Optimal. Leaf size=225 \[ -\frac {13243 \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{1065935 \sqrt {66} \sqrt {2 x-5}}-\frac {26825 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{33257172 (5 x+7)}-\frac {5 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{1794 (5 x+7)^2}+\frac {5365 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{16628586 \sqrt {5-2 x}}-\frac {16369941 \sqrt {5-2 x} \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{3436574440 \sqrt {11} \sqrt {2 x-5}} \]
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Rubi [A] time = 0.31, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {177, 1604, 1607, 168, 538, 537, 158, 114, 113, 121, 119} \[ -\frac {26825 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{33257172 (5 x+7)}-\frac {5 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{1794 (5 x+7)^2}-\frac {13243 \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{1065935 \sqrt {66} \sqrt {2 x-5}}+\frac {5365 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{16628586 \sqrt {5-2 x}}-\frac {16369941 \sqrt {5-2 x} \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{3436574440 \sqrt {11} \sqrt {2 x-5}} \]
Antiderivative was successfully verified.
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Rule 113
Rule 114
Rule 119
Rule 121
Rule 158
Rule 168
Rule 177
Rule 537
Rule 538
Rule 1604
Rule 1607
Rubi steps
\begin {align*} \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx &=-\frac {5 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1794 (7+5 x)^2}-\frac {\int \frac {-1063+1372 x-120 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2} \, dx}{3588}\\ &=-\frac {5 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1794 (7+5 x)^2}-\frac {26825 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{33257172 (7+5 x)}-\frac {\int \frac {-7905051+4073184 x+1931400 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{199543032}\\ &=-\frac {5 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1794 (7+5 x)^2}-\frac {26825 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{33257172 (7+5 x)}-\frac {\int \frac {\frac {1369224}{5}+386280 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{199543032}+\frac {5456647 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{110857240}\\ &=-\frac {5 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1794 (7+5 x)^2}-\frac {26825 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{33257172 (7+5 x)}-\frac {5365 \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx}{5542862}-\frac {13243 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{2131870}-\frac {5456647 \operatorname {Subst}\left (\int \frac {1}{\left (31-5 x^2\right ) \sqrt {\frac {11}{3}-\frac {4 x^2}{3}} \sqrt {-\frac {11}{3}-\frac {2 x^2}{3}}} \, dx,x,\sqrt {2-3 x}\right )}{55428620}\\ &=-\frac {5 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1794 (7+5 x)^2}-\frac {26825 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{33257172 (7+5 x)}-\frac {\left (5456647 \sqrt {\frac {3}{11}} \sqrt {5-2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (31-5 x^2\right ) \sqrt {\frac {11}{3}-\frac {4 x^2}{3}} \sqrt {1+\frac {2 x^2}{11}}} \, dx,x,\sqrt {2-3 x}\right )}{55428620 \sqrt {-5+2 x}}-\frac {\left (13243 \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}{1065935 \sqrt {22} \sqrt {-5+2 x}}-\frac {\left (5365 \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}{5542862 \sqrt {5-2 x}}\\ &=-\frac {5 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1794 (7+5 x)^2}-\frac {26825 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{33257172 (7+5 x)}+\frac {5365 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{16628586 \sqrt {5-2 x}}-\frac {13243 \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{1065935 \sqrt {66} \sqrt {-5+2 x}}-\frac {16369941 \sqrt {5-2 x} \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{3436574440 \sqrt {11} \sqrt {-5+2 x}}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 142, normalized size = 0.63 \[ \frac {-\sqrt {55-22 x} (5 x+7)^2 \left (-64043148 \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )+36589300 E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )+49109823 \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )\right )-17050 \sqrt {2-3 x} (2 x-5) \sqrt {4 x+1} (26825 x+56093)}{113406956520 \sqrt {2 x-5} (5 x+7)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{1000 \, x^{5} + 1950 \, x^{4} - 4195 \, x^{3} - 13111 \, x^{2} - 9849 \, x - 1715}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 461, normalized size = 2.05 \[ -\frac {\sqrt {-3 x +2}\, \sqrt {2 x -5}\, \sqrt {4 x +1}\, \left (10976790000 x^{4}-9062381900 x^{3}-914732500 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x^{2} \EllipticE \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )+1601078700 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x^{2} \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )-1227745575 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x^{2} \EllipticPi \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )-57342304250 x^{2}-2561251000 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x \EllipticE \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )+4483020360 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )-3437687610 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x \EllipticPi \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )+24657761150 x -1792875700 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticE \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )+3138114252 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )-2406381327 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticPi \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )+9563856500\right )}{113406956520 \left (24 x^{3}-70 x^{2}+21 x +10\right ) \left (5 x +7\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {2-3\,x}}{\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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