Optimal. Leaf size=108 \[ -\frac {a^8}{30 x^{30}}-\frac {2 a^7 b}{7 x^{28}}-\frac {14 a^6 b^2}{13 x^{26}}-\frac {7 a^5 b^3}{3 x^{24}}-\frac {35 a^4 b^4}{11 x^{22}}-\frac {14 a^3 b^5}{5 x^{20}}-\frac {14 a^2 b^6}{9 x^{18}}-\frac {a b^7}{2 x^{16}}-\frac {b^8}{14 x^{14}} \]
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Rubi [A] time = 0.05, 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 = {266, 43} \[ -\frac {14 a^6 b^2}{13 x^{26}}-\frac {7 a^5 b^3}{3 x^{24}}-\frac {35 a^4 b^4}{11 x^{22}}-\frac {14 a^3 b^5}{5 x^{20}}-\frac {14 a^2 b^6}{9 x^{18}}-\frac {2 a^7 b}{7 x^{28}}-\frac {a^8}{30 x^{30}}-\frac {a b^7}{2 x^{16}}-\frac {b^8}{14 x^{14}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^8}{x^{31}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^{16}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {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^2\right )\\ &=-\frac {a^8}{30 x^{30}}-\frac {2 a^7 b}{7 x^{28}}-\frac {14 a^6 b^2}{13 x^{26}}-\frac {7 a^5 b^3}{3 x^{24}}-\frac {35 a^4 b^4}{11 x^{22}}-\frac {14 a^3 b^5}{5 x^{20}}-\frac {14 a^2 b^6}{9 x^{18}}-\frac {a b^7}{2 x^{16}}-\frac {b^8}{14 x^{14}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 108, normalized size = 1.00 \[ -\frac {a^8}{30 x^{30}}-\frac {2 a^7 b}{7 x^{28}}-\frac {14 a^6 b^2}{13 x^{26}}-\frac {7 a^5 b^3}{3 x^{24}}-\frac {35 a^4 b^4}{11 x^{22}}-\frac {14 a^3 b^5}{5 x^{20}}-\frac {14 a^2 b^6}{9 x^{18}}-\frac {a b^7}{2 x^{16}}-\frac {b^8}{14 x^{14}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 92, normalized size = 0.85 \[ -\frac {6435 \, b^{8} x^{16} + 45045 \, a b^{7} x^{14} + 140140 \, a^{2} b^{6} x^{12} + 252252 \, a^{3} b^{5} x^{10} + 286650 \, a^{4} b^{4} x^{8} + 210210 \, a^{5} b^{3} x^{6} + 97020 \, a^{6} b^{2} x^{4} + 25740 \, a^{7} b x^{2} + 3003 \, a^{8}}{90090 \, x^{30}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.22, size = 92, normalized size = 0.85 \[ -\frac {6435 \, b^{8} x^{16} + 45045 \, a b^{7} x^{14} + 140140 \, a^{2} b^{6} x^{12} + 252252 \, a^{3} b^{5} x^{10} + 286650 \, a^{4} b^{4} x^{8} + 210210 \, a^{5} b^{3} x^{6} + 97020 \, a^{6} b^{2} x^{4} + 25740 \, a^{7} b x^{2} + 3003 \, a^{8}}{90090 \, x^{30}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 91, normalized size = 0.84 \[ -\frac {b^{8}}{14 x^{14}}-\frac {a \,b^{7}}{2 x^{16}}-\frac {14 a^{2} b^{6}}{9 x^{18}}-\frac {14 a^{3} b^{5}}{5 x^{20}}-\frac {35 a^{4} b^{4}}{11 x^{22}}-\frac {7 a^{5} b^{3}}{3 x^{24}}-\frac {14 a^{6} b^{2}}{13 x^{26}}-\frac {2 a^{7} b}{7 x^{28}}-\frac {a^{8}}{30 x^{30}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 92, normalized size = 0.85 \[ -\frac {6435 \, b^{8} x^{16} + 45045 \, a b^{7} x^{14} + 140140 \, a^{2} b^{6} x^{12} + 252252 \, a^{3} b^{5} x^{10} + 286650 \, a^{4} b^{4} x^{8} + 210210 \, a^{5} b^{3} x^{6} + 97020 \, a^{6} b^{2} x^{4} + 25740 \, a^{7} b x^{2} + 3003 \, a^{8}}{90090 \, x^{30}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.86, size = 92, normalized size = 0.85 \[ -\frac {\frac {a^8}{30}+\frac {2\,a^7\,b\,x^2}{7}+\frac {14\,a^6\,b^2\,x^4}{13}+\frac {7\,a^5\,b^3\,x^6}{3}+\frac {35\,a^4\,b^4\,x^8}{11}+\frac {14\,a^3\,b^5\,x^{10}}{5}+\frac {14\,a^2\,b^6\,x^{12}}{9}+\frac {a\,b^7\,x^{14}}{2}+\frac {b^8\,x^{16}}{14}}{x^{30}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.28, size = 99, normalized size = 0.92 \[ \frac {- 3003 a^{8} - 25740 a^{7} b x^{2} - 97020 a^{6} b^{2} x^{4} - 210210 a^{5} b^{3} x^{6} - 286650 a^{4} b^{4} x^{8} - 252252 a^{3} b^{5} x^{10} - 140140 a^{2} b^{6} x^{12} - 45045 a b^{7} x^{14} - 6435 b^{8} x^{16}}{90090 x^{30}} \]
Verification of antiderivative is not currently implemented for this CAS.
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