Optimal. Leaf size=156 \[ -\frac{4721 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{550 \sqrt{33}}+\frac{7 \sqrt{5 x+3} (3 x+2)^{5/2}}{33 (1-2 x)^{3/2}}-\frac{679 \sqrt{5 x+3} (3 x+2)^{3/2}}{363 \sqrt{1-2 x}}-\frac{4517 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{1210}-\frac{78472 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{275 \sqrt{33}} \]
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Rubi [A] time = 0.0500642, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 154, 158, 113, 119} \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{5/2}}{33 (1-2 x)^{3/2}}-\frac{679 \sqrt{5 x+3} (3 x+2)^{3/2}}{363 \sqrt{1-2 x}}-\frac{4517 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{1210}-\frac{4721 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{550 \sqrt{33}}-\frac{78472 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{275 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{7/2}}{(1-2 x)^{5/2} \sqrt{3+5 x}} \, dx &=\frac{7 (2+3 x)^{5/2} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{1}{33} \int \frac{(2+3 x)^{3/2} \left (\frac{373}{2}+306 x\right )}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{679 (2+3 x)^{3/2} \sqrt{3+5 x}}{363 \sqrt{1-2 x}}+\frac{7 (2+3 x)^{5/2} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{1}{363} \int \frac{\left (-\frac{25065}{2}-\frac{40653 x}{2}\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{4517 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1210}-\frac{679 (2+3 x)^{3/2} \sqrt{3+5 x}}{363 \sqrt{1-2 x}}+\frac{7 (2+3 x)^{5/2} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}+\frac{\int \frac{\frac{1788471}{4}+706248 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{5445}\\ &=-\frac{4517 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1210}-\frac{679 (2+3 x)^{3/2} \sqrt{3+5 x}}{363 \sqrt{1-2 x}}+\frac{7 (2+3 x)^{5/2} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}+\frac{4721 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1100}+\frac{78472 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3025}\\ &=-\frac{4517 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1210}-\frac{679 (2+3 x)^{3/2} \sqrt{3+5 x}}{363 \sqrt{1-2 x}}+\frac{7 (2+3 x)^{5/2} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{78472 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{275 \sqrt{33}}-\frac{4721 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{550 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.232371, size = 120, normalized size = 0.77 \[ -\frac{-158095 \sqrt{2-4 x} (2 x-1) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+10 \sqrt{3 x+2} \sqrt{5 x+3} \left (6534 x^2-70234 x+24051\right )+313888 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{36300 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 238, normalized size = 1.5 \begin{align*}{\frac{1}{ \left ( 1089000\,{x}^{3}+834900\,{x}^{2}-254100\,x-217800 \right ) \left ( 2\,x-1 \right ) } \left ( 316190\,\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}-627776\,\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}-158095\,\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 ) +313888\,\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 ) -980100\,{x}^{4}+9293640\,{x}^{3}+9344770\,{x}^{2}-355650\,x-1443060 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}\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{7}{2}}}{\sqrt{5 \, x + 3}{\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{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{40 \, x^{4} - 36 \, x^{3} - 6 \, x^{2} + 13 \, x - 3}, 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{7}{2}}}{\sqrt{5 \, x + 3}{\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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