Integrand size = 37, antiderivative size = 195 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=\frac {10 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5-2 x}{7+5 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{22}} \sqrt {1+4 x}}{\sqrt {7+5 x}}\right )|\frac {62}{39}\right )}{713 \sqrt {-5+2 x} \sqrt {\frac {2-3 x}{7+5 x}}}+\frac {2 \sqrt {\frac {3}{143}} (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{31 \sqrt {-5+2 x} \sqrt {1+4 x}} \]
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Time = 0.13 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.38, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {177, 176, 429, 182, 433, 506, 422} \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=\frac {6 \sqrt {5 x+7} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{31 \sqrt {253} \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}-\frac {5 \sqrt {\frac {22}{31}} \sqrt {4 x+1} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {\frac {31}{11}} \sqrt {2 x-5}}{\sqrt {5 x+7}}\right ),\frac {39}{62}\right )}{1209 \sqrt {2-3 x} \sqrt {-\frac {4 x+1}{2-3 x}}}+\frac {10 \sqrt {\frac {22}{31}} \sqrt {4 x+1} E\left (\arctan \left (\frac {\sqrt {\frac {31}{11}} \sqrt {2 x-5}}{\sqrt {5 x+7}}\right )|\frac {39}{62}\right )}{897 \sqrt {2-3 x} \sqrt {-\frac {4 x+1}{2-3 x}}}-\frac {10 \sqrt {2 x-5} \sqrt {4 x+1}}{897 \sqrt {2-3 x} \sqrt {5 x+7}} \]
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Rule 176
Rule 177
Rule 182
Rule 422
Rule 429
Rule 433
Rule 506
Rubi steps \begin{align*} \text {integral}& = \frac {3}{31} \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx+\frac {5}{31} \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx \\ & = \frac {\left (5 \sqrt {2} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{7+5 x}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {31 x^2}{11}}}{\sqrt {1+\frac {23 x^2}{22}}} \, dx,x,\frac {\sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )}{1209 \sqrt {1+4 x} \sqrt {-\frac {2-3 x}{7+5 x}}}+\frac {\left (3 \sqrt {\frac {2}{253}} \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 )}{31 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{2-3 x}}} \\ & = \frac {6 \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 )}{31 \sqrt {253} \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}+\frac {\left (5 \sqrt {2} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{7+5 x}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {23 x^2}{22}} \sqrt {1+\frac {31 x^2}{11}}} \, dx,x,\frac {\sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )}{1209 \sqrt {1+4 x} \sqrt {-\frac {2-3 x}{7+5 x}}}+\frac {\left (5 \sqrt {2} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{7+5 x}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {23 x^2}{22}} \sqrt {1+\frac {31 x^2}{11}}} \, dx,x,\frac {\sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )}{429 \sqrt {1+4 x} \sqrt {-\frac {2-3 x}{7+5 x}}} \\ & = -\frac {10 \sqrt {-5+2 x} \sqrt {1+4 x}}{897 \sqrt {2-3 x} \sqrt {7+5 x}}+\frac {6 \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 )}{31 \sqrt {253} \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}-\frac {5 \sqrt {\frac {22}{31}} \sqrt {1+4 x} F\left (\tan ^{-1}\left (\frac {\sqrt {\frac {31}{11}} \sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )|\frac {39}{62}\right )}{1209 \sqrt {2-3 x} \sqrt {-\frac {1+4 x}{2-3 x}}}-\frac {\left (10 \sqrt {2} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{7+5 x}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {23 x^2}{22}}}{\left (1+\frac {31 x^2}{11}\right )^{3/2}} \, dx,x,\frac {\sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )}{897 \sqrt {1+4 x} \sqrt {-\frac {2-3 x}{7+5 x}}} \\ & = -\frac {10 \sqrt {-5+2 x} \sqrt {1+4 x}}{897 \sqrt {2-3 x} \sqrt {7+5 x}}+\frac {10 \sqrt {\frac {22}{31}} \sqrt {1+4 x} E\left (\tan ^{-1}\left (\frac {\sqrt {\frac {31}{11}} \sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )|\frac {39}{62}\right )}{897 \sqrt {2-3 x} \sqrt {-\frac {1+4 x}{2-3 x}}}+\frac {6 \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 )}{31 \sqrt {253} \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}-\frac {5 \sqrt {\frac {22}{31}} \sqrt {1+4 x} F\left (\tan ^{-1}\left (\frac {\sqrt {\frac {31}{11}} \sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )|\frac {39}{62}\right )}{1209 \sqrt {2-3 x} \sqrt {-\frac {1+4 x}{2-3 x}}} \\ \end{align*}
Time = 18.16 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=-\frac {2 \sqrt {-5+2 x} \sqrt {1+4 x} \left (1705 \sqrt {\frac {7+5 x}{-2+3 x}} \left (-5-18 x+8 x^2\right )-55 \sqrt {682} \sqrt {\frac {-5-18 x+8 x^2}{(2-3 x)^2}} \left (-14+11 x+15 x^2\right ) E\left (\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right )|\frac {39}{62}\right )-23 \sqrt {682} \sqrt {\frac {-5-18 x+8 x^2}{(2-3 x)^2}} \left (-14+11 x+15 x^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right ),\frac {39}{62}\right )\right )}{305877 \sqrt {2-3 x} \sqrt {7+5 x} \sqrt {\frac {7+5 x}{-2+3 x}} \left (-5-18 x+8 x^2\right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (155 ) = 310\).
Time = 1.62 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.23
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 {7252 \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 )}{8505521739 \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 {140 \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 )}{654270903 \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 {200 \left (\left (x +\frac {7}{5}\right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )-\frac {\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 )}{80730}\right )}{9269 \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 {10 \left (-120 x^{3}+350 x^{2}-105 x -50\right )}{27807 \sqrt {\left (x +\frac {7}{5}\right ) \left (-120 x^{3}+350 x^{2}-105 x -50\right )}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {7+5 x}}\) | \(435\) |
default | \(-\frac {2 \sqrt {7+5 x}\, \sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (1116 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x^{2} F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-495 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x^{2} E\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-1488 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )+660 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x E\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )+496 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-220 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, E\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-313720 x^{2}+705870 x +196075\right )}{7035171 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right )}\) | \(563\) |
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\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 7\right )}^{\frac {3}{2}} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=\int \frac {1}{\sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1} \left (5 x + 7\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 7\right )}^{\frac {3}{2}} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 7\right )}^{\frac {3}{2}} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=\int \frac {1}{\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^{3/2}} \,d x \]
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