Optimal. Leaf size=108 \[ -\frac{7 a^6 b^2}{6 x^{24}}-\frac{28 a^5 b^3}{11 x^{22}}-\frac{7 a^4 b^4}{2 x^{20}}-\frac{28 a^3 b^5}{9 x^{18}}-\frac{7 a^2 b^6}{4 x^{16}}-\frac{4 a^7 b}{13 x^{26}}-\frac{a^8}{28 x^{28}}-\frac{4 a b^7}{7 x^{14}}-\frac{b^8}{12 x^{12}} \]
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Rubi [A] time = 0.0536761, antiderivative size = 108, 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} \[ -\frac{7 a^6 b^2}{6 x^{24}}-\frac{28 a^5 b^3}{11 x^{22}}-\frac{7 a^4 b^4}{2 x^{20}}-\frac{28 a^3 b^5}{9 x^{18}}-\frac{7 a^2 b^6}{4 x^{16}}-\frac{4 a^7 b}{13 x^{26}}-\frac{a^8}{28 x^{28}}-\frac{4 a b^7}{7 x^{14}}-\frac{b^8}{12 x^{12}} \]
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^{29}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{15}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^8}{x^{15}}+\frac{8 a^7 b}{x^{14}}+\frac{28 a^6 b^2}{x^{13}}+\frac{56 a^5 b^3}{x^{12}}+\frac{70 a^4 b^4}{x^{11}}+\frac{56 a^3 b^5}{x^{10}}+\frac{28 a^2 b^6}{x^9}+\frac{8 a b^7}{x^8}+\frac{b^8}{x^7}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^8}{28 x^{28}}-\frac{4 a^7 b}{13 x^{26}}-\frac{7 a^6 b^2}{6 x^{24}}-\frac{28 a^5 b^3}{11 x^{22}}-\frac{7 a^4 b^4}{2 x^{20}}-\frac{28 a^3 b^5}{9 x^{18}}-\frac{7 a^2 b^6}{4 x^{16}}-\frac{4 a b^7}{7 x^{14}}-\frac{b^8}{12 x^{12}}\\ \end{align*}
Mathematica [A] time = 0.0043779, size = 108, normalized size = 1. \[ -\frac{7 a^6 b^2}{6 x^{24}}-\frac{28 a^5 b^3}{11 x^{22}}-\frac{7 a^4 b^4}{2 x^{20}}-\frac{28 a^3 b^5}{9 x^{18}}-\frac{7 a^2 b^6}{4 x^{16}}-\frac{4 a^7 b}{13 x^{26}}-\frac{a^8}{28 x^{28}}-\frac{4 a b^7}{7 x^{14}}-\frac{b^8}{12 x^{12}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 91, normalized size = 0.8 \begin{align*} -{\frac{{a}^{8}}{28\,{x}^{28}}}-{\frac{4\,{a}^{7}b}{13\,{x}^{26}}}-{\frac{7\,{a}^{6}{b}^{2}}{6\,{x}^{24}}}-{\frac{28\,{a}^{5}{b}^{3}}{11\,{x}^{22}}}-{\frac{7\,{a}^{4}{b}^{4}}{2\,{x}^{20}}}-{\frac{28\,{a}^{3}{b}^{5}}{9\,{x}^{18}}}-{\frac{7\,{a}^{2}{b}^{6}}{4\,{x}^{16}}}-{\frac{4\,a{b}^{7}}{7\,{x}^{14}}}-{\frac{{b}^{8}}{12\,{x}^{12}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.97239, size = 124, normalized size = 1.15 \begin{align*} -\frac{3003 \, b^{8} x^{16} + 20592 \, a b^{7} x^{14} + 63063 \, a^{2} b^{6} x^{12} + 112112 \, a^{3} b^{5} x^{10} + 126126 \, a^{4} b^{4} x^{8} + 91728 \, a^{5} b^{3} x^{6} + 42042 \, a^{6} b^{2} x^{4} + 11088 \, a^{7} b x^{2} + 1287 \, a^{8}}{36036 \, x^{28}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29526, size = 247, normalized size = 2.29 \begin{align*} -\frac{3003 \, b^{8} x^{16} + 20592 \, a b^{7} x^{14} + 63063 \, a^{2} b^{6} x^{12} + 112112 \, a^{3} b^{5} x^{10} + 126126 \, a^{4} b^{4} x^{8} + 91728 \, a^{5} b^{3} x^{6} + 42042 \, a^{6} b^{2} x^{4} + 11088 \, a^{7} b x^{2} + 1287 \, a^{8}}{36036 \, x^{28}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.49153, size = 99, normalized size = 0.92 \begin{align*} - \frac{1287 a^{8} + 11088 a^{7} b x^{2} + 42042 a^{6} b^{2} x^{4} + 91728 a^{5} b^{3} x^{6} + 126126 a^{4} b^{4} x^{8} + 112112 a^{3} b^{5} x^{10} + 63063 a^{2} b^{6} x^{12} + 20592 a b^{7} x^{14} + 3003 b^{8} x^{16}}{36036 x^{28}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.08714, size = 124, normalized size = 1.15 \begin{align*} -\frac{3003 \, b^{8} x^{16} + 20592 \, a b^{7} x^{14} + 63063 \, a^{2} b^{6} x^{12} + 112112 \, a^{3} b^{5} x^{10} + 126126 \, a^{4} b^{4} x^{8} + 91728 \, a^{5} b^{3} x^{6} + 42042 \, a^{6} b^{2} x^{4} + 11088 \, a^{7} b x^{2} + 1287 \, a^{8}}{36036 \, x^{28}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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