Optimal. Leaf size=138 \[ -\frac{3}{50} (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac{37}{160} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{407}{640} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{4477 \sqrt{5 x+3} (1-2 x)^{3/2}}{12800}+\frac{147741 \sqrt{5 x+3} \sqrt{1-2 x}}{128000}+\frac{1625151 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{128000 \sqrt{10}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0369935, 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} \[ -\frac{3}{50} (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac{37}{160} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{407}{640} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{4477 \sqrt{5 x+3} (1-2 x)^{3/2}}{12800}+\frac{147741 \sqrt{5 x+3} \sqrt{1-2 x}}{128000}+\frac{1625151 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{128000 \sqrt{10}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 80
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2} \, dx &=-\frac{3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac{37}{20} \int (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx\\ &=-\frac{37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac{1221}{320} \int (1-2 x)^{3/2} \sqrt{3+5 x} \, dx\\ &=-\frac{407}{640} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac{4477 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{1280}\\ &=\frac{4477 (1-2 x)^{3/2} \sqrt{3+5 x}}{12800}-\frac{407}{640} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac{147741 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{25600}\\ &=\frac{147741 \sqrt{1-2 x} \sqrt{3+5 x}}{128000}+\frac{4477 (1-2 x)^{3/2} \sqrt{3+5 x}}{12800}-\frac{407}{640} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac{1625151 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{256000}\\ &=\frac{147741 \sqrt{1-2 x} \sqrt{3+5 x}}{128000}+\frac{4477 (1-2 x)^{3/2} \sqrt{3+5 x}}{12800}-\frac{407}{640} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac{1625151 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{128000 \sqrt{5}}\\ &=\frac{147741 \sqrt{1-2 x} \sqrt{3+5 x}}{128000}+\frac{4477 (1-2 x)^{3/2} \sqrt{3+5 x}}{12800}-\frac{407}{640} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac{1625151 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{128000 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.042729, size = 70, normalized size = 0.51 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (768000 x^4+745600 x^3-364320 x^2-489340 x+46809\right )-1625151 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1280000} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 121, normalized size = 0.9 \begin{align*}{\frac{1}{2560000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -15360000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-14912000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+7286400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1625151\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +9786800\,x\sqrt{-10\,{x}^{2}-x+3}-936180\,\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 = 1.54407, size = 113, normalized size = 0.82 \begin{align*} -\frac{3}{50} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{37}{80} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{37}{1600} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{13431}{6400} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1625151}{2560000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{13431}{128000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.50964, size = 275, normalized size = 1.99 \begin{align*} -\frac{1}{128000} \,{\left (768000 \, x^{4} + 745600 \, x^{3} - 364320 \, x^{2} - 489340 \, x + 46809\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{1625151}{2560000} \, \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 [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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.25272, size = 317, normalized size = 2.3 \begin{align*} -\frac{1}{6400000} \, \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{23}{1920000} \, \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{7}{24000} \, \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{3}{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]