Optimal. Leaf size=101 \[ -\frac {a^8}{12 x^{12}}-\frac {4 a^7 b}{5 x^{10}}-\frac {7 a^6 b^2}{2 x^8}-\frac {28 a^5 b^3}{3 x^6}-\frac {35 a^4 b^4}{2 x^4}-\frac {28 a^3 b^5}{x^2}+28 a^2 b^6 \log (x)+4 a b^7 x^2+\frac {b^8 x^4}{4} \]
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Rubi [A] time = 0.05, antiderivative size = 101, 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} \[ -\frac {7 a^6 b^2}{2 x^8}-\frac {28 a^5 b^3}{3 x^6}-\frac {35 a^4 b^4}{2 x^4}-\frac {28 a^3 b^5}{x^2}+28 a^2 b^6 \log (x)-\frac {4 a^7 b}{5 x^{10}}-\frac {a^8}{12 x^{12}}+4 a b^7 x^2+\frac {b^8 x^4}{4} \]
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^{13}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^7} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (8 a b^7+\frac {a^8}{x^7}+\frac {8 a^7 b}{x^6}+\frac {28 a^6 b^2}{x^5}+\frac {56 a^5 b^3}{x^4}+\frac {70 a^4 b^4}{x^3}+\frac {56 a^3 b^5}{x^2}+\frac {28 a^2 b^6}{x}+b^8 x\right ) \, dx,x,x^2\right )\\ &=-\frac {a^8}{12 x^{12}}-\frac {4 a^7 b}{5 x^{10}}-\frac {7 a^6 b^2}{2 x^8}-\frac {28 a^5 b^3}{3 x^6}-\frac {35 a^4 b^4}{2 x^4}-\frac {28 a^3 b^5}{x^2}+4 a b^7 x^2+\frac {b^8 x^4}{4}+28 a^2 b^6 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 101, normalized size = 1.00 \[ -\frac {a^8}{12 x^{12}}-\frac {4 a^7 b}{5 x^{10}}-\frac {7 a^6 b^2}{2 x^8}-\frac {28 a^5 b^3}{3 x^6}-\frac {35 a^4 b^4}{2 x^4}-\frac {28 a^3 b^5}{x^2}+28 a^2 b^6 \log (x)+4 a b^7 x^2+\frac {b^8 x^4}{4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 94, normalized size = 0.93 \[ \frac {15 \, b^{8} x^{16} + 240 \, a b^{7} x^{14} + 1680 \, a^{2} b^{6} x^{12} \log \relax (x) - 1680 \, a^{3} b^{5} x^{10} - 1050 \, a^{4} b^{4} x^{8} - 560 \, a^{5} b^{3} x^{6} - 210 \, a^{6} b^{2} x^{4} - 48 \, a^{7} b x^{2} - 5 \, a^{8}}{60 \, x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.21, size = 105, normalized size = 1.04 \[ \frac {1}{4} \, b^{8} x^{4} + 4 \, a b^{7} x^{2} + 14 \, a^{2} b^{6} \log \left (x^{2}\right ) - \frac {2058 \, a^{2} b^{6} x^{12} + 1680 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 560 \, a^{5} b^{3} x^{6} + 210 \, a^{6} b^{2} x^{4} + 48 \, a^{7} b x^{2} + 5 \, a^{8}}{60 \, x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 90, normalized size = 0.89 \[ \frac {b^{8} x^{4}}{4}+4 a \,b^{7} x^{2}+28 a^{2} b^{6} \ln \relax (x )-\frac {28 a^{3} b^{5}}{x^{2}}-\frac {35 a^{4} b^{4}}{2 x^{4}}-\frac {28 a^{5} b^{3}}{3 x^{6}}-\frac {7 a^{6} b^{2}}{2 x^{8}}-\frac {4 a^{7} b}{5 x^{10}}-\frac {a^{8}}{12 x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 94, normalized size = 0.93 \[ \frac {1}{4} \, b^{8} x^{4} + 4 \, a b^{7} x^{2} + 14 \, a^{2} b^{6} \log \left (x^{2}\right ) - \frac {1680 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 560 \, a^{5} b^{3} x^{6} + 210 \, a^{6} b^{2} x^{4} + 48 \, a^{7} b x^{2} + 5 \, a^{8}}{60 \, x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 92, normalized size = 0.91 \[ \frac {b^8\,x^4}{4}-\frac {\frac {a^8}{12}+\frac {4\,a^7\,b\,x^2}{5}+\frac {7\,a^6\,b^2\,x^4}{2}+\frac {28\,a^5\,b^3\,x^6}{3}+\frac {35\,a^4\,b^4\,x^8}{2}+28\,a^3\,b^5\,x^{10}}{x^{12}}+4\,a\,b^7\,x^2+28\,a^2\,b^6\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.62, size = 99, normalized size = 0.98 \[ 28 a^{2} b^{6} \log {\relax (x )} + 4 a b^{7} x^{2} + \frac {b^{8} x^{4}}{4} + \frac {- 5 a^{8} - 48 a^{7} b x^{2} - 210 a^{6} b^{2} x^{4} - 560 a^{5} b^{3} x^{6} - 1050 a^{4} b^{4} x^{8} - 1680 a^{3} b^{5} x^{10}}{60 x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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