Optimal. Leaf size=131 \[ -\frac {11 \sqrt {\frac {22}{3}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{3 \sqrt {2 x-5}}+\frac {1}{3} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}+\frac {55 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{18 \sqrt {5-2 x}} \]
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Rubi [A] time = 0.05, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {101, 158, 114, 113, 121, 119} \[ \frac {1}{3} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}-\frac {11 \sqrt {\frac {22}{3}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{3 \sqrt {2 x-5}}+\frac {55 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{18 \sqrt {5-2 x}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 114
Rule 119
Rule 121
Rule 158
Rubi steps
\begin {align*} \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x}} \, dx &=\frac {1}{3} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {1}{3} \int \frac {-\frac {33}{2}+55 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx\\ &=\frac {1}{3} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {55}{6} \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx-\frac {121}{3} \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx\\ &=\frac {1}{3} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {\left (11 \sqrt {22} \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}{3 \sqrt {-5+2 x}}-\frac {\left (55 \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}{6 \sqrt {5-2 x}}\\ &=\frac {1}{3} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {55 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{18 \sqrt {5-2 x}}-\frac {11 \sqrt {\frac {22}{3}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{3 \sqrt {-5+2 x}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 115, normalized size = 0.88 \[ \frac {-44 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )+12 \sqrt {2-3 x} \sqrt {4 x+1} (2 x-5)+55 \sqrt {66} \sqrt {5-2 x} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{36 \sqrt {2 x-5}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}}, x\right ) \]
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 \[ \int \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 140, normalized size = 1.07 \[ \frac {\sqrt {-3 x +2}\, \sqrt {4 x +1}\, \sqrt {2 x -5}\, \left (144 x^{3}-420 x^{2}+126 x +55 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticE \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )-66 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )+60\right )}{432 x^{3}-1260 x^{2}+378 x +180} \]
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 \[ \int \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}}{\sqrt {2\,x-5}} \,d x \]
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 \[ \int \frac {\sqrt {2 - 3 x} \sqrt {4 x + 1}}{\sqrt {2 x - 5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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