Optimal. Leaf size=577 \[ -\frac {\left (124415-6308 \left (\frac {4}{x}+3\right )^2\right ) x^2}{97344 \sqrt {8 x^4-15 x^3+8 x^2+24 x+8}}+\frac {\left (18932921731-1086525994 \left (\frac {4}{x}+3\right )^2\right ) \left (\frac {4}{x}+3\right ) x^2}{78056941248 \sqrt {8 x^4-15 x^3+8 x^2+24 x+8}}+\frac {543262997 \left (\left (\frac {4}{x}+3\right )^4-38 \left (\frac {4}{x}+3\right )^2+517\right ) \left (\frac {4}{x}+3\right ) x^2}{39028470624 \left (\left (\frac {4}{x}+3\right )^2+\sqrt {517}\right ) \sqrt {8 x^4-15 x^3+8 x^2+24 x+8}}+\frac {\left (11921698-359497 \left (\frac {4}{x}+3\right )^2\right ) \left (\frac {4}{x}+3\right ) x^2}{483912 \left (\left (\frac {4}{x}+3\right )^4-38 \left (\frac {4}{x}+3\right )^2+517\right ) \sqrt {8 x^4-15 x^3+8 x^2+24 x+8}}-\frac {\left (64489-1399 \left (\frac {4}{x}+3\right )^2\right ) x^2}{624 \left (\left (\frac {4}{x}+3\right )^4-38 \left (\frac {4}{x}+3\right )^2+517\right ) \sqrt {8 x^4-15 x^3+8 x^2+24 x+8}}+\frac {\left (4346103976-175318963 \sqrt {517}\right ) \left (\left (\frac {4}{x}+3\right )^2+\sqrt {517}\right ) \sqrt {\frac {\left (\frac {4}{x}+3\right )^4-38 \left (\frac {4}{x}+3\right )^2+517}{\left (\left (\frac {4}{x}+3\right )^2+\sqrt {517}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac {3 x+4}{\sqrt [4]{517} x}\right )|\frac {517+19 \sqrt {517}}{1034}\right )}{1207844352\ 517^{3/4} \sqrt {8 x^4-15 x^3+8 x^2+24 x+8}}-\frac {543262997 \left (\left (\frac {4}{x}+3\right )^2+\sqrt {517}\right ) \sqrt {\frac {\left (\frac {4}{x}+3\right )^4-38 \left (\frac {4}{x}+3\right )^2+517}{\left (\left (\frac {4}{x}+3\right )^2+\sqrt {517}\right )^2}} x^2 E\left (2 \tan ^{-1}\left (\frac {3 x+4}{\sqrt [4]{517} x}\right )|\frac {517+19 \sqrt {517}}{1034}\right )}{75490272\ 517^{3/4} \sqrt {8 x^4-15 x^3+8 x^2+24 x+8}} \]
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Rubi [A] time = 0.69, antiderivative size = 577, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {2069, 12, 6719, 1673, 1678, 1197, 1103, 1195, 1663, 1660, 636} \[ -\frac {\left (124415-6308 \left (\frac {4}{x}+3\right )^2\right ) x^2}{97344 \sqrt {8 x^4-15 x^3+8 x^2+24 x+8}}+\frac {\left (18932921731-1086525994 \left (\frac {4}{x}+3\right )^2\right ) \left (\frac {4}{x}+3\right ) x^2}{78056941248 \sqrt {8 x^4-15 x^3+8 x^2+24 x+8}}+\frac {543262997 \left (\left (\frac {4}{x}+3\right )^4-38 \left (\frac {4}{x}+3\right )^2+517\right ) \left (\frac {4}{x}+3\right ) x^2}{39028470624 \left (\left (\frac {4}{x}+3\right )^2+\sqrt {517}\right ) \sqrt {8 x^4-15 x^3+8 x^2+24 x+8}}+\frac {\left (11921698-359497 \left (\frac {4}{x}+3\right )^2\right ) \left (\frac {4}{x}+3\right ) x^2}{483912 \left (\left (\frac {4}{x}+3\right )^4-38 \left (\frac {4}{x}+3\right )^2+517\right ) \sqrt {8 x^4-15 x^3+8 x^2+24 x+8}}-\frac {\left (64489-1399 \left (\frac {4}{x}+3\right )^2\right ) x^2}{624 \left (\left (\frac {4}{x}+3\right )^4-38 \left (\frac {4}{x}+3\right )^2+517\right ) \sqrt {8 x^4-15 x^3+8 x^2+24 x+8}}+\frac {\left (4346103976-175318963 \sqrt {517}\right ) \left (\left (\frac {4}{x}+3\right )^2+\sqrt {517}\right ) \sqrt {\frac {\left (\frac {4}{x}+3\right )^4-38 \left (\frac {4}{x}+3\right )^2+517}{\left (\left (\frac {4}{x}+3\right )^2+\sqrt {517}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac {3 x+4}{\sqrt [4]{517} x}\right )|\frac {517+19 \sqrt {517}}{1034}\right )}{1207844352\ 517^{3/4} \sqrt {8 x^4-15 x^3+8 x^2+24 x+8}}-\frac {543262997 \left (\left (\frac {4}{x}+3\right )^2+\sqrt {517}\right ) \sqrt {\frac {\left (\frac {4}{x}+3\right )^4-38 \left (\frac {4}{x}+3\right )^2+517}{\left (\left (\frac {4}{x}+3\right )^2+\sqrt {517}\right )^2}} x^2 E\left (2 \tan ^{-1}\left (\frac {3 x+4}{\sqrt [4]{517} x}\right )|\frac {517+19 \sqrt {517}}{1034}\right )}{75490272\ 517^{3/4} \sqrt {8 x^4-15 x^3+8 x^2+24 x+8}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 636
Rule 1103
Rule 1195
Rule 1197
Rule 1660
Rule 1663
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 )^{5/2}} \, dx &=-\left (1024 \operatorname {Subst}\left (\int \frac {1}{128 \sqrt {2} (24-32 x)^2 \left (\frac {2117632-2490368 x^2+1048576 x^4}{(24-32 x)^4}\right )^{5/2}} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )\right )\\ &=-\left (\left (4 \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 )^{5/2}} \, 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)^8}{\left (2117632-2490368 x^2+1048576 x^4\right )^{5/2}} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )}{64 \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 (-1174136684544-14611478740992 x^2-25975962206208 x^4-6597069766656 x^6\right )}{\left (2117632-2490368 x^2+1048576 x^4\right )^{5/2}} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )}{64 \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 {110075314176+5479304527872 x^2+24352464568320 x^4+17317308137472 x^6+1099511627776 x^8}{\left (2117632-2490368 x^2+1048576 x^4\right )^{5/2}} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )}{64 \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}\\ &=\frac {\left (11921698-359497 \left (3+\frac {4}{x}\right )^2\right ) \left (3+\frac {4}{x}\right ) x^2}{483912 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right ) \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 {440718049065141914354843648+960217998469209766653591552 x^2+17853201636393873083203584 x^4}{\left (2117632-2490368 x^2+1048576 x^4\right )^{3/2}} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )}{1089672951440055730176 \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 {-1174136684544-14611478740992 x-25975962206208 x^2-6597069766656 x^3}{\left (2117632-2490368 x+1048576 x^2\right )^{5/2}} \, dx,x,\left (\frac {3}{4}+\frac {1}{x}\right )^2\right )}{128 \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}\\ &=-\frac {\left (64489-1399 \left (3+\frac {4}{x}\right )^2\right ) x^2}{624 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right ) \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}+\frac {\left (18932921731-1086525994 \left (3+\frac {4}{x}\right )^2\right ) \left (3+\frac {4}{x}\right ) x^2}{78056941248 \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}+\frac {\left (11921698-359497 \left (3+\frac {4}{x}\right )^2\right ) \left (3+\frac {4}{x}\right ) x^2}{483912 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right ) \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 {7181233034168885225762315128668638150656-5509341313830694309284922151776578174976 x^2}{\sqrt {2117632-2490368 x^2+1048576 x^4}} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )}{6184308026562927361480897835482981859328 \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 {-310869971478503227392-25292215507312705536 x}{\left (2117632-2490368 x+1048576 x^2\right )^{3/2}} \, dx,x,\left (\frac {3}{4}+\frac {1}{x}\right )^2\right )}{514571441799168 \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}\\ &=-\frac {\left (124415-6308 \left (3+\frac {4}{x}\right )^2\right ) x^2}{97344 \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}-\frac {\left (64489-1399 \left (3+\frac {4}{x}\right )^2\right ) x^2}{624 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right ) \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}+\frac {\left (18932921731-1086525994 \left (3+\frac {4}{x}\right )^2\right ) \left (3+\frac {4}{x}\right ) x^2}{78056941248 \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}+\frac {\left (11921698-359497 \left (3+\frac {4}{x}\right )^2\right ) \left (3+\frac {4}{x}\right ) x^2}{483912 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right ) \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}-\frac {\left (543262997 \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 )}{18872568 \sqrt {517} \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}-\frac {\left (\left (90639903871-4346103976 \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 )}{78056941248 \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}\\ &=-\frac {\left (124415-6308 \left (3+\frac {4}{x}\right )^2\right ) x^2}{97344 \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}-\frac {\left (64489-1399 \left (3+\frac {4}{x}\right )^2\right ) x^2}{624 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right ) \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}+\frac {\left (18932921731-1086525994 \left (3+\frac {4}{x}\right )^2\right ) \left (3+\frac {4}{x}\right ) x^2}{78056941248 \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}+\frac {\left (11921698-359497 \left (3+\frac {4}{x}\right )^2\right ) \left (3+\frac {4}{x}\right ) x^2}{483912 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right ) \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}+\frac {543262997 \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}{39028470624 \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 {543262997 \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 )}{75490272\ 517^{3/4} \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}+\frac {\left (4346103976-175318963 \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 )}{1207844352\ 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 = 6.06, size = 6084, normalized size = 10.54 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8}}{512 \, x^{12} - 2880 \, x^{11} + 6936 \, x^{10} - 4527 \, x^{9} - 8808 \, x^{8} + 16776 \, x^{7} + 5528 \, x^{6} - 17856 \, x^{5} - 384 \, x^{4} + 20160 \, x^{3} + 15360 \, x^{2} + 4608 \, x + 512}, x\right ) \]
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 \[ \int \frac {1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 5477, normalized size = 9.49 \[ \text {output too large to display} \]
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 \[ \int \frac {1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (8\,x^4-15\,x^3+8\,x^2+24\,x+8\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (8 x^{4} - 15 x^{3} + 8 x^{2} + 24 x + 8\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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