Optimal. Leaf size=222 \[ -\frac{16732 \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{84035}-\frac{3946 \sqrt{1-2 x} \sqrt{5 x+3}}{84035 \sqrt{3 x+2}}-\frac{2264 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 (3 x+2)^{3/2}}-\frac{779 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 (3 x+2)^{5/2}}+\frac{124 \sqrt{5 x+3}}{147 \sqrt{1-2 x} (3 x+2)^{5/2}}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{5/2}}+\frac{3946 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035} \]
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Rubi [A] time = 0.0824418, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {98, 152, 158, 113, 119} \[ -\frac{3946 \sqrt{1-2 x} \sqrt{5 x+3}}{84035 \sqrt{3 x+2}}-\frac{2264 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 (3 x+2)^{3/2}}-\frac{779 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 (3 x+2)^{5/2}}+\frac{124 \sqrt{5 x+3}}{147 \sqrt{1-2 x} (3 x+2)^{5/2}}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac{16732 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035}+\frac{3946 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035} \]
Antiderivative was successfully verified.
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Rule 98
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{7/2}} \, dx &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac{1}{21} \int \frac{-\frac{367}{2}-315 x}{(1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\\ &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{124 \sqrt{3+5 x}}{147 \sqrt{1-2 x} (2+3 x)^{5/2}}+\frac{2 \int \frac{\frac{59631}{4}+25575 x}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx}{1617}\\ &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{124 \sqrt{3+5 x}}{147 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{779 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{5/2}}+\frac{4 \int \frac{\frac{109857}{2}+\frac{385605 x}{4}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{56595}\\ &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{124 \sqrt{3+5 x}}{147 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{779 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{5/2}}-\frac{2264 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 (2+3 x)^{3/2}}+\frac{8 \int \frac{\frac{682011}{8}+140085 x}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{1188495}\\ &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{124 \sqrt{3+5 x}}{147 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{779 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{5/2}}-\frac{2264 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 (2+3 x)^{3/2}}-\frac{3946 \sqrt{1-2 x} \sqrt{3+5 x}}{84035 \sqrt{2+3 x}}+\frac{16 \int \frac{\frac{328185}{4}-\frac{976635 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{8319465}\\ &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{124 \sqrt{3+5 x}}{147 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{779 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{5/2}}-\frac{2264 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 (2+3 x)^{3/2}}-\frac{3946 \sqrt{1-2 x} \sqrt{3+5 x}}{84035 \sqrt{2+3 x}}-\frac{3946 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{84035}+\frac{25098 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{84035}\\ &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{124 \sqrt{3+5 x}}{147 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{779 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{5/2}}-\frac{2264 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 (2+3 x)^{3/2}}-\frac{3946 \sqrt{1-2 x} \sqrt{3+5 x}}{84035 \sqrt{2+3 x}}+\frac{3946 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035}-\frac{16732 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035}\\ \end{align*}
Mathematica [A] time = 0.200248, size = 108, normalized size = 0.49 \[ \frac{2 \left (\sqrt{2} \left (39620 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-1973 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{\sqrt{5 x+3} \left (-213084 x^4-356292 x^3+2199 x^2+158902 x+43881\right )}{(1-2 x)^{3/2} (3 x+2)^{5/2}}\right )}{252105} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 406, normalized size = 1.8 \begin{align*}{\frac{2}{252105\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 35514\,\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}-713160\,\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}+29595\,\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}-594300\,\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}-7892\,\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}+158480\,\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}-7892\,\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 ) +158480\,\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 ) -1065420\,{x}^{5}-2420712\,{x}^{4}-1057881\,{x}^{3}+801107\,{x}^{2}+696111\,x+131643 \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{3}{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 (5 \, x + 3\right )}^{\frac{3}{2}} \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{3}{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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