Optimal. Leaf size=189 \[ \frac{152 \sqrt{\frac{2}{33}} \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )}{125 \sqrt{2 x-5}}-\frac{\sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{5 (5 x+7)}+\frac{6 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{25 \sqrt{5-2 x}}+\frac{26859 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{7750 \sqrt{11} \sqrt{2 x-5}} \]
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Rubi [A] time = 0.21081, 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 = {160, 1607, 168, 538, 537, 158, 114, 113, 121, 119} \[ -\frac{\sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{5 (5 x+7)}+\frac{152 \sqrt{\frac{2}{33}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{125 \sqrt{2 x-5}}+\frac{6 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{25 \sqrt{5-2 x}}+\frac{26859 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{7750 \sqrt{11} \sqrt{2 x-5}} \]
Antiderivative was successfully verified.
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Rule 160
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{-5+2 x} \sqrt{1+4 x}}{(7+5 x)^2} \, dx &=-\frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{5 (7+5 x)}+\frac{1}{10} \int \frac{-21+140 x-72 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}}{5 (7+5 x)}+\frac{1}{10} \int \frac{\frac{1204}{25}-\frac{72 x}{5}}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx-\frac{8953}{250} \int \frac{1}{\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}}{5 (7+5 x)}-\frac{18}{25} \int \frac{\sqrt{-5+2 x}}{\sqrt{2-3 x} \sqrt{1+4 x}} \, dx+\frac{152}{125} \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx+\frac{8953}{125} \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}}{5 (7+5 x)}+\frac{\left (152 \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}{125 \sqrt{-5+2 x}}+\frac{\left (8953 \sqrt{\frac{3}{11}} \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 )}{125 \sqrt{-5+2 x}}-\frac{\left (18 \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}{25 \sqrt{5-2 x}}\\ &=-\frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{5 (7+5 x)}+\frac{6 \sqrt{11} \sqrt{-5+2 x} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{25 \sqrt{5-2 x}}+\frac{152 \sqrt{\frac{2}{33}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{1+4 x}\right )|\frac{1}{3}\right )}{125 \sqrt{-5+2 x}}+\frac{26859 \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 )}{7750 \sqrt{11} \sqrt{-5+2 x}}\\ \end{align*}
Mathematica [A] time = 0.751783, size = 132, normalized size = 0.7 \[ \frac{\sqrt{2 x-5} \left (\frac{3 \sqrt{11} \left (9424 \text{EllipticF}\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right ),-\frac{1}{2}\right )+20460 E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )+26859 \Pi \left (\frac{55}{124};-\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )\right )}{\sqrt{5-2 x}}-\frac{51150 \sqrt{2-3 x} \sqrt{4 x+1}}{5 x+7}\right )}{255750} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.025, size = 320, normalized size = 1.7 \begin{align*}{\frac{1}{ \left ( 2046000\,{x}^{3}-5967500\,{x}^{2}+1790250\,x+852500 \right ) \left ( 7+5\,x \right ) }\sqrt{2-3\,x}\sqrt{2\,x-5}\sqrt{4\,x+1} \left ( 47120\,\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+102300\,\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-134295\,\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+65968\,\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 ) +143220\,\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 ) -188013\,\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{2 \, x - 5} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{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 (\frac{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{25 \, x^{2} + 70 \, x + 49}, 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{2 x - 5} \sqrt{4 x + 1}}{\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{2 \, x - 5} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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