Optimal. Leaf size=106 \[ -\frac {a^8}{32 x^{32}}-\frac {4 a^7 b}{15 x^{30}}-\frac {a^6 b^2}{x^{28}}-\frac {28 a^5 b^3}{13 x^{26}}-\frac {35 a^4 b^4}{12 x^{24}}-\frac {28 a^3 b^5}{11 x^{22}}-\frac {7 a^2 b^6}{5 x^{20}}-\frac {4 a b^7}{9 x^{18}}-\frac {b^8}{16 x^{16}} \]
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Rubi [A] time = 0.05, antiderivative size = 106, 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} \[ -\frac {a^6 b^2}{x^{28}}-\frac {28 a^5 b^3}{13 x^{26}}-\frac {35 a^4 b^4}{12 x^{24}}-\frac {28 a^3 b^5}{11 x^{22}}-\frac {7 a^2 b^6}{5 x^{20}}-\frac {4 a^7 b}{15 x^{30}}-\frac {a^8}{32 x^{32}}-\frac {4 a b^7}{9 x^{18}}-\frac {b^8}{16 x^{16}} \]
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^{33}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^{17}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^8}{x^{17}}+\frac {8 a^7 b}{x^{16}}+\frac {28 a^6 b^2}{x^{15}}+\frac {56 a^5 b^3}{x^{14}}+\frac {70 a^4 b^4}{x^{13}}+\frac {56 a^3 b^5}{x^{12}}+\frac {28 a^2 b^6}{x^{11}}+\frac {8 a b^7}{x^{10}}+\frac {b^8}{x^9}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^8}{32 x^{32}}-\frac {4 a^7 b}{15 x^{30}}-\frac {a^6 b^2}{x^{28}}-\frac {28 a^5 b^3}{13 x^{26}}-\frac {35 a^4 b^4}{12 x^{24}}-\frac {28 a^3 b^5}{11 x^{22}}-\frac {7 a^2 b^6}{5 x^{20}}-\frac {4 a b^7}{9 x^{18}}-\frac {b^8}{16 x^{16}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 106, normalized size = 1.00 \[ -\frac {a^8}{32 x^{32}}-\frac {4 a^7 b}{15 x^{30}}-\frac {a^6 b^2}{x^{28}}-\frac {28 a^5 b^3}{13 x^{26}}-\frac {35 a^4 b^4}{12 x^{24}}-\frac {28 a^3 b^5}{11 x^{22}}-\frac {7 a^2 b^6}{5 x^{20}}-\frac {4 a b^7}{9 x^{18}}-\frac {b^8}{16 x^{16}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 92, normalized size = 0.87 \[ -\frac {12870 \, b^{8} x^{16} + 91520 \, a b^{7} x^{14} + 288288 \, a^{2} b^{6} x^{12} + 524160 \, a^{3} b^{5} x^{10} + 600600 \, a^{4} b^{4} x^{8} + 443520 \, a^{5} b^{3} x^{6} + 205920 \, a^{6} b^{2} x^{4} + 54912 \, a^{7} b x^{2} + 6435 \, a^{8}}{205920 \, x^{32}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 92, normalized size = 0.87 \[ -\frac {12870 \, b^{8} x^{16} + 91520 \, a b^{7} x^{14} + 288288 \, a^{2} b^{6} x^{12} + 524160 \, a^{3} b^{5} x^{10} + 600600 \, a^{4} b^{4} x^{8} + 443520 \, a^{5} b^{3} x^{6} + 205920 \, a^{6} b^{2} x^{4} + 54912 \, a^{7} b x^{2} + 6435 \, a^{8}}{205920 \, x^{32}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 91, normalized size = 0.86 \[ -\frac {b^{8}}{16 x^{16}}-\frac {4 a \,b^{7}}{9 x^{18}}-\frac {7 a^{2} b^{6}}{5 x^{20}}-\frac {28 a^{3} b^{5}}{11 x^{22}}-\frac {35 a^{4} b^{4}}{12 x^{24}}-\frac {28 a^{5} b^{3}}{13 x^{26}}-\frac {a^{6} b^{2}}{x^{28}}-\frac {4 a^{7} b}{15 x^{30}}-\frac {a^{8}}{32 x^{32}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 92, normalized size = 0.87 \[ -\frac {12870 \, b^{8} x^{16} + 91520 \, a b^{7} x^{14} + 288288 \, a^{2} b^{6} x^{12} + 524160 \, a^{3} b^{5} x^{10} + 600600 \, a^{4} b^{4} x^{8} + 443520 \, a^{5} b^{3} x^{6} + 205920 \, a^{6} b^{2} x^{4} + 54912 \, a^{7} b x^{2} + 6435 \, a^{8}}{205920 \, x^{32}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 91, normalized size = 0.86 \[ -\frac {\frac {a^8}{32}+\frac {4\,a^7\,b\,x^2}{15}+a^6\,b^2\,x^4+\frac {28\,a^5\,b^3\,x^6}{13}+\frac {35\,a^4\,b^4\,x^8}{12}+\frac {28\,a^3\,b^5\,x^{10}}{11}+\frac {7\,a^2\,b^6\,x^{12}}{5}+\frac {4\,a\,b^7\,x^{14}}{9}+\frac {b^8\,x^{16}}{16}}{x^{32}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.34, size = 99, normalized size = 0.93 \[ \frac {- 6435 a^{8} - 54912 a^{7} b x^{2} - 205920 a^{6} b^{2} x^{4} - 443520 a^{5} b^{3} x^{6} - 600600 a^{4} b^{4} x^{8} - 524160 a^{3} b^{5} x^{10} - 288288 a^{2} b^{6} x^{12} - 91520 a b^{7} x^{14} - 12870 b^{8} x^{16}}{205920 x^{32}} \]
Verification of antiderivative is not currently implemented for this CAS.
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