Optimal. Leaf size=162 \[ \frac {121 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{18 \sqrt {2 x-5}}+\frac {1}{10} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}-\frac {22}{45} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}-\frac {847 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{270 \sqrt {5-2 x}} \]
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Rubi [A] time = 0.06, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {101, 154, 158, 114, 113, 121, 119} \[ \frac {1}{10} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}-\frac {22}{45} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}+\frac {121 \sqrt {\frac {11}{6}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{18 \sqrt {2 x-5}}-\frac {847 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{270 \sqrt {5-2 x}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 114
Rule 119
Rule 121
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \, dx &=\frac {1}{10} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}-\frac {1}{10} \int \frac {\left (\frac {99}{2}-44 x\right ) \sqrt {1+4 x}}{\sqrt {2-3 x} \sqrt {-5+2 x}} \, dx\\ &=-\frac {22}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {1}{10} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {1}{90} \int \frac {-\frac {1815}{2}+1694 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx\\ &=-\frac {22}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {1}{10} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {847}{90} \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx+\frac {1331}{36} \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx\\ &=-\frac {22}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {1}{10} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {\left (121 \sqrt {\frac {11}{2}} \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}{18 \sqrt {-5+2 x}}+\frac {\left (847 \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}{90 \sqrt {5-2 x}}\\ &=-\frac {22}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {1}{10} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}-\frac {847 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{270 \sqrt {5-2 x}}+\frac {121 \sqrt {\frac {11}{6}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{18 \sqrt {-5+2 x}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 120, normalized size = 0.74 \[ \frac {605 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )+6 \sqrt {2-3 x} \sqrt {4 x+1} \left (72 x^2-250 x+175\right )-847 \sqrt {66} \sqrt {5-2 x} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{540 \sqrt {2 x-5}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}, x\right ) \]
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 \[ \int \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 145, normalized size = 0.90 \[ \frac {\sqrt {-3 x +2}\, \sqrt {2 x -5}\, \sqrt {4 x +1}\, \left (5184 x^{4}-20160 x^{3}+19236 x^{2}-2250 x -1694 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticE \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )+1815 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )-2100\right )}{12960 x^{3}-37800 x^{2}+11340 x +5400} \]
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 \[ \int \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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