Optimal. Leaf size=100 \[ a^8 \log (x)+4 a^7 b x^2+7 a^6 b^2 x^4+\frac {28}{3} a^5 b^3 x^6+\frac {35}{4} a^4 b^4 x^8+\frac {28}{5} a^3 b^5 x^{10}+\frac {7}{3} a^2 b^6 x^{12}+\frac {4}{7} a b^7 x^{14}+\frac {b^8 x^{16}}{16} \]
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Rubi [A] time = 0.06, antiderivative size = 100, 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} \begin {gather*} \frac {7}{3} a^2 b^6 x^{12}+\frac {28}{5} a^3 b^5 x^{10}+\frac {35}{4} a^4 b^4 x^8+\frac {28}{3} a^5 b^3 x^6+7 a^6 b^2 x^4+4 a^7 b x^2+a^8 \log (x)+\frac {4}{7} a b^7 x^{14}+\frac {b^8 x^{16}}{16} \end {gather*}
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} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (8 a^7 b+\frac {a^8}{x}+28 a^6 b^2 x+56 a^5 b^3 x^2+70 a^4 b^4 x^3+56 a^3 b^5 x^4+28 a^2 b^6 x^5+8 a b^7 x^6+b^8 x^7\right ) \, dx,x,x^2\right )\\ &=4 a^7 b x^2+7 a^6 b^2 x^4+\frac {28}{3} a^5 b^3 x^6+\frac {35}{4} a^4 b^4 x^8+\frac {28}{5} a^3 b^5 x^{10}+\frac {7}{3} a^2 b^6 x^{12}+\frac {4}{7} a b^7 x^{14}+\frac {b^8 x^{16}}{16}+a^8 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 100, normalized size = 1.00 \begin {gather*} a^8 \log (x)+4 a^7 b x^2+7 a^6 b^2 x^4+\frac {28}{3} a^5 b^3 x^6+\frac {35}{4} a^4 b^4 x^8+\frac {28}{5} a^3 b^5 x^{10}+\frac {7}{3} a^2 b^6 x^{12}+\frac {4}{7} a b^7 x^{14}+\frac {b^8 x^{16}}{16} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^8}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.10, size = 88, normalized size = 0.88 \begin {gather*} \frac {1}{16} \, b^{8} x^{16} + \frac {4}{7} \, a b^{7} x^{14} + \frac {7}{3} \, a^{2} b^{6} x^{12} + \frac {28}{5} \, a^{3} b^{5} x^{10} + \frac {35}{4} \, a^{4} b^{4} x^{8} + \frac {28}{3} \, a^{5} b^{3} x^{6} + 7 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.90, size = 91, normalized size = 0.91 \begin {gather*} \frac {1}{16} \, b^{8} x^{16} + \frac {4}{7} \, a b^{7} x^{14} + \frac {7}{3} \, a^{2} b^{6} x^{12} + \frac {28}{5} \, a^{3} b^{5} x^{10} + \frac {35}{4} \, a^{4} b^{4} x^{8} + \frac {28}{3} \, a^{5} b^{3} x^{6} + 7 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + \frac {1}{2} \, a^{8} \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 89, normalized size = 0.89 \begin {gather*} \frac {b^{8} x^{16}}{16}+\frac {4 a \,b^{7} x^{14}}{7}+\frac {7 a^{2} b^{6} x^{12}}{3}+\frac {28 a^{3} b^{5} x^{10}}{5}+\frac {35 a^{4} b^{4} x^{8}}{4}+\frac {28 a^{5} b^{3} x^{6}}{3}+7 a^{6} b^{2} x^{4}+4 a^{7} b \,x^{2}+a^{8} \ln \relax (x ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 91, normalized size = 0.91 \begin {gather*} \frac {1}{16} \, b^{8} x^{16} + \frac {4}{7} \, a b^{7} x^{14} + \frac {7}{3} \, a^{2} b^{6} x^{12} + \frac {28}{5} \, a^{3} b^{5} x^{10} + \frac {35}{4} \, a^{4} b^{4} x^{8} + \frac {28}{3} \, a^{5} b^{3} x^{6} + 7 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + \frac {1}{2} \, a^{8} \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.62, size = 88, normalized size = 0.88 \begin {gather*} a^8\,\ln \relax (x)+\frac {b^8\,x^{16}}{16}+4\,a^7\,b\,x^2+\frac {4\,a\,b^7\,x^{14}}{7}+7\,a^6\,b^2\,x^4+\frac {28\,a^5\,b^3\,x^6}{3}+\frac {35\,a^4\,b^4\,x^8}{4}+\frac {28\,a^3\,b^5\,x^{10}}{5}+\frac {7\,a^2\,b^6\,x^{12}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 102, normalized size = 1.02 \begin {gather*} a^{8} \log {\relax (x )} + 4 a^{7} b x^{2} + 7 a^{6} b^{2} x^{4} + \frac {28 a^{5} b^{3} x^{6}}{3} + \frac {35 a^{4} b^{4} x^{8}}{4} + \frac {28 a^{3} b^{5} x^{10}}{5} + \frac {7 a^{2} b^{6} x^{12}}{3} + \frac {4 a b^{7} x^{14}}{7} + \frac {b^{8} x^{16}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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