Optimal. Leaf size=187 \[ -\frac{663409 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{13750 \sqrt{33}}+\frac{7 \sqrt{5 x+3} (3 x+2)^{7/2}}{33 (1-2 x)^{3/2}}-\frac{910 \sqrt{5 x+3} (3 x+2)^{5/2}}{363 \sqrt{1-2 x}}-\frac{27271 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{6050}-\frac{317384 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{15125}-\frac{44109377 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{27500 \sqrt{33}} \]
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Rubi [A] time = 0.0655777, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{7/2}}{33 (1-2 x)^{3/2}}-\frac{910 \sqrt{5 x+3} (3 x+2)^{5/2}}{363 \sqrt{1-2 x}}-\frac{27271 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{6050}-\frac{317384 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{15125}-\frac{663409 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{13750 \sqrt{33}}-\frac{44109377 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{27500 \sqrt{33}} \]
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)^{9/2}}{(1-2 x)^{5/2} \sqrt{3+5 x}} \, dx &=\frac{7 (2+3 x)^{7/2} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{1}{33} \int \frac{(2+3 x)^{5/2} \left (\frac{499}{2}+411 x\right )}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{910 (2+3 x)^{5/2} \sqrt{3+5 x}}{363 \sqrt{1-2 x}}+\frac{7 (2+3 x)^{7/2} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{1}{363} \int \frac{\left (-24996-\frac{81813 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{27271 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{6050}-\frac{910 (2+3 x)^{5/2} \sqrt{3+5 x}}{363 \sqrt{1-2 x}}+\frac{7 (2+3 x)^{7/2} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}+\frac{\int \frac{\sqrt{2+3 x} \left (\frac{7044525}{4}+2856456 x\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{9075}\\ &=-\frac{317384 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{15125}-\frac{27271 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{6050}-\frac{910 (2+3 x)^{5/2} \sqrt{3+5 x}}{363 \sqrt{1-2 x}}+\frac{7 (2+3 x)^{7/2} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{\int \frac{-\frac{125663067}{2}-\frac{396984393 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{136125}\\ &=-\frac{317384 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{15125}-\frac{27271 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{6050}-\frac{910 (2+3 x)^{5/2} \sqrt{3+5 x}}{363 \sqrt{1-2 x}}+\frac{7 (2+3 x)^{7/2} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}+\frac{663409 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{27500}+\frac{44109377 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{302500}\\ &=-\frac{317384 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{15125}-\frac{27271 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{6050}-\frac{910 (2+3 x)^{5/2} \sqrt{3+5 x}}{363 \sqrt{1-2 x}}+\frac{7 (2+3 x)^{7/2} \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{44109377 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{27500 \sqrt{33}}-\frac{663409 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{13750 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.263319, size = 125, normalized size = 0.67 \[ -\frac{-22216880 \sqrt{2-4 x} (2 x-1) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+10 \sqrt{3 x+2} \sqrt{5 x+3} \left (294030 x^3+1528956 x^2-9445541 x+3478434\right )+44109377 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{907500 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.022, size = 238, normalized size = 1.3 \begin{align*}{\frac{1}{907500\, \left ( 2\,x-1 \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 44433760\,\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}-88218754\,\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}-22216880\,\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 ) +44109377\,\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 ) -44104500\,{x}^{5}-285209100\,{x}^{4}+1108687710\,{x}^{3}+1181150330\,{x}^{2}-94170000\,x-208706040 \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{9}{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 (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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{9}{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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