Optimal. Leaf size=253 \[ -\frac{65672 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{823543}+\frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}-\frac{98642 \sqrt{1-2 x} \sqrt{5 x+3}}{823543 \sqrt{3 x+2}}-\frac{33778 \sqrt{1-2 x} \sqrt{5 x+3}}{117649 (3 x+2)^{3/2}}-\frac{11433 \sqrt{1-2 x} \sqrt{5 x+3}}{16807 (3 x+2)^{5/2}}-\frac{4545 \sqrt{1-2 x} \sqrt{5 x+3}}{2401 (3 x+2)^{7/2}}+\frac{220 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^{7/2}}+\frac{98642 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{823543} \]
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Rubi [A] time = 0.098521, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}-\frac{98642 \sqrt{1-2 x} \sqrt{5 x+3}}{823543 \sqrt{3 x+2}}-\frac{33778 \sqrt{1-2 x} \sqrt{5 x+3}}{117649 (3 x+2)^{3/2}}-\frac{11433 \sqrt{1-2 x} \sqrt{5 x+3}}{16807 (3 x+2)^{5/2}}-\frac{4545 \sqrt{1-2 x} \sqrt{5 x+3}}{2401 (3 x+2)^{7/2}}+\frac{220 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^{7/2}}-\frac{65672 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{823543}+\frac{98642 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{823543} \]
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)^{9/2}} \, dx &=\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac{1}{21} \int \frac{\left (-\frac{345}{2}-315 x\right ) \sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx\\ &=\frac{220 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{7/2}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac{1}{147} \int \frac{-\frac{34305}{2}-\frac{58275 x}{2}}{\sqrt{1-2 x} (2+3 x)^{9/2} \sqrt{3+5 x}} \, dx\\ &=\frac{220 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{4545 \sqrt{1-2 x} \sqrt{3+5 x}}{2401 (2+3 x)^{7/2}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac{2 \int \frac{-\frac{397335}{4}-\frac{340875 x}{2}}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx}{7203}\\ &=\frac{220 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{4545 \sqrt{1-2 x} \sqrt{3+5 x}}{2401 (2+3 x)^{7/2}}-\frac{11433 \sqrt{1-2 x} \sqrt{3+5 x}}{16807 (2+3 x)^{5/2}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac{4 \int \frac{-\frac{1461615}{4}-\frac{2572425 x}{4}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{252105}\\ &=\frac{220 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{4545 \sqrt{1-2 x} \sqrt{3+5 x}}{2401 (2+3 x)^{7/2}}-\frac{11433 \sqrt{1-2 x} \sqrt{3+5 x}}{16807 (2+3 x)^{5/2}}-\frac{33778 \sqrt{1-2 x} \sqrt{3+5 x}}{117649 (2+3 x)^{3/2}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac{8 \int \frac{-\frac{4326885}{8}-\frac{3800025 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{5294205}\\ &=\frac{220 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{4545 \sqrt{1-2 x} \sqrt{3+5 x}}{2401 (2+3 x)^{7/2}}-\frac{11433 \sqrt{1-2 x} \sqrt{3+5 x}}{16807 (2+3 x)^{5/2}}-\frac{33778 \sqrt{1-2 x} \sqrt{3+5 x}}{117649 (2+3 x)^{3/2}}-\frac{98642 \sqrt{1-2 x} \sqrt{3+5 x}}{823543 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac{16 \int \frac{-\frac{1468575}{8}+\frac{11097225 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{37059435}\\ &=\frac{220 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{4545 \sqrt{1-2 x} \sqrt{3+5 x}}{2401 (2+3 x)^{7/2}}-\frac{11433 \sqrt{1-2 x} \sqrt{3+5 x}}{16807 (2+3 x)^{5/2}}-\frac{33778 \sqrt{1-2 x} \sqrt{3+5 x}}{117649 (2+3 x)^{3/2}}-\frac{98642 \sqrt{1-2 x} \sqrt{3+5 x}}{823543 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac{98642 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{823543}+\frac{361196 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{823543}\\ &=\frac{220 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{4545 \sqrt{1-2 x} \sqrt{3+5 x}}{2401 (2+3 x)^{7/2}}-\frac{11433 \sqrt{1-2 x} \sqrt{3+5 x}}{16807 (2+3 x)^{5/2}}-\frac{33778 \sqrt{1-2 x} \sqrt{3+5 x}}{117649 (2+3 x)^{3/2}}-\frac{98642 \sqrt{1-2 x} \sqrt{3+5 x}}{823543 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}+\frac{98642 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{823543}-\frac{65672 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{823543}\\ \end{align*}
Mathematica [A] time = 0.228198, size = 113, normalized size = 0.45 \[ \frac{2 \left (\sqrt{2} \left (591115 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-49321 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{\sqrt{5 x+3} \left (-15980004 x^5-28748088 x^4-7681599 x^3+10746933 x^2+6524789 x+866085\right )}{(1-2 x)^{3/2} (3 x+2)^{7/2}}\right )}{2470629} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.028, size = 501, normalized size = 2. \begin{align*} -{\frac{2}{2470629\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 31920210\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}-2663334\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}+47880315\,\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}-3995001\,\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}+10640070\,\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}-887778\,\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}-11822300\,\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}+986420\,\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}+79900020\,{x}^{6}-4728920\,\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 ) +394568\,\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 ) +191680452\,{x}^{5}+124652259\,{x}^{4}-30689868\,{x}^{3}-64864744\,{x}^{2}-23904792\,x-2598255 \right ) \left ( 2+3\,x \right ) ^{-{\frac{7}{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{9}{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}}{1944 \, x^{8} + 3564 \, x^{7} + 378 \, x^{6} - 2583 \, x^{5} - 1050 \, x^{4} + 616 \, x^{3} + 336 \, x^{2} - 48 \, x - 32}, 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{9}{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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