Optimal. Leaf size=138 \[ -\frac{3}{50} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{119}{800} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{1309 \sqrt{5 x+3} (1-2 x)^{5/2}}{24000}+\frac{14399 \sqrt{5 x+3} (1-2 x)^{3/2}}{96000}+\frac{158389 \sqrt{5 x+3} \sqrt{1-2 x}}{320000}+\frac{1742279 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{320000 \sqrt{10}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0387869, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \[ -\frac{3}{50} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{119}{800} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{1309 \sqrt{5 x+3} (1-2 x)^{5/2}}{24000}+\frac{14399 \sqrt{5 x+3} (1-2 x)^{3/2}}{96000}+\frac{158389 \sqrt{5 x+3} \sqrt{1-2 x}}{320000}+\frac{1742279 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{320000 \sqrt{10}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 80
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int (1-2 x)^{5/2} (2+3 x) \sqrt{3+5 x} \, dx &=-\frac{3}{50} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac{119}{100} \int (1-2 x)^{5/2} \sqrt{3+5 x} \, dx\\ &=-\frac{119}{800} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{3}{50} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac{1309 \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx}{1600}\\ &=\frac{1309 (1-2 x)^{5/2} \sqrt{3+5 x}}{24000}-\frac{119}{800} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{3}{50} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac{14399 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{9600}\\ &=\frac{14399 (1-2 x)^{3/2} \sqrt{3+5 x}}{96000}+\frac{1309 (1-2 x)^{5/2} \sqrt{3+5 x}}{24000}-\frac{119}{800} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{3}{50} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac{158389 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{64000}\\ &=\frac{158389 \sqrt{1-2 x} \sqrt{3+5 x}}{320000}+\frac{14399 (1-2 x)^{3/2} \sqrt{3+5 x}}{96000}+\frac{1309 (1-2 x)^{5/2} \sqrt{3+5 x}}{24000}-\frac{119}{800} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{3}{50} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac{1742279 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{640000}\\ &=\frac{158389 \sqrt{1-2 x} \sqrt{3+5 x}}{320000}+\frac{14399 (1-2 x)^{3/2} \sqrt{3+5 x}}{96000}+\frac{1309 (1-2 x)^{5/2} \sqrt{3+5 x}}{24000}-\frac{119}{800} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{3}{50} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac{1742279 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{320000 \sqrt{5}}\\ &=\frac{158389 \sqrt{1-2 x} \sqrt{3+5 x}}{320000}+\frac{14399 (1-2 x)^{3/2} \sqrt{3+5 x}}{96000}+\frac{1309 (1-2 x)^{5/2} \sqrt{3+5 x}}{24000}-\frac{119}{800} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{3}{50} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac{1742279 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{320000 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0433537, size = 70, normalized size = 0.51 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (2304000 x^4-931200 x^3-1849760 x^2+1108180 x+355917\right )-5226837 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{9600000} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 121, normalized size = 0.9 \begin{align*}{\frac{1}{19200000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 46080000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-18624000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-36995200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+5226837\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +22163600\,x\sqrt{-10\,{x}^{2}-x+3}+7118340\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.60654, size = 117, normalized size = 0.85 \begin{align*} -\frac{6}{25} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{121}{1000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{1303}{12000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{14399}{16000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1742279}{6400000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{14399}{320000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.51047, size = 279, normalized size = 2.02 \begin{align*} \frac{1}{960000} \,{\left (2304000 \, x^{4} - 931200 \, x^{3} - 1849760 \, x^{2} + 1108180 \, x + 355917\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{1742279}{6400000} \, \sqrt{10} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 86.546, size = 490, normalized size = 3.55 \begin{align*} \frac{242 \sqrt{5} \left (\begin{cases} \frac{121 \sqrt{2} \left (- \frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{121} + \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}\right )}{32} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{3125} + \frac{638 \sqrt{5} \left (\begin{cases} \frac{1331 \sqrt{2} \left (- \frac{\sqrt{2} \left (5 - 10 x\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} - \frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{1936} + \frac{\operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16}\right )}{8} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{3125} - \frac{256 \sqrt{5} \left (\begin{cases} \frac{14641 \sqrt{2} \left (- \frac{\sqrt{2} \left (5 - 10 x\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} - \frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{3872} - \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{128}\right )}{16} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{3125} + \frac{24 \sqrt{5} \left (\begin{cases} \frac{161051 \sqrt{2} \left (\frac{2 \sqrt{2} \left (5 - 10 x\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{805255} - \frac{\sqrt{2} \left (5 - 10 x\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} - \frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{7744} - \frac{3 \sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{3748096} + \frac{7 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{256}\right )}{32} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{3125} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 2.47171, size = 317, normalized size = 2.3 \begin{align*} \frac{1}{16000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{480000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{4800} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{200} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]