Optimal. Leaf size=187 \[ -\frac{992 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{2401 \sqrt{33}}+\frac{338 \sqrt{1-2 x} \sqrt{5 x+3}}{26411 \sqrt{3 x+2}}-\frac{458 \sqrt{1-2 x} \sqrt{5 x+3}}{3773 (3 x+2)^{3/2}}+\frac{326 \sqrt{5 x+3}}{1617 \sqrt{1-2 x} (3 x+2)^{3/2}}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac{338 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2401 \sqrt{33}} \]
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Rubi [A] time = 0.0663539, 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 = {99, 152, 158, 113, 119} \[ \frac{338 \sqrt{1-2 x} \sqrt{5 x+3}}{26411 \sqrt{3 x+2}}-\frac{458 \sqrt{1-2 x} \sqrt{5 x+3}}{3773 (3 x+2)^{3/2}}+\frac{326 \sqrt{5 x+3}}{1617 \sqrt{1-2 x} (3 x+2)^{3/2}}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac{992 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2401 \sqrt{33}}-\frac{338 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2401 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 99
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{3+5 x}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}-\frac{2}{21} \int \frac{-22-\frac{75 x}{2}}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac{326 \sqrt{3+5 x}}{1617 \sqrt{1-2 x} (2+3 x)^{3/2}}+\frac{4 \int \frac{\frac{4203}{4}+\frac{7335 x}{4}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{1617}\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac{326 \sqrt{3+5 x}}{1617 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{458 \sqrt{1-2 x} \sqrt{3+5 x}}{3773 (2+3 x)^{3/2}}+\frac{8 \int \frac{\frac{14247}{8}+\frac{10305 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{33957}\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac{326 \sqrt{3+5 x}}{1617 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{458 \sqrt{1-2 x} \sqrt{3+5 x}}{3773 (2+3 x)^{3/2}}+\frac{338 \sqrt{1-2 x} \sqrt{3+5 x}}{26411 \sqrt{2+3 x}}+\frac{16 \int \frac{\frac{29115}{8}+\frac{7605 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{237699}\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac{326 \sqrt{3+5 x}}{1617 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{458 \sqrt{1-2 x} \sqrt{3+5 x}}{3773 (2+3 x)^{3/2}}+\frac{338 \sqrt{1-2 x} \sqrt{3+5 x}}{26411 \sqrt{2+3 x}}+\frac{338 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{26411}+\frac{496 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2401}\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac{326 \sqrt{3+5 x}}{1617 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{458 \sqrt{1-2 x} \sqrt{3+5 x}}{3773 (2+3 x)^{3/2}}+\frac{338 \sqrt{1-2 x} \sqrt{3+5 x}}{26411 \sqrt{2+3 x}}-\frac{338 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2401 \sqrt{33}}-\frac{992 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2401 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.162322, size = 103, normalized size = 0.55 \[ \frac{2 \left (\sqrt{2} \left (8015 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+169 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{\sqrt{5 x+3} \left (6084 x^3-21264 x^2+727 x+7965\right )}{(1-2 x)^{3/2} (3 x+2)^{3/2}}\right )}{79233} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 311, normalized size = 1.7 \begin{align*} -{\frac{2}{79233\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 1014\,\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}+48090\,\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}+169\,\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}+8015\,\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}-338\,\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 ) -16030\,\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 ) -30420\,{x}^{4}+88068\,{x}^{3}+60157\,{x}^{2}-42006\,x-23895 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3+5\,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{\sqrt{5 \, x + 3}}{{\left (3 \, x + 2\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} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{216 \, x^{6} + 108 \, x^{5} - 198 \, x^{4} - 71 \, x^{3} + 66 \, x^{2} + 12 \, x - 8}, 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{\sqrt{5 \, x + 3}}{{\left (3 \, x + 2\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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