Optimal. Leaf size=108 \[ -\frac {a^8}{45 x^{45}}-\frac {4 a^7 b}{21 x^{42}}-\frac {28 a^6 b^2}{39 x^{39}}-\frac {14 a^5 b^3}{9 x^{36}}-\frac {70 a^4 b^4}{33 x^{33}}-\frac {28 a^3 b^5}{15 x^{30}}-\frac {28 a^2 b^6}{27 x^{27}}-\frac {a b^7}{3 x^{24}}-\frac {b^8}{21 x^{21}} \]
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Rubi [A]
time = 0.04, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45}
\begin {gather*} -\frac {a^8}{45 x^{45}}-\frac {4 a^7 b}{21 x^{42}}-\frac {28 a^6 b^2}{39 x^{39}}-\frac {14 a^5 b^3}{9 x^{36}}-\frac {70 a^4 b^4}{33 x^{33}}-\frac {28 a^3 b^5}{15 x^{30}}-\frac {28 a^2 b^6}{27 x^{27}}-\frac {a b^7}{3 x^{24}}-\frac {b^8}{21 x^{21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^8}{x^{46}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^8}{x^{16}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {a^8}{x^{16}}+\frac {8 a^7 b}{x^{15}}+\frac {28 a^6 b^2}{x^{14}}+\frac {56 a^5 b^3}{x^{13}}+\frac {70 a^4 b^4}{x^{12}}+\frac {56 a^3 b^5}{x^{11}}+\frac {28 a^2 b^6}{x^{10}}+\frac {8 a b^7}{x^9}+\frac {b^8}{x^8}\right ) \, dx,x,x^3\right )\\ &=-\frac {a^8}{45 x^{45}}-\frac {4 a^7 b}{21 x^{42}}-\frac {28 a^6 b^2}{39 x^{39}}-\frac {14 a^5 b^3}{9 x^{36}}-\frac {70 a^4 b^4}{33 x^{33}}-\frac {28 a^3 b^5}{15 x^{30}}-\frac {28 a^2 b^6}{27 x^{27}}-\frac {a b^7}{3 x^{24}}-\frac {b^8}{21 x^{21}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 108, normalized size = 1.00 \begin {gather*} -\frac {a^8}{45 x^{45}}-\frac {4 a^7 b}{21 x^{42}}-\frac {28 a^6 b^2}{39 x^{39}}-\frac {14 a^5 b^3}{9 x^{36}}-\frac {70 a^4 b^4}{33 x^{33}}-\frac {28 a^3 b^5}{15 x^{30}}-\frac {28 a^2 b^6}{27 x^{27}}-\frac {a b^7}{3 x^{24}}-\frac {b^8}{21 x^{21}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 91, normalized size = 0.84
method | result | size |
default | \(-\frac {a^{8}}{45 x^{45}}-\frac {4 a^{7} b}{21 x^{42}}-\frac {28 a^{6} b^{2}}{39 x^{39}}-\frac {14 a^{5} b^{3}}{9 x^{36}}-\frac {70 a^{4} b^{4}}{33 x^{33}}-\frac {28 a^{3} b^{5}}{15 x^{30}}-\frac {28 a^{2} b^{6}}{27 x^{27}}-\frac {a \,b^{7}}{3 x^{24}}-\frac {b^{8}}{21 x^{21}}\) | \(91\) |
risch | \(\frac {-\frac {28}{15} a^{3} b^{5} x^{15}-\frac {28}{27} a^{2} b^{6} x^{18}-\frac {1}{3} a \,b^{7} x^{21}-\frac {1}{21} b^{8} x^{24}-\frac {4}{21} a^{7} b \,x^{3}-\frac {28}{39} a^{6} b^{2} x^{6}-\frac {14}{9} a^{5} b^{3} x^{9}-\frac {70}{33} a^{4} b^{4} x^{12}-\frac {1}{45} a^{8}}{x^{45}}\) | \(92\) |
gosper | \(-\frac {6435 b^{8} x^{24}+45045 a \,b^{7} x^{21}+140140 a^{2} b^{6} x^{18}+252252 a^{3} b^{5} x^{15}+286650 a^{4} b^{4} x^{12}+210210 a^{5} b^{3} x^{9}+97020 a^{6} b^{2} x^{6}+25740 a^{7} b \,x^{3}+3003 a^{8}}{135135 x^{45}}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 92, normalized size = 0.85 \begin {gather*} -\frac {6435 \, b^{8} x^{24} + 45045 \, a b^{7} x^{21} + 140140 \, a^{2} b^{6} x^{18} + 252252 \, a^{3} b^{5} x^{15} + 286650 \, a^{4} b^{4} x^{12} + 210210 \, a^{5} b^{3} x^{9} + 97020 \, a^{6} b^{2} x^{6} + 25740 \, a^{7} b x^{3} + 3003 \, a^{8}}{135135 \, x^{45}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 92, normalized size = 0.85 \begin {gather*} -\frac {6435 \, b^{8} x^{24} + 45045 \, a b^{7} x^{21} + 140140 \, a^{2} b^{6} x^{18} + 252252 \, a^{3} b^{5} x^{15} + 286650 \, a^{4} b^{4} x^{12} + 210210 \, a^{5} b^{3} x^{9} + 97020 \, a^{6} b^{2} x^{6} + 25740 \, a^{7} b x^{3} + 3003 \, a^{8}}{135135 \, x^{45}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.69, size = 99, normalized size = 0.92 \begin {gather*} \frac {- 3003 a^{8} - 25740 a^{7} b x^{3} - 97020 a^{6} b^{2} x^{6} - 210210 a^{5} b^{3} x^{9} - 286650 a^{4} b^{4} x^{12} - 252252 a^{3} b^{5} x^{15} - 140140 a^{2} b^{6} x^{18} - 45045 a b^{7} x^{21} - 6435 b^{8} x^{24}}{135135 x^{45}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.33, size = 92, normalized size = 0.85 \begin {gather*} -\frac {6435 \, b^{8} x^{24} + 45045 \, a b^{7} x^{21} + 140140 \, a^{2} b^{6} x^{18} + 252252 \, a^{3} b^{5} x^{15} + 286650 \, a^{4} b^{4} x^{12} + 210210 \, a^{5} b^{3} x^{9} + 97020 \, a^{6} b^{2} x^{6} + 25740 \, a^{7} b x^{3} + 3003 \, a^{8}}{135135 \, x^{45}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.00, size = 92, normalized size = 0.85 \begin {gather*} -\frac {\frac {a^8}{45}+\frac {4\,a^7\,b\,x^3}{21}+\frac {28\,a^6\,b^2\,x^6}{39}+\frac {14\,a^5\,b^3\,x^9}{9}+\frac {70\,a^4\,b^4\,x^{12}}{33}+\frac {28\,a^3\,b^5\,x^{15}}{15}+\frac {28\,a^2\,b^6\,x^{18}}{27}+\frac {a\,b^7\,x^{21}}{3}+\frac {b^8\,x^{24}}{21}}{x^{45}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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