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 = 0.53, antiderivative size = 434, normalized size of antiderivative = 1.00, 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 12
Rule 636
Rule 1103
Rule 1195
Rule 1197
Rule 1247
Rule 1673
Rule 1678
Rule 2069
Rule 6719
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*}
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Mathematica [C] time = 6.08, size = 6019, normalized size = 13.87 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8}}{64 \, x^{8} - 240 \, x^{7} + 353 \, x^{6} + 144 \, x^{5} - 528 \, x^{4} + 144 \, x^{3} + 704 \, x^{2} + 384 \, x + 64}, x\right ) \]
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 \[ \int \frac {1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 5421, normalized size = 12.49 \[ \text {output too large to display} \]
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 \[ \int \frac {1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (8\,x^4-15\,x^3+8\,x^2+24\,x+8\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (8 x^{4} - 15 x^{3} + 8 x^{2} + 24 x + 8\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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