Optimal. Leaf size=365 \[ \frac{7 \sqrt{\frac{11}{23}} \sqrt{5 x+7} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right ),-\frac{39}{23}\right )}{10 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}+\frac{41 \sqrt{\frac{11}{62}} \sqrt{2-3 x} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{\frac{22}{23}} \sqrt{5 x+7}}{\sqrt{2 x-5}}\right ),\frac{39}{62}\right )}{20 \sqrt{-\frac{2-3 x}{4 x+1}} \sqrt{4 x+1}}+\frac{\sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{5 \sqrt{2 x-5}}-\frac{\sqrt{429} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{10 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}+\frac{943 \sqrt{2-3 x} \Pi \left (\frac{78}{55};\tan ^{-1}\left (\frac{\sqrt{\frac{22}{23}} \sqrt{5 x+7}}{\sqrt{2 x-5}}\right )|\frac{39}{62}\right )}{100 \sqrt{682} \sqrt{-\frac{2-3 x}{4 x+1}} \sqrt{4 x+1}} \]
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Rubi [A] time = 0.216143, antiderivative size = 365, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.216, Rules used = {173, 176, 424, 170, 418, 165, 536, 539} \[ \frac{\sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{5 \sqrt{2 x-5}}+\frac{7 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{10 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}+\frac{41 \sqrt{\frac{11}{62}} \sqrt{2-3 x} F\left (\tan ^{-1}\left (\frac{\sqrt{\frac{22}{23}} \sqrt{5 x+7}}{\sqrt{2 x-5}}\right )|\frac{39}{62}\right )}{20 \sqrt{-\frac{2-3 x}{4 x+1}} \sqrt{4 x+1}}-\frac{\sqrt{429} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{10 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}+\frac{943 \sqrt{2-3 x} \Pi \left (\frac{78}{55};\tan ^{-1}\left (\frac{\sqrt{\frac{22}{23}} \sqrt{5 x+7}}{\sqrt{2 x-5}}\right )|\frac{39}{62}\right )}{100 \sqrt{682} \sqrt{-\frac{2-3 x}{4 x+1}} \sqrt{4 x+1}} \]
Antiderivative was successfully verified.
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Rule 173
Rule 176
Rule 424
Rule 170
Rule 418
Rule 165
Rule 536
Rule 539
Rubi steps
\begin{align*} \int \frac{\sqrt{2-3 x} \sqrt{1+4 x}}{\sqrt{-5+2 x} \sqrt{7+5 x}} \, dx &=\frac{\sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{5 \sqrt{-5+2 x}}-\frac{41}{20} \int \frac{\sqrt{-5+2 x}}{\sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx+\frac{77}{20} \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx+\frac{429}{10} \int \frac{\sqrt{2-3 x}}{(-5+2 x)^{3/2} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx\\ &=\frac{\sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{5 \sqrt{-5+2 x}}-\frac{\left (1599 \sqrt{-\frac{2-3 x}{-5+2 x}} (-5+2 x) \sqrt{\frac{1+4 x}{-5+2 x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (5-2 x^2\right ) \sqrt{1+\frac{11 x^2}{31}} \sqrt{1+\frac{22 x^2}{23}}} \, dx,x,\frac{\sqrt{7+5 x}}{\sqrt{-5+2 x}}\right )}{10 \sqrt{713} \sqrt{2-3 x} \sqrt{1+4 x}}+\frac{\left (7 \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 )}{10 \sqrt{-5+2 x} \sqrt{\frac{7+5 x}{2-3 x}}}-\frac{\left (39 \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 )}{10 \sqrt{-\frac{2-3 x}{-5+2 x}} \sqrt{7+5 x}}\\ &=\frac{\sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{5 \sqrt{-5+2 x}}-\frac{\sqrt{429} \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 )}{10 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{7+5 x}}+\frac{7 \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 )}{10 \sqrt{-5+2 x} \sqrt{\frac{7+5 x}{5-2 x}}}-\frac{\left (41 \sqrt{\frac{23}{31}} \sqrt{-\frac{2-3 x}{-5+2 x}} (-5+2 x) \sqrt{\frac{1+4 x}{-5+2 x}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{22 x^2}{23}}}{\left (5-2 x^2\right ) \sqrt{1+\frac{11 x^2}{31}}} \, dx,x,\frac{\sqrt{7+5 x}}{\sqrt{-5+2 x}}\right )}{20 \sqrt{2-3 x} \sqrt{1+4 x}}-\frac{\left (451 \sqrt{-\frac{2-3 x}{-5+2 x}} (-5+2 x) \sqrt{\frac{1+4 x}{-5+2 x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{11 x^2}{31}} \sqrt{1+\frac{22 x^2}{23}}} \, dx,x,\frac{\sqrt{7+5 x}}{\sqrt{-5+2 x}}\right )}{20 \sqrt{713} \sqrt{2-3 x} \sqrt{1+4 x}}\\ &=\frac{\sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{5 \sqrt{-5+2 x}}-\frac{\sqrt{429} \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 )}{10 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{7+5 x}}+\frac{7 \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 )}{10 \sqrt{-5+2 x} \sqrt{\frac{7+5 x}{5-2 x}}}+\frac{41 \sqrt{\frac{11}{62}} \sqrt{2-3 x} F\left (\tan ^{-1}\left (\frac{\sqrt{\frac{22}{23}} \sqrt{7+5 x}}{\sqrt{-5+2 x}}\right )|\frac{39}{62}\right )}{20 \sqrt{-\frac{2-3 x}{1+4 x}} \sqrt{1+4 x}}+\frac{943 \sqrt{2-3 x} \Pi \left (\frac{78}{55};\tan ^{-1}\left (\frac{\sqrt{\frac{22}{23}} \sqrt{7+5 x}}{\sqrt{-5+2 x}}\right )|\frac{39}{62}\right )}{100 \sqrt{682} \sqrt{-\frac{2-3 x}{1+4 x}} \sqrt{1+4 x}}\\ \end{align*}
Mathematica [A] time = 1.22962, size = 318, normalized size = 0.87 \[ \frac{\sqrt{2-3 x} \left (1984 \sqrt{682} \sqrt{\frac{5-2 x}{5 x+7}} \sqrt{\frac{4 x+1}{5 x+7}} \left (15 x^2+11 x-14\right ) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{155-62 x}{55 x+77}}\right ),\frac{23}{62}\right )-3410 \sqrt{682} \sqrt{\frac{5-2 x}{5 x+7}} \sqrt{\frac{4 x+1}{5 x+7}} \left (15 x^2+11 x-14\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{155-62 x}{55 x+77}}\right )|\frac{23}{62}\right )+\sqrt{\frac{3 x-2}{5 x+7}} \left (17050 \left (24 x^3-70 x^2+21 x+10\right )-1599 \sqrt{682} \sqrt{\frac{4 x+1}{5 x+7}} (5 x+7)^2 \sqrt{\frac{-6 x^2+19 x-10}{(5 x+7)^2}} \Pi \left (-\frac{55}{62};\sin ^{-1}\left (\sqrt{\frac{155-62 x}{55 x+77}}\right )|\frac{23}{62}\right )\right )\right )}{34100 \sqrt{2 x-5} \sqrt{4 x+1} \left (\frac{3 x-2}{5 x+7}\right )^{3/2} (5 x+7)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.021, size = 875, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{\sqrt{5 \, x + 7} \sqrt{2 \, x - 5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{5 \, x + 7} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{10 \, x^{2} - 11 \, x - 35}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 - 3 x} \sqrt{4 x + 1}}{\sqrt{2 x - 5} \sqrt{5 x + 7}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{\sqrt{5 \, x + 7} \sqrt{2 \, x - 5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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