Optimal. Leaf size=191 \[ -\frac{11908 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{127575}+\frac{2}{27} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{46}{567} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{499 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{2835}-\frac{11908 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{25515}-\frac{886499 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{255150} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0673405, 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 = {101, 154, 158, 113, 119} \[ \frac{2}{27} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{46}{567} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{499 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{2835}-\frac{11908 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{25515}-\frac{11908 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{127575}-\frac{886499 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{255150} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{\sqrt{2+3 x}} \, dx &=\frac{2}{27} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{2}{27} \int \frac{\left (-29-\frac{115 x}{2}\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{\sqrt{2+3 x}} \, dx\\ &=\frac{46}{567} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{27} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{4 \int \frac{\left (-\frac{685}{4}-\frac{7485 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{2835}\\ &=-\frac{499 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{2835}+\frac{46}{567} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{27} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{4 \int \frac{\sqrt{3+5 x} \left (\frac{263745}{8}+44655 x\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{42525}\\ &=-\frac{11908 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{25515}-\frac{499 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{2835}+\frac{46}{567} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{27} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{4 \int \frac{-\frac{8371455}{8}-\frac{13297485 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{382725}\\ &=-\frac{11908 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{25515}-\frac{499 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{2835}+\frac{46}{567} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{27} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{65494 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{127575}+\frac{886499 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{255150}\\ &=-\frac{11908 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{25515}-\frac{499 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{2835}+\frac{46}{567} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{27} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{886499 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{255150}-\frac{11908 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{127575}\\ \end{align*}
Mathematica [A] time = 0.254763, size = 105, normalized size = 0.55 \[ \frac{886499 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (98707 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (94500 x^3+14400 x^2-62325 x-10259\right )\right )}{382725 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.012, size = 155, normalized size = 0.8 \begin{align*}{\frac{1}{22963500\,{x}^{3}+17605350\,{x}^{2}-5358150\,x-4592700}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -85050000\,{x}^{6}+493535\,\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 ) -886499\,\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 ) -78165000\,{x}^{5}+66001500\,{x}^{4}+72271350\,{x}^{3}-3417540\,{x}^{2}-13372890\,x-1846620 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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}}}{\sqrt{3 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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{-2 \, x + 1}}{\sqrt{3 \, x + 2}}, x\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 [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}}}{\sqrt{3 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]