Integrand size = 37, antiderivative size = 330 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{7/2}} \, dx=\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{195 (7+5 x)^{5/2}}-\frac {3646 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{16267095 (7+5 x)^{3/2}}-\frac {20464840 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{90467822133 \sqrt {7+5 x}}+\frac {8185936 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{90467822133 \sqrt {-5+2 x}}-\frac {4092968 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{2319687747 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {111628 \sqrt {\frac {11}{23}} \sqrt {7+5 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{74828637 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}} \]
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Time = 0.27 (sec) , antiderivative size = 330, 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 = {170, 1618, 1613, 1616, 12, 176, 429, 182, 435} \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{7/2}} \, dx=-\frac {4092968 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{2319687747 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {111628 \sqrt {\frac {11}{23}} \sqrt {5 x+7} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{74828637 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}+\frac {8185936 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{90467822133 \sqrt {2 x-5}}-\frac {20464840 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{90467822133 \sqrt {5 x+7}}-\frac {3646 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{16267095 (5 x+7)^{3/2}}+\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{195 (5 x+7)^{5/2}} \]
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Rule 12
Rule 170
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}}{195 (7+5 x)^{5/2}}-\frac {1}{195} \int \frac {-41+90 x+48 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{5/2}} \, dx \\ & = \frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{195 (7+5 x)^{5/2}}-\frac {3646 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{16267095 (7+5 x)^{3/2}}-\frac {\int \frac {-489390+1112210 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx}{16267095} \\ & = \frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{195 (7+5 x)^{5/2}}-\frac {3646 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{16267095 (7+5 x)^{3/2}}-\frac {20464840 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{90467822133 \sqrt {7+5 x}}-\frac {\int \frac {-1235106290-1862300440 x+2455780800 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{452339110665} \\ & = \frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{195 (7+5 x)^{5/2}}-\frac {3646 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{16267095 (7+5 x)^{3/2}}-\frac {20464840 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{90467822133 \sqrt {7+5 x}}+\frac {8185936 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{90467822133 \sqrt {-5+2 x}}+\frac {\int \frac {890724463200}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{108561386559600}+\frac {45022648 \int \frac {\sqrt {2-3 x}}{(-5+2 x)^{3/2} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{2319687747} \\ & = \frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{195 (7+5 x)^{5/2}}-\frac {3646 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{16267095 (7+5 x)^{3/2}}-\frac {20464840 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{90467822133 \sqrt {7+5 x}}+\frac {8185936 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{90467822133 \sqrt {-5+2 x}}+\frac {613954 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{74828637}-\frac {\left (4092968 \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 )}{2319687747 \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}}{195 (7+5 x)^{5/2}}-\frac {3646 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{16267095 (7+5 x)^{3/2}}-\frac {20464840 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{90467822133 \sqrt {7+5 x}}+\frac {8185936 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{90467822133 \sqrt {-5+2 x}}-\frac {4092968 \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 )}{2319687747 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {\left (55814 \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 )}{74828637 \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}}{195 (7+5 x)^{5/2}}-\frac {3646 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{16267095 (7+5 x)^{3/2}}-\frac {20464840 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{90467822133 \sqrt {7+5 x}}+\frac {8185936 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{90467822133 \sqrt {-5+2 x}}-\frac {4092968 \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 )}{2319687747 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {111628 \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 )}{74828637 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}} \\ \end{align*}
Time = 30.35 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{7/2}} \, dx=-\frac {2 \sqrt {-5+2 x} \sqrt {1+4 x} \left (31 \sqrt {\frac {7+5 x}{-2+3 x}} \left (-374624540-2271416114 x-2953846743 x^2+643813106 x^3+370051256 x^4\right )-2046484 \sqrt {682} (-2+3 x) (7+5 x)^3 \sqrt {\frac {-5-18 x+8 x^2}{(2-3 x)^2}} E\left (\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right )|\frac {39}{62}\right )+958111 \sqrt {682} (-2+3 x) (7+5 x)^3 \sqrt {\frac {-5-18 x+8 x^2}{(2-3 x)^2}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right ),\frac {39}{62}\right )\right )}{90467822133 \sqrt {2-3 x} (7+5 x)^{5/2} \sqrt {\frac {7+5 x}{-2+3 x}} \left (-5-18 x+8 x^2\right )} \]
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Time = 1.62 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.49
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}}{24375 \left (x +\frac {7}{5}\right )^{3}}-\frac {3646 \sqrt {-120 x^{4}+182 x^{3}+385 x^{2}-197 x -70}}{406677375 \left (x +\frac {7}{5}\right )^{2}}-\frac {4092968 \left (-120 x^{3}+350 x^{2}-105 x -50\right )}{90467822133 \sqrt {\left (x +\frac {7}{5}\right ) \left (-120 x^{3}+350 x^{2}-105 x -50\right )}}+\frac {44912956 \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 )}{2515638730052331 \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 {5209232 \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 )}{193510671542487 \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 {81859360 \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 )}{30155940711 \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}}\) | \(493\) |
default | \(\frac {2 \left (460458900 \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 ) x^{4}-389302650 \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 ) x^{4}+675339720 \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 ) x^{3}-570977220 \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 ) x^{3}-611898716 \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 )+517339966 \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 )-630317072 \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 )+532912072 \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 )+401110864 \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 )-339125864 \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 )+263846545528 x^{4}+459038744578 x^{3}-2106092727759 x^{2}-1619519689282 x -267107297020\right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {2-3 x}}{2080759909059 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right ) \left (7+5 x \right )^{\frac {3}{2}}}\) | \(913\) |
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\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{7/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {7}{2}} \sqrt {2 \, x - 5}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{7/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {7}{2}} \sqrt {2 \, x - 5}} \,d x } \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{7/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {7}{2}} \sqrt {2 \, x - 5}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{7/2}} \, dx=\int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}}{\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^{7/2}} \,d x \]
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