Optimal. Leaf size=391 \[ -\frac {1368371 \sqrt {\frac {11}{23}} \sqrt {5 x+7} \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{43200 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}-\frac {1}{9} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7} (2-3 x)^{3/2}+\frac {23}{240} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7} \sqrt {2-3 x}-\frac {13027 \sqrt {4 x+1} \sqrt {5 x+7} \sqrt {2-3 x}}{4800 \sqrt {2 x-5}}+\frac {13027 \sqrt {\frac {143}{3}} \sqrt {\frac {5 x+7}{5-2 x}} \sqrt {2-3 x} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{3200 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {65750101 \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} (2-3 x) \Pi \left (-\frac {69}{55};\sin ^{-1}\left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{216000 \sqrt {429} \sqrt {2 x-5} \sqrt {4 x+1}} \]
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Rubi [A] time = 0.43, antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.270, Rules used = {161, 1600, 1602, 1598, 170, 418, 165, 537, 176, 424} \[ -\frac {1}{9} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7} (2-3 x)^{3/2}+\frac {23}{240} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7} \sqrt {2-3 x}-\frac {13027 \sqrt {4 x+1} \sqrt {5 x+7} \sqrt {2-3 x}}{4800 \sqrt {2 x-5}}-\frac {1368371 \sqrt {\frac {11}{23}} \sqrt {5 x+7} F\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{43200 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}+\frac {13027 \sqrt {\frac {143}{3}} \sqrt {\frac {5 x+7}{5-2 x}} \sqrt {2-3 x} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{3200 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}-\frac {65750101 \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} (2-3 x) \Pi \left (-\frac {69}{55};\sin ^{-1}\left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{216000 \sqrt {429} \sqrt {2 x-5} \sqrt {4 x+1}} \]
Antiderivative was successfully verified.
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Rule 161
Rule 165
Rule 170
Rule 176
Rule 418
Rule 424
Rule 537
Rule 1598
Rule 1600
Rule 1602
Rubi steps
\begin {align*} \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x} \, dx &=-\frac {1}{9} (2-3 x)^{3/2} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {1}{18} \int \frac {\sqrt {2-3 x} \left (617+1042 x-138 x^2\right )}{\sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx\\ &=\frac {23}{240} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {1}{9} (2-3 x)^{3/2} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {\int \frac {170254+143540 x-468972 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{2880}\\ &=-\frac {13027 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{4800 \sqrt {-5+2 x}}+\frac {23}{240} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {1}{9} (2-3 x)^{3/2} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}+\frac {\int \frac {-154352184+50903304 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{691200}-\frac {1862861 \int \frac {\sqrt {2-3 x}}{(-5+2 x)^{3/2} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{3200}\\ &=-\frac {13027 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{4800 \sqrt {-5+2 x}}+\frac {23}{240} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {1}{9} (2-3 x)^{3/2} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {2120971 \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{86400}-\frac {15052081 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{86400}+\frac {\left (169351 \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 )}{3200 \sqrt {-\frac {2-3 x}{-5+2 x}} \sqrt {7+5 x}}\\ &=-\frac {13027 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{4800 \sqrt {-5+2 x}}+\frac {23}{240} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {1}{9} (2-3 x)^{3/2} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}+\frac {13027 \sqrt {\frac {143}{3}} \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 )}{3200 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {\left (65750101 (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 )}{43200 \sqrt {897} \sqrt {-5+2 x} \sqrt {1+4 x}}-\frac {\left (1368371 \sqrt {\frac {11}{46}} \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 )}{43200 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{2-3 x}}}\\ &=-\frac {13027 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{4800 \sqrt {-5+2 x}}+\frac {23}{240} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {1}{9} (2-3 x)^{3/2} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}+\frac {13027 \sqrt {\frac {143}{3}} \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 )}{3200 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {1368371 \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 )}{43200 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}-\frac {65750101 (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 )}{216000 \sqrt {429} \sqrt {-5+2 x} \sqrt {1+4 x}}\\ \end {align*}
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Mathematica [A] time = 4.27, size = 340, normalized size = 0.87 \[ -\frac {\sqrt {2 x-5} \sqrt {4 x+1} \left (-4532324 \sqrt {682} \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right ),\frac {39}{62}\right )+7269066 \sqrt {682} \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) E\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right )|\frac {39}{62}\right )+\sqrt {\frac {5 x+7}{3 x-2}} \left (186 \left (1152000 x^5-2184000 x^4-7723240 x^3+7278862 x^2+17658613 x+3848705\right )-2120971 \sqrt {682} (2-3 x)^2 \sqrt {\frac {4 x+1}{3 x-2}} \sqrt {\frac {10 x^2-11 x-35}{(2-3 x)^2}} \Pi \left (\frac {117}{62};\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right )|\frac {39}{62}\right )\right )\right )}{5356800 \sqrt {2-3 x} \sqrt {5 x+7} \sqrt {\frac {5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.18, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}, x\right ) \]
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 \[ \int \sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 885, normalized size = 2.26 \[ -\frac {\sqrt {-3 x +2}\, \sqrt {2 x -5}\, \sqrt {4 x +1}\, \sqrt {5 x +7}\, \left (-4942080000 x^{5}+9369360000 x^{4}+33132699600 x^{3}+2682519840 \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 )-454813040 \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 )+780517328 \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} \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 )-49668641880 x^{2}+1341259920 \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 )-227406520 \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 )+390258664 \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 \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 )-55468893480 x +167657490 \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 )-28425815 \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 )+48782333 \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}}\, \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 )+48037189200\right )}{123552000 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right )} \]
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 \[ \int \sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,\sqrt {5\,x+7} \,d x \]
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 \[ \int \sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1} \sqrt {5 x + 7}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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