Optimal. Leaf size=189 \[ -\frac{2 \sqrt{\frac{6}{11}} \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )}{25 \sqrt{2 x-5}}+\frac{\sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{39 (5 x+7)}-\frac{2 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{195 \sqrt{5-2 x}}-\frac{6101 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{20150 \sqrt{11} \sqrt{2 x-5}} \]
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Rubi [A] time = 0.214234, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {164, 1607, 168, 538, 537, 158, 114, 113, 121, 119} \[ \frac{\sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{39 (5 x+7)}-\frac{2 \sqrt{\frac{6}{11}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{25 \sqrt{2 x-5}}-\frac{2 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{195 \sqrt{5-2 x}}-\frac{6101 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{20150 \sqrt{11} \sqrt{2 x-5}} \]
Antiderivative was successfully verified.
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Rule 164
Rule 1607
Rule 168
Rule 538
Rule 537
Rule 158
Rule 114
Rule 113
Rule 121
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{2-3 x} \sqrt{1+4 x}}{\sqrt{-5+2 x} (7+5 x)^2} \, dx &=\frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{39 (7+5 x)}-\frac{1}{78} \int \frac{-29+120 x-24 x^2}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)} \, dx\\ &=\frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{39 (7+5 x)}-\frac{1}{78} \int \frac{\frac{768}{25}-\frac{24 x}{5}}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx+\frac{6101 \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)} \, dx}{1950}\\ &=\frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{39 (7+5 x)}+\frac{2}{65} \int \frac{\sqrt{-5+2 x}}{\sqrt{2-3 x} \sqrt{1+4 x}} \, dx-\frac{6}{25} \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx-\frac{6101}{975} \operatorname{Subst}\left (\int \frac{1}{\left (31-5 x^2\right ) \sqrt{\frac{11}{3}-\frac{4 x^2}{3}} \sqrt{-\frac{11}{3}-\frac{2 x^2}{3}}} \, dx,x,\sqrt{2-3 x}\right )\\ &=\frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{39 (7+5 x)}-\frac{\left (6 \sqrt{\frac{2}{11}} \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}{25 \sqrt{-5+2 x}}-\frac{\left (6101 \sqrt{5-2 x}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (31-5 x^2\right ) \sqrt{\frac{11}{3}-\frac{4 x^2}{3}} \sqrt{1+\frac{2 x^2}{11}}} \, dx,x,\sqrt{2-3 x}\right )}{325 \sqrt{33} \sqrt{-5+2 x}}+\frac{\left (2 \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}{65 \sqrt{5-2 x}}\\ &=\frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{39 (7+5 x)}-\frac{2 \sqrt{11} \sqrt{-5+2 x} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{195 \sqrt{5-2 x}}-\frac{2 \sqrt{\frac{6}{11}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{1+4 x}\right )|\frac{1}{3}\right )}{25 \sqrt{-5+2 x}}-\frac{6101 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{20150 \sqrt{11} \sqrt{-5+2 x}}\\ \end{align*}
Mathematica [A] time = 0.623383, size = 132, normalized size = 0.7 \[ \frac{3 \sqrt{55-22 x} \left (14508 \text{EllipticF}\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right ),-\frac{1}{2}\right )+6820 E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )+18303 \Pi \left (\frac{55}{124};-\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )\right )+\frac{51150 \sqrt{2-3 x} \sqrt{4 x+1} (2 x-5)}{5 x+7}}{1994850 \sqrt{2 x-5}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 320, normalized size = 1.7 \begin{align*} -{\frac{1}{ \left ( 15958800\,{x}^{3}-46546500\,{x}^{2}+13963950\,x+6649500 \right ) \left ( 7+5\,x \right ) }\sqrt{2-3\,x}\sqrt{2\,x-5}\sqrt{4\,x+1} \left ( 72540\,\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 ) x+34100\,\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 ) x-91515\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticPi} \left ( 2/11\,\sqrt{22-33\,x},{\frac{55}{124}},i/2\sqrt{2} \right ) x+101556\,\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 ) +47740\,\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 ) -128121\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticPi} \left ( 2/11\,\sqrt{22-33\,x},{\frac{55}{124}},i/2\sqrt{2} \right ) -409200\,{x}^{3}+1193500\,{x}^{2}-358050\,x-170500 \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 \frac{\sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{2} \sqrt{2 \, x - 5}}\,{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 (\frac{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{50 \, x^{3} + 15 \, x^{2} - 252 \, x - 245}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 - 3 x} \sqrt{4 x + 1}}{\sqrt{2 x - 5} \left (5 x + 7\right )^{2}}\, dx \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 \frac{\sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{2} \sqrt{2 \, x - 5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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