Optimal. Leaf size=187 \[ -\frac{230 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1331 \sqrt{33}}-\frac{2960 \sqrt{1-2 x} \sqrt{3 x+2}}{43923 \sqrt{5 x+3}}-\frac{575 \sqrt{1-2 x} \sqrt{3 x+2}}{3993 (5 x+3)^{3/2}}+\frac{26 \sqrt{3 x+2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{7 \sqrt{3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}+\frac{592 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1331 \sqrt{33}} \]
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Rubi [A] time = 0.0669503, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{2960 \sqrt{1-2 x} \sqrt{3 x+2}}{43923 \sqrt{5 x+3}}-\frac{575 \sqrt{1-2 x} \sqrt{3 x+2}}{3993 (5 x+3)^{3/2}}+\frac{26 \sqrt{3 x+2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{7 \sqrt{3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{230 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1331 \sqrt{33}}+\frac{592 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1331 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{3/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac{7 \sqrt{2+3 x}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{1}{33} \int \frac{-\frac{159}{2}-114 x}{(1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{7 \sqrt{2+3 x}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{26 \sqrt{2+3 x}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{2 \int \frac{\frac{17157}{4}+\frac{12285 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx}{2541}\\ &=\frac{7 \sqrt{2+3 x}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{26 \sqrt{2+3 x}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{575 \sqrt{1-2 x} \sqrt{2+3 x}}{3993 (3+5 x)^{3/2}}-\frac{4 \int \frac{-\frac{27951}{4}-\frac{36225 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{83853}\\ &=\frac{7 \sqrt{2+3 x}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{26 \sqrt{2+3 x}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{575 \sqrt{1-2 x} \sqrt{2+3 x}}{3993 (3+5 x)^{3/2}}-\frac{2960 \sqrt{1-2 x} \sqrt{2+3 x}}{43923 \sqrt{3+5 x}}+\frac{8 \int \frac{-\frac{32193}{8}-23310 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{922383}\\ &=\frac{7 \sqrt{2+3 x}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{26 \sqrt{2+3 x}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{575 \sqrt{1-2 x} \sqrt{2+3 x}}{3993 (3+5 x)^{3/2}}-\frac{2960 \sqrt{1-2 x} \sqrt{2+3 x}}{43923 \sqrt{3+5 x}}-\frac{592 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{14641}+\frac{115 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1331}\\ &=\frac{7 \sqrt{2+3 x}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{26 \sqrt{2+3 x}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{575 \sqrt{1-2 x} \sqrt{2+3 x}}{3993 (3+5 x)^{3/2}}-\frac{2960 \sqrt{1-2 x} \sqrt{2+3 x}}{43923 \sqrt{3+5 x}}+\frac{592 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1331 \sqrt{33}}-\frac{230 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1331 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.149289, size = 104, normalized size = 0.56 \[ \frac{\sqrt{2} \left (4387 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-592 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{2 \sqrt{3 x+2} \left (29600 x^3+810 x^2-13572 x-1775\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}}{43923} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 311, normalized size = 1.7 \begin{align*} -{\frac{1}{43923\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 43870\,\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}-5920\,\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}+4387\,\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}-592\,\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}-13161\,\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 ) +1776\,\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 ) +177600\,{x}^{4}+123260\,{x}^{3}-78192\,{x}^{2}-64938\,x-7100 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{2+3\,x}}}} \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 (3 \, x + 2\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{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{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}{1000 \, x^{6} + 300 \, x^{5} - 870 \, x^{4} - 179 \, x^{3} + 261 \, x^{2} + 27 \, x - 27}, 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 (3 \, x + 2\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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