Optimal. Leaf size=160 \[ -\frac{1196 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{15435}+\frac{5594 \sqrt{1-2 x} \sqrt{5 x+3}}{15435 \sqrt{3 x+2}}-\frac{404 \sqrt{1-2 x} \sqrt{5 x+3}}{2205 (3 x+2)^{3/2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{105 (3 x+2)^{5/2}}-\frac{5594 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15435} \]
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Rubi [A] time = 0.0504849, 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 = {98, 152, 158, 113, 119} \[ \frac{5594 \sqrt{1-2 x} \sqrt{5 x+3}}{15435 \sqrt{3 x+2}}-\frac{404 \sqrt{1-2 x} \sqrt{5 x+3}}{2205 (3 x+2)^{3/2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{105 (3 x+2)^{5/2}}-\frac{1196 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15435}-\frac{5594 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15435} \]
Antiderivative was successfully verified.
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Rule 98
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^{7/2}} \, dx &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^{5/2}}-\frac{2}{105} \int \frac{-248-\frac{845 x}{2}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^{5/2}}-\frac{404 \sqrt{1-2 x} \sqrt{3+5 x}}{2205 (2+3 x)^{3/2}}-\frac{4 \int \frac{-\frac{2279}{4}-505 x}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{2205}\\ &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^{5/2}}-\frac{404 \sqrt{1-2 x} \sqrt{3+5 x}}{2205 (2+3 x)^{3/2}}+\frac{5594 \sqrt{1-2 x} \sqrt{3+5 x}}{15435 \sqrt{2+3 x}}-\frac{8 \int \frac{-2920-\frac{13985 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{15435}\\ &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^{5/2}}-\frac{404 \sqrt{1-2 x} \sqrt{3+5 x}}{2205 (2+3 x)^{3/2}}+\frac{5594 \sqrt{1-2 x} \sqrt{3+5 x}}{15435 \sqrt{2+3 x}}+\frac{5594 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{15435}+\frac{6578 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{15435}\\ &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^{5/2}}-\frac{404 \sqrt{1-2 x} \sqrt{3+5 x}}{2205 (2+3 x)^{3/2}}+\frac{5594 \sqrt{1-2 x} \sqrt{3+5 x}}{15435 \sqrt{2+3 x}}-\frac{5594 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15435}-\frac{1196 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15435}\\ \end{align*}
Mathematica [A] time = 0.130809, size = 99, normalized size = 0.62 \[ \frac{2 \left (\sqrt{2} \left (7070 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+2797 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (25173 x^2+29322 x+8507\right )}{(3 x+2)^{5/2}}\right )}{46305} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 314, normalized size = 2. \begin{align*} -{\frac{2}{463050\,{x}^{2}+46305\,x-138915} \left ( 63630\,\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}+25173\,\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}+84840\,\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}+33564\,\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}+28280\,\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 ) +11188\,\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 ) -755190\,{x}^{4}-955179\,{x}^{3}-116619\,{x}^{2}+238377\,x+76563 \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{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{7}{2}} \sqrt{-2 \, x + 1}}\,{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 (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, 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 (3 \, x + 2\right )}^{\frac{7}{2}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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