Optimal. Leaf size=434 \[ -\frac{\left (172-7 \left (\frac{4}{x}+3\right )^2\right ) x^2}{208 \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left (50896-2455 \left (\frac{4}{x}+3\right )^2\right ) \left (\frac{4}{x}+3\right ) x^2}{322608 \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{2455 \left (\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517\right ) \left (\frac{4}{x}+3\right ) x^2}{322608 \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left (4910-203 \sqrt{517}\right ) \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{\frac{\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517}{\left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac{3 x+4}{\sqrt [4]{517} x}\right )|\frac{517+19 \sqrt{517}}{1034}\right )}{2496\ 517^{3/4} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}-\frac{2455 \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{\frac{\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517}{\left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right )^2}} x^2 E\left (2 \tan ^{-1}\left (\frac{3 x+4}{\sqrt [4]{517} x}\right )|\frac{517+19 \sqrt{517}}{1034}\right )}{624\ 517^{3/4} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}} \]
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Rubi [A] time = 0.52758, antiderivative size = 434, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {2069, 12, 6719, 1673, 1678, 1197, 1103, 1195, 1247, 636} \[ -\frac{\left (172-7 \left (\frac{4}{x}+3\right )^2\right ) x^2}{208 \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left (50896-2455 \left (\frac{4}{x}+3\right )^2\right ) \left (\frac{4}{x}+3\right ) x^2}{322608 \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{2455 \left (\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517\right ) \left (\frac{4}{x}+3\right ) x^2}{322608 \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left (4910-203 \sqrt{517}\right ) \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{\frac{\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517}{\left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac{3 x+4}{\sqrt [4]{517} x}\right )|\frac{517+19 \sqrt{517}}{1034}\right )}{2496\ 517^{3/4} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}-\frac{2455 \left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right ) \sqrt{\frac{\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517}{\left (\left (\frac{4}{x}+3\right )^2+\sqrt{517}\right )^2}} x^2 E\left (2 \tan ^{-1}\left (\frac{3 x+4}{\sqrt [4]{517} x}\right )|\frac{517+19 \sqrt{517}}{1034}\right )}{624\ 517^{3/4} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}} \]
Antiderivative was successfully verified.
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Rule 2069
Rule 12
Rule 6719
Rule 1673
Rule 1678
Rule 1197
Rule 1103
Rule 1195
Rule 1247
Rule 636
Rubi steps
\begin{align*} \int \frac{1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{3/2}} \, dx &=-\left (1024 \operatorname{Subst}\left (\int \frac{1}{16 \sqrt{2} (24-32 x)^2 \left (\frac{2117632-2490368 x^2+1048576 x^4}{(24-32 x)^4}\right )^{3/2}} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )\right )\\ &=-\left (\left (32 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{(24-32 x)^2 \left (\frac{2117632-2490368 x^2+1048576 x^4}{(24-32 x)^4}\right )^{3/2}} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )\right )\\ &=-\frac{\left (\sqrt{2117632-2490368 \left (\frac{3}{4}+\frac{1}{x}\right )^2+1048576 \left (\frac{3}{4}+\frac{1}{x}\right )^4} x^2\right ) \operatorname{Subst}\left (\int \frac{(24-32 x)^4}{\left (2117632-2490368 x^2+1048576 x^4\right )^{3/2}} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )}{8 \sqrt{8+24 x+8 x^2-15 x^3+8 x^4}}\\ &=-\frac{\left (\sqrt{2117632-2490368 \left (\frac{3}{4}+\frac{1}{x}\right )^2+1048576 \left (\frac{3}{4}+\frac{1}{x}\right )^4} x^2\right ) \operatorname{Subst}\left (\int \frac{x \left (-1769472-3145728 x^2\right )}{\left (2117632-2490368 x^2+1048576 x^4\right )^{3/2}} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )}{8 \sqrt{8+24 x+8 x^2-15 x^3+8 x^4}}-\frac{\left (\sqrt{2117632-2490368 \left (\frac{3}{4}+\frac{1}{x}\right )^2+1048576 \left (\frac{3}{4}+\frac{1}{x}\right )^4} x^2\right ) \operatorname{Subst}\left (\int \frac{331776+3538944 x^2+1048576 x^4}{\left (2117632-2490368 x^2+1048576 x^4\right )^{3/2}} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )}{8 \sqrt{8+24 x+8 x^2-15 x^3+8 x^4}}\\ &=\frac{\left (50896-2455 \left (3+\frac{4}{x}\right )^2\right ) \left (3+\frac{4}{x}\right ) x^2}{322608 \sqrt{8+24 x+8 x^2-15 x^3+8 x^4}}-\frac{\left (\sqrt{2117632-2490368 \left (\frac{3}{4}+\frac{1}{x}\right )^2+1048576 \left (\frac{3}{4}+\frac{1}{x}\right )^4} x^2\right ) \operatorname{Subst}\left (\int \frac{29541080280760057856-22112674170389135360 x^2}{\sqrt{2117632-2490368 x^2+1048576 x^4}} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )}{45403039643335655424 \sqrt{8+24 x+8 x^2-15 x^3+8 x^4}}-\frac{\left (\sqrt{2117632-2490368 \left (\frac{3}{4}+\frac{1}{x}\right )^2+1048576 \left (\frac{3}{4}+\frac{1}{x}\right )^4} x^2\right ) \operatorname{Subst}\left (\int \frac{-1769472-3145728 x}{\left (2117632-2490368 x+1048576 x^2\right )^{3/2}} \, dx,x,\left (\frac{3}{4}+\frac{1}{x}\right )^2\right )}{16 \sqrt{8+24 x+8 x^2-15 x^3+8 x^4}}\\ &=-\frac{\left (172-7 \left (3+\frac{4}{x}\right )^2\right ) x^2}{208 \sqrt{8+24 x+8 x^2-15 x^3+8 x^4}}+\frac{\left (50896-2455 \left (3+\frac{4}{x}\right )^2\right ) \left (3+\frac{4}{x}\right ) x^2}{322608 \sqrt{8+24 x+8 x^2-15 x^3+8 x^4}}-\frac{\left (2455 \sqrt{2117632-2490368 \left (\frac{3}{4}+\frac{1}{x}\right )^2+1048576 \left (\frac{3}{4}+\frac{1}{x}\right )^4} x^2\right ) \operatorname{Subst}\left (\int \frac{1-\frac{16 x^2}{\sqrt{517}}}{\sqrt{2117632-2490368 x^2+1048576 x^4}} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )}{156 \sqrt{517} \sqrt{8+24 x+8 x^2-15 x^3+8 x^4}}-\frac{\left (\left (104951-4910 \sqrt{517}\right ) \sqrt{2117632-2490368 \left (\frac{3}{4}+\frac{1}{x}\right )^2+1048576 \left (\frac{3}{4}+\frac{1}{x}\right )^4} x^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2117632-2490368 x^2+1048576 x^4}} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )}{161304 \sqrt{8+24 x+8 x^2-15 x^3+8 x^4}}\\ &=-\frac{\left (172-7 \left (3+\frac{4}{x}\right )^2\right ) x^2}{208 \sqrt{8+24 x+8 x^2-15 x^3+8 x^4}}+\frac{\left (50896-2455 \left (3+\frac{4}{x}\right )^2\right ) \left (3+\frac{4}{x}\right ) x^2}{322608 \sqrt{8+24 x+8 x^2-15 x^3+8 x^4}}+\frac{2455 \left (517-38 \left (3+\frac{4}{x}\right )^2+\left (3+\frac{4}{x}\right )^4\right ) \left (3+\frac{4}{x}\right ) x^2}{322608 \left (\sqrt{517}+\left (3+\frac{4}{x}\right )^2\right ) \sqrt{8+24 x+8 x^2-15 x^3+8 x^4}}-\frac{2455 \left (\sqrt{517}+\left (3+\frac{4}{x}\right )^2\right ) \sqrt{\frac{517-38 \left (3+\frac{4}{x}\right )^2+\left (3+\frac{4}{x}\right )^4}{\left (\sqrt{517}+\left (3+\frac{4}{x}\right )^2\right )^2}} x^2 E\left (2 \tan ^{-1}\left (\frac{4+3 x}{\sqrt [4]{517} x}\right )|\frac{517+19 \sqrt{517}}{1034}\right )}{624\ 517^{3/4} \sqrt{8+24 x+8 x^2-15 x^3+8 x^4}}+\frac{\left (4910-203 \sqrt{517}\right ) \left (\sqrt{517}+\left (3+\frac{4}{x}\right )^2\right ) \sqrt{\frac{517-38 \left (3+\frac{4}{x}\right )^2+\left (3+\frac{4}{x}\right )^4}{\left (\sqrt{517}+\left (3+\frac{4}{x}\right )^2\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac{4+3 x}{\sqrt [4]{517} x}\right )|\frac{517+19 \sqrt{517}}{1034}\right )}{2496\ 517^{3/4} \sqrt{8+24 x+8 x^2-15 x^3+8 x^4}}\\ \end{align*}
Mathematica [C] time = 6.04897, size = 6019, normalized size = 13.87 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.017, size = 5421, normalized size = 12.5 \begin{align*} \text{output 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{1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{\frac{3}{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{\sqrt{8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8}}{64 \, x^{8} - 240 \, x^{7} + 353 \, x^{6} + 144 \, x^{5} - 528 \, x^{4} + 144 \, x^{3} + 704 \, x^{2} + 384 \, x + 64}, 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{1}{\left (8 x^{4} - 15 x^{3} + 8 x^{2} + 24 x + 8\right )^{\frac{3}{2}}}\, 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{1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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