Integrand size = 37, antiderivative size = 370 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{9/2}} \, dx=-\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{35 (7+5 x)^{7/2}}+\frac {2558 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{695175 (7+5 x)^{5/2}}+\frac {23758016 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{57992193675 (7+5 x)^{3/2}}+\frac {32843987836 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{451524900265803 \sqrt {7+5 x}}-\frac {65687975672 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{2257624501329015 \sqrt {-5+2 x}}+\frac {32843987836 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{57887807726385 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {1212290288 \sqrt {\frac {11}{23}} \sqrt {7+5 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{1867348636335 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}} \]
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Time = 0.34 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{9/2}} \, dx=\frac {32843987836 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{57887807726385 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {1212290288 \sqrt {\frac {11}{23}} \sqrt {5 x+7} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{1867348636335 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}-\frac {65687975672 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2257624501329015 \sqrt {2 x-5}}+\frac {32843987836 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{451524900265803 \sqrt {5 x+7}}+\frac {23758016 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{57992193675 (5 x+7)^{3/2}}+\frac {2558 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{695175 (5 x+7)^{5/2}}-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{35 (5 x+7)^{7/2}} \]
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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}}{35 (7+5 x)^{7/2}}+\frac {1}{35} \int \frac {-21+140 x-72 x^2}{\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}}{35 (7+5 x)^{7/2}}+\frac {2558 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{695175 (7+5 x)^{5/2}}+\frac {\int \frac {-548842+1382130 x-429744 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{5/2}} \, dx}{4866225} \\ & = -\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{35 (7+5 x)^{7/2}}+\frac {2558 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{695175 (7+5 x)^{5/2}}+\frac {23758016 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{57992193675 (7+5 x)^{3/2}}+\frac {\int \frac {-6576343950+7032607120 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx}{405945355725} \\ & = -\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{35 (7+5 x)^{7/2}}+\frac {2558 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{695175 (7+5 x)^{5/2}}+\frac {23758016 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{57992193675 (7+5 x)^{3/2}}+\frac {32843987836 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{451524900265803 \sqrt {7+5 x}}+\frac {\int \frac {-20435008709500-14944014465380 x+19706392701600 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{11288122506645075} \\ & = -\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{35 (7+5 x)^{7/2}}+\frac {2558 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{695175 (7+5 x)^{5/2}}+\frac {23758016 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{57992193675 (7+5 x)^{3/2}}+\frac {32843987836 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{451524900265803 \sqrt {7+5 x}}-\frac {65687975672 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{2257624501329015 \sqrt {-5+2 x}}-\frac {\int \frac {9673349124067200}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{2709149401594818000}-\frac {361283866196 \int \frac {\sqrt {2-3 x}}{(-5+2 x)^{3/2} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{57887807726385} \\ & = -\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{35 (7+5 x)^{7/2}}+\frac {2558 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{695175 (7+5 x)^{5/2}}+\frac {23758016 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{57992193675 (7+5 x)^{3/2}}+\frac {32843987836 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{451524900265803 \sqrt {7+5 x}}-\frac {65687975672 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{2257624501329015 \sqrt {-5+2 x}}-\frac {6667596584 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{1867348636335}+\frac {\left (32843987836 \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 )}{57887807726385 \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}}{35 (7+5 x)^{7/2}}+\frac {2558 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{695175 (7+5 x)^{5/2}}+\frac {23758016 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{57992193675 (7+5 x)^{3/2}}+\frac {32843987836 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{451524900265803 \sqrt {7+5 x}}-\frac {65687975672 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{2257624501329015 \sqrt {-5+2 x}}+\frac {32843987836 \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 )}{57887807726385 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {\left (606145144 \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 )}{1867348636335 \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}}{35 (7+5 x)^{7/2}}+\frac {2558 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{695175 (7+5 x)^{5/2}}+\frac {23758016 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{57992193675 (7+5 x)^{3/2}}+\frac {32843987836 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{451524900265803 \sqrt {7+5 x}}-\frac {65687975672 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{2257624501329015 \sqrt {-5+2 x}}+\frac {32843987836 \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 )}{57887807726385 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {1212290288 \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 )}{1867348636335 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 17.71 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.54 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{9/2}} \, dx=\frac {2 \left (\frac {90675 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \left (15395515423270+113490310442229 x+54668919175710 x^2+10263746198750 x^3\right )}{(7+5 x)^{7/2}}+11 \sqrt {15} \left (\frac {27073896336630 \sqrt {1+4 x} \sqrt {7+5 x} \sqrt {-75+30 x}}{\sqrt {2-3 x}}-\frac {27073896336630 \sqrt {715} \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{\sqrt {\frac {5-2 x}{2-3 x}} \sqrt {1+4 x}}+\frac {39800941623080 \sqrt {715} \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{\sqrt {\frac {5-2 x}{2-3 x}} \sqrt {1+4 x}}-\frac {46692478872404 \sqrt {715} \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{\sqrt {\frac {5-2 x}{2-3 x}} \sqrt {1+4 x}}+\frac {3535063529751 i \sqrt {10230} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{-5+2 x}} \operatorname {EllipticPi}\left (-\frac {23}{55},i \text {arcsinh}\left (\frac {\sqrt {\frac {22}{23}} \sqrt {7+5 x}}{\sqrt {-5+2 x}}\right ),\frac {23}{62}\right )}{\sqrt {\frac {2-3 x}{5-2 x}} \sqrt {1+4 x}}-\frac {4405470235335 \sqrt {10230} \sqrt {2-3 x} \sqrt {\frac {-5+2 x}{1+4 x}} \operatorname {EllipticPi}\left (\frac {78}{55},\arcsin \left (\frac {\sqrt {\frac {22}{39}} \sqrt {7+5 x}}{\sqrt {1+4 x}}\right ),\frac {39}{62}\right )}{\sqrt {-5+2 x} \sqrt {\frac {-2+3 x}{1+4 x}}}\right )\right )}{204710101658008435125} \]
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Time = 1.63 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.41
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}}{21875 \left (x +\frac {7}{5}\right )^{4}}+\frac {2558 \sqrt {-120 x^{4}+182 x^{3}+385 x^{2}-197 x -70}}{86896875 \left (x +\frac {7}{5}\right )^{3}}+\frac {23758016 \sqrt {-120 x^{4}+182 x^{3}+385 x^{2}-197 x -70}}{1449804841875 \left (x +\frac {7}{5}\right )^{2}}+\frac {-\frac {262751902688}{150508300088601} x^{3}+\frac {328439878360}{64503557180829} x^{2}-\frac {32843987836}{21501185726943} x -\frac {328439878360}{451524900265803}}{\sqrt {\left (x +\frac {7}{5}\right ) \left (-120 x^{3}+350 x^{2}-105 x -50\right )}}-\frac {21231177880 \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 )}{1793650414527312003 \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 {5971634152 \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 )}{689865544048966155 \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 {131375951344 \left (x +\frac {7}{5}\right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}{150508300088601}-\frac {65687975672 \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 )}{6075267533076379365}}{\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}}\) | \(522\) |
default | \(\text {Expression too large to display}\) | \(1088\) |
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\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{9/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{9/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{9/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{9/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{9/2}} \, dx=\int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}}{{\left (5\,x+7\right )}^{9/2}} \,d x \]
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