Integrand size = 35, antiderivative size = 225 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx=-\frac {25 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{55614 (7+5 x)^2}-\frac {223825 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1030972332 (7+5 x)}+\frac {44765 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{515486166 \sqrt {5-2 x}}-\frac {24007 \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{6608797 \sqrt {66} \sqrt {-5+2 x}}-\frac {48493305 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{21306761528 \sqrt {11} \sqrt {-5+2 x}} \]
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Time = 0.19 (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 = {178, 1618, 1621, 174, 552, 551, 164, 115, 114, 122, 120} \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx=-\frac {24007 \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{6608797 \sqrt {66} \sqrt {2 x-5}}+\frac {44765 \sqrt {11} \sqrt {2 x-5} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{515486166 \sqrt {5-2 x}}-\frac {48493305 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{21306761528 \sqrt {11} \sqrt {2 x-5}}-\frac {223825 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{1030972332 (5 x+7)}-\frac {25 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{55614 (5 x+7)^2} \]
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Rule 114
Rule 115
Rule 120
Rule 122
Rule 164
Rule 174
Rule 178
Rule 551
Rule 552
Rule 1618
Rule 1621
Rubi steps \begin{align*} \text {integral}& = -\frac {25 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{55614 (7+5 x)^2}+\frac {\int \frac {16079-6860 x+600 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2} \, dx}{111228} \\ & = -\frac {25 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{55614 (7+5 x)^2}-\frac {223825 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1030972332 (7+5 x)}+\frac {\int \frac {123236883-38449440 x-16115400 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{6185833992} \\ & = -\frac {25 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{55614 (7+5 x)^2}-\frac {223825 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1030972332 (7+5 x)}+\frac {\int \frac {-3177576-3223080 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{6185833992}+\frac {16164435 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{687314888} \\ & = -\frac {25 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{55614 (7+5 x)^2}-\frac {223825 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1030972332 (7+5 x)}-\frac {44765 \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx}{171828722}-\frac {24007 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{13217594}-\frac {16164435 \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 )}{343657444} \\ & = -\frac {25 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{55614 (7+5 x)^2}-\frac {223825 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1030972332 (7+5 x)}-\frac {\left (16164435 \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 )}{343657444 \sqrt {-5+2 x}}-\frac {\left (24007 \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}{6608797 \sqrt {22} \sqrt {-5+2 x}}-\frac {\left (44765 \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}{171828722 \sqrt {5-2 x}} \\ & = -\frac {25 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{55614 (7+5 x)^2}-\frac {223825 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1030972332 (7+5 x)}+\frac {44765 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{515486166 \sqrt {5-2 x}}-\frac {24007 \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{6608797 \sqrt {66} \sqrt {-5+2 x}}-\frac {48493305 \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 )}{21306761528 \sqrt {11} \sqrt {-5+2 x}} \\ \end{align*}
Time = 6.18 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx=\frac {-17050 \sqrt {2-3 x} (-5+2 x) \sqrt {1+4 x} (81209+44765 x)-\sqrt {55-22 x} (7+5 x)^2 \left (61059460 E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )-116097852 \operatorname {EllipticF}\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )+145479915 \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )\right )}{703123130424 \sqrt {-5+2 x} (7+5 x)^2} \]
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Time = 1.66 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.21
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (-\frac {25 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{55614 \left (7+5 x \right )^{2}}-\frac {223825 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{1030972332 \left (7+5 x \right )}-\frac {44133 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{10395637681 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {44765 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{10395637681 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {16164435 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )}{478199333326 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(273\) |
risch | \(\frac {25 \left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (81209+44765 x \right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{1030972332 \left (7+5 x \right )^{2} \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}+\frac {\left (\frac {14711 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{10395637681 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {44765 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{31186913043 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {16164435 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \Pi \left (\frac {2 \sqrt {22-33 x}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )}{2578118144888 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(310\) |
default | \(\frac {\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (510436700 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x^{2}-283138625 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x^{2}-2424665250 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right ) x^{2}+1429222760 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x -792788150 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x -6789062700 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right ) x +1000455932 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-554951705 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-4752343890 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )-6795327000 x^{4}+7492177550 x^{3}+30009373625 x^{2}-13617971675 x -5136469250\right )}{260835999996 \left (24 x^{3}-70 x^{2}+21 x +10\right ) \left (7+5 x \right )^{2}}\) | \(434\) |
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\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx=\int { \frac {1}{{\left (5 \, x + 7\right )}^{3} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx=\int \frac {1}{\sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1} \left (5 x + 7\right )^{3}}\, dx \]
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\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx=\int { \frac {1}{{\left (5 \, x + 7\right )}^{3} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx=\int { \frac {1}{{\left (5 \, x + 7\right )}^{3} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx=\int \frac {1}{\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^3} \,d x \]
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