Optimal. Leaf size=330 \[ -\frac {48884 \sqrt {\frac {11}{23}} \sqrt {5 x+7} \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{9593415 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}-\frac {2852696 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{11598438735 \sqrt {2 x-5}}+\frac {1426348 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{2319687747 \sqrt {5 x+7}}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{2085525 (5 x+7)^{3/2}}-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{25 (5 x+7)^{5/2}}+\frac {1426348 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{297395865 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}} \]
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Rubi [A] time = 0.39, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {160, 1604, 1599, 1602, 12, 170, 418, 176, 424} \[ -\frac {2852696 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{11598438735 \sqrt {2 x-5}}+\frac {1426348 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{2319687747 \sqrt {5 x+7}}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{2085525 (5 x+7)^{3/2}}-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{25 (5 x+7)^{5/2}}-\frac {48884 \sqrt {\frac {11}{23}} \sqrt {5 x+7} F\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{9593415 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}+\frac {1426348 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{297395865 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 160
Rule 170
Rule 176
Rule 418
Rule 424
Rule 1599
Rule 1602
Rule 1604
Rubi steps
\begin {align*} \int \frac {\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}}{25 (7+5 x)^{5/2}}+\frac {1}{25} \int \frac {-21+140 x-72 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}}{25 (7+5 x)^{5/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2085525 (7+5 x)^{3/2}}+\frac {\int \frac {-254100+327910 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx}{2085525}\\ &=-\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{25 (7+5 x)^{5/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2085525 (7+5 x)^{3/2}}+\frac {1426348 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2319687747 \sqrt {7+5 x}}+\frac {\int \frac {-762330250-648988340 x+855808800 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{57992193675}\\ &=-\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{25 (7+5 x)^{5/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2085525 (7+5 x)^{3/2}}+\frac {1426348 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2319687747 \sqrt {7+5 x}}-\frac {2852696 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{11598438735 \sqrt {-5+2 x}}-\frac {\int \frac {390064989600}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{13918126482000}-\frac {15689828 \int \frac {\sqrt {2-3 x}}{(-5+2 x)^{3/2} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{297395865}\\ &=-\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{25 (7+5 x)^{5/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2085525 (7+5 x)^{3/2}}+\frac {1426348 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2319687747 \sqrt {7+5 x}}-\frac {2852696 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{11598438735 \sqrt {-5+2 x}}-\frac {268862 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{9593415}+\frac {\left (1426348 \sqrt {\frac {11}{23}} \sqrt {2-3 x} \sqrt {-\frac {7+5 x}{-5+2 x}}\right ) \operatorname {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 )}{297395865 \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}}{25 (7+5 x)^{5/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2085525 (7+5 x)^{3/2}}+\frac {1426348 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2319687747 \sqrt {7+5 x}}-\frac {2852696 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{11598438735 \sqrt {-5+2 x}}+\frac {1426348 \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 )}{297395865 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {\left (24442 \sqrt {\frac {22}{23}} \sqrt {-\frac {-5+2 x}{2-3 x}} \sqrt {7+5 x}\right ) \operatorname {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 )}{9593415 \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}}{25 (7+5 x)^{5/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2085525 (7+5 x)^{3/2}}+\frac {1426348 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2319687747 \sqrt {7+5 x}}-\frac {2852696 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{11598438735 \sqrt {-5+2 x}}+\frac {1426348 \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 )}{297395865 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {48884 \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 )}{9593415 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}\\ \end {align*}
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Mathematica [A] time = 2.46, size = 251, normalized size = 0.76 \[ -\frac {2 \sqrt {2 x-5} \sqrt {4 x+1} \left (-236555 \sqrt {682} (3 x-2) \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} (5 x+7)^3 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right ),\frac {39}{62}\right )+713174 \sqrt {682} (3 x-2) \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} (5 x+7)^3 E\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right )|\frac {39}{62}\right )+31 \sqrt {\frac {5 x+7}{3 x-2}} \left (50105384 x^4-729949210 x^3+1137407943 x^2+880765228 x+137502130\right )\right )}{11598438735 \sqrt {2-3 x} (5 x+7)^{5/2} \sqrt {\frac {5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{625 \, x^{4} + 3500 \, x^{3} + 7350 \, x^{2} + 6860 \, x + 2401}, x\right ) \]
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 \[ \int \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 973, normalized size = 2.95 \[ -\frac {2 \left (285269600 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x^{4} \EllipticE \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+17811200 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x^{4} \EllipticF \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )-3514495404 x^{4}+941389680 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x^{3} \EllipticE \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+58776960 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x^{3} \EllipticF \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+19294337060 x^{3}+976335206 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x^{2} \EllipticE \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+60958832 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x^{2} \EllipticF \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )-26198770563 x^{2}+329486388 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x \EllipticE \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+20571936 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x \EllipticF \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )-3855274122 x +34945526 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, \EllipticE \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+2181872 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, \EllipticF \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+9191461480\right ) \sqrt {4 x +1}\, \sqrt {2 x -5}\, \sqrt {-3 x +2}}{11598438735 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right ) \left (5 x +7\right )^{\frac {3}{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 \[ \int \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}}{{\left (5\,x+7\right )}^{7/2}} \,d x \]
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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