Optimal. Leaf size=191 \[ -\frac{7388 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{46305}+\frac{74 \sqrt{1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^{5/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}+\frac{119732 \sqrt{1-2 x} \sqrt{5 x+3}}{46305 \sqrt{3 x+2}}-\frac{3632 \sqrt{1-2 x} \sqrt{5 x+3}}{6615 (3 x+2)^{3/2}}-\frac{119732 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0665454, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {97, 150, 152, 158, 113, 119} \[ \frac{74 \sqrt{1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^{5/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}+\frac{119732 \sqrt{1-2 x} \sqrt{5 x+3}}{46305 \sqrt{3 x+2}}-\frac{3632 \sqrt{1-2 x} \sqrt{5 x+3}}{6615 (3 x+2)^{3/2}}-\frac{7388 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305}-\frac{119732 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 97
Rule 150
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{9/2}} \, dx &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac{2}{21} \int \frac{\left (-\frac{3}{2}-30 x\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^{7/2}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac{74 \sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}-\frac{4}{315} \int \frac{\sqrt{3+5 x} \left (-369+\frac{255 x}{2}\right )}{\sqrt{1-2 x} (2+3 x)^{5/2}} \, dx\\ &=-\frac{3632 \sqrt{1-2 x} \sqrt{3+5 x}}{6615 (2+3 x)^{3/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac{74 \sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}-\frac{8 \int \frac{-\frac{30243}{4}-\frac{465 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{19845}\\ &=-\frac{3632 \sqrt{1-2 x} \sqrt{3+5 x}}{6615 (2+3 x)^{3/2}}+\frac{119732 \sqrt{1-2 x} \sqrt{3+5 x}}{46305 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac{74 \sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}-\frac{16 \int \frac{-\frac{599745}{8}-\frac{448995 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{138915}\\ &=-\frac{3632 \sqrt{1-2 x} \sqrt{3+5 x}}{6615 (2+3 x)^{3/2}}+\frac{119732 \sqrt{1-2 x} \sqrt{3+5 x}}{46305 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac{74 \sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}+\frac{40634 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{46305}+\frac{119732 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{46305}\\ &=-\frac{3632 \sqrt{1-2 x} \sqrt{3+5 x}}{6615 (2+3 x)^{3/2}}+\frac{119732 \sqrt{1-2 x} \sqrt{3+5 x}}{46305 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac{74 \sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}-\frac{119732 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305}-\frac{7388 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305}\\ \end{align*}
Mathematica [A] time = 0.157817, size = 104, normalized size = 0.54 \[ \frac{2 \left (\sqrt{2} \left (1085 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+59866 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (1616382 x^3+3385161 x^2+2314860 x+519367\right )}{(3 x+2)^{7/2}}\right )}{138915} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.02, size = 409, normalized size = 2.1 \begin{align*} -{\frac{2}{1389150\,{x}^{2}+138915\,x-416745} \left ( 29295\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1616382\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+58590\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+3232764\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+39060\,\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}+2155176\,\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}+8680\,\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 ) +478928\,\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 ) -48491460\,{x}^{5}-106403976\,{x}^{4}-65053845\,{x}^{3}+7940859\,{x}^{2}+19275639\,x+4674303 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{7}{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{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{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 (10 \, x^{2} + x - 3\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32}, 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{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]