Optimal. Leaf size=243 \[ \frac{5592499 \sqrt{\frac{11}{6}} \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )}{3888 \sqrt{2 x-5}}+\frac{2}{45} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^3-\frac{61}{270} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^2-\frac{8141 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)}{2700}-\frac{5256763 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{97200}-\frac{17746949 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{29160 \sqrt{5-2 x}} \]
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Rubi [A] time = 0.299877, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{2}{45} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^3-\frac{61}{270} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^2-\frac{8141 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)}{2700}-\frac{5256763 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{97200}+\frac{5592499 \sqrt{\frac{11}{6}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{3888 \sqrt{2 x-5}}-\frac{17746949 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{29160 \sqrt{5-2 x}} \]
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)^2 \, dx &=\frac{2}{45} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3+\frac{1}{45} \int \frac{(7+5 x)^2 \left (-3-1190 x+854 x^2\right )}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx\\ &=-\frac{61}{270} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^2+\frac{2}{45} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3-\frac{\int \frac{(7+5 x) \left (299866+1013390 x-1367688 x^2\right )}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{7560}\\ &=-\frac{8141 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)}{2700}-\frac{61}{270} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^2+\frac{2}{45} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3+\frac{\int \frac{-589706376-121654680 x+1766272368 x^2}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{907200}\\ &=-\frac{5256763 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{97200}-\frac{8141 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)}{2700}-\frac{61}{270} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^2+\frac{2}{45} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3+\frac{\int \frac{-119325868200+357778491840 x}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{97977600}\\ &=-\frac{5256763 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{97200}-\frac{8141 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)}{2700}-\frac{61}{270} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^2+\frac{2}{45} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3+\frac{17746949 \int \frac{\sqrt{-5+2 x}}{\sqrt{2-3 x} \sqrt{1+4 x}} \, dx}{9720}+\frac{61517489 \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{7776}\\ &=-\frac{5256763 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{97200}-\frac{8141 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)}{2700}-\frac{61}{270} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^2+\frac{2}{45} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3+\frac{\left (5592499 \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}{3888 \sqrt{-5+2 x}}+\frac{\left (17746949 \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}{9720 \sqrt{5-2 x}}\\ &=-\frac{5256763 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{97200}-\frac{8141 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)}{2700}-\frac{61}{270} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^2+\frac{2}{45} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3-\frac{17746949 \sqrt{11} \sqrt{-5+2 x} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{29160 \sqrt{5-2 x}}+\frac{5592499 \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 )}{3888 \sqrt{-5+2 x}}\\ \end{align*}
Mathematica [A] time = 0.267728, size = 130, normalized size = 0.53 \[ \frac{27962495 \sqrt{66} \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )+6 \sqrt{2-3 x} \sqrt{4 x+1} \left (216000 x^4+147600 x^3-1649952 x^2-2933650 x+6902575\right )-35493898 \sqrt{66} \sqrt{5-2 x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{116640 \sqrt{2 x-5}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 155, normalized size = 0.6 \begin{align*}{\frac{1}{2799360\,{x}^{3}-8164800\,{x}^{2}+2449440\,x+1166400}\sqrt{2-3\,x}\sqrt{2\,x-5}\sqrt{4\,x+1} \left ( 15552000\,{x}^{6}+83887485\,\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 ) -70987796\,\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 ) +4147200\,{x}^{5}-125816544\,{x}^{4}-163495440\,{x}^{3}+604794324\,{x}^{2}-171873450\,x-82830900 \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 )}^{2} \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 (25 \, x^{2} + 70 \, x + 49\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 )}^{2} \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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