Optimal. Leaf size=101 \[ \frac {62 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \Pi \left (-\frac {69}{55};\sin ^{-1}\left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{5 \sqrt {429} \sqrt {2 x-5} \sqrt {4 x+1}} \]
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Rubi [A] time = 0.04, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {165, 537} \[ \frac {62 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \Pi \left (-\frac {69}{55};\sin ^{-1}\left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{5 \sqrt {429} \sqrt {2 x-5} \sqrt {4 x+1}} \]
Antiderivative was successfully verified.
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Rule 165
Rule 537
Rubi steps
\begin {align*} \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx &=\frac {\left (62 (2-3 x) \sqrt {-\frac {-5+2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {11 x^2}{23}} \sqrt {1+\frac {11 x^2}{39}} \left (5+3 x^2\right )} \, dx,x,\frac {\sqrt {7+5 x}}{\sqrt {2-3 x}}\right )}{\sqrt {897} \sqrt {-5+2 x} \sqrt {1+4 x}}\\ &=\frac {62 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}} \Pi \left (-\frac {69}{55};\sin ^{-1}\left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{5 \sqrt {429} \sqrt {-5+2 x} \sqrt {1+4 x}}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 170, normalized size = 1.68 \[ \frac {\sqrt {\frac {4 x+1}{5 x+7}} (5 x+7)^{3/2} \left (117 \sqrt {\frac {-6 x^2+19 x-10}{(5 x+7)^2}} \Pi \left (-\frac {55}{62};\sin ^{-1}\left (\sqrt {\frac {155-62 x}{55 x+77}}\right )|\frac {23}{62}\right )-62 \sqrt {\frac {5-2 x}{5 x+7}} \sqrt {\frac {3 x-2}{5 x+7}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {155-62 x}{55 x+77}}\right ),\frac {23}{62}\right )\right )}{5 \sqrt {682} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{40 \, x^{3} - 34 \, x^{2} - 151 \, x - 35}, x\right ) \]
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 \[ \int \frac {\sqrt {-3 \, x + 2}}{\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 172, normalized size = 1.70 \[ \frac {\left (55 \EllipticF \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+69 \EllipticPi \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {124}{55}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )\right ) \sqrt {\frac {3 x -2}{4 x +1}}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {11}\, \left (4 x +1\right )^{\frac {3}{2}} \sqrt {2 x -5}\, \sqrt {5 x +7}\, \sqrt {-3 x +2}}{128700 x^{3}-227370 x^{2}-356070 x +300300} \]
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 \[ \int \frac {\sqrt {-3 \, x + 2}}{\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5}}\,{d x} \]
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 \[ \int \frac {\sqrt {2-3\,x}}{\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,\sqrt {5\,x+7}} \,d x \]
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 \[ \int \frac {\sqrt {2 - 3 x}}{\sqrt {2 x - 5} \sqrt {4 x + 1} \sqrt {5 x + 7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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