Integrand size = 35, antiderivative size = 225 \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx=-\frac {5 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1794 (7+5 x)^2}-\frac {26825 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{33257172 (7+5 x)}+\frac {5365 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{16628586 \sqrt {5-2 x}}-\frac {13243 \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{1065935 \sqrt {66} \sqrt {-5+2 x}}-\frac {16369941 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{3436574440 \sqrt {11} \sqrt {-5+2 x}} \]
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Time = 0.20 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {183, 1618, 1621, 174, 552, 551, 164, 115, 114, 122, 120} \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx=-\frac {13243 \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{1065935 \sqrt {66} \sqrt {2 x-5}}+\frac {5365 \sqrt {11} \sqrt {2 x-5} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{16628586 \sqrt {5-2 x}}-\frac {16369941 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{3436574440 \sqrt {11} \sqrt {2 x-5}}-\frac {26825 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{33257172 (5 x+7)}-\frac {5 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{1794 (5 x+7)^2} \]
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Rule 114
Rule 115
Rule 120
Rule 122
Rule 164
Rule 174
Rule 183
Rule 551
Rule 552
Rule 1618
Rule 1621
Rubi steps \begin{align*} \text {integral}& = -\frac {5 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1794 (7+5 x)^2}-\frac {\int \frac {-1063+1372 x-120 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2} \, dx}{3588} \\ & = -\frac {5 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1794 (7+5 x)^2}-\frac {26825 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{33257172 (7+5 x)}-\frac {\int \frac {-7905051+4073184 x+1931400 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{199543032} \\ & = -\frac {5 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1794 (7+5 x)^2}-\frac {26825 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{33257172 (7+5 x)}-\frac {\int \frac {\frac {1369224}{5}+386280 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{199543032}+\frac {5456647 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{110857240} \\ & = -\frac {5 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1794 (7+5 x)^2}-\frac {26825 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{33257172 (7+5 x)}-\frac {5365 \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx}{5542862}-\frac {13243 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{2131870}-\frac {5456647 \text {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 )}{55428620} \\ & = -\frac {5 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1794 (7+5 x)^2}-\frac {26825 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{33257172 (7+5 x)}-\frac {\left (5456647 \sqrt {\frac {3}{11}} \sqrt {5-2 x}\right ) \text {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 )}{55428620 \sqrt {-5+2 x}}-\frac {\left (13243 \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}{1065935 \sqrt {22} \sqrt {-5+2 x}}-\frac {\left (5365 \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}{5542862 \sqrt {5-2 x}} \\ & = -\frac {5 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1794 (7+5 x)^2}-\frac {26825 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{33257172 (7+5 x)}+\frac {5365 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{16628586 \sqrt {5-2 x}}-\frac {13243 \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{1065935 \sqrt {66} \sqrt {-5+2 x}}-\frac {16369941 \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 )}{3436574440 \sqrt {11} \sqrt {-5+2 x}} \\ \end{align*}
Time = 5.47 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx=\frac {-17050 \sqrt {2-3 x} (-5+2 x) \sqrt {1+4 x} (56093+26825 x)-\sqrt {55-22 x} (7+5 x)^2 \left (36589300 E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )-64043148 \operatorname {EllipticF}\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )+49109823 \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )\right )}{113406956520 \sqrt {-5+2 x} (7+5 x)^2} \]
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Time = 1.66 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.21
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (-\frac {5 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{1794 \left (7+5 x \right )^{2}}-\frac {26825 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{33257172 \left (7+5 x \right )}-\frac {19017 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{1676715755 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {5365 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{335343151 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {5456647 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )}{77128924730 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(273\) |
risch | \(\frac {5 \left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (56093+26825 x \right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{33257172 \left (7+5 x \right )^{2} \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (-\frac {5365 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{1006029453 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {6339 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{1676715755 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {5456647 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \Pi \left (\frac {2 \sqrt {22-33 x}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )}{415825507240 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(311\) |
default | \(\frac {\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (254612300 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x^{2}-169668125 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x^{2}-818497050 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right ) x^{2}+712914440 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x -475070750 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x -2291791740 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right ) x +499040108 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-332549525 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-1604254218 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )-4072035000 x^{4}+3361851350 x^{3}+21272145125 x^{2}-9147233975 x -3547882250\right )}{42070322580 \left (24 x^{3}-70 x^{2}+21 x +10\right ) \left (7+5 x \right )^{2}}\) | \(434\) |
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\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx=\int { \frac {\sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]
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\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx=\int \frac {\sqrt {2 - 3 x}}{\sqrt {2 x - 5} \sqrt {4 x + 1} \left (5 x + 7\right )^{3}}\, dx \]
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\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx=\int { \frac {\sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]
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\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx=\int { \frac {\sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx=\int \frac {\sqrt {2-3\,x}}{\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^3} \,d x \]
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