Optimal. Leaf size=160 \[ -\frac{12758 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{6615}-\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}-\frac{118 \sqrt{1-2 x} (5 x+3)^{3/2}}{315 (3 x+2)^{3/2}}-\frac{12758 \sqrt{1-2 x} \sqrt{5 x+3}}{6615 \sqrt{3 x+2}}+\frac{31588 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6615} \]
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Rubi [A] time = 0.0506104, 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, 150, 158, 113, 119} \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}-\frac{118 \sqrt{1-2 x} (5 x+3)^{3/2}}{315 (3 x+2)^{3/2}}-\frac{12758 \sqrt{1-2 x} \sqrt{5 x+3}}{6615 \sqrt{3 x+2}}-\frac{12758 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6615}+\frac{31588 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6615} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{7/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac{2}{15} \int \frac{\left (\frac{19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^{5/2}} \, dx\\ &=-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{315 (2+3 x)^{3/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac{4}{945} \int \frac{\left (\frac{999}{4}-\frac{4035 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{3/2}} \, dx\\ &=-\frac{12758 \sqrt{1-2 x} \sqrt{3+5 x}}{6615 \sqrt{2+3 x}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{315 (2+3 x)^{3/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac{8 \int \frac{-\frac{73785}{8}-\frac{118455 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{19845}\\ &=-\frac{12758 \sqrt{1-2 x} \sqrt{3+5 x}}{6615 \sqrt{2+3 x}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{315 (2+3 x)^{3/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}-\frac{31588 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{6615}+\frac{70169 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{6615}\\ &=-\frac{12758 \sqrt{1-2 x} \sqrt{3+5 x}}{6615 \sqrt{2+3 x}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{315 (2+3 x)^{3/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac{31588 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6615}-\frac{12758 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6615}\\ \end{align*}
Mathematica [A] time = 0.2348, size = 99, normalized size = 0.62 \[ \frac{\sqrt{2} \left (242095 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-31588 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{6 \sqrt{1-2 x} \sqrt{5 x+3} \left (87021 x^2+113319 x+36919\right )}{(3 x+2)^{5/2}}}{19845} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.018, size = 314, normalized size = 2. \begin{align*} -{\frac{1}{198450\,{x}^{2}+19845\,x-59535} \left ( 2178855\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-284292\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+2905140\,\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}-379056\,\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}+968380\,\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 ) -126352\,\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 ) +5221260\,{x}^{4}+7321266\,{x}^{3}+1328676\,{x}^{2}-1818228\,x-664542 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{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 (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{7}{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}}{81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16}, 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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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