Optimal. Leaf size=191 \[ \frac{5264 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1215}-\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{15 (3 x+2)^{5/2}}+\frac{46 (5 x+3)^{3/2} (1-2 x)^{3/2}}{27 (3 x+2)^{3/2}}-\frac{316 (5 x+3)^{3/2} \sqrt{1-2 x}}{27 \sqrt{3 x+2}}+\frac{5264}{243} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{19174 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1215} \]
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Rubi [A] time = 0.0623416, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {97, 150, 154, 158, 113, 119} \[ -\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{15 (3 x+2)^{5/2}}+\frac{46 (5 x+3)^{3/2} (1-2 x)^{3/2}}{27 (3 x+2)^{3/2}}-\frac{316 (5 x+3)^{3/2} \sqrt{1-2 x}}{27 \sqrt{3 x+2}}+\frac{5264}{243} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{5264 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1215}-\frac{19174 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1215} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac{2}{15} \int \frac{\left (-\frac{15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^{5/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac{46 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{3/2}}-\frac{4}{135} \int \frac{\left (-\frac{915}{2}-\frac{1965 x}{2}\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac{46 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{3/2}}-\frac{316 \sqrt{1-2 x} (3+5 x)^{3/2}}{27 \sqrt{2+3 x}}+\frac{8}{405} \int \frac{\left (\frac{6705}{4}-9870 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{5264}{243} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac{46 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{3/2}}-\frac{316 \sqrt{1-2 x} (3+5 x)^{3/2}}{27 \sqrt{2+3 x}}-\frac{8 \int \frac{-\frac{42855}{4}-\frac{143805 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3645}\\ &=\frac{5264}{243} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac{46 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{3/2}}-\frac{316 \sqrt{1-2 x} (3+5 x)^{3/2}}{27 \sqrt{2+3 x}}+\frac{19174 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1215}-\frac{28952 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1215}\\ &=\frac{5264}{243} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac{46 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{3/2}}-\frac{316 \sqrt{1-2 x} (3+5 x)^{3/2}}{27 \sqrt{2+3 x}}-\frac{19174 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1215}+\frac{5264 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1215}\\ \end{align*}
Mathematica [A] time = 0.152304, size = 104, normalized size = 0.54 \[ \frac{2 \left (\sqrt{2} \left (9587 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-53015 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (2700 x^3+68913 x^2+83412 x+25927\right )}{(3 x+2)^{5/2}}\right )}{3645} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.019, size = 319, normalized size = 1.7 \begin{align*}{\frac{2}{36450\,{x}^{2}+3645\,x-10935} \left ( 477135\,\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}-86283\,\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}+636180\,\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}-115044\,\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}+212060\,\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 ) -38348\,\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}^{5}+2075490\,{x}^{4}+2684799\,{x}^{3}+407829\,{x}^{2}-672927\,x-233343 \right ) \sqrt{3+5\,x}\sqrt{1-2\,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{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\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 (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\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{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\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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