Optimal. Leaf size=189 \[ -\frac{4364 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{12005 \sqrt{33}}+\frac{5636 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 \sqrt{3 x+2}}-\frac{26 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 (3 x+2)^{3/2}}-\frac{36 \sqrt{1-2 x} \sqrt{5 x+3}}{245 (3 x+2)^{5/2}}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{5636 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005} \]
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Rubi [A] time = 0.0645097, antiderivative size = 189, 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} \[ \frac{5636 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 \sqrt{3 x+2}}-\frac{26 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 (3 x+2)^{3/2}}-\frac{36 \sqrt{1-2 x} \sqrt{5 x+3}}{245 (3 x+2)^{5/2}}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{4364 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005 \sqrt{33}}-\frac{5636 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005} \]
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)^{3/2} (2+3 x)^{7/2}} \, dx &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{2}{7} \int \frac{-22-\frac{75 x}{2}}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{36 \sqrt{1-2 x} \sqrt{3+5 x}}{245 (2+3 x)^{5/2}}-\frac{4}{245} \int \frac{-\frac{347}{4}-135 x}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{36 \sqrt{1-2 x} \sqrt{3+5 x}}{245 (2+3 x)^{5/2}}-\frac{26 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{3/2}}-\frac{8 \int \frac{-\frac{1539}{4}-\frac{195 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{5145}\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{36 \sqrt{1-2 x} \sqrt{3+5 x}}{245 (2+3 x)^{5/2}}-\frac{26 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{3/2}}+\frac{5636 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 \sqrt{2+3 x}}-\frac{16 \int \frac{-\frac{28635}{8}-\frac{21135 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{36015}\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{36 \sqrt{1-2 x} \sqrt{3+5 x}}{245 (2+3 x)^{5/2}}-\frac{26 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{3/2}}+\frac{5636 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 \sqrt{2+3 x}}+\frac{2182 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{12005}+\frac{5636 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{12005}\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{36 \sqrt{1-2 x} \sqrt{3+5 x}}{245 (2+3 x)^{5/2}}-\frac{26 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{3/2}}+\frac{5636 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 \sqrt{2+3 x}}-\frac{5636 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005}-\frac{4364 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.155818, size = 104, normalized size = 0.55 \[ \frac{2 \left (\sqrt{2} \left (455 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+2818 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{3 \sqrt{5 x+3} \left (50724 x^3+41724 x^2-13127 x-11923\right )}{\sqrt{1-2 x} (3 x+2)^{5/2}}\right )}{36015} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 314, normalized size = 1.7 \begin{align*} -{\frac{2}{360150\,{x}^{2}+36015\,x-108045}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 4095\,\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}+25362\,\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}+5460\,\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}+33816\,\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}+1820\,\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 ) +11272\,\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 ) -760860\,{x}^{4}-1082376\,{x}^{3}-178611\,{x}^{2}+296988\,x+107307 \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{\sqrt{5 \, x + 3}}{{\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{\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{\sqrt{5 \, x + 3}}{{\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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