Integrand size = 37, antiderivative size = 330 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{7/2}} \, dx=-\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{25 (7+5 x)^{5/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2085525 (7+5 x)^{3/2}}+\frac {1426348 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2319687747 \sqrt {7+5 x}}-\frac {2852696 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{11598438735 \sqrt {-5+2 x}}+\frac {1426348 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{297395865 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {48884 \sqrt {\frac {11}{23}} \sqrt {7+5 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{9593415 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}} \]
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Time = 0.26 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {166, 1618, 1613, 1616, 12, 176, 429, 182, 435} \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{7/2}} \, dx=\frac {1426348 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{297395865 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {48884 \sqrt {\frac {11}{23}} \sqrt {5 x+7} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{9593415 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}-\frac {2852696 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{11598438735 \sqrt {2 x-5}}+\frac {1426348 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{2319687747 \sqrt {5 x+7}}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{2085525 (5 x+7)^{3/2}}-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{25 (5 x+7)^{5/2}} \]
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Rule 12
Rule 166
Rule 176
Rule 182
Rule 429
Rule 435
Rule 1613
Rule 1616
Rule 1618
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{25 (7+5 x)^{5/2}}+\frac {1}{25} \int \frac {-21+140 x-72 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{5/2}} \, dx \\ & = -\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{25 (7+5 x)^{5/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2085525 (7+5 x)^{3/2}}+\frac {\int \frac {-254100+327910 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx}{2085525} \\ & = -\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{25 (7+5 x)^{5/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2085525 (7+5 x)^{3/2}}+\frac {1426348 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2319687747 \sqrt {7+5 x}}+\frac {\int \frac {-762330250-648988340 x+855808800 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{57992193675} \\ & = -\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{25 (7+5 x)^{5/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2085525 (7+5 x)^{3/2}}+\frac {1426348 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2319687747 \sqrt {7+5 x}}-\frac {2852696 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{11598438735 \sqrt {-5+2 x}}-\frac {\int \frac {390064989600}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{13918126482000}-\frac {15689828 \int \frac {\sqrt {2-3 x}}{(-5+2 x)^{3/2} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{297395865} \\ & = -\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{25 (7+5 x)^{5/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2085525 (7+5 x)^{3/2}}+\frac {1426348 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2319687747 \sqrt {7+5 x}}-\frac {2852696 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{11598438735 \sqrt {-5+2 x}}-\frac {268862 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{9593415}+\frac {\left (1426348 \sqrt {\frac {11}{23}} \sqrt {2-3 x} \sqrt {-\frac {7+5 x}{-5+2 x}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {1-\frac {39 x^2}{23}}} \, dx,x,\frac {\sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )}{297395865 \sqrt {-\frac {2-3 x}{-5+2 x}} \sqrt {7+5 x}} \\ & = -\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{25 (7+5 x)^{5/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2085525 (7+5 x)^{3/2}}+\frac {1426348 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2319687747 \sqrt {7+5 x}}-\frac {2852696 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{11598438735 \sqrt {-5+2 x}}+\frac {1426348 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{297395865 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {\left (24442 \sqrt {\frac {22}{23}} \sqrt {-\frac {-5+2 x}{2-3 x}} \sqrt {7+5 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{2}} \sqrt {1+\frac {31 x^2}{23}}} \, dx,x,\frac {\sqrt {1+4 x}}{\sqrt {2-3 x}}\right )}{9593415 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{2-3 x}}} \\ & = -\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{25 (7+5 x)^{5/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2085525 (7+5 x)^{3/2}}+\frac {1426348 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2319687747 \sqrt {7+5 x}}-\frac {2852696 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{11598438735 \sqrt {-5+2 x}}+\frac {1426348 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{297395865 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {48884 \sqrt {\frac {11}{23}} \sqrt {7+5 x} F\left (\tan ^{-1}\left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{9593415 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 18.89 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.74 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{7/2}} \, dx=\frac {2 \left (\frac {15 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \left (59328580+498566971 x+89146750 x^2\right )}{(7+5 x)^{5/2}}+242 \sqrt {15} \left (\frac {8841 \sqrt {1+4 x} \sqrt {7+5 x} \sqrt {-75+30 x}}{\sqrt {2-3 x}}-\frac {8841 \sqrt {715} \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{\sqrt {\frac {5-2 x}{2-3 x}} \sqrt {1+4 x}}+\frac {506884 \sqrt {\frac {55}{13}} \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{3 \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {1+4 x}}-\frac {32705806 \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{3 \sqrt {715} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {1+4 x}}+\frac {3203187 i \sqrt {\frac {3}{3410}} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{-5+2 x}} \operatorname {EllipticPi}\left (-\frac {23}{55},i \text {arcsinh}\left (\frac {\sqrt {\frac {22}{23}} \sqrt {7+5 x}}{\sqrt {-5+2 x}}\right ),\frac {23}{62}\right )}{\sqrt {\frac {2-3 x}{5-2 x}} \sqrt {1+4 x}}-\frac {512187 \sqrt {\frac {30}{341}} \sqrt {2-3 x} \sqrt {\frac {-5+2 x}{1+4 x}} \operatorname {EllipticPi}\left (\frac {78}{55},\arcsin \left (\frac {\sqrt {\frac {22}{39}} \sqrt {7+5 x}}{\sqrt {1+4 x}}\right ),\frac {39}{62}\right )}{\sqrt {-5+2 x} \sqrt {\frac {-2+3 x}{1+4 x}}}\right )\right )}{173976581025} \]
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Time = 1.63 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.49
method | result | size |
elliptic | \(\frac {\sqrt {-\left (7+5 x \right ) \left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (-\frac {2 \sqrt {-120 x^{4}+182 x^{3}+385 x^{2}-197 x -70}}{3125 \left (x +\frac {7}{5}\right )^{3}}+\frac {17906 \sqrt {-120 x^{4}+182 x^{3}+385 x^{2}-197 x -70}}{52138125 \left (x +\frac {7}{5}\right )^{2}}+\frac {-\frac {11410784}{773229249} x^{3}+\frac {99844360}{2319687747} x^{2}-\frac {9984436}{773229249} x -\frac {14263480}{2319687747}}{\sqrt {\left (x +\frac {7}{5}\right ) \left (-120 x^{3}+350 x^{2}-105 x -50\right )}}-\frac {5544220 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{64503557180829 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}-\frac {1815352 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, \left (\frac {2 F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{3}-\frac {31 \Pi \left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )}{15}\right )}{24809060454165 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}+\frac {\frac {5705392 \left (x +\frac {7}{5}\right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}{773229249}-\frac {2852696 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, \left (\frac {181 F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{341}-\frac {117 E\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{62}+\frac {91 \Pi \left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )}{55}\right )}{31211398635885}}{\sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {7+5 x}}\) | \(493\) |
default | \(-\frac {2 \left (160464150 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, E\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right ) x^{4}-170482950 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right ) x^{4}+235347420 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, E\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right ) x^{3}-250041660 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right ) x^{3}-213239026 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x^{2} E\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )+226552898 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x^{2} F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-219657592 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x E\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )+233372216 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )+139782104 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, E\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-148509592 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-35725138792 x^{4}+520453786730 x^{3}-810971863359 x^{2}-627985607564 x -98039018690\right ) \sqrt {1+4 x}\, \sqrt {-5+2 x}\, \sqrt {2-3 x}}{266764090905 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right ) \left (7+5 x \right )^{\frac {3}{2}}}\) | \(913\) |
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\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{7/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{7/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{7/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{7/2}} \, dx=\int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}}{{\left (5\,x+7\right )}^{7/2}} \,d x \]
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