Optimal. Leaf size=469 \[ \frac{65 \sqrt{\frac{11}{23}} \sqrt{5 x+7} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right ),-\frac{39}{23}\right )}{8 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}-\frac{895 \sqrt{\frac{11}{62}} \sqrt{2-3 x} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{\frac{22}{23}} \sqrt{5 x+7}}{\sqrt{2 x-5}}\right ),\frac{39}{62}\right )}{48 \sqrt{-\frac{2-3 x}{4 x+1}} \sqrt{4 x+1}}-\frac{5 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{12 \sqrt{2 x-5}}+\frac{5 \sqrt{\frac{143}{3}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{8 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}+\frac{23 \sqrt{\frac{31}{22}} \sqrt{\frac{2-3 x}{5 x+7}} \sqrt{\frac{5-2 x}{5 x+7}} (5 x+7) \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{4 x+1}}{\sqrt{5 x+7}}\right )|\frac{39}{62}\right )}{6 \sqrt{2-3 x} \sqrt{2 x-5}}-\frac{4117 \sqrt{2-3 x} \Pi \left (\frac{78}{55};\tan ^{-1}\left (\frac{\sqrt{\frac{22}{23}} \sqrt{5 x+7}}{\sqrt{2 x-5}}\right )|\frac{39}{62}\right )}{48 \sqrt{682} \sqrt{-\frac{2-3 x}{4 x+1}} \sqrt{4 x+1}} \]
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Rubi [A] time = 0.280348, antiderivative size = 469, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.27, Rules used = {166, 173, 176, 424, 170, 418, 165, 536, 539, 537} \[ -\frac{5 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{12 \sqrt{2 x-5}}+\frac{65 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{8 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}-\frac{895 \sqrt{\frac{11}{62}} \sqrt{2-3 x} F\left (\tan ^{-1}\left (\frac{\sqrt{\frac{22}{23}} \sqrt{5 x+7}}{\sqrt{2 x-5}}\right )|\frac{39}{62}\right )}{48 \sqrt{-\frac{2-3 x}{4 x+1}} \sqrt{4 x+1}}+\frac{5 \sqrt{\frac{143}{3}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{8 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}+\frac{23 \sqrt{\frac{31}{22}} \sqrt{\frac{2-3 x}{5 x+7}} \sqrt{\frac{5-2 x}{5 x+7}} (5 x+7) \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{4 x+1}}{\sqrt{5 x+7}}\right )|\frac{39}{62}\right )}{6 \sqrt{2-3 x} \sqrt{2 x-5}}-\frac{4117 \sqrt{2-3 x} \Pi \left (\frac{78}{55};\tan ^{-1}\left (\frac{\sqrt{\frac{22}{23}} \sqrt{5 x+7}}{\sqrt{2 x-5}}\right )|\frac{39}{62}\right )}{48 \sqrt{682} \sqrt{-\frac{2-3 x}{4 x+1}} \sqrt{4 x+1}} \]
Antiderivative was successfully verified.
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Rule 166
Rule 173
Rule 176
Rule 424
Rule 170
Rule 418
Rule 165
Rule 536
Rule 539
Rule 537
Rubi steps
\begin{align*} \int \frac{(7+5 x)^{3/2}}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx &=-\left (\frac{5}{3} \int \frac{\sqrt{2-3 x} \sqrt{7+5 x}}{\sqrt{-5+2 x} \sqrt{1+4 x}} \, dx\right )+\frac{31}{3} \int \frac{\sqrt{7+5 x}}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx\\ &=-\frac{5 \sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{12 \sqrt{-5+2 x}}+\frac{895}{48} \int \frac{\sqrt{-5+2 x}}{\sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx+\frac{715}{16} \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx-\frac{715}{8} \int \frac{\sqrt{2-3 x}}{(-5+2 x)^{3/2} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx+\frac{\left (713 \sqrt{2} \sqrt{\frac{2-3 x}{7+5 x}} \sqrt{-\frac{-5+2 x}{7+5 x}} (7+5 x)\right ) \operatorname{Subst}\left (\int \frac{1}{\left (4-5 x^2\right ) \sqrt{1-\frac{31 x^2}{11}} \sqrt{1-\frac{39 x^2}{22}}} \, dx,x,\frac{\sqrt{1+4 x}}{\sqrt{7+5 x}}\right )}{33 \sqrt{2-3 x} \sqrt{-5+2 x}}\\ &=-\frac{5 \sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{12 \sqrt{-5+2 x}}+\frac{23 \sqrt{\frac{31}{22}} \sqrt{\frac{2-3 x}{7+5 x}} \sqrt{\frac{5-2 x}{7+5 x}} (7+5 x) \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{1+4 x}}{\sqrt{7+5 x}}\right )|\frac{39}{62}\right )}{6 \sqrt{2-3 x} \sqrt{-5+2 x}}+\frac{\left (11635 \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 )}{8 \sqrt{713} \sqrt{2-3 x} \sqrt{1+4 x}}+\frac{\left (65 \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 )}{8 \sqrt{-5+2 x} \sqrt{\frac{7+5 x}{2-3 x}}}+\frac{\left (65 \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 )}{8 \sqrt{-\frac{2-3 x}{-5+2 x}} \sqrt{7+5 x}}\\ &=-\frac{5 \sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{12 \sqrt{-5+2 x}}+\frac{5 \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 )}{8 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{7+5 x}}+\frac{65 \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 )}{8 \sqrt{-5+2 x} \sqrt{\frac{7+5 x}{5-2 x}}}+\frac{23 \sqrt{\frac{31}{22}} \sqrt{\frac{2-3 x}{7+5 x}} \sqrt{\frac{5-2 x}{7+5 x}} (7+5 x) \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{1+4 x}}{\sqrt{7+5 x}}\right )|\frac{39}{62}\right )}{6 \sqrt{2-3 x} \sqrt{-5+2 x}}+\frac{\left (895 \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 )}{48 \sqrt{2-3 x} \sqrt{1+4 x}}+\frac{\left (9845 \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 )}{48 \sqrt{713} \sqrt{2-3 x} \sqrt{1+4 x}}\\ &=-\frac{5 \sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{12 \sqrt{-5+2 x}}+\frac{5 \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 )}{8 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{7+5 x}}+\frac{65 \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 )}{8 \sqrt{-5+2 x} \sqrt{\frac{7+5 x}{5-2 x}}}-\frac{895 \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 )}{48 \sqrt{-\frac{2-3 x}{1+4 x}} \sqrt{1+4 x}}+\frac{23 \sqrt{\frac{31}{22}} \sqrt{\frac{2-3 x}{7+5 x}} \sqrt{\frac{5-2 x}{7+5 x}} (7+5 x) \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{1+4 x}}{\sqrt{7+5 x}}\right )|\frac{39}{62}\right )}{6 \sqrt{2-3 x} \sqrt{-5+2 x}}-\frac{4117 \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 )}{48 \sqrt{682} \sqrt{-\frac{2-3 x}{1+4 x}} \sqrt{1+4 x}}\\ \end{align*}
Mathematica [A] time = 1.2229, size = 347, normalized size = 0.74 \[ \frac{\sqrt{2 x-5} \left (-6969 \sqrt{341} \sqrt{\frac{3 x-2}{4 x+1}} \sqrt{\frac{5 x+7}{4 x+1}} \left (8 x^2-18 x-5\right ) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{22}{39}} \sqrt{\frac{5 x+7}{4 x+1}}\right ),\frac{39}{62}\right )+6820 \sqrt{341} \sqrt{\frac{3 x-2}{4 x+1}} \sqrt{\frac{5 x+7}{4 x+1}} \left (8 x^2-18 x-5\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{22}{39}} \sqrt{\frac{5 x+7}{4 x+1}}\right )|\frac{39}{62}\right )+\sqrt{\frac{2 x-5}{4 x+1}} \left (13640 \sqrt{2} \left (30 x^3-53 x^2-83 x+70\right )+9821 \sqrt{341} \sqrt{\frac{3 x-2}{4 x+1}} \sqrt{\frac{10 x^2-11 x-35}{(4 x+1)^2}} (4 x+1)^2 \Pi \left (\frac{78}{55};\sin ^{-1}\left (\sqrt{\frac{22}{39}} \sqrt{\frac{5 x+7}{4 x+1}}\right )|\frac{39}{62}\right )\right )\right )}{16368 \sqrt{4-6 x} \left (\frac{2 x-5}{4 x+1}\right )^{3/2} (4 x+1)^{3/2} \sqrt{5 x+7}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.023, size = 875, normalized size = 1.9 \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{{\left (5 \, x + 7\right )}^{\frac{3}{2}}}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}\,{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{{\left (5 \, x + 7\right )}^{\frac{3}{2}} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{24 \, x^{3} - 70 \, x^{2} + 21 \, x + 10}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{{\left (5 \, x + 7\right )}^{\frac{3}{2}}}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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