Optimal. Leaf size=84 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^2}{11 \sqrt{1-2 x}}+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} (10380 x+25003)}{8800}-\frac{56421 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{800 \sqrt{10}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0201763, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {98, 147, 54, 216} \[ \frac{7 \sqrt{5 x+3} (3 x+2)^2}{11 \sqrt{1-2 x}}+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} (10380 x+25003)}{8800}-\frac{56421 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{800 \sqrt{10}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 98
Rule 147
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(2+3 x)^3}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx &=\frac{7 (2+3 x)^2 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}-\frac{1}{11} \int \frac{(2+3 x) \left (159+\frac{519 x}{2}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{7 (2+3 x)^2 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{3 \sqrt{1-2 x} \sqrt{3+5 x} (25003+10380 x)}{8800}-\frac{56421 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1600}\\ &=\frac{7 (2+3 x)^2 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{3 \sqrt{1-2 x} \sqrt{3+5 x} (25003+10380 x)}{8800}-\frac{56421 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{800 \sqrt{5}}\\ &=\frac{7 (2+3 x)^2 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{3 \sqrt{1-2 x} \sqrt{3+5 x} (25003+10380 x)}{8800}-\frac{56421 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{800 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0276325, size = 64, normalized size = 0.76 \[ \frac{620631 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (11880 x^2+51678 x-97409\right )}{88000 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 106, normalized size = 1.3 \begin{align*} -{\frac{1}{352000\,x-176000} \left ( 1241262\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-237600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-620631\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -1033560\,x\sqrt{-10\,{x}^{2}-x+3}+1948180\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\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 = 2.07783, size = 88, normalized size = 1.05 \begin{align*} -\frac{56421}{16000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{27}{40} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{2619}{800} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{343 \, \sqrt{-10 \, x^{2} - x + 3}}{44 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.76261, size = 258, normalized size = 3.07 \begin{align*} \frac{620631 \, \sqrt{10}{\left (2 \, x - 1\right )} \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 ) + 20 \,{\left (11880 \, x^{2} + 51678 \, x - 97409\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{176000 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (3 x + 2\right )^{3}}{\left (1 - 2 x\right )^{\frac{3}{2}} \sqrt{5 x + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.20922, size = 96, normalized size = 1.14 \begin{align*} -\frac{56421}{8000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (594 \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} + 63 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 620695 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{220000 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]