Integrand size = 37, antiderivative size = 349 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{3/2}} \, dx=-\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{5 \sqrt {7+5 x}}+\frac {6 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{25 \sqrt {-5+2 x}}-\frac {3 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{25 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {296 \sqrt {\frac {11}{23}} \sqrt {7+5 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{75 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}-\frac {26474 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{375 \sqrt {429} \sqrt {-5+2 x} \sqrt {1+4 x}} \]
[Out]
Time = 0.22 (sec) , antiderivative size = 349, 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, 1616, 1612, 176, 429, 171, 551, 182, 435} \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{3/2}} \, dx=-\frac {3 \sqrt {429} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{25 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {26474 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{375 \sqrt {429} \sqrt {2 x-5} \sqrt {4 x+1}}+\frac {296 \sqrt {\frac {11}{23}} \sqrt {5 x+7} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{75 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}+\frac {6 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{25 \sqrt {2 x-5}}-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{5 \sqrt {5 x+7}} \]
[In]
[Out]
Rule 166
Rule 171
Rule 176
Rule 182
Rule 429
Rule 435
Rule 551
Rule 1612
Rule 1616
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{5 \sqrt {7+5 x}}+\frac {1}{5} \int \frac {-21+140 x-72 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx \\ & = -\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{5 \sqrt {7+5 x}}+\frac {6 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{25 \sqrt {-5+2 x}}-\frac {\int \frac {-12384-20496 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{1200}+\frac {1287}{25} \int \frac {\sqrt {2-3 x}}{(-5+2 x)^{3/2} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx \\ & = -\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{5 \sqrt {7+5 x}}+\frac {6 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{25 \sqrt {-5+2 x}}-\frac {427}{75} \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx+\frac {1628}{75} \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx-\frac {\left (117 \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 )}{25 \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}}{5 \sqrt {7+5 x}}+\frac {6 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{25 \sqrt {-5+2 x}}-\frac {3 \sqrt {429} \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 )}{25 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {\left (26474 (2-3 x) \sqrt {-\frac {-5+2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {11 x^2}{23}} \sqrt {1+\frac {11 x^2}{39}} \left (5+3 x^2\right )} \, dx,x,\frac {\sqrt {7+5 x}}{\sqrt {2-3 x}}\right )}{75 \sqrt {897} \sqrt {-5+2 x} \sqrt {1+4 x}}+\frac {\left (148 \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 )}{75 \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}}{5 \sqrt {7+5 x}}+\frac {6 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{25 \sqrt {-5+2 x}}-\frac {3 \sqrt {429} \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 )}{25 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {296 \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 )}{75 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}-\frac {26474 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}} \Pi \left (-\frac {69}{55};\sin ^{-1}\left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{375 \sqrt {429} \sqrt {-5+2 x} \sqrt {1+4 x}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 14.88 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.62 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{3/2}} \, dx=-\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{5 \sqrt {7+5 x}}-\frac {2 \left (\frac {9 \sqrt {1+4 x} \sqrt {7+5 x} \sqrt {-75+30 x}}{2 \sqrt {2-3 x}}-\frac {9 \sqrt {715} \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{2 \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {1+4 x}}+\frac {86 \sqrt {\frac {55}{13}} \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{\sqrt {\frac {5-2 x}{2-3 x}} \sqrt {1+4 x}}-\frac {5549 \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{\sqrt {715} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {1+4 x}}-\frac {39 i \sqrt {\frac {165}{62}} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{-5+2 x}} \operatorname {EllipticPi}\left (-\frac {23}{55},i \text {arcsinh}\left (\frac {\sqrt {\frac {22}{23}} \sqrt {7+5 x}}{\sqrt {-5+2 x}}\right ),\frac {23}{62}\right )}{\sqrt {\frac {2-3 x}{5-2 x}} \sqrt {1+4 x}}-\frac {23 \sqrt {\frac {165}{62}} \sqrt {2-3 x} \sqrt {\frac {-5+2 x}{1+4 x}} \operatorname {EllipticPi}\left (\frac {78}{55},\arcsin \left (\frac {\sqrt {\frac {22}{39}} \sqrt {7+5 x}}{\sqrt {1+4 x}}\right ),\frac {39}{62}\right )}{\sqrt {-5+2 x} \sqrt {\frac {-2+3 x}{1+4 x}}}\right )}{25 \sqrt {15}} \]
[In]
[Out]
Time = 1.61 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.25
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 \left (-120 x^{3}+350 x^{2}-105 x -50\right )}{25 \sqrt {\left (x +\frac {7}{5}\right ) \left (-120 x^{3}+350 x^{2}-105 x -50\right )}}-\frac {14 \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 )}{509795 \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 {56 \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 )}{305877 \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 {36 \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 )}{5 \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}}\) | \(435\) |
default | \(-\frac {\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {7+5 x}\, \left (146520 \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 )-238266 \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} \Pi \left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )-173745 \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 )-195360 \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 )+317688 \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 \Pi \left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )+231660 \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 )+65120 \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 )-105896 \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}}\, \Pi \left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )-77220 \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 )-17760600 x^{3}-58313970 x^{2}+232219845 x +61422075\right )}{3700125 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right )}\) | \(821\) |
[In]
[Out]
\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{3/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{3/2}} \, dx=\int \frac {\sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1}}{\left (5 x + 7\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{3/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{3/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{3/2}} \, dx=\int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}}{{\left (5\,x+7\right )}^{3/2}} \,d x \]
[In]
[Out]