Optimal. Leaf size=290 \[ \frac {44 \sqrt {\frac {11}{23}} \sqrt {5 x+7} \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{2691 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}+\frac {3740 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{3253419 \sqrt {2 x-5}}-\frac {9350 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{3253419 \sqrt {5 x+7}}+\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{117 (5 x+7)^{3/2}}-\frac {1870 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{83421 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}} \]
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Rubi [A] time = 0.32, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {164, 1599, 1602, 12, 170, 418, 176, 424} \[ \frac {3740 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{3253419 \sqrt {2 x-5}}-\frac {9350 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{3253419 \sqrt {5 x+7}}+\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{117 (5 x+7)^{3/2}}+\frac {44 \sqrt {\frac {11}{23}} \sqrt {5 x+7} F\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{2691 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}-\frac {1870 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{83421 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 164
Rule 170
Rule 176
Rule 418
Rule 424
Rule 1599
Rule 1602
Rubi steps
\begin {align*} \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{5/2}} \, dx &=\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{117 (7+5 x)^{3/2}}-\frac {1}{117} \int \frac {-33+110 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx\\ &=\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{117 (7+5 x)^{3/2}}-\frac {9350 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3253419 \sqrt {7+5 x}}-\frac {\int \frac {-66308-170170 x+224400 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{3253419}\\ &=\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{117 (7+5 x)^{3/2}}-\frac {9350 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3253419 \sqrt {7+5 x}}+\frac {3740 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{3253419 \sqrt {-5+2 x}}+\frac {\int \frac {70218720}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{780820560}+\frac {20570 \int \frac {\sqrt {2-3 x}}{(-5+2 x)^{3/2} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{83421}\\ &=\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{117 (7+5 x)^{3/2}}-\frac {9350 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3253419 \sqrt {7+5 x}}+\frac {3740 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{3253419 \sqrt {-5+2 x}}+\frac {242 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{2691}-\frac {\left (1870 \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 )}{83421 \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}}{117 (7+5 x)^{3/2}}-\frac {9350 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3253419 \sqrt {7+5 x}}+\frac {3740 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{3253419 \sqrt {-5+2 x}}-\frac {1870 \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 )}{83421 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {\left (22 \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 )}{2691 \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}}{117 (7+5 x)^{3/2}}-\frac {9350 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3253419 \sqrt {7+5 x}}+\frac {3740 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{3253419 \sqrt {-5+2 x}}-\frac {1870 \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 )}{83421 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {44 \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 )}{2691 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}\\ \end {align*}
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Mathematica [A] time = 1.95, size = 246, normalized size = 0.85 \[ -\frac {2 \sqrt {2 x-5} \sqrt {4 x+1} \left (506 \sqrt {682} (3 x-2) \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} (5 x+7)^2 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right ),\frac {39}{62}\right )-935 \sqrt {682} (3 x-2) \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} (5 x+7)^2 E\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right )|\frac {39}{62}\right )+31 \sqrt {\frac {5 x+7}{3 x-2}} \left (58928 x^3-94580 x^2-122348 x-23755\right )\right )}{3253419 \sqrt {2-3 x} (5 x+7)^{3/2} \sqrt {\frac {5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{250 \, x^{4} + 425 \, x^{3} - 1155 \, x^{2} - 2989 \, x - 1715}, x\right ) \]
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 \[ \int \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {5}{2}} \sqrt {2 \, x - 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 786, normalized size = 2.71 \[ -\frac {2 \left (-74800 \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 )+20240 \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 )-1312518 x^{3}-142120 \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 )+38456 \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 )+3086255 x^{2}-57035 \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 )+15433 \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 )+1200968 x -6545 \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 )+1771 \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 )-1783420\right ) \sqrt {2 x -5}\, \sqrt {4 x +1}\, \sqrt {-3 x +2}}{3253419 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right ) \sqrt {5 x +7}} \]
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 \[ \int \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {5}{2}} \sqrt {2 \, x - 5}}\,{d x} \]
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 \[ \int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}}{\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^{5/2}} \,d x \]
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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