 119} \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{3 \sqrt{3 x+2}}-\frac{8}{15} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{1076}{675} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{12758 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3375}+\frac{31588 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3375} \]
Antiderivative was successfully verified.
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Rule 97
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{3 \sqrt{2+3 x}}+\frac{2}{3} \int \frac{\left (-\frac{25}{2}-30 x\right ) (1-2 x)^{3/2}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{3 \sqrt{2+3 x}}-\frac{8}{15} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{4}{225} \int \frac{\left (-\frac{2895}{4}-\frac{4035 x}{2}\right ) \sqrt{1-2 x}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{3 \sqrt{2+3 x}}-\frac{1076}{675} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{8}{15} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{8 \int \frac{-\frac{73785}{8}-\frac{118455 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{10125}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{3 \sqrt{2+3 x}}-\frac{1076}{675} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{8}{15} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{31588 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3375}+\frac{70169 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3375}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{3 \sqrt{2+3 x}}-\frac{1076}{675} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{8}{15} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{31588 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3375}-\frac{12758 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3375}\\ \end{align*}
Mathematica [A] time = 0.173506, size = 102, normalized size = 0.64 \[ \frac{242095 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} \left (180 x^2-534 x-1661\right )}{\sqrt{3 x+2}}-31588 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{10125} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.016, size = 145, normalized size = 0.9 \begin{align*} -{\frac{1}{303750\,{x}^{3}+232875\,{x}^{2}-70875\,x-60750}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 242095\,\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 ) -31588\,\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 ) -54000\,{x}^{4}+154800\,{x}^{3}+530520\,{x}^{2}+1770\,x-149490 \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{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\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 (4 \, x^{2} - 4 \, x + 1\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{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\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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