Optimal. Leaf size=220 \[ -\frac{7738 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{84035}+\frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac{27618 \sqrt{1-2 x} \sqrt{5 x+3}}{84035 \sqrt{3 x+2}}-\frac{4437 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 (3 x+2)^{3/2}}-\frac{1432 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 (3 x+2)^{5/2}}+\frac{99 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^{5/2}}+\frac{9206 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035} \]
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Rubi [A] time = 0.0812855, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac{27618 \sqrt{1-2 x} \sqrt{5 x+3}}{84035 \sqrt{3 x+2}}-\frac{4437 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 (3 x+2)^{3/2}}-\frac{1432 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 (3 x+2)^{5/2}}+\frac{99 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{7738 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035}+\frac{9206 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035} \]
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)^{5/2} (2+3 x)^{7/2}} \, dx &=\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac{1}{21} \int \frac{\left (-\frac{147}{2}-150 x\right ) \sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx\\ &=\frac{99 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{5/2}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac{1}{147} \int \frac{-4959-\frac{17025 x}{2}}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\\ &=\frac{99 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{1432 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{5/2}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac{2 \int \frac{-\frac{72609}{4}-32220 x}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{5145}\\ &=\frac{99 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{1432 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{5/2}}-\frac{4437 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 (2+3 x)^{3/2}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac{4 \int \frac{-\frac{91683}{4}-\frac{199665 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{108045}\\ &=\frac{99 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{1432 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{5/2}}-\frac{4437 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 (2+3 x)^{3/2}}-\frac{27618 \sqrt{1-2 x} \sqrt{3+5 x}}{84035 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac{8 \int \frac{\frac{362655}{8}+\frac{621405 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{756315}\\ &=\frac{99 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{1432 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{5/2}}-\frac{4437 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 (2+3 x)^{3/2}}-\frac{27618 \sqrt{1-2 x} \sqrt{3+5 x}}{84035 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac{27618 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{84035}+\frac{42559 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{84035}\\ &=\frac{99 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{1432 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{5/2}}-\frac{4437 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 (2+3 x)^{3/2}}-\frac{27618 \sqrt{1-2 x} \sqrt{3+5 x}}{84035 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{9206 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035}-\frac{7738 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035}\\ \end{align*}
Mathematica [A] time = 0.180134, size = 110, normalized size = 0.5 \[ \frac{3 \sqrt{2} \left (51765 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-9206 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{2 \sqrt{5 x+3} \left (1491372 x^4+1056186 x^3-718167 x^2-640441 x-88623\right )}{(1-2 x)^{3/2} (3 x+2)^{5/2}}}{252105} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.025, size = 406, normalized size = 1.9 \begin{align*} -{\frac{1}{252105\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 2795310\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-497124\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+2329425\,\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}-414270\,\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}-621180\,\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}+110472\,\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}-621180\,\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 ) +110472\,\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 ) +14913720\,{x}^{5}+19510092\,{x}^{4}-844554\,{x}^{3}-10713412\,{x}^{2}-4728876\,x-531738 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{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{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{7}{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{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{648 \, x^{7} + 756 \, x^{6} - 378 \, x^{5} - 609 \, x^{4} + 56 \, x^{3} + 168 \, x^{2} - 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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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