Optimal. Leaf size=218 \[ -\frac{837304 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{924385 \sqrt{33}}+\frac{26062156 \sqrt{1-2 x} \sqrt{5 x+3}}{10168235 \sqrt{3 x+2}}+\frac{349904 \sqrt{1-2 x} \sqrt{5 x+3}}{1452605 (3 x+2)^{3/2}}-\frac{806 \sqrt{1-2 x} \sqrt{5 x+3}}{207515 (3 x+2)^{5/2}}+\frac{1336 \sqrt{5 x+3}}{17787 \sqrt{1-2 x} (3 x+2)^{5/2}}+\frac{4 \sqrt{5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac{26062156 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{924385 \sqrt{33}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0850025, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {104, 152, 158, 113, 119} \[ \frac{26062156 \sqrt{1-2 x} \sqrt{5 x+3}}{10168235 \sqrt{3 x+2}}+\frac{349904 \sqrt{1-2 x} \sqrt{5 x+3}}{1452605 (3 x+2)^{3/2}}-\frac{806 \sqrt{1-2 x} \sqrt{5 x+3}}{207515 (3 x+2)^{5/2}}+\frac{1336 \sqrt{5 x+3}}{17787 \sqrt{1-2 x} (3 x+2)^{5/2}}+\frac{4 \sqrt{5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac{837304 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{924385 \sqrt{33}}-\frac{26062156 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{924385 \sqrt{33}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 104
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{5/2} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx &=\frac{4 \sqrt{3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac{2}{231} \int \frac{-\frac{229}{2}-105 x}{(1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\\ &=\frac{4 \sqrt{3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{1336 \sqrt{3+5 x}}{17787 \sqrt{1-2 x} (2+3 x)^{5/2}}+\frac{4 \int \frac{\frac{32997}{4}+12525 x}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx}{17787}\\ &=\frac{4 \sqrt{3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{1336 \sqrt{3+5 x}}{17787 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{806 \sqrt{1-2 x} \sqrt{3+5 x}}{207515 (2+3 x)^{5/2}}+\frac{8 \int \frac{\frac{137259}{2}+\frac{18135 x}{4}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{622545}\\ &=\frac{4 \sqrt{3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{1336 \sqrt{3+5 x}}{17787 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{806 \sqrt{1-2 x} \sqrt{3+5 x}}{207515 (2+3 x)^{5/2}}+\frac{349904 \sqrt{1-2 x} \sqrt{3+5 x}}{1452605 (2+3 x)^{3/2}}+\frac{16 \int \frac{\frac{14298057}{8}-984105 x}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{13073445}\\ &=\frac{4 \sqrt{3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{1336 \sqrt{3+5 x}}{17787 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{806 \sqrt{1-2 x} \sqrt{3+5 x}}{207515 (2+3 x)^{5/2}}+\frac{349904 \sqrt{1-2 x} \sqrt{3+5 x}}{1452605 (2+3 x)^{3/2}}+\frac{26062156 \sqrt{1-2 x} \sqrt{3+5 x}}{10168235 \sqrt{2+3 x}}+\frac{32 \int \frac{\frac{93140595}{4}+\frac{293199255 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{91514115}\\ &=\frac{4 \sqrt{3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{1336 \sqrt{3+5 x}}{17787 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{806 \sqrt{1-2 x} \sqrt{3+5 x}}{207515 (2+3 x)^{5/2}}+\frac{349904 \sqrt{1-2 x} \sqrt{3+5 x}}{1452605 (2+3 x)^{3/2}}+\frac{26062156 \sqrt{1-2 x} \sqrt{3+5 x}}{10168235 \sqrt{2+3 x}}+\frac{418652 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{924385}+\frac{26062156 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{10168235}\\ &=\frac{4 \sqrt{3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{1336 \sqrt{3+5 x}}{17787 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{806 \sqrt{1-2 x} \sqrt{3+5 x}}{207515 (2+3 x)^{5/2}}+\frac{349904 \sqrt{1-2 x} \sqrt{3+5 x}}{1452605 (2+3 x)^{3/2}}+\frac{26062156 \sqrt{1-2 x} \sqrt{3+5 x}}{10168235 \sqrt{2+3 x}}-\frac{26062156 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{924385 \sqrt{33}}-\frac{837304 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{924385 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.285535, size = 107, normalized size = 0.49 \[ \frac{-24493280 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{2 \sqrt{10 x+6} \left (1407356424 x^4+513206712 x^3-914077314 x^2-176797172 x+165071409\right )}{(1-2 x)^{3/2} (3 x+2)^{5/2}}+52124312 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{30504705 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.027, size = 406, normalized size = 1.9 \begin{align*}{\frac{2}{30504705\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 110219760\,\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}-234559404\,\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}+91849800\,\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}-195466170\,\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}-24493280\,\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}+52124312\,\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}-24493280\,\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 ) +52124312\,\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 ) +7036782120\,{x}^{5}+6788102832\,{x}^{4}-3030766434\,{x}^{3}-3626217802\,{x}^{2}+294965529\,x+495214227 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \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{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{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}}{3240 \, x^{8} + 5724 \, x^{7} + 378 \, x^{6} - 4179 \, x^{5} - 1547 \, x^{4} + 1008 \, x^{3} + 504 \, x^{2} - 80 \, x - 48}, 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{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]