Optimal. Leaf size=370 \[ -\frac {1212290288 \sqrt {\frac {11}{23}} \sqrt {5 x+7} \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{1867348636335 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}-\frac {65687975672 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2257624501329015 \sqrt {2 x-5}}+\frac {32843987836 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{451524900265803 \sqrt {5 x+7}}+\frac {23758016 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{57992193675 (5 x+7)^{3/2}}+\frac {2558 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{695175 (5 x+7)^{5/2}}-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{35 (5 x+7)^{7/2}}+\frac {32843987836 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{57887807726385 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}} \]
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Rubi [A] time = 0.51, antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ -\frac {65687975672 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2257624501329015 \sqrt {2 x-5}}+\frac {32843987836 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{451524900265803 \sqrt {5 x+7}}+\frac {23758016 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{57992193675 (5 x+7)^{3/2}}+\frac {2558 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{695175 (5 x+7)^{5/2}}-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{35 (5 x+7)^{7/2}}-\frac {1212290288 \sqrt {\frac {11}{23}} \sqrt {5 x+7} F\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{1867348636335 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}+\frac {32843987836 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{57887807726385 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}} \]
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)^{9/2}} \, dx &=-\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 ) \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 )}{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 ) \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 )}{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*}
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Mathematica [A] time = 2.81, size = 259, normalized size = 0.70 \[ \frac {2 \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7} \left (\frac {242 \left (19017205 \sqrt {682} (3 x-2) \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right ),\frac {39}{62}\right )+203578437 \sqrt {\frac {5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )-67859479 \sqrt {682} (3 x-2) \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} E\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right )|\frac {39}{62}\right )\right )}{\sqrt {\frac {5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )}-\frac {(3 x-2) \left (10263746198750 x^3+54668919175710 x^2+113490310442229 x+15395515423270\right )}{(5 x+7)^4}\right )}{2257624501329015 \sqrt {2-3 x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{3125 \, x^{5} + 21875 \, x^{4} + 61250 \, x^{3} + 85750 \, x^{2} + 60025 \, x + 16807}, x\right ) \]
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 \[ \int \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 1160, normalized size = 3.14 \[ \text {result too large to display} \]
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 \[ \int \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}}{{\left (5\,x+7\right )}^{9/2}} \,d x \]
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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