Optimal. Leaf size=97 \[ -\frac {a^8}{8 x^8}-\frac {4 a^7 b}{3 x^6}-\frac {7 a^6 b^2}{x^4}-\frac {28 a^5 b^3}{x^2}+70 a^4 b^4 \log (x)+28 a^3 b^5 x^2+7 a^2 b^6 x^4+\frac {4}{3} a b^7 x^6+\frac {b^8 x^8}{8} \]
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Rubi [A] time = 0.06, antiderivative size = 97, 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} \[ -\frac {7 a^6 b^2}{x^4}-\frac {28 a^5 b^3}{x^2}+28 a^3 b^5 x^2+7 a^2 b^6 x^4+70 a^4 b^4 \log (x)-\frac {4 a^7 b}{3 x^6}-\frac {a^8}{8 x^8}+\frac {4}{3} a b^7 x^6+\frac {b^8 x^8}{8} \]
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^9} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^5} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (56 a^3 b^5+\frac {a^8}{x^5}+\frac {8 a^7 b}{x^4}+\frac {28 a^6 b^2}{x^3}+\frac {56 a^5 b^3}{x^2}+\frac {70 a^4 b^4}{x}+28 a^2 b^6 x+8 a b^7 x^2+b^8 x^3\right ) \, dx,x,x^2\right )\\ &=-\frac {a^8}{8 x^8}-\frac {4 a^7 b}{3 x^6}-\frac {7 a^6 b^2}{x^4}-\frac {28 a^5 b^3}{x^2}+28 a^3 b^5 x^2+7 a^2 b^6 x^4+\frac {4}{3} a b^7 x^6+\frac {b^8 x^8}{8}+70 a^4 b^4 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 97, normalized size = 1.00 \[ -\frac {a^8}{8 x^8}-\frac {4 a^7 b}{3 x^6}-\frac {7 a^6 b^2}{x^4}-\frac {28 a^5 b^3}{x^2}+70 a^4 b^4 \log (x)+28 a^3 b^5 x^2+7 a^2 b^6 x^4+\frac {4}{3} a b^7 x^6+\frac {b^8 x^8}{8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 94, normalized size = 0.97 \[ \frac {3 \, b^{8} x^{16} + 32 \, a b^{7} x^{14} + 168 \, a^{2} b^{6} x^{12} + 672 \, a^{3} b^{5} x^{10} + 1680 \, a^{4} b^{4} x^{8} \log \relax (x) - 672 \, a^{5} b^{3} x^{6} - 168 \, a^{6} b^{2} x^{4} - 32 \, a^{7} b x^{2} - 3 \, a^{8}}{24 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 105, normalized size = 1.08 \[ \frac {1}{8} \, b^{8} x^{8} + \frac {4}{3} \, a b^{7} x^{6} + 7 \, a^{2} b^{6} x^{4} + 28 \, a^{3} b^{5} x^{2} + 35 \, a^{4} b^{4} \log \left (x^{2}\right ) - \frac {1750 \, a^{4} b^{4} x^{8} + 672 \, a^{5} b^{3} x^{6} + 168 \, a^{6} b^{2} x^{4} + 32 \, a^{7} b x^{2} + 3 \, a^{8}}{24 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 90, normalized size = 0.93 \[ \frac {b^{8} x^{8}}{8}+\frac {4 a \,b^{7} x^{6}}{3}+7 a^{2} b^{6} x^{4}+28 a^{3} b^{5} x^{2}+70 a^{4} b^{4} \ln \relax (x )-\frac {28 a^{5} b^{3}}{x^{2}}-\frac {7 a^{6} b^{2}}{x^{4}}-\frac {4 a^{7} b}{3 x^{6}}-\frac {a^{8}}{8 x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 94, normalized size = 0.97 \[ \frac {1}{8} \, b^{8} x^{8} + \frac {4}{3} \, a b^{7} x^{6} + 7 \, a^{2} b^{6} x^{4} + 28 \, a^{3} b^{5} x^{2} + 35 \, a^{4} b^{4} \log \left (x^{2}\right ) - \frac {672 \, a^{5} b^{3} x^{6} + 168 \, a^{6} b^{2} x^{4} + 32 \, a^{7} b x^{2} + 3 \, a^{8}}{24 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 92, normalized size = 0.95 \[ \frac {b^8\,x^8}{8}-\frac {\frac {a^8}{8}+\frac {4\,a^7\,b\,x^2}{3}+7\,a^6\,b^2\,x^4+28\,a^5\,b^3\,x^6}{x^8}+\frac {4\,a\,b^7\,x^6}{3}+28\,a^3\,b^5\,x^2+7\,a^2\,b^6\,x^4+70\,a^4\,b^4\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 100, normalized size = 1.03 \[ 70 a^{4} b^{4} \log {\relax (x )} + 28 a^{3} b^{5} x^{2} + 7 a^{2} b^{6} x^{4} + \frac {4 a b^{7} x^{6}}{3} + \frac {b^{8} x^{8}}{8} + \frac {- 3 a^{8} - 32 a^{7} b x^{2} - 168 a^{6} b^{2} x^{4} - 672 a^{5} b^{3} x^{6}}{24 x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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