Optimal. Leaf size=160 \[ -\frac{164}{945} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )-\frac{2 \sqrt{5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^{5/2}}+\frac{3896 \sqrt{5 x+3} \sqrt{1-2 x}}{945 \sqrt{3 x+2}}+\frac{82 \sqrt{5 x+3} \sqrt{1-2 x}}{135 (3 x+2)^{3/2}}-\frac{3896}{945} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0533367, antiderivative size = 160, 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 = {97, 150, 152, 158, 113, 119} \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^{5/2}}+\frac{3896 \sqrt{5 x+3} \sqrt{1-2 x}}{945 \sqrt{3 x+2}}+\frac{82 \sqrt{5 x+3} \sqrt{1-2 x}}{135 (3 x+2)^{3/2}}-\frac{164}{945} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{3896}{945} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^{7/2}} \, dx &=-\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^{5/2}}+\frac{2}{15} \int \frac{\left (-\frac{13}{2}-20 x\right ) \sqrt{1-2 x}}{(2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^{5/2}}+\frac{82 \sqrt{1-2 x} \sqrt{3+5 x}}{135 (2+3 x)^{3/2}}-\frac{4}{135} \int \frac{-134+\frac{85 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^{5/2}}+\frac{82 \sqrt{1-2 x} \sqrt{3+5 x}}{135 (2+3 x)^{3/2}}+\frac{3896 \sqrt{1-2 x} \sqrt{3+5 x}}{945 \sqrt{2+3 x}}-\frac{8}{945} \int \frac{-\frac{6295}{4}-2435 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^{5/2}}+\frac{82 \sqrt{1-2 x} \sqrt{3+5 x}}{135 (2+3 x)^{3/2}}+\frac{3896 \sqrt{1-2 x} \sqrt{3+5 x}}{945 \sqrt{2+3 x}}+\frac{902}{945} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{3896}{945} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^{5/2}}+\frac{82 \sqrt{1-2 x} \sqrt{3+5 x}}{135 (2+3 x)^{3/2}}+\frac{3896 \sqrt{1-2 x} \sqrt{3+5 x}}{945 \sqrt{2+3 x}}-\frac{3896}{945} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{164}{945} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.132987, size = 99, normalized size = 0.62 \[ \frac{2 \left (\sqrt{2} \left (1948 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-595 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (17532 x^2+24363 x+8303\right )}{(3 x+2)^{5/2}}\right )}{2835} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.018, size = 314, normalized size = 2. \begin{align*}{\frac{2}{28350\,{x}^{2}+2835\,x-8505} \left ( 5355\,\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}-17532\,\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}+7140\,\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}-23376\,\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}+2380\,\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 ) -7792\,\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 ) +525960\,{x}^{4}+783486\,{x}^{3}+164391\,{x}^{2}-194358\,x-74727 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \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 (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{7}{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}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, 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 (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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