Optimal. Leaf size=150 \[ -\frac{16 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{\sqrt{33}}+\frac{2660 \sqrt{1-2 x} \sqrt{3 x+2}}{33 \sqrt{5 x+3}}-\frac{40 \sqrt{1-2 x} \sqrt{3 x+2}}{3 (5 x+3)^{3/2}}+\frac{2 \sqrt{1-2 x}}{\sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{532 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0534975, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {99, 152, 158, 113, 119} \[ \frac{2660 \sqrt{1-2 x} \sqrt{3 x+2}}{33 \sqrt{5 x+3}}-\frac{40 \sqrt{1-2 x} \sqrt{3 x+2}}{3 (5 x+3)^{3/2}}+\frac{2 \sqrt{1-2 x}}{\sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{16 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}}-\frac{532 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 99
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x}}{(2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac{2 \sqrt{1-2 x}}{\sqrt{2+3 x} (3+5 x)^{3/2}}-2 \int \frac{-13+15 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{\sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{40 \sqrt{1-2 x} \sqrt{2+3 x}}{3 (3+5 x)^{3/2}}+\frac{4}{33} \int \frac{-\frac{1067}{2}+330 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{\sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{40 \sqrt{1-2 x} \sqrt{2+3 x}}{3 (3+5 x)^{3/2}}+\frac{2660 \sqrt{1-2 x} \sqrt{2+3 x}}{33 \sqrt{3+5 x}}-\frac{8}{363} \int \frac{-\frac{13893}{2}-\frac{21945 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{\sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{40 \sqrt{1-2 x} \sqrt{2+3 x}}{3 (3+5 x)^{3/2}}+\frac{2660 \sqrt{1-2 x} \sqrt{2+3 x}}{33 \sqrt{3+5 x}}+8 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{532}{11} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{\sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{40 \sqrt{1-2 x} \sqrt{2+3 x}}{3 (3+5 x)^{3/2}}+\frac{2660 \sqrt{1-2 x} \sqrt{2+3 x}}{33 \sqrt{3+5 x}}-\frac{532 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}}-\frac{16 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.202085, size = 99, normalized size = 0.66 \[ \frac{2}{33} \left (2 \sqrt{2} \left (133 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-67 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{\sqrt{1-2 x} \left (19950 x^2+24610 x+7573\right )}{\sqrt{3 x+2} (5 x+3)^{3/2}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.021, size = 219, normalized size = 1.5 \begin{align*}{\frac{2}{198\,{x}^{2}+33\,x-66}\sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 670\,\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}-1330\,\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}+402\,\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 ) -798\,\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 ) +39900\,{x}^{3}+29270\,{x}^{2}-9464\,x-7573 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \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{\sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{1125 \, x^{5} + 3525 \, x^{4} + 4415 \, x^{3} + 2763 \, x^{2} + 864 \, x + 108}, 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{\sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]