Optimal. Leaf size=191 \[ \frac{249448 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{138915}+\frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{315 (3 x+2)^{5/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}+\frac{2108 \sqrt{1-2 x} (5 x+3)^{3/2}}{6615 (3 x+2)^{3/2}}+\frac{249448 \sqrt{1-2 x} \sqrt{5 x+3}}{138915 \sqrt{3 x+2}}-\frac{962678 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{138915} \]
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Rubi [A] time = 0.0650683, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {97, 150, 158, 113, 119} \[ \frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{315 (3 x+2)^{5/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}+\frac{2108 \sqrt{1-2 x} (5 x+3)^{3/2}}{6615 (3 x+2)^{3/2}}+\frac{249448 \sqrt{1-2 x} \sqrt{5 x+3}}{138915 \sqrt{3 x+2}}+\frac{249448 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{138915}-\frac{962678 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{138915} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac{2}{21} \int \frac{\left (\frac{7}{2}-40 x\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{315 (2+3 x)^{5/2}}-\frac{4}{315} \int \frac{\left (-\frac{343}{2}-\frac{1305 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^{5/2}} \, dx\\ &=\frac{2108 \sqrt{1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{3/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{315 (2+3 x)^{5/2}}-\frac{8 \int \frac{\left (-9882-\frac{152835 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{3/2}} \, dx}{19845}\\ &=\frac{249448 \sqrt{1-2 x} \sqrt{3+5 x}}{138915 \sqrt{2+3 x}}+\frac{2108 \sqrt{1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{3/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{315 (2+3 x)^{5/2}}-\frac{16 \int \frac{-\frac{2274105}{8}-\frac{7220085 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{416745}\\ &=\frac{249448 \sqrt{1-2 x} \sqrt{3+5 x}}{138915 \sqrt{2+3 x}}+\frac{2108 \sqrt{1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{3/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{315 (2+3 x)^{5/2}}+\frac{962678 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{138915}-\frac{1371964 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{138915}\\ &=\frac{249448 \sqrt{1-2 x} \sqrt{3+5 x}}{138915 \sqrt{2+3 x}}+\frac{2108 \sqrt{1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{3/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{315 (2+3 x)^{5/2}}-\frac{962678 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{138915}+\frac{249448 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{138915}\\ \end{align*}
Mathematica [A] time = 0.247343, size = 104, normalized size = 0.54 \[ \frac{2 \left (\sqrt{2} \left (481339 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2539285 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (10680903 x^3+20067219 x^2+12594615 x+2640643\right )}{(3 x+2)^{7/2}}\right )}{416745} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 409, normalized size = 2.1 \begin{align*}{\frac{2}{4167450\,{x}^{2}+416745\,x-1250235} \left ( 68560695\,\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}-12996153\,\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}+137121390\,\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}-25992306\,\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}+91414260\,\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}-17328204\,\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}+20314280\,\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 ) -3850712\,\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 ) +320427090\,{x}^{5}+634059279\,{x}^{4}+341911980\,{x}^{3}-63601836\,{x}^{2}-105429606\,x-23765787 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{7}{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{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{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 (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, 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 (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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