Optimal. Leaf size=191 \[ -\frac{28174 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{108045}+\frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{81164 \sqrt{1-2 x} \sqrt{5 x+3}}{108045 \sqrt{3 x+2}}-\frac{15601 \sqrt{1-2 x} \sqrt{5 x+3}}{15435 (3 x+2)^{3/2}}+\frac{163 \sqrt{1-2 x} \sqrt{5 x+3}}{735 (3 x+2)^{5/2}}+\frac{81164 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045} \]
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Rubi [A] time = 0.0637846, 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 = {98, 150, 152, 158, 113, 119} \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{81164 \sqrt{1-2 x} \sqrt{5 x+3}}{108045 \sqrt{3 x+2}}-\frac{15601 \sqrt{1-2 x} \sqrt{5 x+3}}{15435 (3 x+2)^{3/2}}+\frac{163 \sqrt{1-2 x} \sqrt{5 x+3}}{735 (3 x+2)^{5/2}}-\frac{28174 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045}+\frac{81164 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045} \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx &=\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{1}{7} \int \frac{\left (-\frac{159}{2}-160 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{7/2}} \, dx\\ &=\frac{163 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^{5/2}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{2}{735} \int \frac{-\frac{28873}{4}-\frac{25555 x}{2}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\\ &=\frac{163 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^{5/2}}-\frac{15601 \sqrt{1-2 x} \sqrt{3+5 x}}{15435 (2+3 x)^{3/2}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{4 \int \frac{-9619-\frac{78005 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{15435}\\ &=\frac{163 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^{5/2}}-\frac{15601 \sqrt{1-2 x} \sqrt{3+5 x}}{15435 (2+3 x)^{3/2}}-\frac{81164 \sqrt{1-2 x} \sqrt{3+5 x}}{108045 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{8 \int \frac{\frac{88535}{8}+\frac{101455 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{108045}\\ &=\frac{163 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^{5/2}}-\frac{15601 \sqrt{1-2 x} \sqrt{3+5 x}}{15435 (2+3 x)^{3/2}}-\frac{81164 \sqrt{1-2 x} \sqrt{3+5 x}}{108045 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{81164 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{108045}+\frac{154957 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{108045}\\ &=\frac{163 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^{5/2}}-\frac{15601 \sqrt{1-2 x} \sqrt{3+5 x}}{15435 (2+3 x)^{3/2}}-\frac{81164 \sqrt{1-2 x} \sqrt{3+5 x}}{108045 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{5/2}}+\frac{81164 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045}-\frac{28174 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045}\\ \end{align*}
Mathematica [A] time = 0.149179, size = 104, normalized size = 0.54 \[ \frac{\sqrt{2} \left (546035 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-81164 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{6 \sqrt{5 x+3} \left (730476 x^3+936351 x^2+292777 x-4877\right )}{\sqrt{1-2 x} (3 x+2)^{5/2}}}{324135} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.026, size = 314, normalized size = 1.6 \begin{align*}{\frac{1}{3241350\,{x}^{2}+324135\,x-972405}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 730476\,\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}-4914315\,\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}+973968\,\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}-6552420\,\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}+324656\,\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 ) -2184140\,\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 ) -21914280\,{x}^{4}-41239098\,{x}^{3}-25637628\,{x}^{2}-5123676\,x+87786 \right ) \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}}}{{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\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 (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, 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}}}{{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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