Integrand size = 35, antiderivative size = 203 \[ \int \frac {(7+5 x)^4}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=-\frac {120355}{288} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {305}{24} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)-\frac {25}{84} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2-\frac {5109835 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{756 \sqrt {5-2 x}}+\frac {392989907 \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{2016 \sqrt {66} \sqrt {-5+2 x}} \]
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Time = 0.14 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {173, 1614, 1629, 164, 115, 114, 122, 120} \[ \int \frac {(7+5 x)^4}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {392989907 \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{2016 \sqrt {66} \sqrt {2 x-5}}-\frac {5109835 \sqrt {11} \sqrt {2 x-5} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{756 \sqrt {5-2 x}}-\frac {25}{84} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2-\frac {305}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)-\frac {120355}{288} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \]
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Rule 114
Rule 115
Rule 120
Rule 122
Rule 164
Rule 173
Rule 1614
Rule 1629
Rubi steps \begin{align*} \text {integral}& = -\frac {25}{84} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {1}{168} \int \frac {(7+5 x) \left (48949+134855 x+128100 x^2\right )}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx \\ & = -\frac {305}{24} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)-\frac {25}{84} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2-\frac {\int \frac {-9476460-227834100 x-303294600 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{20160} \\ & = -\frac {120355}{288} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {305}{24} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)-\frac {25}{84} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2-\frac {\int \frac {8530322220-88297948800 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{2177280} \\ & = -\frac {120355}{288} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {305}{24} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)-\frac {25}{84} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {5109835}{252} \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx+\frac {392989907 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{4032} \\ & = -\frac {120355}{288} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {305}{24} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)-\frac {25}{84} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {\left (392989907 \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}{2016 \sqrt {22} \sqrt {-5+2 x}}+\frac {\left (5109835 \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}{252 \sqrt {5-2 x}} \\ & = -\frac {120355}{288} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {305}{24} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)-\frac {25}{84} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2-\frac {5109835 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{756 \sqrt {5-2 x}}+\frac {392989907 \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{2016 \sqrt {66} \sqrt {-5+2 x}} \\ \end{align*}
Time = 22.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.62 \[ \int \frac {(7+5 x)^4}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {-1650 \sqrt {2-3 x} \sqrt {1+4 x} \left (-210245+50078 x+10608 x^2+1200 x^3\right )-449665480 \sqrt {66} \sqrt {5-2 x} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )+392989907 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{133056 \sqrt {-5+2 x}} \]
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Time = 1.64 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.71
method | result | size |
default | \(\frac {\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (449665480 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-279638761 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-23760000 x^{5}-200138400 x^{4}-900068400 x^{3}+4611000900 x^{2}-1569263850 x -693808500\right )}{3193344 x^{3}-9313920 x^{2}+2794176 x +1330560}\) | \(144\) |
elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (-\frac {675 x \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{8}-\frac {150175 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{288}-\frac {752233 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{23232 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {5109835 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{15246 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {625 x^{2} \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{84}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(228\) |
risch | \(\frac {25 \left (600 x^{2}+6804 x +42049\right ) \left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{2016 \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}+\frac {\left (\frac {752233 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{69696 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {5109835 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{45738 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(251\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.29 \[ \int \frac {(7+5 x)^4}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=-\frac {25}{2016} \, {\left (600 \, x^{2} + 6804 \, x + 42049\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} - \frac {184083109}{31104} \, \sqrt {-6} {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right ) + \frac {5109835}{756} \, \sqrt {-6} {\rm weierstrassZeta}\left (\frac {847}{108}, \frac {6655}{2916}, {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right )\right ) \]
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\[ \int \frac {(7+5 x)^4}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int \frac {\left (5 x + 7\right )^{4}}{\sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1}}\, dx \]
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\[ \int \frac {(7+5 x)^4}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{4}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
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\[ \int \frac {(7+5 x)^4}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{4}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
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Timed out. \[ \int \frac {(7+5 x)^4}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int \frac {{\left (5\,x+7\right )}^4}{\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}} \,d x \]
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