Optimal. Leaf size=281 \[ \frac{522167393 \sqrt{\frac{11}{6}} \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )}{23328 \sqrt{2 x-5}}+\frac{2}{55} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^4-\frac{427 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^3}{2970}-\frac{17561 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^2}{8910}-\frac{12243139 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)}{356400}-\frac{1182926269 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{1603800}-\frac{6489123157 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{699840 \sqrt{5-2 x}} \]
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Rubi [A] time = 0.389425, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {161, 1600, 1615, 158, 114, 113, 121, 119} \[ \frac{2}{55} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^4-\frac{427 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^3}{2970}-\frac{17561 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^2}{8910}-\frac{12243139 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)}{356400}-\frac{1182926269 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{1603800}+\frac{522167393 \sqrt{\frac{11}{6}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{23328 \sqrt{2 x-5}}-\frac{6489123157 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{699840 \sqrt{5-2 x}} \]
Antiderivative was successfully verified.
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Rule 161
Rule 1600
Rule 1615
Rule 158
Rule 114
Rule 113
Rule 121
Rule 119
Rubi steps
\begin{align*} \int \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3 \, dx &=\frac{2}{55} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^4+\frac{1}{55} \int \frac{(7+5 x)^3 \left (-3-1190 x+854 x^2\right )}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx\\ &=-\frac{427 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3}{2970}+\frac{2}{55} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^4-\frac{\int \frac{(7+5 x)^2 \left (386274+1593290 x-1966832 x^2\right )}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{11880}\\ &=-\frac{17561 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^2}{8910}-\frac{427 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3}{2970}+\frac{2}{55} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^4+\frac{\int \frac{(7+5 x) \left (-1136748928-1303270640 x+4113694704 x^2\right )}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{1995840}\\ &=-\frac{12243139 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)}{356400}-\frac{17561 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^2}{8910}-\frac{427 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3}{2970}+\frac{2}{55} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^4-\frac{\int \frac{1970951691408-958810283760 x-6359411622144 x^2}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{239500800}\\ &=-\frac{1182926269 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{1603800}-\frac{12243139 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)}{356400}-\frac{17561 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^2}{8910}-\frac{427 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3}{2970}+\frac{2}{55} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^4-\frac{\int \frac{413184248769600-1439027951296320 x}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{25866086400}\\ &=-\frac{1182926269 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{1603800}-\frac{12243139 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)}{356400}-\frac{17561 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^2}{8910}-\frac{427 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3}{2970}+\frac{2}{55} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^4+\frac{6489123157 \int \frac{\sqrt{-5+2 x}}{\sqrt{2-3 x} \sqrt{1+4 x}} \, dx}{233280}+\frac{5743841323 \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{46656}\\ &=-\frac{1182926269 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{1603800}-\frac{12243139 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)}{356400}-\frac{17561 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^2}{8910}-\frac{427 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3}{2970}+\frac{2}{55} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^4+\frac{\left (522167393 \sqrt{\frac{11}{2}} \sqrt{5-2 x}\right ) \int \frac{1}{\sqrt{2-3 x} \sqrt{\frac{10}{11}-\frac{4 x}{11}} \sqrt{1+4 x}} \, dx}{23328 \sqrt{-5+2 x}}+\frac{\left (6489123157 \sqrt{-5+2 x}\right ) \int \frac{\sqrt{\frac{15}{11}-\frac{6 x}{11}}}{\sqrt{2-3 x} \sqrt{\frac{3}{11}+\frac{12 x}{11}}} \, dx}{233280 \sqrt{5-2 x}}\\ &=-\frac{1182926269 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{1603800}-\frac{12243139 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)}{356400}-\frac{17561 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^2}{8910}-\frac{427 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3}{2970}+\frac{2}{55} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^4-\frac{6489123157 \sqrt{11} \sqrt{-5+2 x} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{699840 \sqrt{5-2 x}}+\frac{522167393 \sqrt{\frac{11}{6}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{1+4 x}\right )|\frac{1}{3}\right )}{23328 \sqrt{-5+2 x}}\\ \end{align*}
Mathematica [A] time = 0.416079, size = 135, normalized size = 0.48 \[ \frac{57438413230 \sqrt{66} \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )+24 \sqrt{2-3 x} \sqrt{4 x+1} \left (29160000 x^5+67338000 x^4-167736600 x^3-670058262 x^2-797747975 x+3325071575\right )-71380354727 \sqrt{66} \sqrt{5-2 x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{15396480 \sqrt{2 x-5}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 160, normalized size = 0.6 \begin{align*}{\frac{1}{184757760\,{x}^{3}-538876800\,{x}^{2}+161663040\,x+76982400}\sqrt{2-3\,x}\sqrt{2\,x-5}\sqrt{4\,x+1} \left ( 4199040000\,{x}^{7}+7947072000\,{x}^{6}+86157619845\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticF} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ) -71380354727\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticE} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ) -28894190400\,{x}^{5}-88040305728\,{x}^{4}-70646534280\,{x}^{3}+542756583588\,{x}^{2}-180358343100\,x-79801717800 \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 + 7\right )}^{3} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 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 (125 \, x^{3} + 525 \, x^{2} + 735 \, x + 343\right )} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}, 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 + 7\right )}^{3} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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