Optimal. Leaf size=218 \[ -\frac{12996374 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{15946875 \sqrt{33}}+\frac{2}{55} (1-2 x)^{5/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{326 (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}}{7425}+\frac{30362 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{779625}-\frac{78797 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{3898125}-\frac{12996374 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{35083125}-\frac{829177897 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{31893750 \sqrt{33}} \]
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Rubi [A] time = 0.0811639, 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 = {101, 154, 158, 113, 119} \[ \frac{2}{55} (1-2 x)^{5/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{326 (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}}{7425}+\frac{30362 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{779625}-\frac{78797 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{3898125}-\frac{12996374 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{35083125}-\frac{12996374 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15946875 \sqrt{33}}-\frac{829177897 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{31893750 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{3/2} \, dx &=\frac{2}{55} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{2}{55} \int \frac{\left (-\frac{111}{2}-\frac{163 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{\sqrt{2+3 x}} \, dx\\ &=\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}}{7425}+\frac{2}{55} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{4 \int \frac{\left (-2809-\frac{15181 x}{4}\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{\sqrt{2+3 x}} \, dx}{7425}\\ &=\frac{30362 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{779625}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}}{7425}+\frac{2}{55} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{8 \int \frac{\left (-\frac{466273}{8}-\frac{236391 x}{8}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{779625}\\ &=-\frac{78797 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{3898125}+\frac{30362 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{779625}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}}{7425}+\frac{2}{55} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{8 \int \frac{\sqrt{3+5 x} \left (\frac{48347127}{16}+\frac{19494561 x}{4}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{11694375}\\ &=-\frac{12996374 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{35083125}-\frac{78797 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{3898125}+\frac{30362 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{779625}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}}{7425}+\frac{2}{55} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{8 \int \frac{-\frac{1578296283}{16}-\frac{2487533691 x}{16}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{105249375}\\ &=-\frac{12996374 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{35083125}-\frac{78797 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{3898125}+\frac{30362 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{779625}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}}{7425}+\frac{2}{55} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{6498187 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{15946875}+\frac{829177897 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{350831250}\\ &=-\frac{12996374 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{35083125}-\frac{78797 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{3898125}+\frac{30362 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{779625}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}}{7425}+\frac{2}{55} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{829177897 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{31893750 \sqrt{33}}-\frac{12996374 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15946875 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.255298, size = 107, normalized size = 0.49 \[ \frac{-400297555 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (127575000 x^4-51502500 x^3-95024250 x^2+48272535 x+22517617\right )+829177897 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{526246875 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.011, size = 160, normalized size = 0.7 \begin{align*}{\frac{1}{31574812500\,{x}^{3}+24207356250\,{x}^{2}-7367456250\,x-6314962500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 114817500000\,{x}^{7}+41674500000\,{x}^{6}+400297555\,\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 ) -829177897\,\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 ) -147849300000\,{x}^{5}-34269426000\,{x}^{4}+82799446950\,{x}^{3}+22504288380\,{x}^{2}-13417755870\,x-4053171060 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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