Optimal. Leaf size=106 \[ -\frac{a^6 b^2}{x^{28}}-\frac{28 a^5 b^3}{13 x^{26}}-\frac{35 a^4 b^4}{12 x^{24}}-\frac{28 a^3 b^5}{11 x^{22}}-\frac{7 a^2 b^6}{5 x^{20}}-\frac{4 a^7 b}{15 x^{30}}-\frac{a^8}{32 x^{32}}-\frac{4 a b^7}{9 x^{18}}-\frac{b^8}{16 x^{16}} \]
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Rubi [A] time = 0.0517498, antiderivative size = 106, normalized size of antiderivative = 1., 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 266
Rule 43
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*}
Mathematica [A] time = 0.0040941, size = 106, normalized size = 1. \[ -\frac{a^6 b^2}{x^{28}}-\frac{28 a^5 b^3}{13 x^{26}}-\frac{35 a^4 b^4}{12 x^{24}}-\frac{28 a^3 b^5}{11 x^{22}}-\frac{7 a^2 b^6}{5 x^{20}}-\frac{4 a^7 b}{15 x^{30}}-\frac{a^8}{32 x^{32}}-\frac{4 a b^7}{9 x^{18}}-\frac{b^8}{16 x^{16}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 91, normalized size = 0.9 \begin{align*} -{\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*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.34983, size = 124, normalized size = 1.17 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25686, size = 254, normalized size = 2.4 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.83298, size = 99, normalized size = 0.93 \begin{align*} - \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.86592, size = 124, normalized size = 1.17 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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